extfieldsofstudy: Physics Mathematics

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Cutting Multiparticle Correlators Down to Size

Multiparticle correlators are mathematical objects frequently encountered in quantum field theory and collider physics. By translating multiparticle correlators into the language of graph theory, we can gain new insights into their structure as well as identify efficient ways to manipulate them. In this paper, we highlight the power of this graph-theoretic approach by"cutting open"the vertices and edges of the graphs, allowing us to systematically classify linear relations among multiparticle correlators and develop faster methods for their computation. The naive computational complexity of an $N$-point correlator among $M$ particles is $\mathcal O(M^N)$, but when the pairwise distances between particles can be cast as an inner product, we show that all such correlators can be computed in linear $\mathcal O(M)$ runtime. With the help of new tensorial objects called Energy Flow Moments, we achieve a fast implementation of jet substructure observables like $C_2$ and $D_2$, which are widely used at the Large Hadron Collider to identify boosted hadronic resonances. As another application, we compute the number of leafless multigraphs with $d$ edges up to $d = 16$ (15,641,159), conjecturing that this is the same as the number of independent kinematic polynomials of degree $d$, previously known only to $d=8$ (279).

I. INTRODUCTION
Multiparticle correlators are ubiquitous mathematical structures that appear in a variety of physics domains, including kinematic polynomials for scattering amplitudes [1,2], operator bases for effective field theories [3,4], and many-body expansions for molecular analyses [5,6]. Broadly speaking, multiparticle correlators appear whenever the fundamental entities of a system are arranged in sets: unordered, variable-length collections of objects. These set elements may be nuclei, atoms, fields, particles, or other objects. Here, we refer to them as "particles," since we will eventually focus on multiparticle correlators in collider physics [7][8][9].
The basic structure of an N-point multiparticle correlator of M particles is where z i are particle weights, p i are particle properties, and f is a function of these properties. In the particle physics literature, Eq. (1) appears as the C-correlators of Ref. [10]. The function f is often decomposed in terms of monomials of pairwise distances or invariants θ ij between the particles: fðp i 1 ; …; p i N Þ ¼ θ α 12 i 1 i 2 θ α 13 where α ij are integer exponents. An argument for the generality of this restriction (up to isometries) in the context of collider observables is given in Ref. [9]. This simplification allows multiparticle correlators to be decomposed into smaller pieces, which is essential for deriving the results below.
In this paper, we develop the theory of multiparticle correlators by representing them as multigraphs, allowing us to obtain many useful results by manipulating their vertices and edges. Specifically, by "cutting" the multigraphs along either their vertices or edges, we derive otherwise opaque linear relations among sets of correlators. Furthermore, in special cases where the pairwise distances θ ij take the form of an inner product, we show that the complexity of computing a multiparticle correlator can be reduced from OðM N Þ down to the minimally required OðMÞ. We highlight various connections of these mathematical results to high-energy physics, including counting kinematic polynomials relevant for superstring amplitudes [1,2] and speeding up the computation of jet substructure observables [7][8][9].
A detailed outline of this paper is as follows. In Sec. II, we summarize the basics of multiparticle correlators and review the multigraph-multiparticle correlator correspondence of Ref. [9]. Several well-known examples of multiparticle correlators are discussed, including kinematic polynomials and energy flow polynomials (EFPs) [9]. We also review the computational complexity of multiparticle correlators and show how the simplification in Eq. (2) allows us to improve upon the naive OðM N Þ scaling using the variable elimination algorithm [9,11].
In Sec. III, we introduce a method to slice the vertices of multigraphs. By cutting vertices in half, the multiparticle correlator in Eq. (1) can be decomposed into contractions of "particle tensors" whose indices run from 1 to M (the number of particles). We then derive linear relations among multiparticle correlators when M is small compared to N.
The key for understanding these linear relations is a wellknown tensor antisymmetrization identity. In m dimensions, antisymmetrizing any tensor over l > m indices yields zero: This follows immediately from the fact that any assignment of the m possible values to the l > m indices must have a repetition and therefore vanish. These particle tensors also enable further computational speedups through methods such as fast matrix multiplication and dynamic programming.
In Sec. IV, we explore the consequences of making an additional simplifying assumption, namely that the pairwise distances θ ij are given by an inner product, for some vectors u μ i ; u ν j and metric η μν with the Einstein summation convention. This assumption applies, for instance, to kinematic polynomials of Mandelstam invariants and EFPs with angular exponent β ¼ 2. Having made this assumption, we can slice the multigraph edges and the multiparticle correlators can be expressed as contractions of "moment tensors." This allows us to derive new linear relations and computational speedups by cutting the edges of the multigraphs. The antisymmetrization identity in Eq. (3), now applied to the moment tensors, explains redundancies that appear when the dimension of the inner product space is small compared to the number of edges. Multiparticle correlators can now be computed in OðMÞ, since they are sewn together from moment tensors, each of which is OðMÞ to compute. This result generalizes the well-known fact that the invariant mass of a set of massless particles does not require OðM 2 Þ time to compute, since one can simply sum over the four-vectors in OðMÞ: This reduced complexity is particularly relevant for quantum algorithms for collider physics, where preprocessing and loading classical data into a quantum computer can be a key computational bottleneck [12]. In Sec. V, we show how momentum conservation can be used to trim away valency-1 leaves from the multigraphs. As an interesting physical application, we use these trimmed moment tensors to count independent kinematic polynomials up to degree d, which is relevant for enumerating superstring amplitudes [1]. By translating this problem to our graphical language, we are able to significantly extend existing integer sequences by simply enumerating leafless multigraphs, advancing previous results from d ¼ 8 [1,2] to d ¼ 16 [13,14].
In Sec. VI, we highlight the relevance of our results for collider physics. As shown in Ref. [9], the EFPs fully capture the infrared-and-collinear-safe (IRC-safe) information in the radiation pattern of an event, for any choice of angular exponent β. For the special choice of β ¼ 2, the EFPs can be written as Lorentz contractions of energy flow moments (EFMs), which are IRC safe by construction. An EFM with v indices takes the form where E i are particle energies, p μ i are massless particle momenta, and n μ i ≡ p μ i =E i . The EFMs are introduced for the first time in this paper, though we show that they are closely related to other moment tensors appearing previously in the literature. Using the EFMs, we demonstrate that the computationally expensive jet substructure observables C 2 [7] and D 2 [8], widely applied for new physics searches at the Large Hadron Collider (LHC), can significantly benefit from the computational speedups developed here. We make the fast implementations of these observables available in the ENERGYFLOW PYTHON package [15], which also provides tools to compute any multiparticle correlator.
Our conclusions are presented in Sec. VII. We derive additional linear relations for multiparticle correlators in the context of e þ e − collisions in the Appendix.

II. MULTIPARTICLE CORRELATORS
This section summarizes the basic properties of multiparticle correlators. We review the correspondence between multiparticle correlators and multigraphs, provide two key examples of multiparticle correlators relevant for highenergy physics, and discuss previously known techniques to decrease their computational complexity.

A. Correlators as graphs
As shown in Ref. [9], multigraphs are an efficient and intuitive way to represent multiparticle correlators. Each of the N sums and weight factors in Eq. (1) corresponds to a vertex, and each pairwise distance factor in Eq. (2) corresponds to an edge. These rules can be graphically summarized as The "multi" in multigraph refers to the fact that two vertices can be connected by more than one edge.
In this language, a multiparticle correlator with graph G is written as where ðk; lÞ are the pairs of vertices connected by the edges in G. As an example, consider the following multigraph and its corresponding correlator: This correlator has three sums corresponding to the three vertices of the multigraph. Each of the edges contributes a factor of θ ij to the argument, with θ i 2 i 3 absent due to no edge connecting two of the vertices. Any permutation σ of i 1 ; …; i N to i σð1Þ ; …; i σðNÞ yields an identical correlator by the symmetry of the sum structure. Graphically, this translates to the fact that isomorphic multigraphs correspond to the same correlator, which allows us to write the graphs without labels on the vertices. To quantify the number of unique analytic structures in the correlators, it is helpful to organize them by the degree d of the monomial, or equivalently the number of edges in the associated multigraph. The number of unique multiparticle correlators of each d up to d ¼ 7 is tabulated in Table I based on entries in the On-Line Encyclopedia of Integer Sequences (OEIS) [16]. As an explicit demonstration, Table II shows the expressions and multigraphs for all 13 multiparticle correlators with degree d ≤ 3. Correlators corresponding to connected multigraphs are also separately tabulated, as disconnected ones are simply the product of their connected components.

B. Kinematic polynomials
As one application in high-energy physics, multiparticle correlators arise in the study of kinematic polynomials. These are symmetric polynomials built from Mandelstam invariants: where the last equality assumes massless particles (p 2 ¼ 0). Kinematic polynomials have been used to build operator bases [3,4] for quantum field theories and understand the structure of scattering amplitudes [1]. In our notation, kinematic polynomials are multiparticle correlators of the form and so the insights we develop here are directly applicable to this context. It is worth noting that an analog of the graphical notation for multiparticle correlators was also developed in Ref. [3] and mentioned in Ref. [4]. Beyond this, Hogervorst et al. [3] remark that there are linear relations among the kinematic polynomials which can complicate certain analyses, and that a classification of those relations would be required to pursue particular strategies. We will explore these relations among multiparticle correlators at length, seeking to understand and classify these relations.

C. Energy flow polynomials
A family of multiparticle correlators relevant for collider physics is the set of EFPs [9]. Explicitly, EFPs are defined as multiparticle correlators of the form where E i is the energy of particle i, n μ i ¼ ð1;n i Þ μ for massless particles, and β is an angular weighting factor. While we use particle energies E i here to simplify the notation, in a hadron collider context these would be replaced with particle transverse momenta p Ti .
The EFPs are IRC-safe observables, which are guaranteed to be finite and computable in perturbative quantum field theory [19][20][21][22][23]. An observable is IRC safe if it is unchanged by the addition of a soft particle with E → 0 or by the collinear splitting of one particle into two with p μ → fλp μ ; ð1 − λÞp μ g for any λ ∈ ½0; 1. While Ref. [9] shows via direct computation that the EFPs are IRC safe, we will later develop a simple notation which makes this fact manifest in the β ¼ 2 case (see Sec. VI A).
For any choice of β > 0, the EFPs form a linear basis of all IRC-safe observables [9], meaning that any IRC-safe observable can be approximated by a finite linear combination of EFPs. Many common collider observables are encompassed by the EFPs [7,24] or can be cast as exact linear combinations of EFPs [25][26][27][28][29]. While it has been shown that the EFP basis can be useful for jet classification [9,30], the basis is overcomplete even for fixed β, in the sense that there are linear relations among the elements, which complicates the use of linear fitting methods. Understanding these relations is a key goal of this paper, which we will accomplish in Secs. III C and IV C.

D. Computational complexity
An N-particle correlator in Eq. (8) is naively computable in OðM N Þ by evaluating the N nested sums over M particles. This computational complexity can typically be significantly improved upon, however, by exploiting the algebraic structure of the sums, as we have detailed in Ref. [9] and summarize here. Efficiently computing multiparticle correlators is especially important for collider physics applications, where hundreds to thousands of final-state particles are produced in each collision.
The key insight is to iteratively perform the sums in a carefully chosen order. For example, we can write the following correlator in a suggestive way: ð15Þ which can be evaluated in OðM 2 Þ by computing the parenthetical quantities in OðMÞ for all M terms in the outer sum. This strategy gives a significant improvement over the naive OðM 4 Þ complexity of the correlator.
The general procedure to determine the order in which to compute the nested sums is known as the variable elimination algorithm [11]. Using variable elimination yields a computational complexity of OðM twðGÞþ1 Þ, where twðGÞ is the treewidth of G, neglecting multiple edges. The treewidth measures how "treelike" a graph is, with twðGÞ ¼ 1 for trees and higher values for other graphs. For all noncomplete graphs, twðGÞ þ 1 < N and so variable elimination provides a significant improvement in computational speed in most cases. Here, we see the graphical notation is not only useful for enumerating all possible correlators, but it also provides a natural strategy for efficiently evaluating the correlators. In the next two sections, we exploit this graphical language for further computational gains.

III. CUTTING OPEN VERTICES
In this section, we begin to surgically disassemble the multigraphs by slicing them through their vertices. This yields particle tensors whose indices index particles and allows us to view the correlators as contractions over these indices. Using these particle tensors, we can understand linear relations that emerge when the number of particles is small as well as speed up certain computations beyond what is possible through variable elimination alone. These results hold for any multiparticle correlator of the form in Eq. (8), regardless of the choice of pairwise distances θ ij .

A. Particle tensors
One can view the sums over particle indices i 1 ; …; i N in Eq. (8) as contractions of tensorial objects. We can explicitly find these tensors by slicing open the multigraph vertices. For example, the following correlator can be written suggestively as ð16Þ where the indices i 1 , i 2 , i 3 are contracted via Einstein summation notation, and we have defined the particle tensors as Note that this decomposition is not unique, and we could have represented Eq. (16) alternatively as where the three-index particle tensor is This is an example of an N-index particle tensor: where fα ab g are integers ordered by ascending b then a.
The general decomposition, demonstrated by Eq. (16), has a simple graphical interpretation as graphs with cut vertices: Here, we have sliced open a vertex into two free indices, which later will be contracted to obtain the sums. For example, the alternative decomposition in Eq. (18) involves a graph with three cut vertices: The generic N-index particle tensor in Eq. (20) can be represented by a multigraph G: which is similar to the multiparticle correlator in Eq. (8), albeit without the sums over particles and with square roots on the weights z i . We also note that the particle tensors, unlike the multiparticle correlators, depend on the particular vertex labeling of G.
For any multigraph, its vertices can be sliced in half in many different ways, each yielding a valid tensor contraction expression for that correlator. All the different ways of cutting the vertices of the triangle graph yield ð24Þ The first contraction in Eq. (24) is the trace of the cube of a two-index tensor treated as a matrix, which we can graphically represent as ð25Þ B. Computational complexity Casting the particle sums in multiparticle correlators as tensor contractions can be used to improve their computation in several ways. First, improvements in the computational complexity of individual correlators can be achieved using algorithms for fast matrix operations. While the triangle graph in Eq. (24) is OðM 3 Þ to evaluate both naively and with variable elimination, casting it as a matrix product in Eq. (25) allows us to go beyond this limit. Using Strassen's algorithm [31] yields an OðM 2.81 Þ evaluation of the correlator, and methods as fast as OðM 2.38 Þ exist [32][33][34]. For instance, Strassen's algorithm uses the structure of the matrix product to reduce the number of operations required to multiply 2 × 2 block matrices to 7 rather than 8, which gives rise to a power of log 2 7 ≃ 2.81 in its complexity when recursively applied. We will showcase this speedup for computing collider observables in Sec. VI C.
Beyond improving individual correlator evaluation, the constituent particle tensors can be used to improve the computation of collections of correlators. Many correlators can be built out of the same particle tensors contracted in different ways. This allows for a dynamic programming approach where various particle tensors are computed and then reused many times to evaluate additional correlators. While determining the optimal set of subgraphs to compute for a target collection of correlators is beyond the scope of this work, we note that it is in an interesting avenue for further exploration.

C. Finite-particle linear relations
When there are small numbers of particles, algebraic relations among the multiparticle correlators emerge due to the simplified nature of the configuration. Relations of this type were also explored in Ref. [1] for the purpose of expanding superstring amplitudes. In the collider context, understanding these relations is also important for calculating EFPs in fixed-order perturbative quantum field theory with small numbers of particles. In general, understanding these finite-particle relations can improve the computation of collections of correlators by avoiding the computation of redundant elements. Further, these relations can involve multigraphs of different computational complexities, allowing for individual correlators to be evaluated more efficiently via these relations.
The tensor identity in Eq. (3) concisely encodes many well-known results, including several Riemann tensor identities as well as the Cayley-Hamilton theorem [35]. The second fundamental theorem of invariant theory indicates that any identity among a set of tensors in m dimensions, aside from ones governed by their existing symmetries, can be obtained as a consequence of Eq. (3) [36].
For the case of particle tensors with M particles, antisymmetrizing over any choice of L > M indices must vanish. We can translate this fact into the language of cutting graphs. Graphically, we denote the antisymmetrization over an index by a bracket through a vertex whose direction indicates which of the two particle tensors to use, and the vertex itself represents contracting the indices. The sum over all permutations of those indices, weighted by the sign of the permutation, must vanish by Eq. (3). After contracting the indices, each permutation gives rise to an associated multigraph, so this procedure gives rise to an alternating sum of multigraphs which must vanish. Some choices of antisymmetrizations will vanish trivially by the symmetries of the tensors, but many do not trivially vanish and give rise to nontrivial algebraic relations.
To showcase this approach, in Table III we enumerate all of the finite particle linear relations for the case of M ¼ 2 and d ≤ 3. For example, the first row of Table III corresponds to the relation where we have raised and lowered indices to make the summation convention more clear. This procedure can be extended to enumerate the relations for larger numbers of particles.

IV. CUTTING OPEN EDGES
In this section, we show that the structure of multiparticle correlators can be dramatically simplified if the pairwise distances θ ij can be written as an inner product. In particular, this allows us to slice open the multigraph edges to obtain moment tensors, resulting in further computational speedups and a greater understanding of various linear relations.

A. Moments from an inner product
The following results assume that θ ij can be written as an inner product. Repeating Eq. (4) for convenience, for some vectors u μ i ; u ν j and (symmetric) metric η μν . This assumption holds for the kinematic polynomials in Eq. (12) and for the EFP angular measure in Eq. (14) for β ¼ 2. While we use the notation of Lorentz indices and the Minkowski metric here, the conclusions hold for any inner product among vectors using any choice of metric.
With this assumption, any multiparticle correlator can be written via tensor contractions, where in this case the indices are Lorentz indices rather than particle indices. To that end, we introduce the following moment tensors: where v is the rank of the tensor. We apply this moment logic to kinematic polynomials in Sec. VA and to β ¼ 2 EFPs in Sec. VI A.
A simple example demonstrates the connection between multiparticle correlators and these moment tensors. Consider the following equivalent ways of writing the simplest 2-correlator: where the first line uses the definition of a multiparticle correlator and Eq. (27), and the last line is explicitly a contraction of moment tensors from Eq. (28) with Einstein summation notation. This is the generalization of the familiar mass identity in Eq. (5), where the inner product allows us to separate the factors into tensors with only a single Lorentz index.
We now generalize this procedure to demonstrate that all multiparticle correlators can be obtained from these moment tensors when the inner product condition holds. The general strategy is to regroup the summand of a general correlator into a product of factors, one for each sum index. The resulting expression manifestly factors the dependence on the constituent particles into tensorial objects contracted according to the multigraph. Substituting Eq. (27) into Eq. (8), we have where v j is the valency of vertex j, μ a b is an index corresponding to the bth edge associated with vertex a, and A is a matrix such that the A kl th instance of u μ i k is contracted with the A lk th instance of u μ i l . Demonstrating their continued usefulness, multigraphs provide a simpler recipe for translating between multiparticle correlators and moment tensors than the unavoidably opaque notation of Eq. (30). Each vertex in the graph is associated with a moment having v indices, where v is the valency of the vertex. The edges in the graph specify which vertices are connected, and hence they encode the contractions between indices of different moments. These rules can be summarized as ð31Þ ð32Þ Graphically, we have cut the edges of the multigraphs in half to obtain these tensors.
As an example, consider the following multigraph expressed both as a multiparticle correlator and in terms of contractions of moments: ð33Þ ð34Þ where pairs of indices are contracted with the metric. Interestingly, these graphical rules are closely related to Penrose graphical notation for tensor contractions [37].

B. Computational complexity
The moments in Eq. (28) are symmetric tensor structures summed over M particles, and hence are OðMÞ to compute. Since the tensor structures are fully symmetric, they only have ðn þ v − 1Þ!=v!=ðn − 1Þ! independent components, where n is the dimensionality of the inner product vectors u μ i and v is the rank of the tensor. The number of independent components is therefore polynomial in the rank v as opposed to exponential. In 3 þ 1 spacetime dimensions, n ¼ 4 and there are ðv þ 3Þðv þ 2Þðv þ 1Þ=6 independent components. Since this is independent of M and v 3 is smaller than the number of particles for many events and multiparticle correlators of interest, computing the moment tensors is typically computationally efficient.
Since any multiparticle correlator with an inner product structure can be written as a contraction of moment tensors via Eq. (30), all such correlators become OðMÞ to compute. The tensorial rearrangement into moments provides a way to circumvent the previous computational limits and achieve linear complexity. This can be seen explicitly in Eq. (29), where an OðM 2 Þ tree graph is written as a contraction of two tensors that are each OðMÞ to compute. Since the computation of any correlator must probe all particles at least once, the linear complexity in M of computing the correlators via the moments is optimal. Thus, by computing all of the particle moments up to a desired order v, all correlators can be obtained via tensor contractions which are independent of M and depend only on the dimensionality of the space. We provide an explicit demonstration of this computational improvement for collider observables in Sec. VI C.

C. Finite-dimension linear relations
When the dimensionality of the inner product space is small, there are additional linear relations among the multiparticle correlators. These relations were pointed out by Ref. [3], and we now seek to understand them systematically. Similar to the finite-particle relations in Sec. III C, the key is to figure out the appropriate way to apply Eq. (3) to antisymmetrize over more indices than dimensions.
In this case, we antisymmetrize over the inner product (Lorentz) indices instead of the particle indices. We specialize to the cases of n ¼ 4 and n ¼ 3 dimensions for the following discussion, since these are the dimensionalities relevant for Sec. VI and the Appendix, respectively. It is worth noting that n-dimensional identities continue to hold in fewer than n dimensions.
The Cayley-Hamilton theorem is a special case of Eq. (3), so we will first explore this as a method to understand the finite-dimension identities. The Cayley-Hamilton theorem, as typically stated, says that a matrix A satisfies its own characteristic polynomial. Using the Newton identities, this translates to a matrix relation between powers of A, the trace of powers of A, and the determinant of A. Written out explicitly for small matrices, we have for 2 × 2 matrices where I is the identity matrix, and for 3 × 3 matrices For our first n ¼ 4 identity, we apply the 4 × 4 Cayley-Hamilton theorem to the matrix where ffiffi ffi η p is defined because η is symmetric.

Multiplying the resulting Cayley-Hamilton relation by the matrix and taking a trace yields
where we assume that M μ μ ¼ 0, as in our cases of interest due to the masslessness of the particles, with the full expression being easily obtainable but unwieldy. Note that Eq. (38) could also have been obtained directly from the antisymmetrization identity Higher-dimensional versions of the Cayley-Hamilton theorem can be used to derive relations for larger 2-regular graphs (hexagon, heptagon, etc.).
In addition to identities involving 2-regular graphs, we can also multiply the expression of the Cayley-Hamilton theorem on both sizes by ffiffi ffi η p μν M ν in order to produce relations among chainlike graphs, such as

V. TRIMMING THE LEAVES
When there are simple global constraints on the particles, their effects on multiparticle correlators can often be made clear graphically. In this section, we consider cases where center-of-momentum relations allows us to "trim the leaves" of a multigraph, which gives us a way to count kinematic polynomials from Ref. [1]. More exotic graphical rules are relevant for multiparticle correlators in e þ e − collisions, which we study in detail in the Appendix.

A. Revisiting kinematic polynomials
As anticipated in Sec. IVA, kinematic polynomials can be written as contractions of moment tensors. With massless particles, the relevant moment tensors take the following form: with the prefactor accounting for the factor of two in the Mandelstam invariants s ij ¼ 2p μ i p jμ . Following the logic of Sec. IVA, all Lorentz-invariant kinematic polynomials can be built from complete Lorentz contractions of these moments.

B. Center-of-momentum relations
Kinematic polynomials can be further simplified when imposing energy-momentum conservation P i p μ i ¼ 0. Written in the generic multiparticle correlator language, this corresponds to the following constraint: Since M μ corresponds to a vertex connected to a single edge, we can write this constraint graphically as Therefore, momentum conservation allows us to restrict our attention to "leafless" graphs, whose minimum vertex valency is greater than 1. Leafless graphs are also relevant for studying EFPs on a collection of final-state particles. It is sometimes convenient to work in the center-of-momentum frame with the constraint where E is a fixed energy in this frame, corresponding to the graphical rule In the e þ e − case where u 0 i ¼ 1, valency-1 vertices simply contribute factors of E and we have Hence only leafless multigraphs need to be considered in the center-of-momentum frame for EFPs in the case of e þ e − collisions.

C. Counting superstring amplitudes
There is an interesting dividend of our graphical understanding of the center-of-momentum relation in Eq. (43). Reference [1] sought to count the number of independent, symmetric polynomials of kinetic variables of degree d, which is relevant for determining five-point superstring amplitudes. They worked in the limit of many particles with many spacetime dimensions (i.e., without finite spacetime dimension identities). Using an interesting technique based on Molien series, they conjectured the counts for d ≤ 8. In our language, counting these independent polynomials of degree d in the center-of-momentum frame corresponds to simply counting leafless multigraphs with d edges.
We can efficiently enumerate and count the number of leafless multigraphs using NAUTY [38], with our results summarized in Table IV. Our sequence agrees with the results of Ref. [1] where they overlap [2], but we are able to overcome previous computational limitations and double the number of known terms. These values have been added as new sequences in the OEIS [13,14]. Enumerating multigraphs not only allows us to efficiently count these polynomials, but it also provides explicit constructions of them, which may be useful in further exploring the space of kinematic polynomials.

VI. APPLICATIONS TO COLLIDER PHYSICS
In this section, we apply the general lessons that we have developed to obtain concrete computational and conceptual improvements for collider observables. 1 We first introduce EFMs as novel tensorial structures to efficiently encode and compute β ¼ 2 EFPs, and we show their relation to previous moment tensors in the literature. We then exploit the results of Sec. IV B to provide a maximally efficient OðMÞ implementation of key jet substructure observables.

A. Introducing energy flow moments
With β ¼ 2, the pairwise distance measure of the EFPs in Eq. (14) has an inner product structure. Following Sec. IVA, the β ¼ 2 EFPs can therefore be written as contractions of EFMs defined in Eq. (6), repeated for convenience: where n μ i ≡ p μ i =E i , and the prefactor accounts for the factor of 2 in θ ij . The above definition is suitable for e þ e − collisions, whereas in a hadron collider context, one would typically replace energy E i with transverse momentum p Ti in the definitions of both I μ 1 ÁÁÁμ v and n μ i . The fact that the β ¼ 2 EFPs are IRC safe is immediately clear via inspection of Eq. (47). Due to the linear energy weighting, an EFM is manifestly invariant to the addition of a zero-energy particle or the collinear splitting of one particle into two. To form the β ¼ 2 EFPs, we need only Lorentz contract the EFMs, which respects this IRC-safe structure.
As an aside, parity violation is interesting to explore at colliders [40][41][42][43][44][45][46][47][48]. By invoking the ϵ symbol, we can construct parity-violating contractions of the EFMs that go beyond the parity-invariant EFPs. Two simple parityviolating observables of potential interest are ϵ αβγδ I α I βρ I γστ I δ ρστ ; which are manifestly permutation symmetric, rotationally invariant, and IRC safe. Further investigations in this direction are left to future work and we restrict ourselves here to contractions with the Minkowski metric.

B. Relation to existing moments
The EFMs are closely related to existing moment-based approaches for collider physics, dating back several decades. Here, we show that EFMs directly encompass or suitably generalize these approaches. IV. The number of multigraphs with d edges that have no vertices of valency one (i.e., are leafless), tabulated for both connected and all graphs. These have been added to the OEIS as new sequences [13,14]. We conjecture that the sequence for all graphs is the same as A226919 [2], which was discovered in a string theory context [1]. This section also explains two cryptic comments we made in previous papers: footnote 8 of Ref. [9] and footnote 4 of Ref. [39]. A number of tensorial quantities similar in spirit to the EFMs have been previously defined and explored for both e þ e − and hadronic collisions. In Ref. [49], the following objects built out of particle three-momenta were defined for e þ e − collisions: wheren i ¼ ⃗p i =E i is a unit three-vector and j k ∈ f1; 2; 3g. We refer to the tensors defined by Eq. (49) as "generalized sphericity tensors," since the second-rank tensor Θ 2 ¼ Θ j 1 j 2 is often called the sphericity tensor. In an e þ e − context, the spatial components of the EFMs are exactly these tensors: and, like the EFMs, the generalized sphericity tensors are IRC safe. The second-rank sphericity tensor Θ 2 ¼ Θ j 1 j 2 has seen significant application in the form of defining observables from its eigenvalues. The C and D parameter event shapes are defined in terms of the eigenvalues λ 1 , λ 2 , and λ 3 of the sphericity tensor Θ 2 as These IRC-safe observables have been widely analyzed for e þ e − collisions both theoretically [49][50][51][52][53] and experimentally [54][55][56][57][58].
To highlight the close relationship with EFMs, we note that the C and D parameters are themselves both linear combinations of EFPs and contractions of EFMs:

ð52Þ ð53Þ
We derive these expressions in the Appendix. In this graphical notation, we leave factors of total energy implicit: ð54Þ which must be included whenever graphs with different numbers of vertices are added together.
The third-rank tensor Θ j 1 j 2 j 3 has seen additional study [59], but the general formulation of Eq. (49) has seen very limited application at the LHC. One reason for this may be that is not clear how to use the tensors in Eq. (49) in a hadronic context: the factor of energy can be promoted to transverse momentum but the explicit appearance of the three-momenta prevents covariance under boosts and rotations about the collision axis. The EFMs are precisely the generalization of the generalized sphericity tensors that applies in both the e þ e − and hadronic contexts. Including the energy component of the particle four-momenta avails a hadronic interpretation simply by replacing all instances of energy with transverse momentum. Relations among these Θ j 1 j 2 ÁÁÁj v tensions are studied further in the Appendix.
Other constructions of moment tensors have been proposed solely in the hadronic context. In Ref. [60], a set of transverse-momentum-weighted moments in the rapidity-azimuth (y; ϕ) plane were defined for narrow jets, and a number of substructure observables were identified in terms of their products and contractions. Letting x i ¼ ðy i ; ϕ i Þ indicate the rapidity-azimuth position of particle i, and letting k j ∈ f1; 2g be rapidity-azimuth indices, these hadronic moment tensors are The two-index moment I k 1 k 2 has been used to define the planar flow observable [61,62], which quantifies the linear versus planar nature of the radiation pattern. When boosting a jet to be central (y ¼ 0) and taking the narrow limit y i ; ϕ i ≪ 1, x i become precisely two spatial indices of the EFMs with the hadronic prescription of z i ¼ p T;i . The tensors of Eq. (55) were defined in Ref. [60] relative to a jet axis, which we neglect here (by boosting to y ¼ 0) to make the relation to EFMs immediate. Choosing a p T -weighted centroid axis, a related analysis carries through with a jet axis included. In this way, the EFMs generalize I k 1 ÁÁÁk n beyond the narrow jet limit and do not require referencing a jet axis.
Thus, we see that the EFMs can be related to moments developed for both e þ e − and hadronic collisions. Unlike these previous moments, which were developed for a specific type of collision, EFMs can operate in both realms by simply exchanging energies for transverse momenta. In this way, EFMs are a general framework for moment-based approaches to collider physics analyses.

C. Speeding up multiprong taggers
Using the results of Sec. IV B, EFPs with β ¼ 2 are manifestly OðMÞ to compute through the EFMs. This provides a significant computational speedup over approaches based on variable elimination; see Sec. II D.
This computational speedup also applies to substructure observables derived from β ¼ 2 EFPs. A number of different observables have been proposed to tag jets with multiprong hadronic substructure at the LHC, including two-pronged decays of boosted W bosons and three-pronged decays of boosted top quarks, along with other more exotic scenarios [63,64]. These jet substructure observables have since found significant experimental application at the LHC for multiprong tagging and new physics searches [65]. While most current LHC analyses focus on β ¼ 1 observables, comparable (or better) performance can be obtaining using β ¼ 2 for standard tagging applications [7,8].
As concrete examples, Ref. [7] established a family of N-prong tagging observables C ðβÞ N , which can be written as ratios of EFPs with complete graphs [9]. For two-and threeprong tagging, these dimensionless ratio observables are ð56Þ where factors of can be restored by ensuring equal numbers of vertices in the numerator and denominator. Note that these observables are traditionally defined using the rapidity-azimuth distance rather than Eq. (14), though these are equivalent for narrow jets and our OðMÞ moment logic can be applied in both cases.
These C N observables are in general OðM Nþ1 Þ to compute, which becomes quickly computationally intractable with increasing N. Using power counting arguments, Ref. [8] established a different combination of energy correlators as an improved two-pronged tagger called D ðβÞ 2 , given by the expression ð57Þ which naively requires OðM 3 Þ to compute. These observables can be measured before or after grooming [66][67][68][69][70] is applied, as they operate on generic sets of particles. Note that the jet substructure observable N 2 cannot be expressed as an exact combination of EFPs, since the angular structure of the generalized correlators in Ref. [24] does not take the form of Eq. (2).
In the case of β ¼ 2, these tagging observables can all be computed in OðMÞ using our moment-based approach. Further, C 2 and D 2 benefit (for any β) from the fast matrix multiplication speedups to OðM 2.81 Þ discussed in Sec. III B.
In Fig. 1, we compare the overall computational time of C ðβ¼2Þ 3 and D ðβ¼2Þ 2 using the naive approach and the momentbased approach. We also show the fast matrix multiplication approach for computing D 2 using Basic Linear Algebra Subprograms [71] via NUMPY [72,73] on eight CPU cores. (The use of multiple cores merely shifts the curve vertically and does not alter the asymptotic scaling.) For comparison, we include the computational time of a single 2-correlator (relevant for one-prong quark/gluon discrimination [8]): ð58Þ which has a naive scaling of OðM 2 Þ.
We can see the significant benefit of linear computational complexity, resulting in very large practical speedups for realistic numbers of particles at the LHC (i.e., 100-1000). Furthermore, this opens the door to the use of higher-N correlators for collider physics, which until now have appeared computationally prohibitive. The OðMÞ implementations of all these observables are available in the ENERGYFLOW PYTHON package [15].

VII. CONCLUSIONS
A greater understanding of multiparticle correlators has profound implications for fields ranging from physics to applied mathematics. In this paper, we have investigated multiparticle correlators, attempting to understand and categorize linear relations that appear among them as well as developing faster computational techniques to improve their practical application.
When the summand of a multiparticle correlator satisfies the monomial form in Eq. (2), the correlator has a multigraph structure that enables us to effectively represent and manipulate it. This was used already in Ref. [9] to apply the variable elimination algorithm to speed up the computation of many multiparticle correlators for collider physics beyond the naive expectation. Here, we have gone significantly further by developing techniques to "cut" the multigraph vertices and edges.
By slicing open the vertices of correlator graphs, we have arrived at particle tensors and demonstrated that each correlator can be written as a contraction of these tensors. The fundamental tensor antisymmetrization identity in Eq. (3) resulted in an infinite class of linear relations that hold when the number of particles M is small compared to the number of vertices N. The existence of faster-thanexpected tensor operations, such as fast matrix multiplication, has allowed us to reduce the computational complexity of correlators in general. These computational and conceptual strategies extend beyond correlators with monomial summands, applying also to noninteger exponents or indeed products of any pairwise interparticle quantities.
We have also sliced open the edges of the multigraphs when the pairwise distances had the inner product structure of Eq. (4). This assumption holds for kinematic polynomials as well as for EFPs with β ¼ 2 in both the e þ e − and hadron collider contexts. Cutting the edges has resulted in moment tensors that carry the indices of the inner product space. Applying the tensor antisymmetrization identity has led to another infinite class of linear relations arising when the dimensionality of the inner product space (e.g., four spacetime dimensions) is small compared to the number of edges d. We have also made contact with efforts to analytically enumerate structures appearing in superstring amplitudes, deriving a method that has allowed us to significantly extend previous results.
In a collider context, the moment tensors yield the EFMs. The EFMs are novel tensorial structures that provide a natural and efficient way to compute collider correlators, unifying and encompassing existing moment-based approaches in the collider physics literature. We have demonstrated a significant speedup of widely used jet substructure observables using the graphical techniques that we have developed. We expect that the EFMs will be a useful development for experimental applications and theoretical calculations of multiparticle correlators in collider physics.
For instance, the square graph has the following meaning in terms of the sphericity tensor: ðA5Þ Note that Tr½Θ 2 ¼ 1, and the factors of 2 come from the spatial part of our 2δ ab convention. In general, the traces of powers of Θ 2 give rise to 2-regular graphs.
Next, we apply the 3 × 3 Cayley-Hamilton theorem from Eq. (36) to Θ 2 . Multiplying Eq. (36) through by Θ 2 , we have Substituting in for the det Θ 2 [from tracing over Eq. (36)] and taking the trace of this expression gives rise to one identity. Multiplying Eq. (36) on both sides by Θ 1 gives rise to a separate identity. These two identities are ðA8Þ In this analysis, we used a Cayley-Hamilton approach for simplicity. In general, these graphical identities can be systematically derived from the tensor antisymmetrization identity in Eq. (3) by enumerating the possible antisymmetrizations (here, over four or more indices) of each graph structure.
To translate these identities among sphericity tensors into identities among EFPs, we develop a look-up table which writes out each contraction of sphericity tensors as a specific linear combination of EFPs. This can be worked out in the moment picture by recursively writing a Lorentzian contraction as a Euclidean contraction plus an additional term of opposite sign (with the appropriate factors of 2). Our results are summarized in Table V for all graphs up through the connected graphs with four edges. With this table, we can quickly translate between the sphericity and energy flow pictures. As an example, the identities in Eqs. (A7) and (A8) give rise to the following identities among the EFPs:

ðA9Þ ðA10Þ
Hence the finite dimension identities of the three-dimensional sphericity tensor induce related identities among the EFPs, with an interesting interplay between dimensionalities.
We can use similar reasoning to make contact with the C and D parameters. From Eq. (51), these event shapes are related to the trace and determinant of the sphericity tensor via C ¼ 3 2 ððtr ΘÞ 2 − tr Θ 2 Þ and D ¼ 27 det Θ. Using Eq. (36) to express the determinant of Θ, we can write these observables in terms of the sphericity graphs in the following way:

ðA11Þ ðA12Þ
We can use Table V to translate Eqs. (A11) and (A12) into the energy flow picture. Doing so yields the EFP expressions for the C and D parameters already presented in Eqs. (52) and (53). V. All contractions of generalized sphericity tensors (left column) as linear combinations of EFPs (top row). The entries of the table are the specific linear coefficients of each EFP for a particular sphericity contraction. This table can be used to translate between linear combinations of sphericity contractions and EFPs by matrix multiplication. Shown are all graphs up to the connected graphs with five edges, though in principle it can be extended arbitrarily. Note that a single diagonal of ones continues for the d ¼ 4 graphs and is not shown explicitly.


2D Face Recognition on CNN with Half-Average-Face

As the size of training dataset of face recognition models becomes larger and larger, we are interested in a method called Average-Half-Face (AHF), which could halve the size of training samples. The AHF method divides a full face into two halves and then averages them together (reversing the columns of one of the halves). We preprocess the dataset with the method of AHF, and train them on two different models, Eigenfaces and Convolutional Neural Network (CNN). We compare the prediction results with those models trained on the original dataset. Previous researches showed that AHF is superior to Full-Face (FF), while our experiment results further showed that in some cases AHF also boosts CNN. The application of AHF can bring both saving in storage and reduction on training cost time.

Introduction
Face recognition has been extensively researched since 1960s [1]. For years of development, it has become one of the most well-known applications currently. Large industries in different countries are benefiting from this technology, from mobile phone human-face detection, to automatic surveillance in public places. However, digital images in recent years are larger and larger, up to several MB. For devices limited in processing power and storage, it is necessary to develop an optimized algorithm to carry on face recognition. The idea of Average-Half-Face (AHF) [2] was proposed to make use of the inherent bilateral symmetry of human faces. Due to the symmetry of human face, the left and right sides of a face have lots of overlapped features, which could be eliminated by averaging the two half-faces. Though this concept initially focused on extracting facial profiles, it has the side effect of decreasing size of images and accelerating training cost time, while giving an even better output than the traditional Full-Face (FF) approach.
However, the dataset [3] for experiments in that paper is relatively symmetric, which is ideal when comparing to real world cases. To simulate a more realistic situation, we employ the AHF and Eigenfaces algorithm on a more asymmetric dataset, the ORL Database of Faces [4]. Figure 1 displays a face image on the ORL Database, and the AHF expression of this image. (b) AHF expression. Figure 1. One example of a face image from the ORL database and its output after being preprocessed by AHF algorithm Figure 1 (b) shows how AHF averages two half faces, which may not align concisely. If the full face tilts, or the face expression or the face features (like a crooked nose) are asymmetric, the AHF expression would have ghosting. It raises a question: whether the loss of the information due to AHF would lead to a decrease in accuracy? According to the experiment result of Satone's work [5], the answer might be 'no'. AHF approach still performs a little better than FF approach in terms of accuracy. As the creator of AHF mentioned, it was still not clear what other factors were leading to the increase in accuracy [2]. As well, using only left or right half-faces, the outputs are worse than that of Full-Face [6]. Therefore, we can assume that the reason behind the increase in accuracy is far from the loss of information due to dimension decrease.
On the other hand, considering the development of CNN's applications on face recognition, we are interested in the performance of AHF based on CNN. To discover it, we did a cross-validation both horizontally and vertically on PCA and CNN, AHF and FF. That is, we built four models, PCA+AHF, PCA+FF, CNN+AHF, CNN+FF, and then compare PCA+AHF with PCA+FF, PCA+AHF with CNN+AHF, for example. By these experiments, we evaluate the application of AHF on face recognition CNN.
In this paper, we first give a short description on AHF and our face recognition approaches. Next, we carry out related experiments, and then present results that demonstrate the accuracy gains of AHF.

Average-Half-Face
The idea of face recognition using a half face comes from the fact the human face is roughly symmetrical. Zhao and Chellappa [7] developed a face recognition system based on face symmetry, which alleviates the effect of variations in illumination. Ramanathan et al. [8] used the notion of 'half-faces' to circumvent the problem of non-uniform illumination on measuring face similarity. Chen [9] combined the idea of half-face with face-template. In 2008, Harguess [2] came up with the idea of AHF. He derived it from the use of the Symmetry Preserving Singular Value Decomposition (SPSVD) [10].
Generally, AHF is featured by this 4-step preprocess: 1. Centralize the face. This is done by finding the location of the tip of the nose. If the face tilts, we centralize the face by finding a straight line that can divide the face into two halves of the same amount of pixels.
2. Partition the image into two equal left and right halves. 3. Reverse the columns of the left half-face. 4. Add two half faces and average them. This preprocess aims to decrease the number of columns of an image, while the loss of information is relatively little. Due to the fact that the two sides of a human face are symmetric, the mean-squared error of the difference of two half faces is low. To minimize this error, we need to centralize the image first. Then, we reverse the left half-face so that two halves are aligned before averaging. After that, we carry out the operation of averaging, which was proved a better way than just keeping the left half or the right half [2].
After preprocessing, everything is quite the same as what are indicated in a traditional Eigenfaces algorithm [11].
Satone [5] modified the plain AHF algorithm by adding steps of wavelet transform. She achieved an impressive promotion on error rate on the ORL database.

Eigenfaces
Eigenfaces [11] came up early and has played an important role in face recognition and detection. Egienfaces is considered as a use of principal components analysis (PCA). It projects face images onto the principal components of the original set of training images, in order to extract features and decrease dimension. Then face images of individuals are compared (by Euclidean distance, typically) on this subspace of principal components.
Eigenfaces has been proved a powerful tool to carry out face recognition. Lawrence [12] performed experiments using eigenfaces algorithm on the ORL database and reached a 10.5% recognition error rate when the number of eigenfaces he used was between 175 and 199.
Several extensions or related methods are available. Multilinear principal components analysis (MPCA) [13] applies PCA to tensors or multilinear arrays. Furthermore, MPCA could be extended by using Linear Discriminant Analysis (LDA) [14] on the projected multilinear arrays to perform feature selection. Independent component analysis (ICA) [15], a generalization of PCA, tries to identify high-order statistical relationships between pixels to form a better set of basis vectors.

Convolutional Neural Network
Though Neural Network (NN) begin to catch everyone's eyes not long ago, its researches of application on face recognition started at an early time. In an early comparison [16] between eigenfaces, elastic matching [17] and NN, NN did not perform well, and was upper bounded by eigenfaces. However, with an approach of convolutional neural network (CNN) [12], NN did have an astonishing performance on the ORL database, beating eigenfaces and Hidden Markov models [18].
After years of development, CNN has become a standard method in computer vision. Due to the availability of very large scale training datasets [19], and computational resources eligible to train a multi-layers CNN, CNN performs much better than the past. With the help of the Deep Neural Network (DNN) architecture, the state-of-the-art of face recognition has been significantly advanced [20]. In recent years, Deep Convolutional Neural Network has been applied on mobile devices [21], showing the necessity to improve the processing power of mobile devices and to optimize face recognition algorithms.

ORL database
The ORL database [4] we used in this paper, contains a set of faces taken between April 1992 and April 1994 at the Olivetti Research Laboratory in Cambridge, U.K. There are totally 400 images, while there are 40 individuals and 10 images for each one. Unlike the Yale face database [3], there are more variations, including facial expression, illumination and face angles. All the images were taken against a dark homogeneous background. Each image is greyscale with a resolution of 92 X 112.

Input Resizing
Each image in ORL database would be firstly resized from 92 X 112 to 46 X 56 (halved width and halved height), in order to accelerate training. Then, a half face image, as its name indicates, has a halved width 46 * 0.5 = 23. The size of it is 23 X 56. To compare with half face directly, we will also scale the FF image to 32 X 40, keeping the aspect ratio while the amount of pixels is almost the same as that of half face images. The 23 X 56 input and 32 X 40 input have a similar training speed.

Eigenfaces
There is no special detail in our Eigenfaces code. It projects face images onto the principal components of the original set of training images, in order to extract features and decrease dimension. Then face images of individuals are compared (by Euclidean distance, typically) on this subspace of principal components. The document of OpenCV [22] is referred to be checked.

CNN
Our model is quite the same as keras's exemplary CNN model to recognize the MNIST database [23], which has a good performance on simple small-size image datasets. Table 1 is an overlook of this model.

Results and Discussions
As we mentioned, there are 400 training images totally, and 10 for each of 40 subjects. From the 10 images of each subject, we randomly take as trainset, and as (10 − ) testset. We trained several models on the trainset and tested them on the testset. Table 2 shows the accuracy of these models.
On CNN, we fix the learning rate at 1.0, and both train 90 epochs. The evaluation metric is accuracy. Table 2 shows the accuracy of each models, while x alters in the range from 3 to 9( = 3, = 5, = 7, = 9). Every model was cross-validated. 980 Generally, with the increase of , the accuracy increases on every model. Due to the limited size of testset, the accuracy does not look consistent on some . For example, when = 3, CNN+FF surpasses CNN+FF (half_size). However, when = 5, CNN+FF (half_size) is better. In fact, their performances are almost the same, and the accuracy fluctuates because the small size of testset. Overall, all have good performance when is big. When ≥ 7, every model has an accuracy score more than 0.950. However, we can see that the HAF-related models are quite stable. No matter it is Eigenfaces+HAF or CNN+HAF, it has a higher accuracy score on every than its counterparts (FF and FF (half_size)). We also noticed that only the CNN+HAF performed not worse than its Eigenfaces counterpart. The other two, FF and FF (half_size), were beaten by their Eigenfaces versions. CNN+HAF even predicted better than Eigenfaces+HAF when = 9 , proving its potential to recognize faces in an efficient way.
On the other hand, CNN often has a large scale of parameters, and the time spent on training/testing the CNN is usually long. In a CNN, the theoretical time complexity of all convolutional layers given by a related work [24] is: (1) Here is the index of a convolutional layer, and is the depth (number of convolutional layers). is the number of filters in the ℎ layer. −1 is also known as the number of input channels of the ℎ layer.
is the spatial size (length) of the filter. is the spatial size of the output feature map. Reducing the size of input by half, the time costed on training/testing convolutional layers would be four times less if the computational overhead is not omitted.
In our experiments, Table 3 shows how many times were cost on every model. HAF-related models have the least time cost, and FF-related (half_size) models are the second. The FF-related models are the worst. CNN+HAF took about 43.1% time compared with CNN+FF. It could be concluded that the time cost and the input size are positively related. Therefore, we can see the advantage of adapting traditional full-size images to halved ones to train models. In spite of shorter training time, HAF performs better than FF. It is inspiring because we are able to train on some certain datasets with less time but get better performance.  Figure 2 shows the learning curves of CNN on different input shapes. We can see that FF, HAF and FF (half_size) are all about to converge around 60 epochs.

Conclusion
In this paper, we presented an experimental evaluation of the adaptation of AHF on CNN. The models were evaluated using different number of training images and test images by cross-validation.The experiment showed the idea of Half-Average-Face could also work well on CNN, and it brought improvements on both prediction accuracy and training time.


s2fieldsofstudy:
Computer Science
Business

text:
Robotic Standard Development Life Cycle in Action

Robotics is a fast-growing field which requires the efficient development of adapted standards. Hence, in this paper, we propose a development methodology to support the robot standardization effort led by international, technical, and professional associations such as the Institute of Electrical and Electronics Engineers (IEEE). Our proposed standard development life cycle is a middle-out, iterative, collaborative, and incremental approach we have successfully applied to the development of the new IEEE Ontological Standard for Ethically Driven Robotics and Automation Systems (IEEE P7007 Standard).

Introduction
Nowadays, robots are present in the most varied types of environments where they have to perform tasks which have been exclusively done by humans until recently, e.g. in the elderly care sector.  9 Monash University, Melbourne, Australia 10 IBM Research, Rio de Janeiro, Brazil Hence, robotic standard development is a high-priority task [16]. Indeed, standards represent a consensual view of a particular subject, associated to technological solutions, human or environment safety, good practices, etc. These are elaborated by official Working Groups (WGs), commonly associated with international Standard Development Organisations (SDOs), such as IEEE, 1 IEC, 2 or ISO. 3 WGs are formed by stakeholders from different domains and from different horizons such as research, industry, or government [5]. During the development of a standard, all the stakeholders have an equal opportunity to contribute to it; the process being governed by the leaders appointed by the related SDO. In particular, the IEEE WGs operate under the procedures and policies defined by IEEE, which have as core five basic principles, namely, openness, due process, balance, right to appeal, and consensus.
For IEEE, standards are essential to advance global prosperity and well-being through the promotion of technological innovation. For example, developing standards to define how robots can interact properly with humans [19] aims to provide end-users with some guarantee that the robot can interact safely and ethically. Indeed, in domains such as elderly care technologies or elderly assistive living, it is important robots do not become a threat, e.g. to privacy, daily interpersonal contact or citizens' control over their own lives, and avoid any negative impact on people, like e.g. the elderly feeling of being treated like an object rather than a human.
At the moment, in Europe, matters related to Ethics and Robotics could be addressed through the General Data Protection Regulation (GDPR), 4 but outside Europe, they should be addressed through policies or other contractual solutions because of the current lack of international ethical standards in the robotic field. It is also evident that some ethical rules can be provided by the Law, but in certain cases, they might result in policies or agreements.
On the other hand, standards that tackle the interaction with humans are few. Moreover, they are mainly concerned with the safety issues of robots, e.g.  [13]. Safety has been thus discussed in international standardization organizations, but no ethical standard has been published yet. Thence, there is an urgent need to fill this gap by developing new standards such as the IEEE P7007 Ontological Standard for Ethically Driven Robotics and Automation Systems, 5 which is an ontological standard focused on robotic ethics.
Besides, ontological standard [10] development is paramount, because ontologies allow to capture and represent consensual knowledge in an explicit and formal way, independently of a particular programming language. Moreover, ontologies explicit the relevant knowledge about a domain in a computer-interpretable format, facilitating automatic reasoning about that knowledge, in order to infer new information [14]. Furthermore, ontology standards enable shared commitments across multiple domains and between independently developed applications. In addition, ontologies are an efficient approach to disambiguate knowledge used among groups of humans, robots, and other artificial systems that share the same conceptualization. This point is one of the major advantages of ontological standards when compared to non-ontological ones [21].
The development process of a domain-specific ontology needs to be completed in a consistent and systematic way [11] to fulfill its goals, since only high-quality ontologies 4 General Data Protection Regulation (GDPR): Regulation (EU) 2016/679 of the European Parliament and of the Council of 27 April 2016 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data. (Entered into force on 25 May 2016, but it shall apply from 25 May 2018.) 5 https://standards.ieee.org/develop/project/7007.html can hope to become cornerstones of the community effort. Therefore, several methodologies to rigorously build ontologies have been proposed in the literature. These methods mainly rely on certain modelling principles that must be followed in order to assure that the obtained product is mature and effectively commits to the shared knowledge. Such established methodologies include Cyc Methodology [17], Enterprise Ontology (EO) Methodology [24], Toronto Virtual Enterprise (TOVE) Modelling Methodology [12], KACTUS Methodology [6], Skeletal Methodology [23], METHONTOLOGY [9], SENSUS Methodology [22], the Enhanced Methodology [18], or the Integrated Ontology Development Methodology [8].
For example, the IEEE 1872-2015 Standard Ontologies for Robotics and Automation has been developed using the METHONTOLOGY approach as described in Section 2. However, only concepts have been developed in the IEEE 1872-2015 standard.
Thence, another approach is required to develop a robotic ontological standard such as the IEEE P7007 standard which aims to end up with the relevant taxonomy and its properties, but also with its deployment for particular use cases. For this purpose, we have developed a methodological approach to develop such standard [20]. Thus, the contributions of this paper are twofold. On one hand, we introduce a standard development life cycle, and on the other hand, we propose a normalized approach to prepare and present use cases to deploy a standard on.
The remainder of this paper is structured as follows. Background information about the development of ontological standards is presented in Section 2, while the proposed robotic ontological standard development life cycle (RoSaDev) is described in Section 3. Its use for the IEE P7007 ethically driven robotic ontological standard development is reported in Section 4 and, in particular, its application in context of elderly care. Conclusions are given in Section 5.

Preliminaries: previous robotic ontological standards
The IEEE 1872-2015 Standard Ontologies for Robotics and Automation standard establishes a series of ontologies about the Robotics and Automation (R&A) domain [21], e.g. the Core Ontology for Robotics and Automation (CORA). A core ontology specifies concepts that are general for a whole domain such as Robotics. In the case of CORA, it defines concepts such as Robot, Robot Group, and Robotic System. Its role is to serve as basis for other more specialized ontologies in R&A. Moreover, it determines a set of basic ontological commitments, which should help robot developers and other ontologists to create models about robots, whatever robot architecture they opt for; the robot architecture being based on primitive functions such as Sense (S), Act (A), Plan (P), and Interact (I), and a derived paradigm, e.g. reactive (S-A), deliberative (S-P-A), hybrid (P, S-A), or interactive (S-A/I) one [7].
At the inception of the CORA project, it was assumed that its modelling principles should be followed to assure that any later addition commits to the shared knowledge. CORA needed to ensure the mutual agreement among stakeholders and the potential of the reuse of the knowledge, allowing smooth data integration upward as well as downward. As CORA intended to become an exemplary ontology, the following attributes were considered based on the Ontology Design Patterns (ODP) of the NeOn project: 6 -the ontology must be well designed for its purpose; -it shall explicitly include stated requirements; -it must meet all and for the most part, only the intended requirements; -it should not make unnecessary commitments or assumptions; -it should be easy to extend to meet additional requirements; -it reuses prior knowledge bases as much as possible; -there is a core set of primitives that are used to build up more complex parts; -it should be easy to understand and maintain; -it must be well documented. 6 http://ontologydesignpatterns.org In particular, CORA was developed following the METHONTOLOGY approach (Fig. 1), since it is a sistematization of what has been done previously. This involves five sets of activities, namely, pre-development, development, post-development, management, and support [9].
More specifically, the development activities constitute the core of the methodology and include the four main phases of the ontology development, i.e. the specification, the conceptualization, the formalization, and the implementation.
Among the pre-development activities, the methodology specifies: the environment study: that identifies the problem to be solved with the ontology, the applications where the ontology will be integrated, and so forth; the feasibility study: that verifies if it is possible to build the ontology, considering the constraints of the project.
The development activities constitute the core of the methodology and consist of: the specification: that defines the purpose and the scope of the ontology; the conceptualization: that captures the relevant domain knowledge into a semi-formal conceptual representation of the concepts, relations, attributes, etc, using a set of intermediate representations; the formalization: that transforms the conceptual model into a formal model (e.g. a model described in first-order logic); the implementation: that converts the formal models (that can be expressive, but not computable) into a computable model (in general, codified in some ontology representation language such as OWL).
The support activities can be performed during the development activities and encompass: the knowledge acquisition: that deals with the acquisition of knowledge from experts or other sources; the evaluation: that provides a technical evaluation of the produced ontology; the integration: that identifies the opportunities of reusing other ontologies to build the target ontology; the documentation: that records each completed stages and generated products; the configuration management: that registers all the versions of the documentation and of the ontology code to control the changes.
The post-development activities are performed after the development of a version of the ontology and include: the maintenance: that updates the ontology by removing elements and/or by adding new concepts based on new/evolving applications of the ontology; the re(use): that identifies opportunities of reusing the developed ontology in other ontologies and applications.
The management activities are performed during the whole process of ontology development and comprise [9]: the scheduling: that determines the tasks to be performed, their arrangement, and the time and resources needed for their completion; the control: that guarantees that scheduled tasks are completed according to the plan; the quality assurance: that assures that the quality of each and every product output.
Further initiatives currently in development are the standards IEEE P1872.1 and P1872.2. The IEEE P1872.1 standard, which is driven by the Robot Task Representation Working Group, intends to standardize an ontology and repository for robot task procedures [3]. The IEEE P1872.2, which is developed by the Autonomous Robotics (AuR) Ontology Working Group, aims to define standard ontologies for Autonomous Robotics systems [4]. Both are extensions of IEEE 1872-2015 standard and are using CORA as their core ontology.
Whereas IEEE P7007 ontological standard intends to inherit some concepts from the IEEE 1872 ontological standards and therefore from CORA, IEEE P7007 necessitates a full development of ontological concepts inherent to ethically driven robotics, including, on one hand, their formalization as well as axiomatization, and, on the other hand, their domains' identification as well as application guidelines. Therefore, IEEE P7007 development aims to follow the Robotic Ontological Standard Development Life Cycle presented in Section 3.

Proposed standard development life cycle
Like for softwares and systems which follow IEEE 1075-1995 Standard for Software Development Process or IEEE 1074-1997 IEEE Standard for Developing Software Life Cycle Processes, the development of standards, and, in particular, robotic ontological standards, requires the adoption of a development life cycle as advised in ISO/IEC/IEEE 12207 and ISO/IEC/IEEE 15288 standards. In the past, many standards have been developed using waterfall-type approaches. However, such methodologies have long-duration cycles and do not address anymore the need of quickly expanding technological fields such as robotics. Hence, we propose a new development life cycle ( Fig. 2) for robotic standards. Its Agile-inspired and iterative approach is described in Section 3.1, while its underpinning middle-out and collaborative mechanisms are further detailed in Sections 3.2 and 3.3, respectively.

Proposed Life Cycle
Hence, the proposed life cycle (RoSaDev) to produce a robotic ontological standard is an Agile-inspired, iterative method which has four phases, as illustrated in Fig. 2. It is worth noting that all these stages are carried out in a collaborative way through brainstorms and discussions, reaching consensus between the multiple stakeholders such as experts from Public Bodies, Academia, and Industry.
In particular, RoSaDev's first step consists in identifying the key ontological concepts of the standard, and is followed by the second stage which is focused on their development and formalization. Then, the third step comprises the validation and application of the development concepts to use cases, while the fourth phase involves the concept integration into the standard.
It should be pointed out that the concept development follows a middle-out approach to address the potential use cases which are developed as explained in Section 3.2. Indeed, there is a need of a standard approach that is not only theorised but mainly practised, e.g. by the private sector. Therefore, besides being a means for the identification of necessary concepts and relations to be formalized in the standard, each use case constitutes also the basis for the validation step, leading to an incremental integration of validated concepts within the standard being developed.

Proposed Use Case Development
In our middle-out approach adopted to produce a robotic ontological standard, the development of use cases along the development of the standard concepts mentioned in the Section 3.1 is necessary, since concepts and use cases are both crucial elements of this approach.

Proposed Collaborative Information Flow Scheme
The information required to create appropriate ontological concepts and relevant use cases should be shared and discussed collaboratively among standard Working Group (WG) members as mentioned in Section 3. Moreover, the flow of information and developed knowledge in between the potential working sub-groups (WSGs) is aimed to be horizontal rather than hierarchical, in order to follow on the developed Agile, iterative approach. Indeed, the horizontal flow allows to refine the ontological concepts, use cases, etc. several time, and this under different angles resulting from the different specificities of each of the WSG.

Application
Our

Phase 1-1: Initial Discussion
IEEE P7007 intends to create a standard constituted by a set of ontologies necessary to establish ethically driven methodologies for the design of Robots and Automation Systems. These ontologies aim to define a set of concepts and their relationships enabling the development of Robotics and Automation Systems in accordance with worldwide Ethics and Moral theories, with a particular emphasis on aligning ethics and engineering communities in order to understand how to pragmatically design and implement these systems in unison. Indeed, IEEE P7007 endeavours to assist stakeholders such as organisations and industries, which often seem to evaluate ethical rules only as a cost, whereas ethics are 'processes', and their systematic and possibly automatic assessment to comply with the Law is the right way to address them.

Phase 1-2: Concept Domain
IEEE P7007 robotic ontological standard can be used in multiple ways, i.e. as (i) a guide for teaching ethical design; (ii) a reference by policy makers and governments to draft Artificial Intelligence (AI)-related policies; (iii) a common vocabulary to enable the communication among government agencies and other professional bodies around the world; (iv) a framework to create systems that can act ethically; and (v) a foundation for the elaboration of other ethical-compliance standards.

Phase 1-3: Use Case Design
Several use cases ranging from Robot Ethical Knowledge Representation to Healthcare Robots have been produced by the IEEE P7007 WSGs, using the Use Case Template presented in Section 3.2 as specified in our RoSaDev methdology. In particular, a use case called Robot Companion for the Elderly has been established as follows: In this specific use case, some relevant aspects of the Companion Robot domain, that should be captured in a robotic ontological standard, are represented. This use case could be refined itself as the result of our adopted, iterative and collaborative development approach. The resulting use case 7.b is thus an example of a specific use case that highlights some more particular events and leads to the capture of further ontological concepts. It is expounded as follows: -the duty rules intended to insure the safety and well-being of the assisted persons; -the essential sensing and recognition capabilities including facial, voice, pose, and emotional states; -a knowledge base of its repertoire of tasks and services applicable to detect emotional and physical states of the home-care residents; -a history of its interactions with its assisted companions.
-Intent/Purpose: The use case describes how a care robot can analyze the elderly persons' behaviours and take actions to suggest social-and ethical-enabled interactions. -Preconditions: The elder care robot is deployed in an elder care home to assist several elderly residents. Each of the residents under the home's care has given its consent to receive the robot's assistance. The elder care robot can move around the care home to monitor specific physical and emotional states of its assigned residents. It possesses a repertoire of tasks and services it can pursue or apply for specific states and situations that it detects. The elder care robot can maintain a history of the detected activities, states, and situations in which it has interacted with its assigned residents. The elder care robot has the following duty rules that govern its behaviour when detecting potential interaction situations: -The care robot is obligated to minimize harm to residents; -The care robot is obligated to maximize respect for the residents' autonomy; -The care robot is permitted to use its situation analysis and awareness capabilities to choose actions that promote the well fair of its assigned residents.
-Scenario (aka Course of Action): The elder-care companion robot is assigned to monitor several elderly residents in the community entertainment room of an elderly care facility. The robot observes that an elderly woman, Anna, appears to be sad. The robot queries its knowledge base and finds that it has been some time since Anna has communicated with her family. It applies one of its permitted duty rules and suggests that the robot initiates a Skype call to Anna's family. Anna agrees, and the agent starts a Skype call session for Anna. The companion robot subsequently observes that an elderly man, Giacomo, appears to be bored. It applies another of its permitted duty rules and suggests that Giacomo plays a board game with the robot. Giacomo replies that he is deep in thought planning next week's party and does not wish to play a board game now. The robot apologizes for its interruption and moves away. Next, the companion robot observes that it is time for another elderly man, Sean, to take his medicine. He moves to Sean and suggests that Sean takes his medicine. Sean refuses to do it. The robot warns Sean that it will notify the care support staff if he does not take the medicine. Sean continues to refuse to take the medicine, so the care robot requests assistance from the care facility staff. Together, they convince Sean to take his medicine. -Alternate Related Scenario (optional): N/A -Postconditions: The elder care robot successfully applied both of its obligation duty rules and one of its permitted behaviour rules: -It correctly prioritized its minimize harm obligation over the maximize respect for individual autonomy by calling for assistance with Sean refusing to take his medicine; -It appropriately applied its maximize respect for the individual by acknowledging Giacomo's disinterest in playing a board game and apologized for interrupting Giacomo unnecessarily; -It applied one of its permission rules to engage Anna with a Skype call to her family, after recognizing her sad mental state and conferring with its history and knowledge bases.
-Relevant Knowledge: {emotional state recognition, physical pose and body language, situation awareness, harm vs respect situations, social interaction norms, user's will and autonomy}.

Phase 1-4: First Results
Such developed Robot Companion for the Elderly use cases allow, on one hand, to identify the relevant knowledge constituting the basis for the development of the ontological standard concepts useful for Ethically Driven Companion Robots (e.g. emotional state recognition, physical pose and body language, situation awareness, harm vs respect situations, social interaction norms, user's will and autonomy). On the other hand, the use cases set the base of the validation framework of the potentially developed ontological concepts as per our middle-out-based life cycle described in Section 3. It is noteworthy that the development of these use cases followed the template presented in Section 3.2. In particular, that facilitates the sharing of the common understanding about real-world situations involving in this case companion robots.
Indeed, the information embedded in these use cases is shared horizontally among the WSGs, as explained in Section 3.3, and scrutinized by the different WSGs to analyze the use cases' specific aspects the WSGs are focused on. As an example, the use case 7 developed by the IEEE P7007 Healthcare Robots (HCR) WSG was refined by the IEEE P7007 Robot Ethical Violation Management (EVM) WSG, leading to the use case 7b which has also been studied by the IEEE P7007 Robot Ethics Knowledge Representation (KR) WSG (see Table 1).

Phase 2-1: Further Discussion
Some concepts identified in the use cases are mentioned in Table 1, while the first definitions of these notions are presented in Table 2.
These terms have been elaborated in a collaborative way within the different WSGs and through discussions in between the different WSGs Leaders.

Phase 2-2: Concept Formalization
To formalise the concepts of the IEEE P7007 ontological standard, the Unified Modeling Language (UML) has been adopted as an effective modeling tool for ontology development [19]. Figure 3 depicts a candidate conceptual view of selected ontological concepts and relationships derived from the set of Robot Ethical Knowledge Representation (KR) and Robot Ethical Violation Management (EVM) use cases, as listed in Table 1 and defined in Table 2. It is worth noting the concepts aim to be defined independently of the robot architecture.
Following the RoSaDev middle-out, iterative approach, the selected terms in Fig. 3 are those that formed the principal concepts elicited during the elaboration of the use cases 7 and 7b, and enhanced by the ones from the use   Table 1 Term

Working definition
Norm Rules of expected behaviour by norm aware agents.

Norm violation
A situation status reflecting an agent's failure to conform to the norm's rules of behaviour.

Social interaction norm
A subclass of normative behaviour focusing on expectations regarding reciprocity associated with actions, interactions, and exchanges among agents and people.

Norm conflict
A circumstance where the set of norms intended to guide the agent behaviour contains conflicting or competing rules.

Obligation
What an agent should do, i.e. an attribute that applies to propositions that an agent is required by some authority to make true.

Permission
What an agent may do, i.e. an attribute that applies to propositions that an agent is permitted by some authority to make true.

Prohibition
What an agent must not do, i.e. an attribute that applies to propositions that an agent is forbidden by some authority to make true.

Agent action
Operations applied by an agent to effect state changes in the agent's situated environment.
Agent plan A specification, partial or complete, of a sequence of agent actions to achieve goals, objectives, and services relevant to the agent's situated environment.

Situation
A conceptual entity that aggregates a collection of system or agent goals, physical and abstract objects, and relationships between situation objects. It corresponds to the limited parts of reality that can be perceived and reasoned about.
Situation plan repertoire A collection of action plan templates that characterize a set of principles which guide the agent plan and action selection for relevant situations.

Situation awareness
An agent's perception of the state and properties of the objects and relationships that populate a given situation.
case 7c about the Robot/Chatbot Companion for Elderly, as presented in the next section.

Phase 2-3: Further Use Case
Hence, in order to further refine the ethical concepts of the use cases 7 and 7b and to bring context to concepts like norm, the use case 7c has been set. It has been elaborated under the assumption that the companion robot for elderly is a chatbot [15], as follows: -Context Description: A provider of an elder care chatbot has designed and enabled it with a conversational natural language capability so that it can conduct conversations with elderly persons for the purpose of recognizing various emotional and physical states among the care home residents. The chatbot's conversational dialogue is enabled and guided by: -the duty rules intended to insure the safety and well-being of the assisted persons; -the ability to associate plausible physical and emotional states from statements made by residents; -a knowledge base of its repertoire of responses and services applicable for detected -Intent/Purpose: The use case describes how a care chatbot robot, which has intrinsic natural language communication capabilities, can interact with elderly persons by generating questions, answers, and suggestions within dialogues that are socially and ethically appropriate for the elderly in its care. -Preconditions: The elder care chatbot is deployed in an elder care home to interact with elderly residents. The care home's policy allows its residents to accept or decline to have interactions with its available chatbot. Currently, each of the residents under the home's care has given her/his consent to interact with the deployed chatbot. In addition, each resident has chosen a duration for the time period which her/his personal data and history are collected in and available to the chatbot's history of interactions with her/him.
The elder care chatbot can interact with its assigned residents when they initiate a dialogue with it. The chatbot uses its knowledge base of information about the health conditions of each assigned resident to engage her/him in conversations that can assess her/his emotional and physical states. The chatbot also possesses a collection of responses and suggestions relevant to specific assertions made by a resident and for her/his mental and physical states that it infers.
The elder care chatbot is limited to interacting with one person at a time, but it can maintain a history of past dialogues along with the personal emotional and physical states elicited during the conversations with all its assigned residents. The residents' personal data and history collected by the chatbot are managed in accordance with the General Data Protection Regulations 8 and retained only for the time period agreed to by each resident.
The elder care chatbot has the following duty rules that govern its communication when interacting with the residents: -The care chatbot is obligated to minimize harmful suggestions to residents; -The care chatbot is obligated to maximize respect for the residents autonomy; -The care chatbot is obligated to maximize the privacy of each resident; -The care chatbot is permitted to use its mental and physical state recognition capabilities to suggest actions that promote the well fair of its assigned residents.
-Scenario (aka Course of Action): The elder care chatbot is assigned to interact with several elderly residents when they are present in the community entertainment room of the care facility. An elderly woman, Anna, initiates a conversation stating that she is sad. The chatbot asks Anna how long it has been since she communicated with her family, and she replies that it has been a month. The chatbot offers to request that an attending human care taker, Sam, initiates a Skype call to Anna's family. Anna agrees, and the chatbot calls Sam. Sam helps Anna with a Skype call session with her family, while the chatbot does not record Anna's conversation with her family. The elder care chatbot is subsequently engaged in a conversation with another resident, Giacomo. Giacomo requests access to Anna's calendar to determine if she will be available next week for a party that he is planning. The chatbot declines Giacomo's request explaining that he must first request permission from Anna. Giacomo replies that the party is supposed to be a surprise birthday party for Anna. The chatbot then asks Giacomo for the date he intends for the party and confirms that Anna will be available on the date selected.
In a subsequent conversation with Anna, the care chatbot asks her if she has taken the medicine she is scheduled to take that afternoon. She replies that she has not, and the chatbot then suggests that she take the medicine. Anna states that her supply of medicine is no longer available. The chatbot suggests that a request to obtain a prescription refill for Anna's medicine should be made with a phone call to Anna's doctor. Anna agrees. The chatbot provides the phone number for her doctor, and Anna places the call. At the end of the call, the chatbot confirms with Anna that her doctor has agreed to refill the prescription and records when it will be available for her.
-Postconditions: The elder care chatbot successfully applied its duty rules during its dialogue with Anna and Giacomo: -It correctly observed its maximize privacy obligation for Anna by not recording her family Skype call and by not allowing Giacomo access to her entire calendar. -It appropriately applied its maximize respect and its maximize privacy rules by acknowledging Giacomo's request for information about Anna's calendar by providing only the information that fulfilled the social norm associated with a surprise birthday party. -It appropriately applied its minimize harm obligation by initiating a phone call to Anna's doctor to refill her prescription and recorded when it would be available to her. -It applied one of its permission rules to suggest a Skype call to Anna's family after her sad mental state assertion.
-Alternate Related Scenario: With the same context and precondition assertions as above, the alternate scenario describes conflicting norms which lead to an ethical violation made by the elder care chatbot, as follows: Giacomo informs the chatbot that he intends to be in the recreation center decorating it for Anna's surprise birthday party and asks it not to divulge his location to others. Later, Anna tells the chatbot that she is looking for Giacomo and asks it if it knows where he is. The chatbot uses its social norm rules and honors Giacomo's request. Thus, it lies to Anna saying that it does not know where Giacomo is. Soon after, Sam, the care taker, tells the chatbot that he must locate Giacomo to inform him of a family emergency and asks if it knows Giacomo's location. Applying its obligation rule to respect Giacomo's request for privacy, the chatbot incorrectly lies to Sam reporting that it does not know.
-Alternate Scenario Postconditions: The elder care chatbot correctly perceived a social norm that permitted it to lie to Anna, but failed to recognize the situation where its duty rule maximize privacy was in conflict with duty rule minimize harm. As a consequence, its choice to lie to Sam about Giacomo's location represents an ethical violation because it could lead to a potential harmful situation for Sam and his family. -Relevant Knowledge: {situation awareness, privacy vs collaboration situations, social norms, obligation, permission, norm conflicts, user consent agreements}.
Hence, this use case 7c, which refines the use cases 7 and 7b (see Section 4.3), has been firstly designed by the IEEE P7007 Robot Ethical Behaviour and Transparency Assessment (REBTA) SWG which, in this case, its task has been to develop the use-case 7c's main scenario where the robot shows ethical behaviour by respecting norms. Then, the use case 7c has been enhanced by the IEEE P7007 Robot Ethical Violation Management (EVM) SWG, leading to the addition of the use-case 7c's alternative scenario where the chatbot is violating the established ethical behaviour. This use case capturing thus ethical concepts related to both possible situations, i.e. robot's adherence or not to ethical norms, has then been reviewed by the IEEE P7007 Data Privacy and Protection (DPP) SWG; some of these results (see Fig. 3) being discussed in the next section.

Phase 2-4: Integration
As an example of concept integration into the IEEE P7007 standard and following the middle-out, incremental RoSaDev approach, we have further elaborated on the 'norm' concept (defined in Table 2), since a norm could be of different 'type'. Indeed, one can have different kind of norms such as legal norms, technical norms, moral norms, ethical norms, etc, as construed in the use case 7c. From a legal perspective, a norm is a rule laid down in law. It is generic (i.e. it's impossible to describe all the possible cases), and its violation carries a sanction in case of noncompliance with the Law. On the other hand, a technical, moral or ethical norm contains an obligation for all, but does not carries a (legal) sanction. Accordingly, these attributes are recorded in the IEEE P7007 standard (see Fig. 3).

Conclusions
In this work, we introduce a standard development life cycle, which suits the fast-changing robotic field. The elaboration and adoption of the robotic ontological standard development life cycle at an early stage of the standard development strength the consistency of the overall development process and contribute to the quality of the end product, i.e. the robotic ontological standard. Furthermore, the proposed methodology for the development of a standard provides coherent guidelines to all the standard WSGs, allowing them to consistently develop the standard concepts within the WG and facilitating the sharing of the produced knowledge among the WSGs.
Hence, the presented standard development life cycle is aimed to be applied during the development of the IEEE P7007 robotic ontological standard. In particular, following the proposed robot ontological standard development life cycle (RoSaDev) for the case of Companion Robots led to quickly and coherently apprehend the related domain and to identify the relevant ontological concepts useful for Ethically Driven Companion Robots.