On the homology language of HDA models of transition systems

Journal of Applied and Computational Topology https://doi.org/10.1007/s41468-023-00120-1 On the homology language of HDA models of transition systems Thomas Kahl1 Received: 5 August 2022 / Revised: 15 February 2023 / Accepted: 11 April 2023 © The Author(s) 2023 Abstract Given a transition system with an independence relation on the alphabet of labels, one can associate with it a usually very large symmetric higher-dimensional automaton. The purpose of this paper is to show that by choosing an acyclic relation whose symmetric closure is the given independence relation, it is possible to construct a much smaller nonsymmetric HDA with the same homology language. Keywords Higher-dimensional automata ·Transition system ·Homology language Mathematics Subject Classification 55N35 ·68Q85 1 Introduction Higher-dimensional automata are a powerful combinatorial-topological model for concurrent systems. A higher-dimensional automaton (HDA) is a precubical set (i.e., a cubical set without degeneracies) with an initial state, a set of final states, and a labeling on 1-cubes such that opposite edges of 2-cubes have the same label (van Glabbeek 2006;Pratt1991). An HDA is thus a (labeled) transition system (or an ordinary automaton) with a supplementary structure consisting of twoand higher-dimensional cubes. The transition system represents the states and transitions of a concurrent system. An n-cube in an HDA indicates that the ntransitions starting at its origin are independent in the sense that they may occur in any order, or even simultaneously, without any observable difference. It has been shown in van Glabbeek (2006) that higherdimensional automata are more expressive than the principal traditional models of concurrency. BThomas Kahl [email protected] 1Centro de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal 123 T. Kahl The fact that the twoand higher-dimensional cubes in an HDA represent independence of transitions suggests that the higher-dimensional topology of an HDA contains global information on independence of processes and components of the modeled concurrent system. In Kahl (2021), the homology language has been devised as a homological tool to describe this global independence structure of an HDA. The homology language of an HDA is defined to be the image of the homomorphism induced in homology by a certain labeling chain map that leads from the cubical chain complex of the HDA to the exterior algebra on the alphabet of labels (see Sect.3). The homology language can be computed and analyzed using software (Kahl 2018). Transition systems are arguably the most fundamental model for concurrent systems. A natural way to turn a transition system into an HDA is to fill in empty squares and higher-dimensional cubes (van Glabbeek 2006; Gaucher 2010; Goubault and Mimram 2012; Kahl 2019). This approach requires some concept of independence in order to decide which cubes to fill in. Such a concept of independence may be given by an independence relation—i.e., an irreflexive and symmetric relation—on the alphabet of action labels. Independence relations play a fundamental role in trace theory in the sense of Mazurkiewicz (1987). Asynchronous transition systems (Winskel and Nielsen 1995) are an important example of transition systems that come equipped with an independence relation on the set of labels. HDAs constructed from transition systems using a cube filling procedure based on a symmetric concept of independence, such as an independence relation on the set of labels, are usually very large because an empty n-cube representing the execution of nindependent actions is filled in n!times. From a computational point of view, it is therefore desirable to have a way to produce smaller HDA models of transition systems. It has been shown in Kahl (2019) that using a cube filling rule based on an asymmetric rather than an independence relation on the alphabet, one can construct an HDA model of a transition system where the independence of nactions in a state is represented by a single n-cube (at least if the transition system under consideration is deterministic). It is, of course, not to be expected that two HDAs constructed using different filling rules from the same transition system will be equivalent in a meaningful sense. Every independence relation on an alphabet is the symmetric closure of an acyclic relation, i.e., a relation that is acyclic when seen as a graph (see Sect.4). The purpose of this paper is to establish that the HDAs constructed from a transition system using cube filling rules based, respectively, on an independence relation on the alphabet and a generating acyclic relation have the same homology language. The two HDAs may therefore be regarded as equivalent from the point of view of their independence structures. As we will show in Sect.5, the HDA constructed using the independence relation is the free symmetric HDA generated by the one constructed using the acyclic relation. The latter is thus a significantly smaller HDA model of the given transition system than the former. 123 On the homology language of HDA models of transition systems 2 Precubical sets and HDAs This section presents fundamental material on precubical sets, higher-dimensional automata, and their symmetric variants. 2.1 Precubical sets Aprecubical set is a graded set P=(Pn)n≥0with face maps dk i:Pn→Pn−1(n>0,k=0,1,i=1,...,n) satisfying the cubical identities dk idl j=dl j−1dk i(k,l=0,1,i<j). If x∈Pn, we say that xis of degree or dimension n. The elements of degree nare called the n-cubes of P. The elements of degree 0 are also called the vertices of P, and the 1-cubes are also called the edges of P.Theith starting edge of a cube xof degree n>0 is the edge eix=d0 1...d0 i−1d0 i+1...d0 nx. Aprecubical subset of a precubical set is a graded subset that is stable under the face maps. The n-skeleton of a precubical set Pis the precubical subset P≤ndefined by (P≤n)m=Pmfor m≤nand Pm=∅else. Amorphism of precubical sets is a morphism of graded sets that is compatible with the face maps. The category of precubical sets can be seen as the presheaf category of functors op →Set where is the small subcategory of the category of topological spaces whose objects are the standard n-cubes [0,1]n(n≥0)and whose nonidentity morphisms are composites of the coface maps δk i:[0,1]n→[0,1]n+1(k∈{0,1}, n≥0, i∈{1,...,n+1}) given by δk i(u1,...,un)=(u1,...,ui−1,k,ui...,un). The geometric realization of a precubical set Pis the quotient space |P|=⎛ ⎝ n≥0 Pn×[0,1]n⎞ ⎠/∼ where the sets Pnare given the discrete topology and the equivalence relation is generated by (dk ix,u)∼(x,δk i(u)), x∈Pn+1,u∈[0,1]n,i∈{1,...,n+1},k∈{0,1}. The geometric realization of a morphism of precubical sets f:P→Qis the continuous map |f|: |P|→|Q|given by |f|([x,u])=[f(x), u]. 123 T. Kahl 2.2 The precubical set of permutations The family of symmetric groups S=(Sn)n≥0(with S0={id∅}) is a precubical set with respect to the face maps given by dk iθ(j)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ θ(j), j<θ −1(i), θ( j)<i, θ(j)−1,j<θ −1(i), θ( j)>i, θ(j+1), j≥θ−1(i), θ( j+1)<i, θ(j+1)−1,j≥θ−1(i), θ( j+1)>i (see Kahl (2022) and compare Krasauskas (1987), and Fiedorowicz and Loday (1991)). Since, by definition, d0 iθ=d1 iθ, we may simplify the notation by setting diθ=d0 iθ=d1 iθ. Since S1={id}, the two elements of S2have the same faces. In degrees ≥3, permutations are determined by their faces: Proposition 2.1 Let n ≥3, and let σ, θ ∈Snsuch that diσ=diθfor all i ∈{1,...,n}. Then σ=θ. Proof It follows from Kahl (2022, Lemma 3.3) that it is enough to show that there exists an rsuch that σ(r)=θ(r). Suppose that this is not the case. Set i=σ−1(n) and j=θ−1(n). Then i,j<n. Indeed, suppose that i=n. Then for 1 ≤r≤n−1, dnσ(r)=σ(r)because r<n=σ−1(n),σ(r)≤n, and σ(r)= σ(n)=n. Since θ(r)= σ(r)and θ(r), θ(r+1)≤n,wehaver≥θ−1(n),θ(r+1)<n, and σ(r)=dnσ(r)=dnθ(r)=θ(r+1). In particular, j=θ−1(n)≤1. Hence j=1 and θ(1)=n. Set s=σ−1(1). Since σ(s)= σ(n),s<n. Since θ(s+1)=σ(s)=1, θ−1(1)=s+1. Since the values of σare ≥1, we have d1σ(1)=σ(1)−1,1<σ −1(1)=s, σ(2)−1,1=s. Since n>2, we have σ(1), σ(2)= σ(n)=nand therefore d1σ(1)≤n−2. On the other hand, 1 <s+1=θ−1(1)and θ(1)=n>1 and therefore n−2≥d1σ(1)=d1θ(1)=θ(1)−1=n−1, which is impossible. It follows that i<n. An analogous argument shows that j<n. Hence 1 ≤i,j≤n−1. We show that i<j. Since i≥i=σ−1(n),σ(i+1)≤n, and σ(i+1)= σ(i)=n,wehavednσ(i)=σ(i+1). Hence dnθ(i)=σ(i+1)= θ(i+1). Since θ(i+1)≤n, it follows that i<j. Analogously, j<i. Since this is impossible, there must exist an rsuch that σ(r)=θ(r). 123 On the homology language of HDA models of transition systems For later use, we state the following fact from Kahl (2022): Proposition 2.2 (Kahl 2022, Prop. 3.6(i)) Let P be a precubical set, and let n ≥2, 1≤i<j≤n, k,l∈{0,1},x∈Pn, and θ∈Sn. Then dk dθ(j)θ(i)dl θ(j)x=dl dθ(i)θ(j−1)dk θ(i)x. 2.3 Symmetric precubical sets Asymmetric precubical set is a precubical set Pequipped with a crossed action of S on P, i.e., a morphism of graded sets S×P−→ P,(θ, x)→ θ·xsuch that •for all n≥0 and x∈Pn,id ·x=x; •for all n≥0, σ, θ ∈Sn, and x∈Pn,(σ ·θ)·x=σ·(θ ·x); •for all n≥1, θ∈Sn,x∈Pn,i∈{1,...,n}, and k∈{0,1}, dk i(θ ·x)=diθ·dk θ−1(i)x. Symmetric precubical sets form a category, in which the morphisms are morphisms of precubical sets that are compatible with the crossed actions. We remark that the category of symmetric precubical sets is isomorphic to the presheaf category Setop S where Sis the subcategory of the category of topological spaces whose objects are the standard n-cubes [0,1]n(n≥0)and whose morphisms are composites of the coface maps δk idefined above and the permutation maps tθ:[0,1]n→[0,1]n,(u1,...,un)→ (uθ(1)...,uθ(n))(n≥0,θ∈Sn) (cf. Grandis and Mauri (2003); Fahrenberg (2005); Gaucher (2010); Goubault and Mimram (2012)). The free symmetric precubical set generated by a precubical set Pis the symmetric precubical set SP defined by •SP n=Sn×Pn(n≥0); •dk i(θ, x)=(diθ,dk θ−1(i)x)(n≥1,θ ∈Sn,x∈Pn,1≤i≤n,k∈{0,1}); •σ·(θ, x)=(σ ·θ,x)(n≥0,σ,θ ∈Sn,x∈Pn). The free symmetric precubical set is functorial, and the functor P→ SP from the category of precubical sets to the category of symmetric precubical sets is left adjoint to the forgetful functor. 2.4 Higher-dimensional automata Throughout this paper, we consider a fixed alphabet .Ahigher-dimensional automaton (HDA) over is a tuple Q=(P,ı,F,λ) 123 T. Kahl where Pis a precubical set, ı∈P0is a vertex, called the initial state,F⊆P0is a (possibly empty) set of final states, and λ:P1→is a map, called the labeling function, such that λ(d0 ix)=λ(d1 ix)for all x∈P2and i∈{1,2}(van Glabbeek 2006). We say that an HDA Q=(P,ı,F,λ )is a sub-HDA of Qand write Q⊆Q if Pis a precubical subset of P,ı=ı,F=F∩Q 0, and λ=λ|Q 1.Then-skeleton of Qis the sub-HDA Q≤n=(P≤n,ı,F,λ|(P≤n)1). Higher-dimensional automata form a category, in which a morphism from an HDA Q=(P,ı,F,λ) to an HDA Q=(P,ı,F,λ )is a morphism of precubical sets f:P→Psuch that f(ı)=ı, f(F)⊆F, and λ(f(x)) =λ(x)for all x∈P1. Asymmetric HDA is an HDA Q=(P,ı,F,λ) equipped with a crossed action of Son P. Symmetric HDAs form a category, in which the morphisms are morphisms of HDAs that also are morphisms of symmetric precubical sets. The free symmetric HDA generated by an HDA Q=(P,ı,F,λ) is the symmetric HDA SQ=(SP,(id,ı), S0×F,μ) where μ(id,x)=λ(x)(x∈P1)and the crossed action is the one of SP. The assignment Q→ SQdefines a functor from the category of HDAs to the category of symmetric HDAs, which is left adjoint to the forgetful functor. 3 The homology language and free symmetric HDAs In this section, we recall the definition of the homology language of an HDA from Kahl (2021) and show, as a first contribution of this paper, that an HDA and the free symmetric HDA generated by it have the same homology language. We work over a fixed principal ideal domain, which we suppress from the notation. 3.1 Cubical chains and cubical homology The cubical chain complex of a precubical set Pis the nonnegative chain complex C∗(P)where Cn(P)is the free module generated by Pnand the boundary operator d:Cn(P)→Cn−1(P)is given by dx = n  i=1 (−1)i(d0 ix−d1 ix), x∈Pn(n>0). The cubical homology of P, denoted by H∗(P), is the homology of C∗(P). 3.2 The homology language of an HDA Let Q=(P,ı,F,λ) be an HDA over . Consider the exterior algebra on the free module generated by ,(). Recall that this is the quotient of the tensor algebra on the free module on by the two-sided ideal generated by all elements of the form x⊗xwhere xruns through the free module on (see Bourbaki (1974)formore details). The exterior algebra () is canonically graded by the exterior powers of the free module generated by . We view the graded module () as a chain complex 123 On the homology language of HDA models of transition systems with d=0 and define the labeling chain map l:C∗(P)→() on basis elements x∈Pnby l(x)=1(),n=0, λ(e1x)∧···∧λ(enx), n>0. By Kahl (2018, Prop. 4.4.5), the labeling chain map is indeed a chain map, and therefore it induces a labeling homomorphism in homology: l∗:H∗(P)→H∗(()) =() The homology language of Qis then defined to be the graded module HL(Q)=im l∗={l∗(α) |α∈H∗(P)}. The term reflects an analogy with the ordinary language of an automaton, which is a set of labels of paths. Some fundamental properties of the homology language have been established in Kahl (2021). 3.3 Simple cubical dimaps Asimple cubical dimap from a precubical set Pto a precubical set Pis a continuous map f:|P|→|P|such that for all n≥0 and x∈Pn, there exist elements y∈P n and θ∈Snsuch that for all u∈[0,1]n, f([x,u])=[y,tθ(u)]. By Kahl (2018, Prop. 6.2.4), yand θare uniquely determined by fand x.Wemay therefore slightly abuse notation and write f(x)to denote y. Obviously, the geometric realization of a morphism of precubical sets is a simple cubical dimap. In Kahl (2018), a more general concept of cubical dimap has been defined, hence the adjective simple. Asimple cubical dimap from an HDA Q=(P,ı,F,λ) to an HDA Q= (P,ı,F,λ )is a simple cubical dimap of precubical sets f:P→Pthat preserves the initial and the final states and that satisfies λ(f(x)) =λ(x)for all x∈P1. Our interest in simple cubical dimaps is motivated by the following fact: Proposition 3.1 (Kahl 2021, Prop. 5.7.2) Let Qand Qbe HDAs such that there exists a simple cubical dimap Q→Q. Then H L(Q)⊆HL(Q). 123 T. Kahl 3.4 The homology language of a free symmetric HDA Let Q=(P,ı,F,λ)be an HDA. Since there exists the morphism of HDAs Q→SQ, x→ (id,x), by Proposition 3.1,HL(Q)⊆HL(SQ). We show in Theorem 3.3 below that actually equality holds. Lemma 3.2 Let θ∈Sn(n≥1), and let i ∈{1,...,n}and k ∈{0,1}. Then δk θ−1(i)◦tdiθ=tθ◦δk i:[0,1]n−1→[0,1]n. Proof Let (u1,...,un−1)∈[0,1]n−1.Wehave δk i(u1,...,un−1)=(u1,...,ui−1,k,ui...,un−1). For j∈{1,...,n},set vj=⎧ ⎨ ⎩ uj,1≤j<i, k,j=i, uj−1,i<j≤n. Then we have tθ◦δk i(u1,...,un−1)=tθ(v1,...,v n) =(vθ(1),...,v θ(n)) =(vθ(1),...,v θ(θ−1(i)−1),k,v θ(θ−1(i)+1),...,v θ(n)) =δk θ−1(i)(vθ(1),...,v θ(θ−1(i)−1),v θ(θ−1(i)+1),...,v θ(n)). We have δk θ−1(i)◦tdiθ(u1,...,un−1)=δk θ−1(i)(udiθ(1),...,udiθ(n−1)). Since for j∈{1,...,n−1}, udiθ(j)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ uθ(j),j<θ −1(i), θ( j)<i, uθ(j)−1,j<θ −1(i), θ( j)>i, uθ(j+1),j≥θ−1(i), θ( j+1)<i, uθ(j+1)−1,j≥θ−1(i), θ( j+1)>i =vθ(j),j<θ −1(i), vθ(j+1),j≥θ−1(i), the result follows.  Theorem 3.3 HL(Q)=HL(SQ). Proof We only have to show the inclusion HL(SQ)⊆HL(Q). By Proposition 3.1,it suffices to construct a simple cubical dimap SQ→Q. Consider the continuous map f:|SP|→|P|defined by f([(θ, x), u])=[x,tθ(u)]((θ, x)∈(SP)n,u∈[0,1]n). 123 On the homology language of HDA models of transition systems This is well defined because, by Lemma 3.2,for(θ, x)∈(SP)n,u∈[0,1]n−1, i∈{1,...,n}, and k∈{0,1}, [x,tθ◦δk i(u)]=[x,δk θ−1(i)◦tdiθ(u)]=[dk θ−1(i)x,tdiθ(u)]. By construction, fis a simple cubical dimap of precubical sets. We have f(θ, x)=x for all (θ, x)∈SP. Hence fpreserves the initial and the final states. Moreover, for every edge x∈P1,λ( f(id,x)) =λ(x). It follows that fis a simple cubical dimap of HDAs.  4 HDA models of transition systems The purpose of this section is to define HDA models of transition systems. An HDA model can be constructed with respect to an arbitrary relation on the alphabet of labels. In this paper, we are interested in the case where this relation is an independence or an acyclic relation. Except for the subsection on acyclic relations, the material of this section is taken from Kahl (2019). 4.1 Transition systems and independence relations Atransition system is a 1-truncated extensional HDA, i.e., an HDA with no cubes of dimension ≥2 and no two edges with the same label and the same start and end vertices. An independence relation is an irreflexive and symmetric relation on the alphabet of action labels . An independence relation equips the alphabet with a notion of concurrency: two actions are independent if they may be executed sequentially or simultaneously without any relevant difference. Independence relations play a fundamental role in trace theory (Mazurkiewicz 1987,1995). The results of this paper apply, in particular, to asynchronous transition systems, which are transition systems over an alphabet with an independence relation satisfying certain conditions (see Winskel and Nielsen (1995)). 4.2 Acyclic relations A relation on a set Xis called acyclic if for all n≥1 and x1,...,xn∈X, x1x2,x2x3, ..., xn−1xn⇒xnx1. Note that an acyclic relation is irreflexive (n=1)and, moreover, asymmetric (n=2). Proposition 4.1 Let I be an independence relation on . Consider a totally ordered set (Z,≤)and a map f :→Z such that ∀a,b∈:aIb⇒f(a)= f(b). 123