Petri automata for Kleene allegories

18 Pages·2017·0.69 MB·French

Petri automata for Kleene allegories
Paul Brunet, Damien Pous
To cite this version:
Paul Brunet, Damien Pous. Petri automata for Kleene allegories. Logic in Computer Science, Jul
2015, Kyoto, Japan. pp.68-79, ￿10.1109/LICS.2015.17￿. ￿hal-01073936v3￿
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Petri automata for Kleene allegories
Paul Brunet and Damien Pous
Plume team – LIP, CNRS, ENS de Lyon, Inria, UCBL, Université de Lyon, UMR 5668
{paul.brunet,damien.pous}@ens-lyon.fr
Abstract—Kleenealgebraaxiomsarecompletewithrespectto reflexive-transitive closures are not allowed. They are
both language models and binary relation models. In particular, decidable [8, page 208]; Inequation (3) is known as the
two regular expressions recognise the same language if and only
modularity law in this setting.
iftheyareuniversallyequivalentinthemodelofbinaryrelations.
We consider Kleene allegories, i.e., Kleene algebras with To the best of our knowledge, the decidability of the whole
two additional operations which are natural in binary relation theory, Kleene allegories, is open. Here we obtain several
models: intersection and converse. While regular languages are important steps towards the resolution of this problem:
closedunderthoseoperations,theabovecharacterisationbreaks.
1) wegiveacharacterisationofthefull(in)equationaltheory
Instead, we give a characterisation in terms of languages of
directed and labelled graphs. We then design a finite automata in terms of graph languages;
model allowing to recognise such graphs, by taking inspiration 2) we design an automata model inspired by Petri nets, that
from Petri nets. makes it possible to recognise such graphs;
This model allows us to obtain decidability of identity-free
3) weshowhowtoassociatesuchagraphautomatontoany
relational Kleene lattices, i.e., the equational theory generated
term of Kleene allegories;
by binary relations on the signature of regular expressions with
intersection,butwhereoneforbidsunit.Thisrestrictionisusedto 4) using these graph automata, we give a decision proce-
ensurethatthecorrespondinggraphsareacyclic.Thedecidability dure for the fragment where converse and identity are
of graph-language equivalence in the full model remains open. forbidden.
I. INTRODUCTION ThelatterfragmentwasstudiedrecentlybyAndrékaetal.[1];
its decidability was open as far as we know. The restriction
We consider binary relations and the operations of
to this fragment allows us to exploit simplifying assumptions
union (∪), intersection (∩), composition (⋅), converse (⋅⌣),
about the produced automata, and to obtain a coinductive al-
transitive closure (⋅+), reflexive-transitive closure (⋅⋆), and the
gorithmforlanguageinclusion(SectionV).Weactuallyshow
constantsidentity(1)andemptyrelation(0).Thismodelgives
that language inclusion for these automata is EXPSPACE-
rise to an (in)equational theory: a pair of terms e,f made
complete (Section VI).
from those operations and some variables a,b,... is a valid
The next problem, which remains open, consists in ob-
equation, denoted Rel ⊧ e = f, if the corresponding equality
taining the decidability of language inclusion in the full
holdsuniversally.Similarly,aninequationRel⊧e≤f isvalid
automatamodel:togetherwiththepresentedresults,thiswould
when the corresponding containment holds universally. Here
entail decidability of Kleene allegories. We outline some of
are valid equations and inequations: they hold whatever the
the difficulties arising with converse or unit in presence of
relations we assign to variables a, b, and c.
intersection in Section V-D.
Rel ⊧ (a∪b)⋆⋅b⋅(a∪b)⋆ = (a⋆⋅b⋅a⋆)+ (1)
We continue this introductory section by an informal de-
Rel ⊧ a⋆ ≤ 1 ∪ a⋅a⌣⋅a+ (2) scription of the graph language characterisation and our au-
Rel ⊧ a⋅b ∩ c ≤ a⋅(b ∩ a⌣⋅c) (3) tomata model.
Rel ⊧ a+ ∩ 1 ≤ (a⋅a)+ (4) A. Languages
Various fragments of this theory have been studied in the In the simple case of Kleene algebra, i.e., without converse
literature: and intersection, the (in)equational theory generated by rela-
● Kleene algebra [7], where one removes intersection and tions can be characterised by using regular languages. Write
converse,sothattermsareplainregularexpressions.The e for the language denoted by a regular expression e; for
theory is decidable by Kleene’s work [11], and actually a(cid:74)n(cid:75)y two regular expressions e,f, we have
PSPACE-complete [14], [15]. Equation (1) lies in this
Rel⊧e≤f if and only if e ⊆ f . (5)
fragment, and one can notice that the two expressions (cid:74) (cid:75) (cid:74) (cid:75)
recognise the same language. (This result is easy and folklore; proving that this is also
● Kleene algebra with converse, where one only removes equivalent to provability using Kleene algebra axioms [4],
intersection, is also a decidable fragment [3]. It remains [12], [13] is much harder.)
PSPACE [5]. Inequation (2) belongs to this fragment. While regular languages are closed under intersection and
● (representable, distributive) allegories [8], sometimes converse, the above characterisation does not extend to those
called positive relation algebras, where transitive and operations. For intersection, consider two distinct variables a
G(a): a C. Graph languages
To extend the above graph-theoretical characterisation to
a b
Kleene allegories, we need to handle union and (reflexive-)
G((a⋅(b∩c))∩d): c transitive closures. It suffices for that to consider sets of
d graphs: to each expression e, we associate a set of graphs
a b G(e). This set is most often infinite when the expression e
G((a⋅b)∩(a⋅c)): contains (reflexive-)transitive closures.
a c Writing ◂X for the downward closure of a set of graphs
a X by the preorder ◂ on graphs, we obtain the following
d
G((a ∩ b⋅c)⋅d): generalisation of both (5) and (6): for any two expressions
b c e and f,
Rel⊧e≤f if and only if ◂G(e)⊆◂G(f) . (7)
G(a⋅b ∩ 1): a b
This is Theorem 6 in the sequel, and this result is almost
thereintheworkbyAndrékaetal.[1],[2].Tothebestofour
Figure 1: Graphs associated to some ground terms. knowledge this explicit formulation is new, as well as its use
towards decidability results.
When e and f are ground terms, we recover the charac-
and b. The extended regular expressions a ∩ b and 0 both terisation (6) for representable allegories: G(e) and G(f) are
recognise the empty language, while Rel ⊧/ a∩b = 0: one singleton sets in this case. For plain regular expressions, the
can interpret a and b with intersecting relations. For converse, graphs are just words and the preorder ◂ reduces to isomor-
the extended regular expressions a and a⌣ both recognise the phism. We thus recover the characterisation (5) for Kleene
singleton language consisting of the single-letter word a. Yet algebra. This result also generalises the characterisation pro-
Rel⊧/ a=a⌣, as there are non-symmetric relations. vided by Ésik et al. [3] for Kleene algebra with converse:
graphs of expressions without intersection are just words over
B. Graphs a duplicated alphabet, and the corresponding restriction of the
preorder◂preciselycorrespondstothewordrewritingsystem
Freyd and Scedrov’ decision procedure for representable
they use (see Remark 7).
allegories[8,page208]reliesonanotionofdirected,labelled,
2-pointedgraph.Thesamenotionwasproposedindependently D. Petri automata
by Andréka and Bredikhin [2], in a more comprehensive way. In order to exploit the above characterisation and obtain
Call ground terms the terms in the syntax of allegories decidability results, one has first to represent graph languages
(composition,intersection,converse,andunit).Agroundterm in a finitary way. We propose for that a new finite automata
u can be represented as a labelled directed graph G(u) with model, largely based on Petri nets [16]–[18]. We describe this
two distinguished vertices called the input and the output. We modelbelow,ignoringconverseandunitforthesakeofclarity.
givesomeexamplesinFigure1,seeDefinition1foraprecise Recall that a Petri net consists of
definition.
● a finite set of places, denoted with circles;
These graphs can be endowed with a preorder relation: we ● a set of transitions, denoted with rectangles;
write G◂F when there exists a graph homomorphism from ● foreachtransition,asetofinputplacesandasetofoutput
F to G preserving labels, inputs, and outputs. For instance
places, denoted with arrows;
the graph corresponding to (a⋅(b∩c))∩d is smaller than ● an initial place, denoted by an entrant arrow;
the graph of (a⋅b)∩(a⋅c), thanks to the homomorphism ● a set of final markings, denoted by dotted boxes (a
depicted in Figure 2 using dotted arrows. Notice that the
marking, or configuration, being a set of places).
homomorphism needs not be injective or surjective, so that
The execution model is the following: start by putting a token
this preorder has nothing to do with the respective sizes of
on the initial place; choose a transition whose input places all
the graphs: a graph may very well be smaller than another in
contain a token, remove those tokens and put new tokens in
the sense of ◂, while having more vertices or edges (and vice the output places of the transition; repeat this process until a
versa).
finalmarkingisreached.Theobtainedsequenceoftransitions
The key result from Freyd and Scedrov [8, page 208], or
is called an accepting run. (We actually restrict ourselves to
Andréka and Bredikhin [2, Theorem 1], is that for any two
safePetrinets,toensurethatthereisalwaysatmostonetoken
ground terms u,v, we have
in a given place when playing this game.)
APetriautomatonisjustasafePetrinet1 withvariablesla-
Rel⊧u≤v if and only if G(u)◂G(v) . (6)
bellingtheoutputsofeachtransition.Theautomatondepicted
The graphs are finite so that one can search exhaustively for
1For everyreachable markingin asafe net, thereis atmost onetoken in
a homomorphism, whence the decidability result. anygivenplace.
2
a b same vertex (5). This actually corresponds to composing the
5
functions used to accept F with the homomorphism depicted
G : 4 c 6
with dotted arrows.
d
This automata model is expressive enough for Kleene
▲
allegories: for any expression e, we can construct a Petri
a 1 b automaton A(e) such that L (A(e))=◂G(e) (Section IV).
F : 0 3 We give three other examples of Petri automata to give more
a 2 c intuition on their behaviour.
Figure 2: A graph homomorphism. ThefirsttransitioninthepreviousPetriautomatonsplitsthe
initialtokenintotwotokens,whicharemovedconcurrentlyin
the remainder of the run. This corresponds to an intersection
below is the automaton we will construct for the ground term in the considered expression. This is to be contrasted with the
a⋅b ∩ a⋅c.Anyrunmuststartbyfiringtheleft-mosttransition, behaviour of the following automaton, which we would con-
reaching the marking {B,C}; then we have the choice of structforthenon-groundexpressiona⋅b∪a⋅c.Thisautomaton
firing the upper transition first, reaching the marking {D,C}, has twoaccepting runs: {A},{B},{D} and {A},{C},{E},
or the lower one, reaching the marking {B,E}. In both cases which can be used to accept the (graphs of the) ground terms
we reach the final marking {D,E} by firing the remaining a⋅b and a⋅c.
transition. a b
B D
b
a B D A
A a C c E
a C E
c
In a sense, two transitions competing for the same tokens
represent a non-deterministic choice, i.e., a union in the
To read a graph in such an automaton, we try to find an
starting expression.
accepting run that matches the graph up to homomorphism
Still in the first example, the two tokens created by the first
(Definitions 10 and 11). We do that by using an evolving
transition are later collected in the final marking. Tokens may
function from the tokens to the vertices of the graph. We
also be collected and merged by a transition. Consider for
startwiththefunctionmappingtheuniquetoken,intheinitial
place, to the input vertex of the graph. To fire a transition, we instance the following automaton for (a∩b⋅c)⋅d. It has only
must check that all its input tokens are mapped to the same one accepting run, {A},{B,C},{B,D},{E}, and this run
can be used to read the fourth graph from Figure 1.
vertex in the graph, and that this vertex has several outgoing
edges, labelled according to the outputs of the transition. If a
B
this is the case, we update the function by removing the d
A E
mappings corresponding to the deleted tokens, and by adding
new mappings for each of the created tokens (using the target b C c D
vertices of the aforementioned outgoing edges, according to
the labels). The graph is accepted by the Petri automaton if Asalastexample,considerthefollowingautomatonforthe
we can reach a final marking of the Petri automaton, with all expression a ∩ b+⋅c. The upper transition introduces a loop,
tokens mapped to the output vertex of the graph. sothatthereareinfinitelymanyacceptingruns.Foranyn>0,
For instance, the previous automaton accepts the graph of the graph of the ground term a ∩ bn⋅c is accepted by this
automaton.
a⋅b ∩ a⋅c (F in Figure 2). We start with the function
{A↦0}.Wecanfirethefirsttransition,updatingthefunction
into {B↦1,C ↦2} (We could also choose to update the b
functioninto{B↦2,C ↦1},or{B,C ↦1},or{B,C ↦2} c
B C
butthiswouldleadtoadead-end).Thenwecanfiretheupper
b
transition,evolvingthefunctioninto{D↦3,C ↦2},andwe A a
finishbyfiringtheremainingtransition,obtainingthefunction D
{D,E ↦3}.
We call language of A the set of graphs L (A) accepted According to characterisation (7), the next step is to decide
by a Petri automaton A. This language is downward-closed: whether the containment L (A) ⊆ L (B) holds, for two
L (A)=◂L (A).Forinstance,thepreviousautomatonalso given Petri automata A and B. Several difficulties arise,
accepts the graph G from Figure 2, which is smaller than that do not appear with classical word automata. Our solution
F. Indeed, when we fire the first transition, we can associate nevertheless uses a standard coinductive approach, where we
the two newly created tokens (in places B and C) to the define an appropriate notion of simulation (Section V-C).
3
G(1)∶= relation gives rise to a preorder on ground terms, written ⊲
a and defined by u⊲v if G(u)◂G(v). ∗
G(a)∶=
Given a set S of graphs, we write ◂S for its downward
G(w⌣)∶= G(w) closure: ◂S ∶= {G ∣ G◂G′,G′∈S}. Similarly, we write ⊲S
G(u⋅v)∶= G(u) G(v) for the downward closure of a set of ground terms w.r.t. ⊲.
G(u) As explained in the introduction, the above preorder on
G(u∩v)∶= ground terms precisely characterises inclusion under arbitrary
G(v)
relational interpretations:
Figure3:Inductiveconstructionofthegraphofagroundterm. Theorem3([2,Theorem1],or[8,page208]). Forallground
terms u,v, we have Rel⊧u≤v⇔u⊲v .
Standard notations. For any sets A,B, we write P(S) ∶= To extend this result to Kleene allegories, we introduce
{P ∣ P ⊆A} for the set of subsets of A, A → B for the the following generalisation of the language of a regular
setoffunctionsfromAtoB,andA⇀B forthesetofpartial expression. Sets of words become sets of ground terms.
maps from A to B. The domain of a partial map f is denoted
Definition 4 (Terms and graphs of an expression)
by dom(f). The set of terms of an expression e ∈ Reg⌣X∩, written e , is
the set of ground terms defined inductively as follows:(cid:74) (cid:75)
II. GRAPH-THEORETICALCHARACTERISATION
1 ∶={1} 0 ∶=∅ x ∶={x}
We consider the signature ⟨∩,∪,⋅,⋅+,⋅⌣,0,1⟩ of Kleene (cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75)
allegories, where e⋆ is an abbreviation for 1∪e+. We fix a set (cid:74)e⋅f(cid:75)∶={w⋅w′ ∣ w∈(cid:74)e(cid:75) and w′∈(cid:74)f(cid:75)}
X ofvariables,andwedenotebyReg⌣∩ thesetofexpressions e∩f ∶={w∩w′ ∣ w∈ e and w′∈ f }
X
(cid:74) (cid:75) (cid:74) (cid:75) (cid:74) (cid:75)
e,f... built from variables in X with these connectives. e∪f ∶= e ∪ f
Gwritohunthdetesrumb-ssaigrenathtuereex⟨p∩r,e⋅s,s⋅i⌣o,n1s⟩.uI,fvσ,w∶.X..→buPilt(fSro×mSX) iosnalny (cid:74) (cid:74)e⋆(cid:75)(cid:75)∶=⋃(cid:74) n(cid:75)∈N(cid:74){w(cid:75)1⋅⋯⋅wn ∣ ∀i,wi∈(cid:74)e(cid:75)}
interpretation of the alphabet X into some space of relations, e⌣ ∶={w⌣ ∣ w∈ e } .
(cid:74) (cid:75) (cid:74) (cid:75)
we write ̂σ for the unique homomorphism extending σ into The set of graphs produced by an expression e, denoted by
a function from Reg⌣X∩ to P(S×S). An inequation between G(e) is the set of graphs associated to the ground terms in
two expressions e and f is valid, written Rel ⊧ e ≤ f, if for e :
any relational interpretation σ we have ̂σ(e)⊆̂σ(f). (cid:74) (cid:75)
We let G range over 2-pointed labelled directed graphs, G(e)∶={G(w) ∣ w∈ e } . ∗
(cid:74) (cid:75)
which we simply call graphs in the sequel. Those are tuples
Toobtainthecharacterisationannouncedintheintroduction,
⟨V,E,ι,o⟩withV afinitesetofvertices,E ⊆V ×X×V aset we need a slight refinement of a lemma established by
of edges labelled with X, and ι,o∈V the two distinguished Andréka, Mikulás, and Németi [1]:
vertices, respectively called input and output.
Lemma 5. For all expression e∈Reg⌣X∩X, and all relational
Definition 1 (Graph of a ground term: G(w)) interpretations σ∶X →P(S×S), we have
To each ground term w, we associate a graph G(w), by
induction on w. The graph of a∈X has one edge labelled by ̂σ(e)= ⋃ ̂σ(u)= ⋃ ̂σ(w) .
a linking its input to its output. The graph for 1 has only one u∈ e w∈⊲ e
(cid:74) (cid:75) (cid:74) (cid:75)
vertex, both input and output. The composition of two graphs Proof. The first equality is exactly [1, Lemma 2.1]; for the
with disjoint sets of vertices can be performed by identifying secondone,weusethefactthat̂σ(w)⊆̂σ(u)wheneverw⊲u,
the output of the first graph and the input of the second one. thanks to Theorem 3 (i.e., [2, Theorem 1]).
The intersection on graphs consists in merging their inputs
and merging their outputs. The converse consists simply in Theorem 6. The following properties are equivalent, for all
exchanging the input and the output of a graph. ∗ expressions e,f ∈Reg⌣X∩:
(i) Rel⊧e≤f,
SeeFigure3foragraphicaldescriptionofthisconstruction. (ii) e ⊆⊲ f ,
Those graphs were introduced independently by Freyd and (iii) G(cid:74) ((cid:75)e)⊆(cid:74)◂G(cid:75)(f).
Scedrov [8, page 208], and Andréka and Bredikhin [2].
Proof. We give a detailed proof in Appendix A. The implica-
Definition 2 (Graph homomorphism, preorders ◂ and ⊲) tion (ii) ⇒ (i) follows easily from Lemma 5, and (iii) ⇒ (ii)
A graph homomorphism from ⟨V1,E1,ι1,o1⟩ to is a matter of unfolding definitions. For (i)⇒(iii), we mainly
⟨V2,E2,ι2,o2⟩ is a map ϕ ∶ V1 → V2 such that ϕ(ι1) = ι2, use [2, Lemma 3].
ϕ(o1) = o2, and (p,x,q) ∈ E1 entails (ϕ(p),x,ϕ(q)) ∈ E2.
We denote by ◂ the relation on graphs defined by G◂G′ (The exact characterisation announced in the introduction (7)
if there exists a graph homomorphism from G′ to G. This follows: for any sets X,Y, we have ◂X ⊆◂Y iff X ⊆◂Y.)
4
c C a D t2a t4a
c d t1c C D t3c C D t5d
b B b F t0b B F
A b E E b E E
A b E a G G G G G G
a
G Figure 5: An accepting run in the automaton from Figure 4.
a b
H I
Figure 4: A Petri automaton. The initial place is A, and the Inthesequel,weassumeallconsideredPetriautomatatobe
final configurations are {I} and {F,G}. safe. (I.e., in Petri nets terminology, such that any reachable
marking has at most one token in each place [16]). Formally,
with our definitions: a Petri automaton ⟨P,T,ι,F⟩ is safe if
Also notice that while G(f) only contains graphs emanat- for all configuration ξ ⊆ P reachable from {ι} by firing any
ing from ground terms, this is not the case for its closure number of transitions, if (t,t)∈T is enabled from ξ, p∈ξ,
◂G(f).Forinstance,◂G((a⋅b)∩(c⋅d))containsthefollow- and (x,p)∈t, then p∈t.
ing graph, which is not the graph of any ground term.
Now we explain how to use Petri automata to define
a b languagesofgraphs.Wefirstdefinetherunsofanautomaton.
e
Definition 9 (Run, accepting run, parallel run)
c d A run is a sequence ξ = ⟨(ξk)0⩽k⩽n,(tk)0⩽k<n⟩ of config-
urations and transitions, such that ξk ⊆ P, tk ∈ T and
Remark 7. The graphs associated to ground terms without in- ∀k < n, ξk —t→k ξk+1. When ξ0 = {ι} and ξn ∈ F, we call
tersectionareisomorphictowordsoveraduplicatedalphabet.
ξ an accepting run.
Agraphhomomorphismbetweentwosuchgraphsisprecisely
A parallel run is defined similarly, as a sequence Ξ =
wthhuastbÉessikeeentaasl.acgaellnearnalaisdamtiiosnsibolfe[3m,aTph[e3o]r.emTh5eo.3r]e.m 6 can ⟨(Ξk)0⩽k⩽n,(Tk)0⩽k<n⟩, where the Tk ⊆ T are compatible
sets of transitions such that Ξk —T→k Ξk+1. ∗
III. PETRIAUTOMATA
(Note that a run ξ is uniquely determined by ξ and the
X∪W{exe⌣xt∣exnd∈tXhe}s∪et{X1}.oAfvPaertiraibaleustoimntaotoanseistaX¯Poetfrilanbeetlws:Xh¯os∶=e sdeeqteuremncineis(ttikca)l:lya.l)lsubsequentconfigurationscanbe0computed
transition’s outputs are labelled by X¯.
Consider the following sequence of transitions from the
Definition 8 (Petri Automaton) automaton in Figure 4:
A Petri automaton A over the alphabet X is a tuple
⟨P,T,ι,F⟩ where: ξ=⟨(ξ0,ξ1,ξ2,ξ3,ξ4,ξ5,ξ6),(t0,t1,t2,t3,t4,t5)⟩,
with
● P is a finite set of places,
● T ⊆P(P)×P(X¯ ×P) is a set of transitions, ξ0={A}, t0=({A},{(b,B),(a,G)}),
● ι∈P is the initial place of the automaton, ξ1={B,G}, t1=({B},{(c,C),(b,E)}),
● F ⊆P(P)isasetoffinalconfigurations,aconfiguration ξ2=ξ4={C,E,G}, t2=t4=({C},{(a,D)}),
being a set of places. ξ3=ξ5={D,E,G}, t3=({D,E},{(c,C),(b,E)}),
For each transition t=(t,t)∈T, t and t are assumed to be ξ6={F,G}. t5=({D,E},{(d,F)}).
non-empty;t⊆P istheinputoft;andt⊆X¯×P istheoutput We have
of t. ∗
We use the graphical notation from the introduction to {A} t0(cid:47)(cid:47){B,G} t1(cid:47)(cid:47){C,E,G}(cid:111)(cid:111) t2,t4 (cid:47)(cid:47){D,E,G} t5(cid:47)(cid:47){F,G} ,
represent Petri automata; the Petri automaton from Figure 4 t3
will be used as a running example. andsince{A}istheinitialconfigurationand{F,G}∈F,this
From a configuration ξ ⊆ P, a transition t = (t,t) ∈ T sequenceisanacceptingrun.Itcanberepresentedgraphically
is enabled if t ⊆ ξ. If so, one may fire t, which produces a as in Figure 5.
newconfigurationξ′=ξ∖t∪{p∈P ∣ ∃x∈X¯ ∶(x,p)∈t}.We As for standard finite state automata, we now need to
write ξ—→t A ξ′ in this case. specify how to read a graph in an automaton. As explained
A set of transitions T ⊆T is statically compatible (or just in the introduction, this is done by linking the intermediate
configurations of a run to vertices in the graph, and by
compatible)iftheirinputsarepairwisedisjoint.Iffurthermore
imposing conditions to match transitions with labelled edges
all transitions in T are enabled in a configuration ξ, one can
observe that the configuration ξ′ reached after firing them of the graph.
successively does not depend on the order in which they are Definition 10 (Reading, parallel reading, language of a run)
fired. In that case we write ξ—T→A ξ′. A reading of G = ⟨V,E,ι,o⟩ along a run ξ =
5
c 2 a c 4 a addanedge(k,x,l)wheneverthereissomeplaceq suchthat
b 1 3 5 d (x,q)∈tk, and tl is the first transition after tk in the run with
0 b b 6 q among its inputs, or l = n if there is no such transition in
a the run.
Figure 6: Graph produced by the run depicted in Figure 5.
Definition 12 (Trace of a run)
Let ξ=⟨(ξk)0⩽k⩽n,(tk,tk)0⩽k<n⟩ be run. For an index k⩽n
and a place q, let ν(k,q) be either the smallest index l such
f⟨{o(orξ}k,a)la0ln⩽kdk,⩽∀nρk,k(<itskn,a,tkmth)0ae⩽pfko<flrnloo⟩mwisiξnakgsthoeoqlVudes,n:ρc0e((ξρ0)k)=0⩽{kι⩽}n, ρsunc(hξnt)ha=t tEhξaTt∶=hke{⩽t(rkla,caxen,doνf(qξk∈i+stl1t,h,oeqr)g)nra∣pifh(tx(cid:74)h,ξeq(cid:75)r)e∶=∈is⟨t{kn0}o,..s.u.c,hn}in,dEexξ.,0,n⟩wit∗h
● all tokens in the input of the transition are mapped to the
same vertex in the graph: ∀p,q∈tk, ρk(p)=ρk(q); Togetthefinalgraph,whichislabelledbyX,oneidentifies
● the images oftokens in ξk that are notin the input of the nodes linked by edges labelled by 1, and one replaces each
transition are unchanged: ∀p∈ξk∖tk, ρk(p)=ρk+1(p); edge of the form (i,x⌣,j) by (j,x,i). Formally:
● eachpairintheoutputofthetransitioncanbe“validated”
Definition 13 (Graph produced by a run)
by the graph: ∀p∈tk,∀(x,q)∈tk, Let ξ = ⟨(ξk)0⩽k⩽n,(tk,tk)0⩽k<n⟩ be run. Let ≡ξ be the
x∈X ⇒(ρk(p),x,ρk+1(q))∈E, smallestequivalencerelationon{0,...,n}containingallpairs
x=y⌣ and y∈X ⇒(ρk+1(q),y,ρk(p))∈E, (i,j) such that (i,1,j)∈Eξ.
The graph produced by ξ, is the graph G (ξ) defined by
x=1⇒ρk(p)=ρk+1(q).
Similarly, we define a parallel reading ρ along some parallel G (ξ)∶=⟨{[i]ξ ∣ 0⩽i⩽n},Eξ′,[0]ξ,[n]ξ⟩
run Ξ=⟨(Ξk)0⩽k⩽n,(Tk)0⩽k<n⟩ by requiring that: ρ0(Ξ0)= [i]ξ ∶={k∈{0,...,n} ∣ i≡ξ k}
{ι}●●●, ρ∀∀∀np(((tt∈,,ΞttΞn)))k∈∈=∖TT{⋃kko,,(∀∀}t,,ppt)a,∈∈nqTdtk∈,t∀∀t,,(kρxρk<k,(+qnp1))(t∈ph=)te,ρ=fko(ρlqlko)(w;p)in;g holds: Eξ′ ∶=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩([i]ξ,x,[j]ξ) RRRRRRRRRRRRRRRRRR x∃k∈((∈klX,,[xixa⌣],ξn,l,kd)l)∈∈∈E[Ejξ]ξξor∶ ⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ ∗
x∈X ⇒(ρk(p),x,ρk+1(q))∈E, WewriteG (A)forthesetofgraphsproducedbyaccepting
x=y⌣ and y∈X ⇒(ρk+1(q),y,ρk(p))∈E, runs of a Petri automaton A. To avoid confusions with the
x=1⇒ρk(p)=ρk+1(q). Glan∈gGua(gAeL),(reAse)rvoifnAg l,awngeuwagreiteth“eGoriestipcrotedrumciendobloygAy l”ikweh“eGn
Thelanguageofarunξ,denotedbyL (ξ)isthesetofgraphs is accepted by A” or “A recognises G” to cases where we
that can be read along ξ. ∗ mean G∈L (A).
The graph produced by the run presented in Figure 5
The language of a Petri automaton is finally obtained by
happens to be equal to its trace, since it is labelled in X only.
considering all accepting runs.
A more involved example is given in Figures 7 to 9. Notice
Definition 11 (Language recognised by a Petri automaton) thatalthoughthetraceofarunisacyclicandcanbeendowed
The language recognised by A, written L (A), is the fol- with a partial order structure (simply check that ∀p,ν(_,p)
lowing set of graphs: is increasing), it is not necessarily the case for its produced
graph.
L (A)∶= ⋃ L (ξ) . ∗
ξacceptinginA Lemma 14. For any accepting run ξ, we have G∈L (ξ) if
The language of a run ξ can be characterised by using a and only if G◂G (ξ).
single graph which we call the graph produced by ξ: graphs
Proof. Suppose there exists a graph homomorphism ϕ from
areacceptedbyξ exactlywhentheyaresmallerthanthegraph G (ξ) to G. Then we can build a reading by defining ρk(p)=
producedbyξaccordingto◂(Lemma14below).Forinstance, ϕ([ν(k,p)]ξ) for 0 ⩽ k ⩽ n and p ∈ ξk. On the other
the run presented in Figure 5 produces the graph depicted
hand, if we have a reading (ρk)0⩽k⩽n of G, we can build
in Figure 6.
ahomomorphismϕbylettingϕ([k]ξ)=ρk(p)foranyp∈tk.
This graph is obtained in two steps, by first considering a
notion of trace, which is a graph labelled with X¯ rather than As (ρk)k is a reading, ϕ is well defined. The details of this
proof can be found in Appendix B.
X, and which actually corresponds to the notion of pomset-
tracefromstandardPetrinets(seeSectionVIIformoredetails
As an immediate corollary, we obtain the following charac-
on this correspondence).
terisation of the language of a Petri automaton.
The trace is constructed by creating a vertex k for each
transition tk = (tk,tk) of the run, plus a final vertex n. We Corollary 15. L (A)=◂G (A).
6
c configurations of A to the initial place of A , and then
b C 2 E 1 2
apply some epsilon-elimination procedure.
d⌣
a B 1 3 F Wealsoputthetwoautomatasidebysidefortheintersection,
A 0 1 1 D D but we merge their initial places, their initial transitions, and
their final configurations:
G G G G
● for any pair of initial transitions of the two automata
Figure 7: A run ξ. ({ι1},t1),({ι2},t2), we put in the intersection automa-
2 ton the transition ({ι1},t1∪t2);
b 2 c b c ● a final configuration of this automaton is the union of a
a 1 3 d⌣ 1 final configuration from A1 and a final configuration of
0 1 4 a d A2.
1 0 For the transitive closure (⋅+), we use the ideas for union and
composition, adding loops from the final configurations using
Figure 8: The trace of ξ. Figure 9: The graph
the initial transitions of the automaton.
produced by ξ.
Definition 16
To each expression e∈Reg⌣X∩, we associate a Petri automaton
The left-hand side language is defined through readings A(e) defined inductively as follows:
along accepting runs, which is a local and incremental notion
● x∈X¯,A(x)∶=⟨{0,1},{({0},{(x,1)})},0,{{1}}⟩
and which allows us to define simulations in Section V-C.
● A(0)∶=⟨{0},∅,0,∅⟩
By contrast, the right-hand side language is defined globally,
whicheasesthefollowingconstructionofanautomatonrecog- ● A(e1∪e2)∶=⟨P1∪P2,T,ι1,F1∪F2⟩ with
nising the language of an expression. T ∶=T1∪T2∪{({ι1},t) ∣ ({ι2},t)∈T2}.
● A(e1⋅e2)∶=⟨P1∪P2,T,ι1,F2⟩ with
IV. FROMEXPRESSIONSTOAUTOMATA T ∶=T1∪T2∪{(f,t) ∣ f ∈F1 and ({ι2},t)∈T2}.
faancWtatuehteonmoawuattooshmnoawAtoh(noeww)etthooabattsarsieonccohigaatnseisateonsaetnvhyeenelxsatpnrrogenusgasgieorenc◂oeGn∈n(Reec)et.gio⌣XI∩nn ●● ATA((∶=ee+11T)∩1∶=e∪2⟨){P(∶1=f,,⟨TPt)1,ι∪∣1f,PF2∈,1FT⟩1w,ιai1tnh,dF(⟩{ιw1}it,ht)∈T1}.
wawciecthefipretsi:tngmthoerudngifsryaiptnhhesAeinx(ep)Gre.(sTesio)onmarsaeokeetxhtaahctettlhyceotnohspotesrueracttpoirorond⋅⌣usciimsedopnlbelryy, TF ∶∶=={{f(t1,∪t)f2∣ ∃∣if∈1{∈1F,21},∶f2(t∈,tF)2∈}Tia,ndιi∉t}∪
applied to variables, by using the following rewriting system: {({ι1},t1∪t2) ∣ ∀i∈{1,2}, ({ιi},ti)∈Ti}.
(a∪b)⌣→a⌣∪b⌣ 0⌣→0 (a+)⌣→(a⌣)+ (In the inductive cases, we assume A(ei) = ⟨Pi,Ti,ιi,Fi⟩
(a⋅b)⌣→b⌣⋅a⌣ 1⌣→1 a⌣⌣→a for i∈{1,2}, with P1∩P2=∅.) ∗
(a∩b)⌣→a⌣∩b⌣. WeprovebyinductiononethatA(e)isindeedasafePetri
automaton;forthesafetyrequirement,weaddtotheinduction
(These rules preserve the set of graphs of the expression; also
hypothesis the fact that for any configuration ξ accessible
recall that e⋆ is a shorthand for e++1, so that we do not need in A(e), if there is a final configuration f ∈ F such that
to handle it explicitly.)
f ⊆ ξ, then f = ξ. Another invariant is that the initial place
The formal construction is inductive; it is given in Defi-
never appears in a final configuration, nor in the output of
nition 16 below. We describe it first in informal terms. For
any transition. Note that the place ι becomes unreachable
the case of x ∈ X¯, we simply build an automaton with a by construction in the cases for u2nion, composition and
single transition labelled by x, going from the initial place
intersection, so that it could safely be removed, together with
to a distinct final place2. The union consists in putting both
the associated transitions.
automata side by side, and merging their initial places. For
the composition of A1 and A2 on the other hand we put one Theorem 17 (Correctness). For all expression e ∈ Reg⌣X∩,
automaton in front of the other: L (A(e))=◂G (e).
● the initial place of the resulting automaton is that of A1; Proof. We prove a stronger result: G (A(e)) = G(e) (up
● the final configurations are those of A2; to graph isomorphisms—see Appendix C). This allows us to
● for each final configuration f of A1, and for each initial conclude thanks to Corollary 15.
transition (ι2,t) coming out of the initial place of A2,
we add a transition with input f and output t. This last Corollary 18. The (in)equational theory of Kleene allegories
step amounts to adding epsilon transitions from the final is co-recursively enumerable.
Proof. Construct Petri automata for the two expressions and
2Notethatthiscoversuniformlythecaseoftheunit,ofvariables,andof
conversedvariables. enumerateallpotentialcounter-examples,i.e.,graphs.Agraph
7
is a counter-example if it can be read in one automaton but and the configurations in A , that satisfy some conditions on
2
not in the other, which is a decidable property. theinitialandfinalconfigurations,andsuchthatifξk ≼ξk′ and
Remark19. Ifeisanexpressionwithoutintersection,converse ξk —→t A1 ξk+1, then there is a configuration ξk′+1 in A2 such
or1,itcanbeshownthatthetransitionsinA(e)areallofthe that ξk+1 ≼ξk′+1, ξk′ —t→′ A2 ξk′+1, and these transition steps are
form ({p},{(x,q)}), with only one input, one output and a compatibleinsomesense.However,suchadefinitionwillnot
label in X. As a consequence, the accessible configurations giveustheresultwearelookingfor.Considerthesetworuns:
are singletons, and the resulting Petri automaton has the
a B B
structure of a non-deterministic finite-state automaton (NFA).
A 1
Actually, in that case, the construction we described above is
c
just a variation on Thompson’s construction [20], with inlined b C 2 D
epsilon transition elimination.
c
a X 2′ Z
V. COMPARINGAUTOMATA
W 1′
A. Simple automata
b Y Y
The above results hold for the whole syntax of regular
The graphs produced by the first and the second runs corre-
expressions with converse and intersection. However, in the
remainder of the paper, we have to focus on expressions spondrespectivelytothegroundtermsa∩(b⋅c)and(a⋅c)∩b.
without converse or identity. This is because in combination These two terms are incomparable, but the relation ≼ depicted
below satisfies the previously stated conditions.
with intersection, these two operations introduce cycles in the
graphs associated to ground terms. Consider for instance the
1 2
graphs for a∩1 and a∩b⌣: {A} (cid:47)(cid:47){B,C} (cid:47)(cid:47){B,D}
a
≼ ≼ ≼
1’ 2’
(cid:47)(cid:47) (cid:47)(cid:47)
a {W} {X,Y} {Y,Z}
G(a∩1)= G(a∩b⌣)= .
b The problem here is that in Petri automata, runs are token
firing games. To adequately compare two runs, we need to
Since reflexive-transitive closure (⋅⋆) implicitly contains an closely track the tokens. For this reason, we will relate a
occurrenceoftheidentity,wealsohavetoforbidthisoperator. configuration ξ in A not only to a configuration ξ′ in A ,
Instead,weaddthetransitiveclosure(⋅+).Wethusworkwith buttoamapη kfromξ1′ toξ .Thiswillenableustoakssociat2e
expressions from Reg∩−, defined with the following syntax: k k k
X with each token situated on some place in P another token
2
e,f ∈Reg∩X−∶∶=x∈X ∣ e∩f ∣ e∪f ∣ e⋅f ∣ e+ ∣ 0. plaWceedwoanntAt1o.find a reading of G (ξ) in A2, i.e., a run in A2
Accordingly, ground terms are restricted to the following together with a sequence of maps associating places in A
2
syntax: u,v,w∶∶=x∈X ∣ w⋅w ∣ w∩w. to positions in G (ξ). Consider the picture below. Since we
Automata built using Definition 16 from expressions with- already have a reading of G (ξ) along ξ (by defining ρk(p)=
out converse or unit only have transitions labelled with X. ν(k,p),asintheproofofLemma14),itsufficestofindmaps
This corresponds to the notion of simple automata. from the places in A to the places in A (the maps η ): the
2 1 k
Definition 20 (Simple Petri automaton) readingofG (ξ)inA2 willbeobtainedbycomposingηk with
A Petri automaton A = ⟨P,T,ι,F⟩ is called simple if ρk.
∀(t,t)∈T,∀(x,p)∈t,x∈X. ∗ G (ξ)
For all e∈Reg∩X−, A(e) is simple. Moreover for any run ξ (cid:55)(cid:55) (cid:61)(cid:61) (cid:90)(cid:90) (cid:101)(cid:101)
ofasimplePetriautomaton, ξ =G (ξ)(uptoisomorphism); ρ0 ρn+1
inparticular,asimpleautoma(cid:74)ton(cid:75) onlyproducesacyclicgraphs. ρ1 ρn
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
B. Intuitions ξ0(cid:79)(cid:79) t0 ξ1(cid:79)(cid:79) t1 ⋯ tn−1 ξn(cid:79)(cid:79) tn ξn(cid:79)(cid:79)+1
In this section, we show how the notion of simulation η0 η1 ηn ηn+1
relation, that allows to compare NFA, can be adapted to
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
handle simple Petri automata. Consider two automata A1 = ξ0′ t′ ξ1′ t′ ⋯ t′ ξn′ t′ ξn′+1
⟨P1,T1,ι1,F1⟩ and A2 = ⟨P2,T2,ι2,F2⟩, we try to show 0 1 n−1 n
that for any graph G accepted by A1, G is recognised by A2. We need to impose some constraints on the maps (ηk) to
By Lemma 14, this amounts to proving that for any accepting ensure that (ρk○ηk)0⩽k⩽n is indeed a correct reading in A2.
run ξ in A1, G (ξ) is recognised by some accepting run ξ′ in First, we need to ascertain that a transition t′k in A2 may be
A2. Leaving non-determinism apart, the first idea that comes fired from the reading state ρk○ηk to reach the reading state
to mind is to find a relation between the configurations in A1 ρk+1○ηk+1. Furthermore, as for NFA, we want transitions tk
8
and t′ to be related: specifically, we require t′ to be included
k k a a
(via the homomorphisms ηk and ηk+1) in the transition tk. c C D c C D d
This is meaningful because transition t contains a lot of b B F
k
informationaboutthevertexkofG (ξ)andaboutρ:thelabels A a G b EG EG b EG EG G
of the outgoing edges from k are the labels on the output of
c a b
tk, and the only places that will ever be mapped to k in the b β δ ζ η η d
reading ρ are exactly the places in the input of tk. α b c a κ
This already shows an important dif- a B b γ (cid:15) (cid:15) θ ι
ferencebetweenthesimulationsforNFA A 0 Figure 10: Embedding of a parallel run into the run from Fig-
alantde Paettrrianasuittioomnatpa.—→aForp′NtFoA,awtreanrsei-- b C ure 5.
tion q —→a q′ with the same label a.
Hξk′er—te→′k Ath2eξtk′r+a1nsmitaioynhsavξke—dt→kiffAer1eξnkt+l1abaenlsd. X 0′ aa YZ ● itfhaξt≼doEma(nηd)ξ∈∈FF2.1, then there must be some η∈E suc∗h
Consider the step represented on the We will now prove that the language of A is contained in
1
right, corresponding to a square in the above diagram. The thelanguageofA ifandonlyifthereexistssuchasimulation.
2
output of 0 has a label b that does not appear in 0’ , and We first introduce the following notion of embedding.
0’ has two outputs labelled by a. Nevertheless this satisfies
Definition 22 (Embedding)
the conditions informally stated above, indeed, a∩b ⩽ a∩a Let ξ = ⟨(ξk)0⩽k⩽n,(tk)0⩽i<n⟩ be a run in A1, and Ξ =
holHdos.wever this definition is not yet satisfactory. Consider the ⟨(Ξk)0⩽k⩽n,(Ti)0⩽i<n⟩ a parallel run in A2. An embedding
two runs below: of Ξ into ξ is a sequence (ηi)0⩽i⩽n of maps such that for any
i<n, we have:
a b C ● ηi is a map from Ξi to ξi;
A 0 B 1 ● the image of Ti by ηi is included in ti, meaning that
c D for any (t,t) ∈ Ti, for any p ∈ t and (x,q) ∈ t, ηi(p)
is contained in the input of ti and (x,ηi+1(q)) is in the
b output of t ;
a Y 1′ T T i
● the image of the tokens in Ξi that do not appear in the
X 0′
c input of Ti are preserved (ηi(p) = ηi+1(p)) and their
a Z Z 2′ U image is not in the input of t .
i
Their produced graphs correspond respectively to the ground
terms a⋅(b∩c) and (a⋅b)∩(a⋅c). The problem is that ξ(cid:79)(cid:79)i ti (cid:47)(cid:47)ξi+(cid:79)(cid:79) 1
a⋅(b∩c) ⩽ (a⋅b)∩(a⋅c), but with the previous definition, ηi ηi+1
wecannotrelatetheseruns:theydonothavethesamelength. Ξ (cid:47)(cid:47)Ξ
i i+1
The solution here consists in grouping the transitions 1’ and Ti ∗
2’ together, and consider these two steps as a single step Figure 10 illustrates the embedding of some parallel run,
in a parallel run. This last modification gives us a notion of producingG(((b⋅c⋅a⋅b)∩(b⋅b⋅c⋅a))⋅d),intotherunpre-
simulation that suits our needs. sentedinFigure5.Noticethatisitnecessarytohaveaparallel
run instead of a simple one: to find something that matches
C. Simulations
the second transition in the upper run, we need to fire two
Definition 21 (Simulation) transitions in parallel in the lower run.
A relation ≼⊆P(P1)×P(P2⇀P1) between the configura- There is a close relationship between simulations and em-
tions of A1 and the partial maps from the places of A2 to the beddings:
places of A is called a simulation between A and A if:
1 1 2 Lemma 23. Let A and A be two Petri automata, the
● if ξ≼E and η∈E then the range of η must be included following are equival1ent: 2
in ξ;
● {ι1}≼{[ι2↦ι1]}; 12)) tfhoerreaneyxisatcscaepstiimngularutinonξ≼inbeAtwe,enthAer1e aisndanA2a;ccepting
(t,t) 1
● ifξ≼E andξ——→A ξ′,thenξ′≼E′ whereE′ istheset parallel run Ξ in A that can be embedded into ξ.
1 2
of all η′ such that there is some η∈E and a compatible
se–t odfotmran(sηi)tio—T→nsAT2 d⊆omT2(ηs′u)c;h that: P⟨th(reξoko)df0.e⩽fiIkfn⩽inti,ow(ntek)o0f⩽hksa<ivnme⟩ulabtaeiona,nsiwmaeuclcacetaipontninfignd≼r,uan lsieentquAenξ1c.e Bo=yf
–– ∀∀(pt∈′,dt′o)m∈(Tη,)η,((t∀′)(t⊆′,tt′a)n∈dT∀,(px∉,qt)′)∈⇒t′,(ηx(,pη)′=(qη)′)(p∈)t.; s∀ekts,ξokf ≼maEpsk.(EFku)rt0h⩽ekr⩽mnorseu,chwethactanE0ex=tra{c[tι2fr↦omι1]t}hisanda
9