Weak -Regular Trace Languages
Mazurkiewicz traces describe concurrent behaviors of distributed systems. Trace-closed word languages, which are “linearizations” of trace languages, constitute a weaker notion of concurrency but still give us tools to investigate the latter. In this vein, our contribution is twofold. Firstly, we develop definitions that allow classification of -regular trace languages in terms of the corresponding trace-closed -regular word languages, capturing E-recognizable (reachability) and (deterministically) Büchi recognizable languages. Secondly, we demonstrate the first automata-theoretic result that shows the equivalence of -regular trace-closed word languages and Boolean combinations of deterministically -diamond Büchi recognizable trace-closed languages.
Traces were introduced as models representing partially concurrent behaviors of distributed systems by Mazurkiewicz, who later also provided explicit definition of infinite traces . Zielonka demonstrated the close relation between traces and words that can be viewed as “linearizations” of traces, and also established automata-theoretic results regarding recognizability of languages of finite traces  (alternatively, see  for an introduction). We also refer the reader to  for a comprehensive collection of early results. Subsequently, Gastin-Petit  and Diekert-Muscholl , respectively, demonstrated the direct correspondence between the family of recognizable languages of infinite traces (-regular trace languages), and the families of asynchronous Büchi and deterministic asynchronous Muller automata. As with languages of finite traces, a set of infinite traces is recognizable iff the set of linearizations, i.e. the word language, corresponding to the set of infinite traces is.
It is well known that -regular languages can be obtained by various operations from regular languages of finite words. In general, any -regular language can be represented as , with regular. Languages of this form are recognized by Muller automata. There are also notions of subclasses of -regular languages that are obtained from given regular languages in the following ways:
For regular, languages are referred to as deterministically Büchi recognizable languages, and the corresponding deterministic Büchi automata (DBAs) can be constructed efficiently from the minimal DFA recognizing . The same is true for languages , which are recognized by -automata (reachability automata). Finite Boolean combinations of languages yield the family of weakly recognizable languages. This class can alternatively be characterized in terms of automata, being precisely the class of languages recognizable by deterministic weak automata (DWAs). Finite Boolean combinations of languages result in all -regular languages. For a class of regular languages, we refer to classes .
For both of these operations, we define corresponding operations for recognizable languages of finite traces, and . We show these operations relate to the classical word operations on the language of linearizations of traces in . More precisely, given a language of finite traces with the language of its linearizations, we show how can be modified to a trace-closed , such that the diagram in Fig. (a)a commutes. In particular, for every trace-closed , is trace-closed. Furthermore, for every recognizable , the linearizations of are recognizable by an -diamond E-automaton. Using this, we characterize the class of languages of infinite traces whose linearizations are recognizable by -diamond DWAs, as precisely the Boolean combinations of languages of the form for recognizable languages of finite traces. In the same spirit, we consider and . Here the situation is different, in that not for every recognizable , the language of linearizations of is recognizable by an -diamond DBA. We characterize the subclass of recognizable , where , the minimal DFA for the linearizations , also recognizes the linearizations of as a DBA. For those languages, the diagram (b)b commutes. In particular, for such , is trace-closed. Moreover, we show that every recognizable language of infinite traces is a finite Boolean combination of languages for such . Hence, any trace-closed language of infinite traces is a Boolean combination of -diamond DBA recognizable trace-closed languages.
In related work, Muscholl and Diekert  consider a form of “deterministic” trace languages. In  it is shown that every recognizable language of infinite traces is a Boolean combination of these deterministic languages. However, those languages require modifications to the Büchi acceptance condition in order to obtain a correspondence in terms of -diamond DBAs. The problem of finding a suitable class of languages which has a classical automaton correspondence is left open in .
We begin with presenting definitions that are relevant to the connections between regular and -regular languages. We also formally introduce the notion of regular and -regular trace languages. In Sec. 3, we present definitions that allow construction of various classes of -regular trace languages from regular trace languages. In particular, we classify trace languages whose linearizations are weakly recognizable, and those whose linearizations are DBA recognizable. We establish that every -regular trace language is a Boolean combination of those trace languages whose linearizations are DBA recognizable.
We denote a recognizable language of finite words, or simply a regular language, with the upper case letter and a class of such languages with . Finite words are denoted with lower case letters , , etc. Infinite words are denoted by lower case Greek letters and , and a recognizable language of infinite words, or simply an -regular language, by upper case . For a word or , we denote its infix starting at position and ending at position by or , and the letter with or . For a language , we denote the complement language by .
We assume the reader is familiar with the notions of Deterministic Finite Automata (DFAs) and Deterministic Büchi Automata (DBAs). We say that a language is DBA recognizable iff it is recognized by a DBA. For the class of regular languages, the class coincides with the DBA recognizable languages. Further, the class of finite Boolean combinations of languages from is also the class of -regular languages, and it coincides with the class of languages recognized by nondeterministic Büchi or deterministic Muller automata.
Recall that a Deterministic Weak Automaton (DWA) is a DBA where every strongly connected component of the transition graph has only accepting states or only rejecting states. For a regular language , the minimal DFA recognizing also recognizes as a DBA. Given the minimal DFA recognizing , a DWA recognizing , respectively , can be constructed as follows:
For a symbol and define .
For each , define
Define , respectively
The family of DWAs is closed under Boolean operations. For an -language , define a congruence where iff . If is recognized by a DWA then this congruence has a finite index. We say that an -language is weakly recognizable if it is recognized by a DWA. The class of finite Boolean combinations of languages in is exactly the set of weakly recognizable languages.
Remark 1 (The minimal DWA )
If for a weakly recognizable language , is the index of the congruence defined above, then the language is recognized by a DWA with . Also, for each state there exists a word such that for each iff .
Turning to traces, let denote an irreflexive111A relation is irreflexive if for no we have ., symmetric independence relation over an alphabet , then is the reflexive, symmetric dependence relation over . We refer to the pair as the dependence alphabet. For any letter , we define and . A trace can be identified with a labeled, acyclic, directed dependence graph where is a set of countably many vertices, is a labeling function, and is a countable set of edges such that, firstly, for every ; secondly, every vertex has only finitely many predecessors. and represent the sets of all finite and infinite traces whose dependence graphs satisfy the two conditions above. We denote finite traces with the letter , and an infinite trace with ; the corresponding languages with and respectively. For a trace define , and similarly for a trace . For an infinite trace, define .
For two traces (or ) denotes that is a (proper) prefix of . We denote the prefix relation between words similarly. The least upper bound of two finite traces, whenever it exists, denoted is the smallest trace such that and . Whenever it exists, one can similarly refer to the least upper bound of a finite or an infinite set of traces. The concatenation of two traces is denoted as . Note that for any the concatenation . However, iff .
The canonical morphism associates finite words with finite traces, and the inverse mapping associates finite traces with equivalence classes of words. The morphism can also be extended to a mapping . For a (finite or infinite) trace , the set represents the linearizations of . Two words are equivalent, denoted , iff . We note that for finite traces the relation coincides with the reflexive, transitive closure of the relation . For a word , define the set . Finally, we say that a word language is trace-closed iff , where .
A trace language (resp. ) is recognizable or regular iff (resp. ) is a recognizable word language.
With and we denote the classes of recognizable languages of finite and infinite traces respectively.
Asynchronous cellular automata have been introduced [1, 3] as acceptors of -regular trace languages. However, a global view of their (local) transition relations yields a notion of automata that recognize trace-closed word languages. Throughout this paper, we take this global view of asynchronous automata. Formally, a deterministic asynchronous cellular automaton (DACA) over is a 4-tuple , where , and . Given a state and a letter , the unique -sucessor is given by and for all . That is, the only component that changes its state is the component corresponding to . Given a word the run of on is given as usual by and . This definition extends naturally to infinite runs on infinite . A deterministic asynchronous Muller automaton (DACMA) is an asynchronous automaton with . We define of (a finite or an infinite) run to be the set . Likewise, . A DACMA accepts if for some we have .
A word automaton is called -diamond if for every and every state , . Every (resp. ) is recognized by a DACA  (resp. a DACMA ). Via their global behaviors, they accept the corresponding trace-closed languages, and in particular, every regular trace-closed language (resp. trace-closed -regular language) is recognized by an -diamond DFA (resp. -diamond Muller automaton). In fact for every trace-closed , the minimal DFA accepting is -diamond.
Finally, we want to recall some basic algebraic definitions. Given a language of finite traces, a semigroup , and a morphism , is said to recognize if there exists with . By extension, is said to recognize if such a morphism exists. A linked pair of a semigroup is a tuple with and . We state a well known consequence of Ramsey’s theorem: Let be a (possibly infinite) alphabet, be any finite semigroup and any mapping. Given an infinite sequence and an arbitrary factorization of into words , there exists a linked pair and a strictly monotone sequence of natural numbers with the property that and for all . Let be given by and for . We say this superfactorization is associated with . We will often use Ramsey’s theorem implicitly. Given a semigroup , a morphism is said to saturate if for every linked pair of we have either or . Let be a language of infinite traces, be a finite semigroup, and a saturating morphism. Then recognizes , if for some set of linked pairs of we have . Again, we say recognizes if such a morphism exists. These notions of recognizability coincide with the corresponding notions from Def. 1.
3 From Regular Trace Languages to -Regular Trace Languages
We wish to extend the well-studied relations between regular and -regular languages to the field of finite and infinite traces. We first look at reachability and safety languages, their Boolean combinations, i.e. the weakly recognizable languages, and study how they can be obtained as a result of infinitary operations on regular trace languages. We will later see that the case of Büchi recognizability is not straight forward. Our definitions are consistent with those over word languages; that is, if the dependence relation over the alphabet is complete then these definitions coincide.
3.1 Infinitary Extensions of Regular Trace Languages
In the classification hierarchy of -regular languages, reachability and safety languages occupy the lowest levels. For trace languages we have the following.
Let . The infinitary extension is the -trace language given by .
However, the definition of infinitary extensions of a trace-closed languages is not sound with respect to trace equivalence of -words; i.e. if and , then, in general, .
Let , and . Define . Clearly is trace-closed and, moreover, . Let . Clearly are equivalent words since they induce the same infinite trace which belongs to . However, while , .
Let be trace-closed. Define the -suffix extended trace-closed language (or -suffix extension) of as .
Due to the closure of under concatenation and finite union , we know that is regular whenever is regular.
If , , and is the -suffix extension of , then .
From the definitions of and , we trivially observe that for every it holds that . Therefore, .
To show , we show that: (1) for every infinite trace in , there exists a linearization in ; (2) the language is trace-closed.
(1) Consider . Hence there exist and such that . From the definitions, it follows that for any and and therefore in .
(2) Let , and be a trace such that . Consider any such that . Trace equivalence implies that . Moreover there exists a minimal natural number . Observe that is a maximal symbol appearing in because otherwise we can contradict the minimality of and find such that . Now, let be the finite trace such that .
It must hold that either is the empty trace or , because otherwise . This implies , and hence .
In general . However, iterated -suffix extensions preserve the infinitary extension languages:
Proposition 1 provides us the basis for generating the class of weakly recognizable trace-closed languages corresponding to the recognizable subset of . Henceforth, whenever we speak of the language we refer to . Similarly, for a trace-closed language we always mean whenever we say .
A trace-closed language is recognized by an -diamond DWA if and only if for a finite set of trace-closed regular languages.
Given trace-closed regular languages , we construct -diamond DWA accepting as mentioned previously. Let be the language expressed in disjunctive normal form over (for each is either of the form or ). We define the product DWA where:
if and only if for all
The tuple if and only if it satisfies some conjunct. That is, for some it holds that whenever then , and whenever then for all .
It is easily verified that is an -diamond DWA accepting .
For the other direction, consider the minimal DWA that accepts . Since trace equivalence over finite words is a finer congruence than the language congruence (i.e. for all ), it follows that for any pair of finite trace equivalent words . Thus, is -diamond.
For each SCC of , let trace-closed be the language accepted by . Recall that each SCC of a DWA contains either only accepting states or rejecting states. Then, the language accepted by is given by the following disjunction over all accepting SCC’s , where .
3.2 Infinitary Limits of Regular Trace Languages
We now consider the infinitary limit operator. In the case of word languages, this operator extends regular languages to the family -regular languages that are DBA recognizable. In particular, we seek an effective characterization of languages , such that is recognized by an -diamond DBA.
Let , the infinitary limit is the -trace language containing all such that there exists a sequence satisfying and .
For , it holds that . In fact, if for a finite semigroup , a morphism recognizes , then can be described in terms of a set of linked pairs of , i.e. .
Let , and . Define as the trace-closed language with even number of occurrences of ’s and ’s. The minimal DFA accepting this language is shown in Figure 2. If , then is defined as
The trace-closed language consists of all infinite words that satisfy the same conditions as above.
It is easy to verify that the DFA of Figure 2 does not accept when equipped with a Büchi acceptance condition. For instance, the automaton can loop forever in states , , and , thereby witnessing infinitely many ’s and ’s, without ever visiting state .
There does not exist any -diamond DBA recognizing as described in Example 2.
A proof of this proposition can be found in the appendix.
There exists a family of trace-closed regular languages of finite words, namely over , such that given for any , there exists no -diamond DBA recognizing .
A trace-closed language is -limit-stable (or simply limit-stable) if is also trace-closed. By extension, is limit-stable if is.
Toward characterizing limit-stable languages, we introduce some definitions. Let be a language of traces and let be two traces. The prefix graph of the pair is the directed, acyclic graph with and if for some . A cut of is a set such that each path from to in visits at least one vertex from . Note that if for some , then does not admit a cut. A pair is -separable if admits a cut .
Let . Define an infinite transition-graph with and if for some . Then there is a one to one correspondence between the paths starting from through and the linearizations of . More precisely, for any finite word , there exists a run from on in iff is the linearization of some prefix of . An infinite word is a linearization of iff is a linearization of some prefix of for all . Hence, an -word is a linearization of iff it induces a run in .
Let be a finite semigroup, let , and let be a linked pair of . Let be a morphism from onto . The pair has the -cut property if
either for every factorization with , we have for some ;
or for every factorization with , we have for all .
Let . Then there exists a finite semigroup and a saturating morphism which recognizes both and .
Such a morphism is said to simultaneously recognize and . Given an automaton, we write if some leads from to , and if a final state is also visited.
Given , let be an -diamond automaton. is -cycle closed, if for all and all we have iff .
We can now give an effective characterization of limit-stable languages. Due to space constraints, we only present a part of the following proof here. Lem. 1 ensures that (e) is not trivially satisfied.
Let and let . The following are equivalent:
, and therefore , is limit-stable.
For all sequences and all sequences with , there exists a subsequence and a sequence of proper prefixes with and for all .
For any there exists a strictly monotone such that any infinite path in visits in each segment .
Let be a sequence of traces in . Then there exists a subsequence , such that is -separable for all .
If and are simultaneously recognized by a morphism for some finite semigroup , then every linked pair has the -cut property.
Any DFA recognizing is -cycle closed.
(a)(b): If (b) is false, then we may choose a sequence of traces in with the property that for some sequence of linearizations of , every subsequence , and every sequence of proper prefixes , , we have . Since we have that is a compact space. Hence has a converging subsequence . Because every subsequence of has the properties given in the previous sentence, so does . Let . Then for some with . Hence, . But, by construction, because for some no prefix of length of is in .
(b)(a): Let for traces . We may assume that implies . Let . Then we pick prefixes of , such that is of minimal length with . Consider the subsequence of . Each is a prefix of some linearization of , say . We apply (b) to the sequence and get a sequence of proper prefixes of the , such that and . We now have to show that is already a prefix of . Suppose not, i.e. . Then this would give a trace with .
(a)(f): Suppose is not -cycle closed. Then there exists and with but not . Since is -diamond, this means that the run exists, but does not visit a final state. Now pick with . Then and . But clearly implies that is not trace-closed.
Next, we observe that we find with . This gives with . Conversely, there exists with and therefore with , which implies .
Notice that if and , then (by trace equivalence and the fact that is -diamond) we have . Likewise we have and . Now we can apply (f) to see that iff . However, since , since for all , and since , we have . Hence, . Since furthermore , we have for all whence .
Let for some . Given , it is decidable in time whether or not is limit-stable.
Let be recognizable, trace-closed. Pick a DACMA (c.f. Sec. 2) recognizing . Recall that the global transition behavior of gives an -diamond DFA, which we denote by . Given we define the DBA , where . Note that is -cycle closed, because for any and all with and we have222This can be proven by an induction on the number of swapping operations needed to obtain from . . Now: