The class of -regular languages provides a robust specification language in verification. Every -regular condition can be decomposed into a safety part and a liveness part. The liveness part ensures that something good happens “eventually”. Finitary liveness was proposed by Alur and Henzinger as a stronger formulation of liveness [AH98]. It requires that there exists an unknown, fixed bound such that something good happens within transitions. In this work we consider automata with finitary acceptance conditions defined by finitary Büchi, parity and Streett languages. We give their topological complexity of acceptance conditions, and present a regular-expression characterization of the languages they express. We provide a classification of finitary and classical automata with respect to the expressive power, and give optimal algorithms for classical decisions questions on finitary automata. We (a) show that the finitary languages are -complete; (b) present a complete picture of the expressive power of various classes of automata with finitary and infinitary acceptance conditions; (c) show that the languages defined by finitary parity automata exactly characterize the star-free fragment of -regular languages; and (d) show that emptiness is -complete and universality as well as language inclusion are -complete for finitary automata.
Classical -regular languages: strengths and weakness. The class of -regular languages provides a robust language for specification for solving control and verification problems (see, e.g, [PR89, RW87]). Every -regular specification can be decomposed into a safety part and a liveness part [AS85]. The safety part ensures that the component will not do anything “bad” (such as violate an invariant) within any finite number of transitions. The liveness part ensures that the component will do something “good” (such as proceed, or respond, or terminate) in the long-run. Liveness can be violated only in the limit, by infinite sequences of transitions, as no bound is stipulated on when the “good” thing must happen. This infinitary, classical formulation of liveness has both strengths and weaknesses. A main strength is robustness, in particular, independence from the chosen granularity of transitions. Another main strength is simplicity, allowing liveness to serve as an abstraction for complicated safety conditions. For example, a component may always respond in a number of transitions that depends, in some complicated manner, on the exact size of the stimulus. Yet for correctness, we may be interested only that the component will respond “eventually”. However, these strengths also point to a weakness of the classical definition of liveness: it can be satisfied by components that in practice are quite unsatisfactory because no bound can be put on their response time.
Stronger notion of liveness. For the weakness of the infinitary formulation of liveness, alternative and stronger formulations of liveness have been proposed. One of these is finitary liveness [AH98]: finitary liveness does not insist on a response within a known bound (i.e, every stimulus is followed by a response within transitions), but on response within some unknown bound (i.e, there exists such that every stimulus is followed by a response within transitions). Note that in the finitary case, the bound may be arbitrarily large, but the response time must not grow forever from one stimulus to the next. In this way, finitary liveness still maintains the robustness (independence of step granularity) and simplicity (abstraction of complicated safety) of traditional liveness, while removing unsatisfactory implementations.
Finitary parity and Streett conditions. The classical infinitary notion of fairness is given by the Streett condition: it consists of a set of pairs of requests and corresponding responses (grants) and requires that every request that appears infinitely often must be responded infinitely often. Its finitary counterpart, the finitary Streett condition requires that there is a bound such that in the limit every request is responded within steps. The classical infinitary parity condition consists of a priority function and requires that the minimum priority visited infinitely often is even. Its finitary counterpart, the finitary parity condition requires that there is a bound such that in the limit after every odd priority a lower even priority is visited within steps.
Results on classical automata. There are several robust results on the languages expressible by automata with infinitary Büchi, parity and Streett conditions, as follows: (a) Topological complexity: it is known that Büchi languages are -complete, whereas parity and Streett languages lie in the boolean closure of and [MP92]; (b) Automata expressive power: non-deterministic automata with Büchi conditions have the same expressive power as deterministic and non-deterministic parity and Streett automata [Cho74, Saf92]; and (c) Regular-expression characterization: the class of languages expressed by deterministic parity is exactly defined by -regular expressions (see the handbook [Tho97] for details).
Our results. For finitary Büchi, parity and Streett languages, topological, automata-theoretic, regular-expression and decision problems studies were all missing. In this work we present results in the four directions, as follows:
Topological complexity. We show that finitary Büchi, parity and Streett conditions are -complete.
Automata expressive power. We show that finitary automata are incomparable in expressive power with classical automata. As in the infinitray setting, we show that non-deterministic automata with finitary Büchi, parity and Streett conditions have the same expressive power, as well as deterministic parity and Streett automata, which are strictly more expressive than deterministic finitary Büchi automata. However, in contrast to the infinitary case, for finitary parity condition, non-deterministic automata are strictly more expressive than the deterministic counterpart. As a by-product we derive boolean closure properties for finitary automata.
Regular-expression characterization. We consider the characterization of finitary automata through an extension of -regular languages defined as -regular languages by [BC06]. We show that languages defined by non-deterministic finitary Büchi automata are exactly the star-free fragment of -regular languages.
Decision problems. We show that emptiness is -complete and universality as well as language inclusion are -complete for finitary automata.
Related works. The notion of finitary liveness was proposed and studied in [AH98], and games with finitary objectives was studied in [CHH09]. A generalization of -regular languages as -regular languages was introduced in [BC06] and variants have been studied in [BT09] (also see [Boj10] for a survey); a topological characterization has been given in [HST10]. Our work along with topological and automata-theoretic studies of finitary languages, explores the relation between finitary languages and -regular expressions, rather than identifying a subclass of -regular expressions. We identify the exact subclass of -regular expressions that corresponds to non-deterministic finitary parity automata.
2.1 Languages topological complexity
Let be a finite set, called the alphabet. A word is a sequence of letters, which can be either finite or infinite. A language is a set of words: is a language over finite words and over infinite words.
Cantor topology and Borel hierarchy. Cantor topology on is given by open sets: a language is open if it can be described as where . Let denote the open sets and denote the closed sets (a language is closed if its complement is open): they form the first level of the Borel hierarchy. Inductively, we define: is obtained as countable union of sets; and is obtained as countable intersection of sets. The higher a language is in the Borel hierarchy, the higher its topological complexity.
Since the above classes are closed under continuous preimage, we can define the notion of Wadge reduction [Wad84]: reduces to , denoted by , if there exists a continuous function such , where is the preimage of by . A language is hard with respect to a class if all languages of this class reduce to it. If it additionally belongs to this class, then it is complete.
For , let be the set of finite prefixes of words in . The following property holds:
For all languages , is closed if and only if, for all infinite words , if all finite prefixes of are in , then .
Classical liveness conditions. We now consider three classes of languages that are widespread in verification and specification. They define liveness properties, i.e, intuitively say that something good will happen “eventually”. For an infinite word , let denote the set of letters that appear infinitely often in . The class of Büchi languages is defined as follows, given :
i.e, the Büchi condition requires that some letter in appears infinitely often. The class of parity languages is defined as follows, given a priority function that maps letters to integers (representing priorities):
i.e, the parity condition requires that the lowest priority the appears infinitely often is even. The class of Streett languages is defined as follows, given , where are request-grant pairs:
i.e, the Streett condition requires that for all requests , if it appears infinitely often, then the corresponding grant also appears infinitely often.
The following theorem presents the topological complexity of the classical languages:
Theorem 2.1 (Topological complexity of classical languages [Mp92])
For all , the language is -complete.
The parity and Streett languages lie in the boolean closure of and .
2.2 Finitary languages
The finitary parity and Streett languages have been defined in [CHH09]. We recall their definitions, and also specialize them to finitary Büchi languages. Let , where , the definition for uses distance sequence as follows:
i.e, given a position where is requested, is the time steps (number of transitions) between the request and the corresponding grant . Note that . Then and:
i.e, the finitary Streett condition requires the supremum limit of the distance sequence to be bounded.
Since parity languages can be considered as a particular case of Streett languages, where , the latter allows to define . The same applies to finitary Büchi languages, which is a particular case of finitary parity languages where the letters from the set have priority and others have priority . We get the following definitions. Let a priority function, we define:
i.e, given a position where is odd, is the time steps between the odd priority and a lower even priority. Then . We define similarly the finitary Büchi language: given , let:
i.e, is the time steps before visiting a letter in . Then .
2.3 Automata, -regular and finitary languages
An automaton is a tuple , where is a finite set of states, is the finite input alphabet, is the set of initial states, is the transition relation and is the acceptance condition.
An automaton is deterministic if it has a single initial state and for every state and letter there is at most one transition. The transition relation of deterministic automata are described by functions . An automaton is complete if for every state and letter there is a transition. This is the case when the transition function is total.
Runs. A run is a word over , where . The run is accepting if it is infinite and . We will write to denote . An infinite word induces possibly several runs of : a word induces a run if for all . The language accepted by , denoted by , is:
Acceptance conditions. We will consider various acceptance conditions for automata obtained from the last section by considering as the alphabet. For example, given , the languages and define Büchi and finitary Büchi acceptance conditions, respectively. Automata with finitary acceptance conditions are referred as finitary automata; classical automata are those equipped with infinitary acceptance conditions.
We use a standard notation to denote the set of languages recognized by some class of automata. The first letter is either or , where stands for “non-deterministic” and stands for “deterministic”. The last letter refers to the acceptance condition: stands for “Büchi”, stands for “parity” and stands for “Streett”. The acceptance condition may be prefixed by for “finitary”. For example, denotes non-deterministic parity automata, and denotes deterministic finitary Streett automata. We have the following combination:
Theorem 2.2 (Expressive power results for classical automata)
3 Topological complexity
In this section we define a finitary operator that allows us to describe finitary Büchi, finitary parity and finitary Streett languages topologically and to relate them to the classical Büchi, parity and Streett languages; we then give their topological complexity.
Union-closed-omega-regular operator on languages. Given a language , the language is the union of the languages that are subsets of , -regular and closed, i.e, .
For all languages we have .
Since the set of finite automata can be enumerated in sequence, it follows that is countable. So for all languages , the set is described as a countable union of closed sets. Hence .
We present a pumping lemma for -regular languages that we will use to prove the topological complexity of finitary languages.
Lemma 1 (A pumping lemma)
Let be an -regular language. There exists such that for all words , for all positions , there exist such that for all we have .
Given is a -regular language, let be a complete and deterministic parity automata that recognizes , and let be the number of states of . Consider a word such that , and let be the unique run induced by in . Consider a position in such that . Then there exist such that , this must happen as has states. For , if we consider the word , then the unique run induced by in is . Since the parity condition is independent of finite prefixes and the run is accepted by , it follows that is accepted by . Since recognizes , we have .
The following lemma shows that is obtained by applying the operator to .
For all , where , we have
We present the two directions of the proof.
We first show that . Let such that is closed and -regular. Let , and assume towards contradiction, that . Hence for all , there exists such that and . Let given by the pumping lemma on , from above given we obtain such that and . By the pumping lemma we obtain the witness . Let , and . Since , by the pumping lemma for all we have . This entails that all finite prefixes of the infinite word are in . Since is closed, it follows that . Since it follows that there is some request in position , and there is no corresponding grant for the next steps. Hence there is a position in such that there is request at and no corresponding grant in , and thus it follows that the word . This contradicts that . Hence it follows that .
We now show the converse: . We have:
The language is closed, -regular, and included in . Hence .
The result follows.
Lemma 2 naturally extends to finitary parity and finitary Büchi languages:
The following assertions hold:
For all , we have ;
For all , we have .
Büchi languages are a special case of parity languages, and parity languages are in turn a special case of Streett languages. Since distance sequences for parity and Büchi languages have been defined as a special case of Streett languages, Corollary 1 follows from Lemma 2.
The following lemma states that finitary Büchi languages are -complete.
Theorem 3.1 (Topological characterization of finitary languages)
The finitary Büchi, finitary parity and finitary Streett are -complete.
We show that if , then is -complete. It follows from Corollary 1 that . We now show that is -hard. By Theorem 2.1 we have that is -complete, hence is -complete. We present a topological reduction to show that . Let be the stuttering function defined as follows:
The function is continuous. We check that the following holds:
Left to right direction: assume that from the position of , letters belong to .
Then from the position , letters of belong to , then for .
Right to left direction: let and be integers such that for all we have . Assume and , then the letter in position in is repeated times, thus is either or higher than . The latter is not possible since it must be less than . It follows that the letter in position in belongs to . Hence we get , so is -complete. From this we deduce the two other claims as special cases.
4 Expressive power of finitary automata
In this section we consider the finitary automata, and compare their expressive power to classical automata. We then address the question of determinization. Deterministic finitary automata enjoy nice properties that allows to describe languages they recognize using the operator. As a by-product we get boolean closure properties of finitary automata.
4.1 Comparison with classical automata
Finitary conditions allow to express bounds requirements:
Example 1 ()
Consider the finitary Büchi automaton shown in Fig. 1, the state labeled 0 being its only final state. Its language is . Indeed, -labeled state is visited while reading the letter , and the -labeled state is visited while reading the letter . An infinite word is accepted iff the -labeled state is visited infinitely often and there is a bound between two consecutive visits of the -labeled state. We can easily see that is not -regular, using proof ideas from [BC06]: its complement would be -regular, so it would contain ultimately periodic words, which is not the case.
However, finitary automata cannot distinguish between “many b’s” and “only b’s”:
Example 2 ()
Consider the language of infinitely many ’s, i.e, . The language is recognized by a simple deterministic Büchi automaton. However, we can show that there is no finitary Büchi automata that recognizes . Intuitively, such an automaton would, while reading the infinite word , have to distinguish between all b’s, otherwise it would accept a word with only b’s at the end. Assume towards contradiction that there exists a non-deterministic finitary Büchi automaton with states recognizing . Let us consider the infinite word . Since must be accepted by , there must be an accepting run , represented as follows:
Since is accepting, there exists , and , such that for all we have . Let be the lowest priority infinitely visited in . As is accepting, is even. The state is in position in . Let be an integer such that (a) and (b) . Let us consider the set of states . Since the distance function is bounded by from the -th position, the priority appears at least once in each set of consecutively visited states of size . Since and is the state following , the latter holds from . Since , it appears at least times in . Since there is states in , at least one state has been reached twice. We can thus iterate: the infinite word , and the word is accepted by . However, and hence we have a contradiction.
We summarize the results in the following theorem.
The following assertions hold: (a) ; (b) .
4.2 Deterministic finitary automata
Given a deterministic complete automaton with accepting condition , we will consider the language obtained by using as acceptance condition. Treating the automaton as a transducer, we consider the following function: which maps an infinite word to the unique run of on (there is a unique run since is deterministic and complete). Then:
We will focus on the following property: , which follows from the following lemma. Deterministic complete automata, regarded as transducers, preserve topology and -regularity. Hence applying the finitary operator to the input (the language ) or to the acceptance condition is equivalent.
For all deterministic complete automaton, we have:
for all , is closed closed ( is continuous).
for all , is closed closed ( is closed).
for all , is -regular -regular.
for all , is -regular -regular.
We prove all the cases below.
Let such that is closed. Let be such that for all we have . We define the run and show that . Since is closed, we will show for all we have . From the hypothesis we have , and then there exists an infinite word such that . Let , then we have . Since is deterministic, we get , and hence .
Let such that is closed. Let such that for all we have . Then for all , there exists a word such that , and . We define by induction on an infinite nested sequence of finite words . We denote by the limit of this nested sequence of finite words. We have that . Since is closed, .
Let such that recognized by a Büchi automaton . We define the Büchi automaton , where iff in and in . We now show the correctness of our construction. Let accepted by , then there exists an accepting run , as follows:
where the second component visits infinitely often. Hence:
Hence from , we have , and it follows that . Conversely, let , then we have . Then the above statement holds, which entails that is accepted by . It follows that recognizes .
Let such that is recognized by a Büchi automaton . We define the Büchi automaton , where iff there exists , such that in and in . A proof similar to above show that recognizes .
The desired result follows.
For any deterministic complete automaton recognizing a language , the finitary restriction of this automaton recognizes .
A word is accepted by iff .
Theorem 4.2 allows to extend all known results on deterministic classes to finitary deterministic classes: as a corollary, we have and .
We now show that non-deterministic finitary parity automata are more expressive than deterministic finitary parity automata. However, for every language there exists such that recognizes , and by Theorem 4.2 the deterministic finitary parity automaton recognizes .
For every language there is a deterministic finitary parity automata such that .
Example 3 ()
As for Example 1 we consider the languages and . It follows from Example 1 that both and belong to , hence to . A finitary parity automaton, relying on non-determinism, is easily built to recognize , hence . We can show that we cannot bypass this non-determinism, as by reading a word we have to decide well in advance which sequence will be bounded: a’s or b’s, i.e, . To prove it, we interleave words of the form and , and use a pumping argument to reach a contradiction. Assume towards contradiction that , and let be a deterministic complete finitary parity automaton with states that recognizes . Let be the starting state. Since belongs to , its unique run on is accepting, and can be decomposed as follows: where is the lowest priority visited infinitely often while reading . Then, belongs to this , its unique run on is accepting, and has the following shape: where is the lowest priority visited infinitely often while reading . Repeating this construction and by induction we have, as shown in Fig 2:
where is the lowest priority visited infinitely often while reading and is the lowest priority visited infinitely often while reading . There must be , such that . Let and , we have:
Consider the words and
must be accepted by since it belongs to . Hence is accepted as well, but does not belong to . We have a contradiction, and the result follows.
We have .
Observe that Theorem 4.2 does not hold for non-deterministic automata, since we have but .
4.3 Non-deterministic finitary automata
We can show that non-deterministic finitary Streett automata can be reduced to non-deterministic finitary Büchi automata, and this would complete the picture of expressive power comparison. We first show that non-deterministic finitary Büchi automata are closed under intersection, and use it to show Theorem 4.4.
is closed under intersection.
Let and be two non-deterministic finitary Büchi automata. Without loss of generality we assume both and to be complete. We will define a construction similar to the synchronous product construction, where a switch between copies will happen while visiting or . The finitary Büchi automaton is . We define the transition relation below: