357 research outputs found
The disjunctivities of ω-languages
An ω-language over a finite alphabet X is a set of infinite sequences of letters of X. Consider congruences IL
and Pω, L on X* and a congruence OL on Xω introduced by an ω-language L. IL, Pω, L, and OL are called the
infinitary syntactic-congruence, the principal congruence and the ω-syntactic congruence of L, respectively. If
IL (Pω, L, OL) is the equality then L is called an I-disjunctive (P-disjunctive, O-disjunctive, respectively) ω-
language. Properties concerning such ω-languages are explored and relations between these ω-languages are
also studied
The FO^2 alternation hierarchy is decidable
We consider the two-variable fragment FO^2[<] of first-order logic over
finite words. Numerous characterizations of this class are known. Th\'erien and
Wilke have shown that it is decidable whether a given regular language is
definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<]
is interesting since its satisfiability problem is in NP. Restricting the
number of quantifier alternations yields an infinite hierarchy inside the class
of FO^2[<]-definable languages. We show that each level of this hierarchy is
decidable. For this purpose, we relate each level of the hierarchy with a
decidable variety of finite monoids. Our result implies that there are many
different ways of climbing up the FO^2[<]-quantifier alternation hierarchy:
deterministic and co-deterministic products, Mal'cev products with definite and
reverse definite semigroups, iterated block products with J-trivial monoids,
and some inductively defined omega-term identities. A combinatorial tool in the
process of ascension is that of condensed rankers, a refinement of the rankers
of Weis and Immerman and the turtle programs of Schwentick, Th\'erien, and
Vollmer
Regular Methods for Operator Precedence Languages
The operator precedence languages (OPLs) represent the largest known subclass of the context-free languages which enjoys all desirable closure and decidability properties. This includes the decidability of language inclusion, which is the ultimate verification problem. Operator precedence grammars, automata, and logics have been investigated and used, for example, to verify programs with arithmetic expressions and exceptions (both of which are deterministic pushdown but lie outside the scope of the visibly pushdown languages). In this paper, we complete the picture and give, for the first time, an algebraic characterization of the class of OPLs in the form of a syntactic congruence that has finitely many equivalence classes exactly for the operator precedence languages. This is a generalization of the celebrated Myhill-Nerode theorem for the regular languages to OPLs. As one of the consequences, we show that universality and language inclusion for nondeterministic operator precedence automata can be solved by an antichain algorithm. Antichain algorithms avoid determinization and complementation through an explicit subset construction, by leveraging a quasi-order on words, which allows the pruning of the search space for counterexample words without sacrificing completeness. Antichain algorithms can be implemented symbolically, and these implementations are today the best-performing algorithms in practice for the inclusion of finite automata. We give a generic construction of the quasi-order needed for antichain algorithms from a finite syntactic congruence. This yields the first antichain algorithm for OPLs, an algorithm that solves the ExpTime-hard language inclusion problem for OPLs in exponential time
Constructing Deterministic Parity Automata from Positive and Negative Examples
We present a polynomial time algorithm that constructs a deterministic parity
automaton (DPA) from a given set of positive and negative ultimately periodic
example words. We show that this algorithm is complete for the class of
-regular languages, that is, it can learn a DPA for each regular
-language. For use in the algorithm, we give a definition of a DPA,
that we call the precise DPA of a language, and show that it can be constructed
from the syntactic family of right congruences for that language (introduced by
Maler and Staiger in 1997). Depending on the structure of the language, the
precise DPA can be of exponential size compared to a minimal DPA, but it can
also be a minimal DPA. The upper bound that we obtain on the number of examples
required for our algorithm to find a DPA for is therefore exponential in
the size of a minimal DPA, in general. However we identify two parameters of
regular -languages such that fixing these parameters makes the bound
polynomial.Comment: Changes from v1: - integrate appendix into paper - extend
introduction to cover related work in more detail - add a second (more
involved) example - minor change
On Canonical Models for Rational Functions over Infinite Words
This paper investigates canonical transducers for rational functions over infinite words, i.e. functions of infinite words defined by finite transducers. We first consider sequential functions, defined by finite transducers with a deterministic underlying automaton. We provide a Myhill-Nerodelike characterization, in the vein of Choffrut’s result over finite words, from which we derive an algorithm that computes a transducer realizing the function which is minimal and unique (up to the automaton for the domain). The main contribution of the paper is the notion of a canonical transducer for rational functions over infinite words, extending the notion of canonical bimachine due to Reutenauer and Schützenberger from finite to infinite words. As an application, we show that the canonical transducer is aperiodic whenever the function is definable by some aperiodic transducer, or equivalently, by a first-order transduction. This allows to decide whether a rational function of infinite words is first-order definable.SCOPUS: cp.pinfo:eu-repo/semantics/publishe
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