510 research outputs found
Efficient Algorithms for Morphisms over Omega-Regular Languages
Morphisms to finite semigroups can be used for recognizing omega-regular
languages. The so-called strongly recognizing morphisms can be seen as a
deterministic computation model which provides minimal objects (known as the
syntactic morphism) and a trivial complementation procedure. We give a
quadratic-time algorithm for computing the syntactic morphism from any given
strongly recognizing morphism, thereby showing that minimization is easy as
well. In addition, we give algorithms for efficiently solving various decision
problems for weakly recognizing morphisms. Weakly recognizing morphism are
often smaller than their strongly recognizing counterparts. Finally, we
describe the language operations needed for converting formulas in monadic
second-order logic (MSO) into strongly recognizing morphisms, and we give some
experimental results.Comment: Full version of a paper accepted to FSTTCS 201
Ehrenfeucht-Fraisse Games on Omega-Terms
Fragments of first-order logic over words can often be characterized in terms
of finite monoids or finite semigroups. Usually these algebraic descriptions
yield decidability of the question whether a given regular language is
definable in a particular fragment. An effective algebraic characterization can
be obtained from identities of so-called omega-terms. In order to show that a
given fragment satisfies some identity of omega-terms, one can use
Ehrenfeucht-Fraisse games on word instances of the omega-terms. The resulting
proofs often require a significant amount of book-keeping with respect to the
constants involved. In this paper we introduce Ehrenfeucht-Fraisse games on
omega-terms. To this end we assign a labeled linear order to every omega-term.
Our main theorem shows that a given fragment satisfies some identity of
omega-terms if and only if Duplicator has a winning strategy for the game on
the resulting linear orders. This allows to avoid the book-keeping. As an
application of our main result, we show that one can decide in exponential time
whether all aperiodic monoids satisfy some given identity of omega-terms,
thereby improving a result of McCammond (Int. J. Algebra Comput., 2001)
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
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
Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions
What is the common link, if there is any, between Church-Rosser systems,
prefix codes with bounded synchronization delay, and local Rees extensions? The
first obvious answer is that each of these notions relates to topics of
interest for WORDS: Church-Rosser systems are certain rewriting systems over
words, codes are given by sets of words which form a basis of a free submonoid
in the free monoid of all words (over a given alphabet) and local Rees
extensions provide structural insight into regular languages over words. So, it
seems to be a legitimate title for an extended abstract presented at the
conference WORDS 2017. However, this work is more ambitious, it outlines some
less obvious but much more interesting link between these topics. This link is
based on a structure theory of finite monoids with varieties of groups and the
concept of local divisors playing a prominent role. Parts of this work appeared
in a similar form in conference proceedings where proofs and further material
can be found.Comment: Extended abstract of an invited talk given at WORDS 201
Two-Variable Logic over Countable Linear Orderings
We study the class of languages of finitely-labelled countable linear orderings definable in two-variable first-order logic. We give a number of characterisations, in particular an algebraic one in terms of circle monoids, using equations. This generalises the corresponding characterisation, namely variety DA, over finite words to the countable case. A corollary is that the membership in this class is decidable: for instance given an MSO formula it is possible to check if there is an equivalent two-variable logic formula over countable linear orderings. In addition, we prove that the satisfiability problems for two-variable logic over arbitrary, countable, and scattered linear orderings are NEXPTIME-complete
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