24 research outputs found
Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages
The purpose of this paper is to provide efficient algorithms that decide
membership for classes of several Boolean hierarchies for which efficiency (or
even decidability) were previously not known. We develop new forbidden-chain
characterizations for the single levels of these hierarchies and obtain the
following results: - The classes of the Boolean hierarchy over level
of the dot-depth hierarchy are decidable in (previously only the
decidability was known). The same remains true if predicates mod for fixed
are allowed. - If modular predicates for arbitrary are allowed, then
the classes of the Boolean hierarchy over level are decidable. - For
the restricted case of a two-letter alphabet, the classes of the Boolean
hierarchy over level of the Straubing-Th\'erien hierarchy are
decidable in . This is the first decidability result for this hierarchy. -
The membership problems for all mentioned Boolean-hierarchy classes are
logspace many-one hard for . - The membership problems for quasi-aperiodic
languages and for -quasi-aperiodic languages are logspace many-one complete
for
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
EF+EX Forest Algebras
We examine languages of unranked forests definable using the temporal
operators EF and EX. We characterize the languages definable in this logic, and
various fragments thereof, using the syntactic forest algebras introduced by
Bojanczyk and Walukiewicz. Our algebraic characterizations yield efficient
algorithms for deciding when a given language of forests is definable in this
logic. The proofs are based on understanding the wreath product closures of a
few small algebras, for which we introduce a general ideal theory for forest
algebras. This combines ideas from the work of Bojanczyk and Walukiewicz for
the analogous logics on binary trees and from early work of Stiffler on wreath
product of finite semigroups
Comparing Incomparable Frameworks: A Model Theoretic Approach to Phonology
In previous work, we used techniques from mathematical logic and model theory to study and compare two phonological theories, SPE and Government Phonology. The surprising result was that Government Phonology corresponds to a very weak fragment of SPE, yet it can attain the full expressivity of the latter through more powerful mechanisms of feature spreading. An issue that we didn\u27t elaborate on, however, is the question of what this increase in expressivity buys us in terms of empirical coverage, which we pick up in this paper. Again making good use of our model theoretic techniques, we investigate two phonological phenomena --- Sanskrit n-retroflexion and primary stress assignment in Creek and Cairene Arabic --- and show how much power feature spreading has to be granted in any descriptively adequate account which does not invoke additional technical machinery. These technical results are accompanied by reflections on the relation between empirically minded theory comparisons and the model theoretic approach
Algebraic recognizability of regular tree languages
We propose a new algebraic framework to discuss and classify recognizable
tree languages, and to characterize interesting classes of such languages. Our
algebraic tool, called preclones, encompasses the classical notion of syntactic
Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The
main result in this paper is a variety theorem \`{a} la Eilenberg, but we also
discuss important examples of logically defined classes of recognizable tree
languages, whose characterization and decidability was established in recent
papers (by Benedikt and S\'{e}goufin, and by Bojanczyk and Walukiewicz) and can
be naturally formulated in terms of pseudovarieties of preclones. Finally, this
paper constitutes the foundation for another paper by the same authors, where
first-order definable tree languages receive an algebraic characterization
The expressive power of existential first order sentences of BĂĽchi's sequential calculus
The aim of this paper is to study the first order theory of the successor, interpreted on finite words. More specifically, we complete the study of the hierarchy based on quantifier alternations (or ?n-hierarchy). It was known (Thomas, 1982) that this hierarchy collapses at level 2, but the expressive power of the lower levels was not characterized effectively. We give a semigroup theoretic description of the expressive power of B?1-, the boolean combinations of existential formulas. Our characterization is algebraic and makes use of the syntactic semigroup, but contrary to a number of results in this field, is not in the scope of Eilenberg's variety theorem, since B?1-definable languages are not closed under residuals. An important consequence is the following: given one of the levels of the hierarchy, there is polynomial time algorithm to decide whether the language accepted by a deterministic n-state automaton is expressible by a sentence of this level
Logic on words
First published in the Bulletin of the European Association of Theoretical Computer Science 54 (1994), 145-165.This dialog between Quisani, Yuri Gurevich's imaginary student, and the author, was published in the "Logic in Computer Science Column" of the EATCS Bulletin. It is first addressed to logicians. This dialog is an occasion to present the connections between Büchi's sequential calculus and the theory of finite automata. In particular, the essential role of first order formulæ is emphasized. The quantifier hierarchies on these formulæ are an occasion to present open problems
Extended Temporal Logic on Finite Words and Wreath Product of Monoids with Distinguished Generators
We associate a modal operator with each language belonging to a given class of regular languages and use the (reverse) wreath product of monoids with distinguished generators to characterize the expressive power of the resulting logic
Wreath Products of Forest Algebras, with Applications to Tree Logics
We use the recently developed theory of forest algebras to find algebraic
characterizations of the languages of unranked trees and forests definable in
various logics. These include the temporal logics CTL and EF, and first-order
logic over the ancestor relation. While the characterizations are in general
non-effective, we are able to use them to formulate necessary conditions for
definability and provide new proofs that a number of languages are not
definable in these logics
Temporal logic with cyclic counting and the degree of aperiodicity of finite automata
We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QAM of finite automata whose degree of aperiodicity belongs to the division ideal generated by M is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings. These closure conditions define q-varieties of finite automata. We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages. We also characterize QA M as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata. We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set M of moduli. It follows that when M is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in M