The Sch\"utzenberger product for syntactic spaces

Starting from Boolean algebras of languages closed under quotients and using duality theoretic insights, we derive the notion of Boolean spaces with internal monoids as recognisers for arbitrary formal languages of finite words over finite alphabets. This leads to a setting that is well-suited for applying existing tools from Stone duality as applied in semantics. The main focus of the paper is the development of topo-algebraic constructions pertinent to the treatment of languages given by logic formulas. In particular, using the standard semantic view of quantification as projection, we derive a notion of Sch\"utzenberger product for Boolean spaces with internal monoids. This makes heavy use of the Vietoris construction, and its dual functor, which is central to the coalgebraic treatment of classical modal logic. We show that the unary Sch\"utzenberger product for spaces, when applied to a recogniser for the language associated to a formula with a free first-order variable, yields a recogniser for the language of all models of the corresponding existentially quantified formula. Further, we generalise global and local versions of the theorems of Sch\"utzenberger and Reutenauer characterising the languages recognised by the binary Sch\"utzenberger product. Finally, we provide an equational characterisation of Boolean algebras obtained by local Sch\"utzenberger product with the one element space based on an Egli-Milner type condition on generalised factorisations of ultrafilters on words.Comment: 21 page

Varieties

This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and motivate the fundamental concepts. We do this for the most part without proofs, and often without precise definitions, leaving these to the formal development of the theory that begins in Section 2. Our presentation of the theory draws heavily on the work of Gehrke, Grigorieff and Pin (2008) on the equational theory of lattices of regular languages. In the subsequent sections we consider in more detail aspects of varieties that were only briefly evoked in the introduction: Decidability, operations on languages, and characterizations in formal logic.Comment: This is a chapter in an upcoming Handbook of Automata Theor

Abstract Transducers

Several abstract machines that operate on symbolic input alphabets have been proposed in the last decade, for example, symbolic automata or lattice automata. Applications of these types of automata include software security analysis and natural language processing. While these models provide means to describe words over infinite input alphabets, there is no considerable work on symbolic output (as present in transducers) alphabets, or even abstraction (widening) thereof. Furthermore, established approaches for transforming, for example, minimizing or reducing, finite-state machines that produce output on states or transitions are not applicable. A notion of equivalence of this type of machines is needed to make statements about whether or not transformations maintain the semantics. We present abstract transducers as a new form of finite-state transducers. Both their input alphabet and the output alphabet is composed of abstract words, where one abstract word represents a set of concrete words. The mapping between these representations is described by abstract word domains. By using words instead of single letters, abstract transducers provide the possibility of lookaheads to decide on state transitions to conduct. Since both the input symbol and the output symbol on each transition is an abstract entity, abstraction techniques can be applied naturally. We apply abstract transducers as the foundation for sharing task artifacts for reuse in context of program analysis and verification, and describe task artifacts as abstract words. A task artifact is any entity that contributes to an analysis task and its solution, for example, candidate invariants or source code to weave

Equations defining the polynomial closure of a lattice of regular languages

The polynomial closure Pol(L) of a class of languages L of A ∗ is the set of languages that are finite unions of marked products of the form L0a1L1 · · ·anLn, where the ai are letters and the Li are elements of L. The main result of this paper gives an equational description of Pol(L), given an equational description of L, when L is a lattice of regular languages closed under quotients, or a quotienting algebra of languages, as we call it in the sequel. The term “equational description ” refers to a recent paper [5], where it was shown that any lattice of regular languages can be defined by a set of profinite equations. More formally, our main result can be stated as follows: If L is a quotienting algebra of languages, then Pol(L) is defined by the set of equations of the form x ω yx ω � x ω, where x, y are profinite words such that the equations x = x 2 and y � x are satisfied by L. As an application of this result, we establish a set of profinite equations defining the class of languages of the form L0a1L1 · · ·anLn, where each language Li is either of the form u ∗ (where u is a word) or A ∗ (where A is the alphabet) and we prove that this class is decidable. Let us now give the motivations of our work and a brief survey of the previously known results. Motivations. The polynomial closure occurs in several difficult problems on regular languages. For instance, a language has star-height one if and only if it belongs to the polynomial closure of the set of languages of the form F or F ∗, where F is a finite language. Although this class is known to be decidable, it is still an open problem to find profinite equations for this class. Such a result could serve, in turn, to discover a language of generalized star-height> 1, a widely open problem. The polynomial closure is also one of the two operations appearing in the definition of the concatenation hierarchy over a given set L of regular languages, defined by induction on n as follows. The level 0 is L and, for each n � 0, the level 2n + 1 is the polynomial closure of the level 2n and the level 2n + 2 i

Author manuscript, published in "ICALP 2009, Rhodes, Grèce: France (2009)" Equations defining the polynomial closure of a lattice of regular languages

The polynomial closure Pol(L) of a class of languages L of A ∗ is the set of languages that are finite unions of marked products of the form L0a1L1 · · · anLn, where the ai are letters and the Li are elements of L. The main result of this paper gives an equational description of Pol(L), given an equational description of L, when L is a lattice of regular languages closed under quotients, or a quotienting algebra of languages, as we call it in the sequel. The term “equational description ” refers to a recent paper [5], where it was shown that any lattice of regular languages can be defined by a set of profinite equations. More formally, our main result can be stated as follows: If L is a quotienting algebra of languages, then Pol(L) is defined by the set of equations of the form x ω yx ω � x ω, where x, y are profinite words such that the equations x = x 2 and y � x are satisfied by L. As an application of this result, we establish a set of profinite equations defining the class of languages of the form L0a1L1 · · · anLn, where each language Li is either of the form u ∗ (where u is a word) or A ∗ (where A is the alphabet) and w