The omega-inequality problem for concatenation hierarchies of star-free languages

The problem considered in this paper is whether an inequality of omega-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing-Th\'erien hierarchy

A duality theoretic view on limits of finite structures

A systematic theory of structural limits for finite models has been developed by Nesetril and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of measures arises --- via Stone-Priestley duality and the notion of types from model theory --- by enriching the expressive power of first-order logic with certain ``probabilistic operators''. We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction. The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality-theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively.Comment: 19 page

A duality theoretic view on limits of finite structures: Extended version

A systematic theory of structural limits for finite models has been developed by Nešetřil and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of (finitely additive) measures arises—via Stone-Priestley duality and the notion of types from model theory—by enriching the expressive power of first-order logic with certain “probabilistic operators”. We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction. The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively

A DUALITY THEORETIC VIEW ON LIMITS OF FINITE STRUCTURES

A systematic theory of structural limits for finite models has been developed by Nešetřil and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of (finitely additive) measures arises—via Stone-Priestley duality and the notion of types from model theory—by enriching the expressive power of first-order logic with certain “probabilistic operators”. We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction. The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively

Stone pseudovarieties

Profinite algebras have been extensively considered in the literature. They are the residually finite Stone topological algebras. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. In particular, we are interested in free structures relative to Stone pseudovarieties. In the unary case, the \v{C}ech-Stone compactification of the term algebra provides a construction of the absolutely free Stone topological algebra. A natural representation in a free profinite monoid using the Polish notation turns out also to be faithful for free profinite unary algebras. Looking at Stone spaces as the dual spaces of Boolean algebras, which may be taken as subalgebras of the Boolean algebra of subsets of the algebra of terms, we find a simple characterization of when the dual space admits a natural structure of topological algebra. This provides a new approach to duality theory which, in the case of a Stone signature, culminates in the proof that a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is also residually in it, thereby extending the corresponding result of Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety. A Birkhoff type theorem for residually closed Stone pseudovarieties is also established

The linear nature of pseudowords

Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.The work of the first, third, and fourth authors was partly supported by the Pessoa French-Portuguese project “Separation in automata theory: algebraic, logical, and combinatorial aspects”. The work of the first three authors was also partially supported respectively by CMUP (UID/MAT/ 00144/2019), CMUC (UID/MAT/00324/2019), and CMAT (UID/MAT/ 00013/2013), which are funded by FCT (Portugal) with national (MCTES) and European structural funds (FEDER), under the partnership agreement PT2020. The work of the fourth author was partly supported by ANR 2010 BLAN 0202 01 FREC and by the DeLTA project ANR-16-CE40-000

Pro-aperiodic monoids via saturated models

We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids. Stone duality implies that elements of the free pro-aperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for omega-terms that avoids using McCammond\u27s normal forms, as well as new proofs and extensions of other structural results concerning free pro-aperiodic monoids