Uniform interpolation for propositional and modal team logics

In this paper we consider modal team logic, a generalization of classical modal logic in which it is possible to describe dependence phenomena between data. We prove that most known fragments of full modal team logic allow the elimination of the so called 'existential bisimulation quantifiers', where the existence of a certain set is required only modulo bisimulation (i.e. not in the model itself but possibly in a bisimilar model). As a consequence, we prove that these fragments enjoy the uniform interpolation property

On P-transitive graphs and applications

We introduce a new class of graphs which we call P-transitive graphs, lying between transitive and 3-transitive graphs. First we show that the analogue of de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we show that the mu-calculus fixpoint hierarchy is infinite for P-transitive graphs. Both results contrast with the case of transitive graphs. We give also an undecidability result for an enriched mu-calculus on P-transitive graphs. Finally, we consider a polynomial time reduction from the model checking problem on arbitrary graphs to the model checking problem on P-transitive graphs. All these results carry over to 3-transitive graphs.Comment: In Proceedings GandALF 2011, arXiv:1106.081

Regular Languages meet Prefix Sorting

Indexing strings via prefix (or suffix) sorting is, arguably, one of the most successful algorithmic techniques developed in the last decades. Can indexing be extended to languages? The main contribution of this paper is to initiate the study of the sub-class of regular languages accepted by an automaton whose states can be prefix-sorted. Starting from the recent notion of Wheeler graph [Gagie et al., TCS 2017]-which extends naturally the concept of prefix sorting to labeled graphs-we investigate the properties of Wheeler languages, that is, regular languages admitting an accepting Wheeler finite automaton. Interestingly, we characterize this family as the natural extension of regular languages endowed with the co-lexicographic ordering: when sorted, the strings belonging to a Wheeler language are partitioned into a finite number of co-lexicographic intervals, each formed by elements from a single Myhill-Nerode equivalence class. Moreover: (i) We show that every Wheeler NFA (WNFA) with $n$ states admits an equivalent Wheeler DFA (WDFA) with at most $2n-1-|\Sigma|$ states that can be computed in $O(n^3)$ time. This is in sharp contrast with general NFAs. (ii) We describe a quadratic algorithm to prefix-sort a proper superset of the WDFAs, a $O(n\log n)$-time online algorithm to sort acyclic WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By contribution (i), our algorithms can also be used to index any WNFA at the moderate price of doubling the automaton's size. (iii) We provide a minimization theorem that characterizes the smallest WDFA recognizing the same language of any input WDFA. The corresponding constructive algorithm runs in optimal linear time in the acyclic case, and in $O(n\log n)$ time in the general case. (iv) We show how to compute the smallest WDFA equivalent to any acyclic DFA in nearly-optimal time.Comment: added minimization theorems; uploaded submitted version; New version with new results (W-MH theorem, linear determinization), added author: Giovanna D'Agostin

Type Inference for Bimorphic Recursion

This paper proposes bimorphic recursion, which is restricted polymorphic recursion such that every recursive call in the body of a function definition has the same type. Bimorphic recursion allows us to assign two different types to a recursively defined function: one is for its recursive calls and the other is for its calls outside its definition. Bimorphic recursion in this paper can be nested. This paper shows bimorphic recursion has principal types and decidable type inference. Hence bimorphic recursion gives us flexible typing for recursion with decidable type inference. This paper also shows that its typability becomes undecidable because of nesting of recursions when one removes the instantiation property from the bimorphic recursion.Comment: In Proceedings GandALF 2011, arXiv:1106.081

The logic of the reverse mathematics zoo

Building on previous work by Mummert, Saadaoui and Sovine, we study the logic underlying the web of implications and nonimplications which constitute the so called reverse mathematics zoo. We introduce a tableaux system for this logic and natural deduction systems for important fragments of the language

Bisimulation Quantifiers and Uniform Interpolation for Guarded First Order Logic

The idea that the good model-theoretic and algorithmic properties of Modal Logics are due to the guarded nature of their quantification was put forward by Andreka, van Benthem and Nemeti in a series of papers in the 1990s, exploiting the satisfiability problem, the tree model property, and other similar properties of the Guarded Fragment of First Order Logic (GF). Since then, further work on the Guarded Fragment has been done by various authors, in some cases reinforcing this idea, in some others not. At least at first sight, Craig interpolation is on the negative side: there are implications in GF without an interpolant in GF, while Modal Logic (and even the μ-calculus, a powerful extension of Modal Logic) enjoys a much stronger form of interpolation, the uniform one, in which the interpolant of a valid implication not only exists, but only depends on the antecedent and on the common language of antecedent and consequent. However, Hoogland and Marx proved that Craig interpolation is restored in GF if we consider the modal character of GF with more attention, that is, if relations appearing on guards are viewed as “modalities” and the rest as “propositions”, and only the latter enter in the common language. In this paper we strengthen this result by showing that GF enjoys a Modal Uniform Interpolation Theorem (in the sense of Hoogland and Marx)

Bisimulation quantifiers and uniform interpolation for guarded first order logic

The idea that the good model-theoretic and algorithmic properties of Modal Logics are due to the guarded nature of their quantification was put forward by Andreka, van Benthem and Nemeti in a series of papers in the 1990s, exploiting the satisfiability problem, the tree model property, and other similar properties of the Guarded Fragment of First Order Logic(GF). Since then, further work on the Guarded Fragment has been done by various authors, in some cases reinforcing this idea, in some others not. At least at first sight, Craig interpolation is on the negative side: there are implications in GF without an interpolant in GF, while Modal Logic (and even the \u3bc-calculus, a powerful extension of Modal Logic) enjoys a much stronger form of interpolation, the uniform one, in which the interpolant of a valid implication not only exists, but only depends on the antecedent and on the common language of antecedent and consequent. However, Hoogland and Marx proved that Craig interpolation is restored in GFif we consider the modal character of GFwith more attention, that is, if relations appearing on guards are viewed as \u201cmodalities\u201d and the rest as \u201cpropositions\u201d, and only the latter enter in the common language. In this paper we strengthen this result by showing that GF enjoys a Modal Uniform Interpolation Theorem (in the sense of Hoogland and Marx)

On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components

For every positive integer k we consider the class SCCk of all finite graphs whose strongly connected components have size at most k. We show that for every k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1 and Pi1). This contrasts with the class of all graphs, where Delta2=Comp(Sigma1,Pi1)