Model completions and r-Heyting categories

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

2016

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The main focus of this paper is on bisimulation-invariant MSO, and more particularly on giving a novel model-theoretic approach to it. In model theory, a model companion of a theory is a firstorder description of the class of models in which all potentially solvable systems of equations and non-equations have solutions. We show that bisimulation-invariant MSO on trees gives the model companion for a new temporal logic, "fair CTL", an enrichment of CTL with local fairness constraints. To achieve this, we give a completeness proof for the logic fair CTL which combines tableaux and Stone duality, and a fair CTL encoding of the automata for the modal µ-calculus. Moreover, we also show that MSO on binary trees is the model companion of binary deterministic fair CTL.

Section: Solving Equations and Model Companionsmentioning

confidence: 62%

Monadic second order logic as the model companion of temporal logic

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

2016

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“…As it is clear from the title of that work, this property amounts to an internal existential and universal quantification. This result was further refined in [18] to show that any morphism between finitely presented Heyting algebras has a left and a right adjoint.…”

Section: Bisimulation Quantifiers and Fixed-pointsmentioning

confidence: 97%

Fixed-point Elimination in the Intuitionistic Propositional Calculus

ACM Trans. Comput. Logic

Gouveia

Santocanale

2019

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It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the µ-calculus based on intuitionistic logic is trivial, every µ-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given µ-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixedpoint is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal.

“…The reader interested in Ruitenburg's Theorem might wish to proceed directly to Section 5. While the material in this Section is adapted from [16], Theorem 16, generalizing the duality to some subvarieties of Heyting algebras, is new.…”

Section: Proof Of the Duality Theoremmentioning

confidence: 99%

Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

Math. Struct. Comp. Sci.

Santocanale

2020

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Ruitenburg's Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N ≥ 0 such that f N+2 = f N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms between free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.