Sabotage Modal Logic: Some Model and Proof Theoretic Aspects

International audienceWe investigate some model and proof theoretic aspects of sabotage modal logic. The first contribution is to prove a characterization theorem for sabotage modal logic as the fragment of first-order logic which is invariant with respect to a suitably defined notion of bisimulation (called sabotage bisimulation). The second contribution is to provide a sound and complete tableau method for sabotage modal logic. We also chart a number of open research questions concerning sabotage modal logic, aiming at integrating it within the current landscape of logics of model update

A note on ultraproducts of Veltman models

We consider ultraproducts of Veltman models, and show that a version of Łos theorem is true

Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

A formal proof of modal completeness for provability logic

This work presents a formalized proof of modal completeness for G\"odel-L\"ob provability logic (GL) in the HOL Light theorem prover. We describe the code we developed, and discuss some details of our implementation, focusing on our choices in structuring proofs which make essential use of the tools of HOL Light and which differ in part from the standard strategies found in main textbooks covering the topic in an informal setting. Moreover, we propose a reflection on our own experience in using this specific theorem prover for this formalization task, with an analysis of pros and cons of reasoning within and about the formal system for GL we implemented in our code

Coalgebraic fixpoint logic:Expressivity and completeness results

This dissertation studies the expressivity and completeness of the coalgebraic μ-calculus. This logic is a coalgebraic generalization of the standard μ-calculus, which creates a uniform framework to study different modal fixpoint logics. Our main objective is to show that several important results, such as uniform interpolation, expressive completeness and axiomatic completeness of the standard μ-calculus can be generalized to the level of coalgebras. To achieve this goal we develop automata and game-theoretic tools to study properties of coalgebraic μ-calculus.In Chapter 3, we prove a uniform interpolation theorem for the coalgebraic μ-calculus. This theorem generalizes a result by D’Agostino and Hollenberg (2000) to a wider class of fixpoint logics including the monotone μ-calculus, which is the extension of monotone modal logic with fixpoint operators. In Chapter 4, we generalize the Janin-Walukiewicz theorem (1996), which states that the modal μ-calculus captures exactly the bisimulation invariant fragment of monadic second-order logic, to the level of coalgebras. We obtain a partly new proof of the Janin-Walukiewicz theorem, bisimulation invariance results for the bag functor (graded modal logic), and all exponential polynomial functors. We also derive a characterization theorem for the monotone modal μ-calculus, with respect to a natural monadic second-order language for monotone neighborhood models. In Chapter 5, we prove an axiomatic completeness result for the coalgebraic μ-calculus. Applying ideas from automata theory and coalgebra, we generalize Walukiewicz’ proof of completeness for the modal μ-calculus (2000) to the level of coalgebras. Our main contribution is to bring automata explicitly into the proof theory and distinguish two key aspects of the coalgebraic μ-calculus (and the standard μ-calculus): the one-step dynamic encoded in the semantics of the modal operators, and the combinatorics involved in dealing with nested fixpoints. We provide a generalization of Walukiewicz’ main technical result, which states that every formula of the modal μ-calculus provably implies the translation of a disjunctive automaton, to the level of coalgebras. From this the completeness theorem is almost immediate

Coalgebraic fixpoint logic:Expressivity and completeness results

This dissertation studies the expressivity and completeness of the coalgebraic μ-calculus. This logic is a coalgebraic generalization of the standard μ-calculus, which creates a uniform framework to study different modal fixpoint logics. Our main objective is to show that several important results, such as uniform interpolation, expressive completeness and axiomatic completeness of the standard μ-calculus can be generalized to the level of coalgebras. To achieve this goal we develop automata and game-theoretic tools to study properties of coalgebraic μ-calculus.In Chapter 3, we prove a uniform interpolation theorem for the coalgebraic μ-calculus. This theorem generalizes a result by D’Agostino and Hollenberg (2000) to a wider class of fixpoint logics including the monotone μ-calculus, which is the extension of monotone modal logic with fixpoint operators. In Chapter 4, we generalize the Janin-Walukiewicz theorem (1996), which states that the modal μ-calculus captures exactly the bisimulation invariant fragment of monadic second-order logic, to the level of coalgebras. We obtain a partly new proof of the Janin-Walukiewicz theorem, bisimulation invariance results for the bag functor (graded modal logic), and all exponential polynomial functors. We also derive a characterization theorem for the monotone modal μ-calculus, with respect to a natural monadic second-order language for monotone neighborhood models. In Chapter 5, we prove an axiomatic completeness result for the coalgebraic μ-calculus. Applying ideas from automata theory and coalgebra, we generalize Walukiewicz’ proof of completeness for the modal μ-calculus (2000) to the level of coalgebras. Our main contribution is to bring automata explicitly into the proof theory and distinguish two key aspects of the coalgebraic μ-calculus (and the standard μ-calculus): the one-step dynamic encoded in the semantics of the modal operators, and the combinatorics involved in dealing with nested fixpoints. We provide a generalization of Walukiewicz’ main technical result, which states that every formula of the modal μ-calculus provably implies the translation of a disjunctive automaton, to the level of coalgebras. From this the completeness theorem is almost immediate