Monadic Second-Order Logic and Bisimulation Invariance for Coalgebras

Generalizing standard monadic second-order logic for Kripke models, we introduce monadic second-order logic interpreted over coalgebras for an arbitrary set functor. Similar to well-known results for monadic second-order logic over trees, we provide a translation of this logic into a class of automata, relative to the class of coalgebras that admit a tree-like supporting Kripke frame. We then consider invariance under behavioral equivalence of formulas; more in particular, we investigate whether the coalgebraic mu-calculus is the bisimulation-invariant fragment of monadic second-order logic. Building on recent results by the third author we show that in order to provide such a coalgebraic generalization of the Janin-Walukiewicz Theorem, it suffices to find what we call an adequate uniform construction for the functor. As applications of this result we obtain a partly new proof of the Janin-Walukiewicz Theorem, and bisimulation invariance results for the bag functor (graded modal logic) and all exponential polynomial functors. Finally, we consider in some detail the monotone neighborhood functor, which provides coalgebraic semantics for monotone modal logic. It turns out that there is no adequate uniform construction for this functor, whence the automata-theoretic approach towards bisimulation invariance does not apply directly. This problem can be overcome if we consider global bisimulations between neighborhood models: one of our main technical results provides a characterization of the monotone modal mu-calculus extended with the global modalities, as the fragment of monadic second-order logic for the monotone neighborhood functor that is invariant for global bisimulations

Expressiveness of the modal mu-calculus on monotone neighborhood structures

We characterize the expressive power of the modal mu-calculus on monotone neighborhood structures, in the style of the Janin-Walukiewicz theorem for the standard modal mu-calculus. For this purpose we consider a monadic second-order logic for monotone neighborhood structures. Our main result shows that the monotone modal mu-calculus corresponds exactly to the fragment of this second-order language that is invariant for neighborhood bisimulations

Uniform Interpolation for Coalgebraic Fixpoint Logic

We use the connection between automata and logic to prove that a wide class of coalgebraic fixpoint logics enjoys uniform interpolation. To this aim, first we generalize one of the central results in coalgebraic automata theory, namely closure under projection, which is known to hold for weak-pullback preserving functors, to a more general class of functors, i.e.; functors with quasi-functorial lax extensions. Then we will show that closure under projection implies definability of the bisimulation quantifier in the language of coalgebraic fixpoint logic, and finally we prove the uniform interpolation theorem

An expressive completeness theorem for coalgebraic modal mu-calculi

Generalizing standard monadic second-order logic for Kripke models, we introduce monadic second-order logic interpreted over coalgebras for an arbitrary set functor. We then consider invariance under behavioral equivalence of MSO-formulas. More specifically, we investigate whether the coalgebraic mu-calculus is the bisimulation-invariant fragment of the monadic second-order language for a given functor. Using automatatheoretic techniques and building on recent results by the third author, we show that in order to provide such a characterization result it suffices to find what we call an adequate uniform construction for the coalgebraic type functor. As direct applications of this result we obtain a partly new proof of the Janin-Walukiewicz Theorem for the modal mu-calculus, avoiding the use of syntactic normal forms, and bisimulation invariance results for the bag functor (graded modal logic) and all exponential polynomial functors (including the "game functor"). As a more involved application, involving additional non-trivial ideas, we also derive a characterization theorem for the monotone modal mu-calculus, with respect to a natural monadic second-order language for monotone neighborhood models.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0721

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

Bio self-healing nanoconcretes

Concrete as the most widely used construction material is susceptible to crack when exposes to stresses. Various types of passive techniques have been introduced to fill the generated cracks, however the majority of them are not permanent, effective, and environmentally friendly. Over recent years, the embedment of an active mechanism called “self-healing mechanism” in concrete has been proposed as an alternative approach to ineffective passive techniques. Among the investigated self-healing mechanisms, bio self-healing approach has drawn considerable attention as it can address the cracking issues by inducing calcium carbonate in a sustainable way. However, the effectiveness of bio self-healing concrete highly depends on the successful protection of bacteria in concrete matrix. So far, various types of micro-scale and recently nano-scale protecting careers have been tested to evaluate the effectiveness of bio self-healing concrete. In this chapter, the latest nanobiotechnological self-healing approaches for protection of bacteria in a harsh concrete pH are discussed

Uniform Interpolation in Coalgebraic Modal Logic

A logic has uniform interpolation if its formulas can be projected down to given subsignatures, preserving all logical consequences that do not mention the removed symbols; the weaker property of (Craig) interpolation allows the projected formula - the interpolant - to be different for each logical consequence of the original formula. These properties are of importance, e.g., in the modularization of logical theories. We study interpolation in the context of coalgebraic modal logics, i.e. modal logics axiomatized in rank 1, restricting for clarity to the case with finitely many modalities. Examples of such logics include the modal logics K and KD, neighbourhood logic and its monotone variant, finite-monoid-weighted logics, and coalition logic. We introduce a notion of one-step (uniform) interpolation, which refers only to a restricted logic without nesting of modalities, and show that a coalgebraic modal logic has uniform interpolation if it has one-step interpolation. Moreover, we identify preservation of finite surjective weak pullbacks as a sufficient, and in the monotone case necessary, condition for one-step interpolation. We thus prove or reprove uniform interpolation for most of the examples listed above