Reasoning with global assumptions in arithmetic modal logics

We establish a generic upper bound ExpTime for reasoning with global assumptions in coalgebraic modal logics. Unlike earlier results of this kind, we do not require a tractable set of tableau rules for the in- stance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that offers potential for practical reasoning

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems

The paper presents probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof systems for them. The completeness results are a strong completeness theorem for the system of probabilistic ITL with respect to an abstract semantics and a relative completeness theorem for the system of probabilistic DC with respect to real-time semantics. The proposed systems subsume probabilistic real-time DC as known from the literature. A correspondence between the proposed systems and a system of probabilistic interval temporal logic with finite intervals and expanding modalities is established too.Comment: 43 page

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

A generic logic environment

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Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics

We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka's linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e. in coalgebraic hybrid logic.Comment: Extended version of conference paper in FCT 201

A Complete Axiomatization of Quantified Differential Dynamic Logic for Distributed Hybrid Systems

We address a fundamental mismatch between the combinations of dynamics that occur in cyber-physical systems and the limited kinds of dynamics supported in analysis. Modern applications combine communication, computation, and control. They may even form dynamic distributed networks, where neither structure nor dimension stay the same while the system follows hybrid dynamics, i.e., mixed discrete and continuous dynamics. We provide the logical foundations for closing this analytic gap. We develop a formal model for distributed hybrid systems. It combines quantified differential equations with quantified assignments and dynamic dimensionality-changes. We introduce a dynamic logic for verifying distributed hybrid systems and present a proof calculus for this logic. This is the first formal verification approach for distributed hybrid systems. We prove that our calculus is a sound and complete axiomatization of the behavior of distributed hybrid systems relative to quantified differential equations. In our calculus we have proven collision freedom in distributed car control even when an unbounded number of new cars may appear dynamically on the road