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We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we obtain completeness and identify a common (modal) fragment of a large class of modal intuitionistic logics. We recognize this logic as a di algebraic logic, and as a consequence obtain expressivity somewhere-else. Within the di algebraic framework, we then investigate the extension of the meet-implication fragment of propositional intuitionistic logic with a monotone modality and prove completeness and expressivity-somewhere-else for it. [ABSTRACT FROM AUTHOR]

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 logicwith 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. [ABSTRACT FROM AUTHOR]

Coalgebraic modal logic serves as a unifying framework to study a wide range of modal logics beyond the relational realm, including probabilistic and graded logics as well as conditional logics and logics based on neighbourhoods and games. Coalgebraic predicate logic (CPL), a generalization of a neighbourhood-based first-order logic introduced by Chang, has been identified as a natural first-order extension of coalgebraic modal logic, which in particular coincides with the standard first-order correspondence language when instantiated to Kripke-style relational modal operators. Here, we generalize to the CPL setting the classical van Benthem/Rosen theorem stating that both over arbitrary and over finite models, modal logic is precisely the bisimulation-invariant fragment of first-order logic. As instances of this generic result, we obtain corresponding characterizations for, e.g. conditional logic, neighbourhood logic (i.e. classical modal logic) and monotone modal logic. [ABSTRACT FROM AUTHOR]

State-based systems and modal logics for reasoning about them often heterogeneously combine a number of features such as non-determinism and probabilities. In this paper, we show that the combination of features can be reflected algorithmically, and we develop modular decision procedures for heterogeneous modal logics. The modularity is achieved by formalising the underlying state-based systems as multi-sorted coalgebras and associating both a logical and algorithmic description with a number of basic building blocks. Our main result is that logics arising as combinations of these building blocks can be decided in polynomial space provided this is also the case for the components. By instantiating the general framework to concrete cases, we obtain PSpace decision procedures for a wide variety of structurally different logics, describing, for example, Segala systems and games with uncertain information. [ABSTRACT FROM AUTHOR]

Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can, moreover, be combined in a modular way. In particular, this facilitates a pick-and-choose approach to domain-specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement and to maintain. This paper substantiates the authors’ firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility. [ABSTRACT FROM AUTHOR]

Abstract: We discuss recent work generalising the basic hybrid logic with the difference modality to any reasonable notion of transition. This applies equally to both subrelational transitions such as monotone neighbourhood frames or selection function models as well as those with more structure such as Markov chains and alternating temporal frames. We provide a generic canonical cut-free sequent system and a terminating proof-search strategy for the fragment without the difference modality but including the global modality. [ABSTRACT FROM AUTHOR]

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACE-bounds for a number of logics, including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant proof-theoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way. [ABSTRACT FROM AUTHOR]

We present a domain-theoretic version of Picard's theorem for solving classical initial value problems in ℝn. For the case of vector fields that satisfy a Lipschitz condition, we construct an iterative algorithm that gives two sequences of piecewise linear maps with rational coefficients, which converge, from below and above respectively, exponentially fast, to the unique solution of the initial value problem. We provide a detailed analysis of the speed of convergence and the complexity of computing the iterates. The algorithm uses proper data types based on rational arithmetic, where no rounding of real numbers is required. Thus we obtain a sound implementation framework to solve initial value problems. In particular, the use of rational arithmetic guarantees that implementations of our technique will adhere to the bounds on convergence speed and algebraic complexity. [ABSTRACT FROM PUBLISHER]

An introduction is presented in which the editor discusses articles in the issue on topics including intuitionistic ancestral logic, program extraction applied to monadic parsing, and well-definedness and observation equivalence for inductive-coinductive programs.

The article discusses various reports published within the issue, including topics related to Douglas Bridges and the papers presented at the workshop "Trends in Constructive Mathematics" that was held from June 19-23, 2006 at the Chiemsee island Frauenwörth.