This paper unveils and motivates an ambitious programme of hidden algebraic research in software engineering, beginning with our general goals, continuing with an overview of results, and including some future plans. The main contribution is powerful hidden coinduction techniques for proving behavioral correctness of concurrent systems; several mechanical proofs are given using OBJ3. We also show how modularization, bisimulation, transition systems, concurrency and combinations of the functional, constraint, logic and object paradigms fit into hidden algebra. 1. Introduction

In this paper we present history-dependent automata (HD-automata in brief). They are an extension of ordinary automata that overcomes their limitations in dealing with history-dependent formalisms. In a history-dependent formalism the actions that a system can perform carry information generated in the past history of the system. The most interesting example is -calculus: channel names can be created by some actions and they can then be referenced by successive actions. Other examples are CCS with localities and the history-preserving semantics of Petri nets. Ordinary

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 prooftheoretic 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.

In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatization of rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard Hennessy-Milner logic, graded modal logic and probabilistic modal logic.

We give an axiomatic account of what structure on a category C and an endofunctor H on C yield similar structure on the category H0Coalg of H-coalgebras. We give conditions under which completeness, cocompleteness, symmetric monoidal closed structure, local presentability, and subobject classifiers lift. Our proof of the latter uses a general result about the existence of a subobject classifier in a category containing a small dense subcategory. Our leading example has C = Set with H the endofunctor for which a coalgebra is a finitely branching (labelled) transition system. We explain that example in detail. 1 Introduction Given an endofunctor H on the category Set, an H-coalgebra is a set X together with a function x : X 0! HX. A leading example of such an H is given by the functor P ! that takes a set X to the set of finite subsets of X , with the behaviour of H on maps given by direct image. An H-coalgebra is then a finitely branching transition system. A variant, is given by sta...

Abstract. The theory of hidden algebras combines standard algebraic techniques with coalgebraic techniques to provide a semantic foundation for the object paradigm. This paper focuses on the coalgebraic aspect of hidden algebra, concerned with signatures of destructors at the syntac-tic level and with finality and coffee constructions at the semantic level. Our main result shows the existence of cofree constructions induced by maps between coalgebraic hidden specifications. Their use in giving a se-mantics to parameterised objects and inheritance is then illustrated. The cofreeness result for hidden algebra is generalised to abstract coalgebra and a universal construction for building object systems over existing subsystems is obtained. Finally, existence of final/cofree constructions for arbitrary hidden specifications is discussed. 1

Abstract. We present a modular approach to defining logics for a wide variety of state-based systems. We use coalgebras to model the behaviour of systems, and modal logics to specify behavioural properties of systems. We show that the syntax, semantics and proof systems associated to such logics can all be derived in a modular way. Moreover, we show that the logics thus obtained inherit soundness, completeness and expressiveness properties from their building blocks. We apply these techniques to derive sound, complete and expressive logics for a wide variety of probabilistic systems. 1

This paper forms a step in the development of the recently emerged connection between coalgebra and modal logic. It introduces (back-and-forth) transformations between coalgebras of simple polynomial functors and certain Boolean algebras with operators (BAOs). Categorically, these transformations take the form of an adjunction. The BAO associated with a coalgebra can be used for specification, e.g. of classes in object-oriented languages.

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The use of coalgebras for the specification of dynamical systems with a hidden state space is receiving more and more attention in the years, as a valid alternative to algebraic methods based on observational equivalences. However, to our knowledge, the coalgebraic framework is still lacking a complete equational deduction calculus which enjoys properties similar to those stated in Birkhoff's completeness theorem for the algebraic case. In this paper we present a sound and complete equational calculus for coalgebras of a restricted class of polynomial functors. This restriction allows us to borrow some "algebraic" notions for the formalization of the calculus. Additionally, we discuss the notion of colours as a suitable dualization of variables in the algebraic case. Then the completeness result is extended to the "non-ground" or "coloured" case, which is shown to be expressive enough to deal with equations of hidden sort. Finally we discuss some weaknesses of the proposed results wit...