Most knowledge based systems in artificial intelligence (AI), with a commitment to asymbolic representation, support one basic operation: ''matching of descriptions''. This operation, called unification in work on deduction, is the ''addition-and-multiplication'' of AI-systems and is consequently often supported by special purpose hardware or by a fast instruction set on most AI-machines. Unification theory provides the formal framework for investigations into the properties of this operation. This article surveys what is presently known in unification theory and records its early history.

Unification theory

Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.

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Expressivity of coalgebraic modal logic: the limits and beyond

Expressivity of coalgebraic modal logic: The limits and beyond

Chapter 9 – Rewriting

Publisher Summary This chapter discusses completion and its applications. The ability to reason with equations is important in many computer science applications, including algebraic specifications, high level programming languages, and automated deduction. Reasoning about equations may, for example, involve deciding whether an equation follows as a consequence of a given set of equations, or whether an equation is true in a given model, or what values of the variables satisfy a given equation. The chapter presents the completion procedure, some of its multifaceted extensions, and its main areas of application. The procedure and its extensions are viewed as inference engines that use a limited amount of forward reasoning, trying to churn out enough equational consequences of the given axioms to guarantee that for any equational proof, there will eventually be one that uses only backward reasoning. The applications highlight the advantages inherent in the sense of direction gained from the incorporation of a well-defined notion of simplicity. The completion concept has been further extended to conditional equations and rules, where the applicability of an identity or rewrite is predicated on the fulfillment of some conditions. Conditional equations are very important in applications, such as data type specification and functional programming languages.

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Completion and Its Applications

Let B be a semigroup with the additional relation

$$\begin{gathered} xx \Rightarrow x \hfill \\ xyz \Rightarrow xz if x \mathop {CI}\limits_ = z and xy\mathop {CI}\limits_ = z \hfill \\ \end{gathered} $$

B is called aband or anidempotent semigroup [3].

J. Siekmann

Papers

A Noetherian and confluent rewrite system for idempotent semigroups.