語用論(Pragmatics)を考える

What is modal logic

コンピュータ・サイエンス : ACM computing surveys

Logic, Semantics, Metamathematics. By A. Tarski. Translated by J. H. Woodger. Pp. 471. 60s. 1956. (Clarendon Press)

Alternating-time temporal logic

Mathematics never saw much of a reason to deal with something called nondeterminism. It works with values, functions, sets, and relations. In computing science, however, nondeterminism has been an issue from the very beginning, if only in the form of nondeterministic Turing machines or nondeterministic finite state machines. Early references to nondeterminism in computer science go back to the 1960s [Floyd 1967; McCarthy 1963]. A great variety of theories and formalisms dealing with it have been developed during the last two decades. There are the denotational models based on power domains, the predicate transformers for the choice construct, and modifications of the l-calculus [de Liguoro and Piperno 1992; Astesiano and Costa 1979; Hennessy 1980]. Nondeterminism arises in a natural way when discussing concurrency, and models of concurrency typically also model nondeterminism. There are numerous variants of process languages and algebra, event structures, state transition systems [Manna and Pnueli 1992; Lehmann and Shelah 1983], and Petri nets [Petri 1977; Reisig 1985]. In terms of modeling, nondeterminism may be considered a purely operational notion. However, one of the main reasons for considering nondeterminism in computer science is the need for abstraction, allowing one to disregard irrelevant aspects of actual computations. Typically, we prefer to work with models that do not include all the details of the physical environment of computations such as timing, temperature, representation on hardware, and the like. Since we do not want to model all these complex dependencies, we may instead represent them by nondeterministic choices. The nondeterminism of concurrent systems usually arises as an ab-

Algebraic approaches to nondeterminism—an overview

The current algebraic models for nondeterminism focus on the notion of possibility rather than necessity and consequently equate (nondeterministic) terms that one would intuitively not consider equal. Furthermore, existing models for nondeterminism depart radically from the standard models for (equational) specifications of deterministic operators. One would prefer that a specification language for nondeterministic operators be based on an extension of the standard model concepts, preferably in such a way that the reasoning system for (possibly nondeterministic) operators becomes the standard equational one whenever restricted to the deterministic operators—the objective should be to minimize the departure from the standard frameworks. In this article we define a specification language for nondeterministic operators and multialgebraic semantics. The first complete reasoning system for such specifications is introduced. We also define a transformation of specifications of nondeterministic operators into derived specifications of deterministic ones, obtaining a “computational” semantics of nondeterministic specification by adopting the standard semantics of the derived specification as the semantics of the original one. This semantics turns out to be a refinement of multialgebra semantics. The calculus is shown to be sound and complete also with respect to the new semantics.

A complete calculus for the multialgebraic and functional semantics of nondeterminism

Finding the case id in unlabeled event logs is arguably one of the hardest challenges in process mining research. While this problem has been addressed with greedy approaches, these usually converge to sub-optimal solutions. In this work, we describe an approach to perform complete search over the search space. We formulate the problem as a matter of finding the minimal set of patterns contained in a sequence, where patterns can be interleaved but do not have repeating symbols. This represents a new problem that has not been previously addressed in the literature, with NP-hard variants and conjectured NP-completeness. We solve it in a stepwise manner, by generating and verifying a list of candidate solutions. The techniques, introduced to address various subtasks, can be applied independently for solving more specific problems. The approach has been implemented and applied in a case study with real data from a business process supported in a software application.

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Sequence partitioning for process mining with unlabeled event logs

After motivating the introduction of nondeterministic operators into algebraic specifications, a language L with two primitive predicates, identity and inclusion, for specifying nondeterministic operations is introduced. It is given a multialgebraic semantics which captures the singular (call-time-choice) strategy of passing nondeterministic parameters. A calculus NEQ, with restricted substitutivity rules, is introduced. NEQ is sound and complete wrt. the multialgebraic semantics.

Michal Walicki

Papers

Algebraic approaches to nondeterminism—an overview

A complete calculus for the multialgebraic and functional semantics of nondeterminism

Sequence partitioning for process mining with unlabeled event logs

Multialgebras, Power Algebras and Complete Calculi of Identities and Inclusions

Expressive power of digraph solvability