Sequential operators in computability logic

• Published in: Information and computation Vol. 206; no. 12; pp. 1443 - 1475
• Main Author: Japaridze, Giorgi
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.12.2008; Elsevier
• Subjects: Applied sciences; Computability logic; Computer science; control theory; systems; Constructive logics; Exact sciences and technology; Game semantics; General logic; Interactive computation; Linear logic; Logic and foundations; Mathematical logic, foundations, set theory; Mathematics; Miscellaneous; Model theory; Recursion theory; Sciences and techniques of general use; Theoretical computing; Linear logic; Interactive computation; Computability logic; Game semantics; Constructive logics; Branching; Recurrence; Logical operator; Computer theory; Negation; Recurrence relation; Algorithmics; Research; Computability; Disjunction; Sequential computation; Sequential method; First order; Environment; Parallel computation; Quantifier; Behavior
• ISSN: 0890-5401; 1090-2651
• DOI: 10.1016/j.ic.2008.10.001

Computability logic (CL) is a semantical platform and research program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth which it has more traditionally been. Formulas in CL stand for (interactive) computational problems, understood as games between a machine and its environment; logical operators represent operations on such entities; and “truth” is understood as existence of an effective solution, i.e., of an algorithmic winning strategy. The formalism of CL is open-ended, and may undergo series of extensions as the study of the subject advances. The main groups of operators on which CL has been focused so far are the parallel, choice, branching, and blind operators, with the logical behaviors of the first three groups resembling those of the multiplicatives, additives and exponentials of linear logic, respectively. The present paper introduces a new important group of operators, called sequential. The latter come in the form of sequential conjunction and disjunction, sequential quantifiers, and sequential recurrences (“exponentials”). As the name may suggest, the algorithmic intuitions associated with this group are those of sequential computations, as opposed to the intuitions of parallel computations associated with the parallel group of operations. Specifically, while playing a parallel combination of games means playing all components of the combination simultaneously, playing a sequential combination means playing the components in a sequential fashion, one after one. The main technical result of the present paper is a sound and complete axiomatization of the propositional fragment of computability logic whose vocabulary, together with negation, includes all three — parallel, choice and sequential — sorts of conjunction and disjunction. An extension of this result to the first-order level is also outlined.