Applications of Algebraic Logic and Universal Algebra to Computer Science

1989 ◽  
Author(s):  
IOWA STATE UNIV AMES DEPT OF MATHEMATICS
Keyword(s):  
Computer Science ◽  
Universal Algebra ◽  
Algebraic Logic

Vaughan R. Pratt. Semantical considerations on Floyd–Hoare logic. 17th Annual Symposium on Foundations of Computer Science, Institute of Electrical and Electronics Engineers, New York1976, pp. 109–121. - Michael J. Fischer and Richard E. Ladner. Propositional dynamic logic of regular programs. Journal of computer and system sciences, vol. 18 (1979), pp. 194–211. - Krister Segerberg. A completeness theorem in the modal logic of programs. Universal algebra and applications. Papers presented at Stefan Banach International Mathematical Center at the semester “Universal algebra and applications” held February 15–June 9, 1978, edited by Tadeuz Traczyk, Banach Center Publications, vol. 9, PWN—Polish Scientific Publishers, Warsaw1982, pp. 31–46. - Rohit Parikh. The completeness of propositional dynamic logic. Mathematical foundations of computer science 1978, Proceedings, 7th symposium, Zakopane, Poland, September 4–8, 1978, edited by J. Winkowski, Lecture notes in computer science, vol. 64, Springer-Verlag, Berlin, Heidelberg, and New York, 1978, pp. 403–415. - Dexter Kozen and Rohit Parikh. An elementary proof of the completeness of PDL. Theoretical computer science, vol. 14(1981), pp. 113–118.

1986 ◽  
Vol 51 (1) ◽  
pp. 225-227 ◽  
Author(s):  
Robert Goldblatt
Keyword(s):  
New York ◽  
Computer Science ◽  
Universal Algebra ◽  
Dynamic Logic ◽  
Completeness Theorem ◽  
Theoretical Computer Science ◽  
Propositional Dynamic Logic ◽  
Theoretical Computer ◽  
Mathematical Foundations ◽  
Scientific Publishers

scholarly journals Arithmetical Datatypes, Fracterms, and the Fraction Definition Problem

2020 ◽  
Author(s):  
Jan Aldert Bergstra
Keyword(s):  
Software Engineering ◽  
Computer Science ◽  
Universal Algebra ◽  
Opposite Direction ◽  
Rational Numbers ◽  
Application Area ◽  
Algebraic Specifications ◽  
Implicit Definitions

Datatypes and abstract datatypes are positioned as results of importing aspects of universal algebra into computer science and software engineering. It is suggested that 50 years later, by way of a transfer in the opposite direction, outcomes of research on datatypes can be made available via elementary arithmetic. This idea leads to the notions of an arithmetical signature, an arithmetical datatype and an arithmetical abstract datatype and to algebraic specifications for such entities. The area of fractions in elementary arithmetic is chosen as an application area and while taking a common meadow of rational numbers as the basis, an arithmetical datatype equipped with an absorptive element. The use of datatypes and signatures makes syntax available for giving precise definitions in cases where lack of precision is common place. Fracterm is coined as the name for a fraction when primarily understood as a syntactic entity. The main contribution of the paper is to provide a detailed terminology of fracterms. Subsequently the fraction definition problem is stated, a distinction between explicit definitions of fractions and implicit definitions of fractions is made, and an outline of a survey of both forms of definitions of the notion of a fraction is given.

scholarly journals Non-deterministic algebraization of logics by swap structures1

2018 ◽  
Vol 28 (5) ◽  
pp. 1021-1059 ◽  
Author(s):  
Marcelo E Coniglio ◽  
Aldo Figallo-Orellano ◽  
Ana Claudia Golzio
Keyword(s):  
Computer Science ◽  
Representation Theorem ◽  
Algebraic Logic ◽  
Decomposition Theorem ◽  
Boolean Algebras ◽  
Abstract Algebraic Logic ◽  
Algebraic Models ◽  
Algebraizable Logics ◽  
Semantic Framework ◽  
Finite Matrix

Abstract Multialgebras (or hyperalgebras or non-deterministic algebras) have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency (or LFIs) that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics (as independently introduced by M. Fidel and D. Vakarelov). This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI (which is closely connected with Kalman’s functor), suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics.

scholarly journals Categorical Abstract Algebraic Logic: Meet-Combination of Logical Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
George Voutsadakis
Keyword(s):  
Computer Science ◽  
Algebraic Logic ◽  
Abstract Algebraic Logic ◽  
Combined Logics ◽  
Component Systems ◽  
Combining Logics

The widespread and rapid proliferation of logical systems in several areas of computer science has led to a resurgence of interest in various methods for combining logical systems and in investigations into the properties inherited by the resulting combinations. One of the oldest such methods isfibring. In fibring the shared connectives of the combined logics inherit properties frombothcomponent logical systems, and this leads often to inconsistencies. To deal with such undesired effects, Sernadas et al. (2011, 2012) have recently introduced a novel way of combining logics, calledmeet-combination, in which the combined connectives share only thecommonlogical properties they enjoy in the component systems. In their investigations they provide a sound and concretely complete calculus for the meet-combination based on available sound and complete calculi for the component systems. In this work, an effort is made to abstract those results to a categorical level amenable tocategorical abstract algebraic logictechniques.

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