Tarski's influence on computer science

The influence of Alfred Tarski on computer science was indirect but significant in a number of directions and was in certain respects fundamental. Here surveyed is the work of Tarski on the decision procedure for algebra and geometry, the method of elimination of quantifiers, the semantics of formal languages, modeltheoretic preservation theorems, and algebraic logic; various connections of each with computer science are taken up

Computation on abstract data types. The extensional approach, with an application to streams

AbstractIn this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functional of type level ⩽ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type

UNFOLDING FINITIST ARITHMETIC

The concept of the (full) unfolding \user1{{\cal U}}(S) of a schematic system $S$ is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted $S$? The program to determine \user1{{\cal U}}(S) for various systems $S$ of foundational significance was previously carried out for a system of nonfinitist arithmetic, $NFA$; it was shown that \user1{{\cal U}}(NFA) is proof-theoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system of finitist arithmetic, $FA$, and for an extension of that by a form $BR$ of the so-called Bar Rule. It is shown that \user1{{\cal U}}(FA) and \user1{{\cal U}}(FA + BR) are proof-theoretically equivalent, respectively, to Primitive Recursive Arithmetic, $PRA$, and to Peano Arithmetic, $PA

Logic and Methodology, Center Stage

The first international Congress for Logic, Methodology and Philosophy of Science was held at Stanford University in August of 1960. Occupying the vacuum created by the demise of the Unity of Science movement, it was the culminating event, on an international scale, of a long process of reorganization of communities of the philosophy of science and of logic that took place in the fifteen years following World War II—a process that involved many competing interests and personalities. Alfred Ta..

A new foundational crisis in mathematics, is it really happening?

The article reconsiders the position of the foundations of mathematics after the discovery of HoTT. Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one. We reiterate a pluralist vision of the foundations of mathematics. The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.Comment: Final versio