Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. Introduction.One task of metamathematics is to relate suggestive but nonelementary modeltheoretic concepts to more elementary prooftheoretic concepts, thereby opening up modeltheoretic problems to prooftheoretic methods of attack. Herbrand's Theorem (see [8] or also [9], vol. 2) or Gentzen's Extended Hauptsatz (see [5] or also [10]) was first used along these lines by Beth [1]. Using a modified version he showed that for all first-order systems a certain modeltheoretic notion of definability coincides with a certain proof theoretic notion. I In the present paper the Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms.This may be interpreted as showing that (apart from some relatively minor exceptions which will be made apparent below) the expressive power of each first-order system is rounded out, or the system is functionally complete, in the following sense: Any functional relationship which obtains between concepts that are expressible in the system is itself expressible and provable in the system. A second application is concerned with the hierarchy of second-order formulas. A certain relationship is shown to hold between first-order formulas and those second-order formulas which are of the form (3T,)... (3Tk)A or (T,) . .. (Tk)A with A being a first-order formula. Modeltheoretically this can be regarded as a relationship between the class AC and the class PC4, of sets of models, 2 investigated by Tarski in [12] and [13].A third application is concerned with a problem of axiomatizability. For a system with the extralogical constants R, S, and T, it may sometimes be desirable, for example, to have an axiomatization in which each axiom involves only R and S or only S and T. More generally, given a system and given certain proper subsets of the set of extralogical constants of the system, the question arises whether or not there is an axiomatization such that each extralogical axiom involves only the constants of one of the subsets. It will be shown that for a first-order system an axiom system of the desired kind exists if and only if a certain modeltheoretic condition is satisfied.

TRACER: A Symbolic Execution Tool for Verification

“…The first key aspect of TRACER, originally proposed in [13] for symbolic execution, is the avoidance of full enumeration of symbolic paths by learning from infeasible paths computing interpolants [8]. Preliminary versions of TRACER [12,13] computed interpolants based on strongest postconditions (sp).…”

SMTInterpol: An Interpolating SMT Solver

“…The main goal of SMTInterpol is the computation of Craig interpolants [6] as used by modern model checkers. It extends the SMTLIB standard with the get-interpolants command.…”

From Under-Approximations to Over-Approximations and Back

“…Hence, there is no feasible execution in the ARG that can reach v 8 . At this point, Craig interpolants [13] are used to relabel the ARG. Given a pair of formulas (A, B) s.t.…”