A new method for undecidability proofs of first order theories

Ralf Treinen
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
We claim that the reduction of Post's Correspondence Problem to the decision problem of a theory provides a useful tool for proving undecidability of first order theories given by some interpretation. The goal of this paper is to define a framework for such reduction proofs . The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering . .
more &raquo; ... on The interest of this paper is twofold . First it proposes a general methodology for proving results of the kind : The first order theory of the predicate logic model I = . . . is undecidable . Second, besides examples that serve just for the illustration of the method proposed, we show some applications that are interesting in their own right . We only consider theories of given models, in contrast to theories defined by some sets of axioms that are not necessarily complete (for instance the theory defined by the axioms of Boolean algebras is not complete) . When applied to (the theory of) a given model Z, the method leads to an effective mechanism that yields for each instance P of the Post Correspondence Problem over the alphabet {a, b} a formula, denoted solvablep, such that P is solvable 4' 1 = solvablep (1 .1) Because of the effectiveness of the construction of this formula we immediately get the undecidability result for the theory from the well-known undecidability of Post's Correspondence Problem . Furthermore we are interested in showing not only undecidability of the whole theory of 1, but of a smallest possible fragment of this theory. In the construction of solvablep we will therefore try to avoid alternations of quantifiers as far as possible . The basic principle of the proof method proposed is the simulation of the two data types involved in Post's Correspondence Problem : strings and sequences (resp . sets) . The
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