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## Halting and Equivalence of Program Schemes in Models of Arbitrary Theories

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**Author**

Kozen, Dexter

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**Abstract**

In this note we consider the following decision problems. Let S be a fixed first-order signature.
(i) Given a first-order theory or ground theory T over S of Turing degree A, a program scheme p over S, and input values specified by ground terms t1,...,tn, does p halt on input t1,...,tn in all models of T?
(ii) Given a first-order theory or ground theory T over S of Turing degree A and two program schemes p and q over S, are p and q equivalent in all models of T?
When T is empty, these two problems are the classical halting and equivalence problems for program schemes, respectively. We show that problem (i) is Sigma^A_1-complete and problem (ii) is Pi^A_2-complete. Both problems remain hard for their respective complexity classes even if S is restricted to contain only a single constant, a single unary function symbol, and a single monadic predicate. It follows from (ii) that there can exist no relatively complete deductive system for scheme equivalence over models of theories of any Turing degree.

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**Date Issued**

2010-05-19#####
**Subject**

dynamic model theory; program scheme; scheme equivalence

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**Type**

Technical Report