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**1 - 3**of**3**### TERMINATION OF NON-SIMPLE REWRITE SYSTEMS

"... Rewriting is a computational process in which one term is derived from another by replacing a subterm with another subterm in accordance with a set of rules. If such a set of rules (rewrite system) has the property that no derivation can continue inde nitely, it is said to be terminating. Showing te ..."

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Rewriting is a computational process in which one term is derived from another by replacing a subterm with another subterm in accordance with a set of rules. If such a set of rules (rewrite system) has the property that no derivation can continue inde nitely, it is said to be terminating. Showing termination is an important component of theorem proving and of great interest in programming languages. Two methods of showing termination for rewrite systems that are self-embedding are pre-sented. These \non-simple &quot; rewrite systems can not be shown terminating by any of what are called simpli cation orderings. The rst method of termination employs lexicographic combina-tions of quasi-orderings including the ordering itself applied to multisets of immediate subterms in a general path ordering. Twoversions are presented. The well-founded and well-quasi general path orderings respectively require their component orderings to be well-founded and well-quasi orderings. The de nitions are shown to result in well-founded and well-quasi orderings, respec-tively. A general condition is presented for showing termination of a rewrite system with a quasi-ordering. Conditions on the component orderings are presented which guarantee that the general conditions are satis ed. The well-quasi general path ordering is applied to several

### Type Reconstruction in the Presence of Polymorphic Recursion and Recursive Types

, 1993

"... We establish the equivalence of type reconstruction with polymorphic recursion and recursivetypes is equivalent to regular semiunification whichproves the undecidability of the corresponding type reconstruction problem. We also establish the equivalence of type reconstruction with polymorphic re ..."

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We establish the equivalence of type reconstruction with polymorphic recursion and recursivetypes is equivalent to regular semiunification whichproves the undecidability of the corresponding type reconstruction problem. We also establish the equivalence of type reconstruction with polymorphic recursion and positive recursivetypes to a special case of regular semi-unification whichwe call positive regular semi-unification. The decidability of positive regular semiunification is an open problem.

### A General Theory of Semi-Unification

, 1993

"... Various restrictions on the terms allowed for substitution give rise to different cases of semi-unification. Semi-unification on finite and regular terms has already been considered in the literature. We introduce a general case of semi-unification where substitutions are allowed on non-regular term ..."

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Various restrictions on the terms allowed for substitution give rise to different cases of semi-unification. Semi-unification on finite and regular terms has already been considered in the literature. We introduce a general case of semi-unification where substitutions are allowed on non-regular terms, and we prove the equivalence of this general case to a well-known undecidable data base dependency problem , thus establishing the undecidability of general semi-unification. We present a unified way of looking at the various problems of semi-unification. We give some properties that are common to all the cases of semi-unification. We also the principality property and the solution set for those problems. We prove that semi-unification on general terms has the principality property. Finally, we present a recursive inseparability result between semi-unification on regular terms and semi-unification on general terms. Partly supported by NSF grant CCR-9113196. Address: Department of Compu...