In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #-calA#Ik enriched by pattern-matching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.

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We present techniques to prove termination and innermost termination of term rewriting systems automatically. In contrast to previous approaches, we do not compare left- and right-hand sides of rewrite rules, but introduce the notion of dependency pairs to compare left-hand sides with special subterms of the right-hand sides. This results in a technique which allows to apply existing methods for automated termination proofs to term rewriting systems where they failed up to now. In particular, there are numerous term rewriting systems where a direct termination proof with simplification orderings is not possible, but in combination with our technique, well-known simplification orderings (such as the recursive path ordering, polynomial orderings, or the Knuth-Bendix ordering) can now be used to prove termination automatically. Unlike previous methods, our technique for proving innermost termination automatically can also be applied to prove innermost termination of term rewriting systems that are not terminating. Moreover, as innermost termination implies termination for certain classes of term rewriting systems, this technique can also be used for termination proofs of such systems.

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We present an "unfailing" extension of the standard KnuthBendix completion procedure that is guaranteed to produce a desired canonical system, provided certain conditions are met. Weprove that this unfailing completion method is refutationally complete for theorem proving in equational theories. The method can also be applied to Horn clauses with equality, in which case it corresponds to positive unit resolution plus oriented paramodulation, with unrestricted simplification.

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Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1

Automated deduction techniques are being used in a system called Amphion to derive, from graphical specifications, programs composed from a subroutine library. The system has been applied to construct software for the planning and analysis of interplanetary missions. The library for that application is a collection of subroutines written in FORTRAN-77 at JPL to perform computations in solar-system kinematics. An application domain theory has been developed that describes A preliminary version of this appears in the proceedings of the Twelfth International Conference on Automated Deduction, Nancy, France, June 1994, pages 341-355. y fstickel,waldingerg@ai.sri.com z flowry, pressburger,underwoodg@ptolomy.arc.nasa.gov the procedures in a portion of the library, as well as some basic properties of solar-system astronomy, in the form of first-order axioms. Specifications are elicited from the user through a menu-driven graphical user interface; space scientists have found the graph...

This paper describes a system implemented in SICStus Prolog for automatically checking left termination of logic programs. Given a program and query, the system answers either that the query terminates or that there may be non-termination. The system can use any norm of a wide family of norms. It can handle automatically most of the examples found in the literature on termination of logic programs, and about half of the programs in the benchmarks of [5]. The algorithm employed by the system consists of three main parts: instantiation analysis (i.e., rigidity analysis), constraint inference, and construction of the query-mapping pairs associated with the program and query. Each of these parts generalizes earlier work related to termination analysis. 1 Introduction Termination analysis of logic programs has been the focus of intensive research in recent years. Systems for automatic termination analysis are proposed, for example, in [15, 20, 22]. Methodologies and techniques for proving ...

Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By means of a careful analysis of potential counterexamples we prove the following abstract result. Whenever the disjoint union R1 \Phi R2 of two (finitely branching) terminating term rewriting systems R1 , R2 is non-terminating, then one of the systems, say R1 , enjoys an interesting (undecidable) property, namely it is not termination preserving under non-deterministic collapses, i.e. R1 \Phi fG(x; y) ! x; G(x; y) ! yg is non-terminating, and the other system R2 is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient criteria for modular termination of rewriting and provides the basis for a couple of derived modularity results. Furthermore, we prove that the minimal rank of pote...