C. Walther. Argument-bounded algorithms as a basis for automated termination proofs. In Proceedings of the 9th International Conference on Automated Deduction, Lecture Notes in Computer Science 310, 1988. 28  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and strictly positive inductive types that allows the definition of recursive terms. Their termination checker foetus ensures that all such terms are structurally recursive, i.e. recursive calls appear only with arguments structurally smaller than the input parameters of terms considered. Walter [84] has described reduction checking, which is sometimes referred to as Walter recursion. His estimation calculus examines whether functions are terminating and also whether the output of the function is smaller than the input. This information can be used to check termination of nested recursive ....

C. Walther. Argument-Bounded Algorithms as a Basis for Automated Termination Proofs. In Proceedings of of the 9th International Conference on Automated Deduction, number 310 in Lecture Notes in Computer Science, pages 602--621. Springer-Verlag, 1988.

....types, a greater class of functions is accepted in Telford Turner s ESFP [34] Their termination checker also incorporates a limited form of size change and dataflow analysis which recognizes certain functions as reducers or preservers. Functions in these classes first described by Walther [35] have the property that the size of their output is bound by the size of some input argument (strictly smaller in the case of reducers) Finally, Pientka [29] has implemented termination and reduction checking for higher order logic programs based on the subterm ordering. Unfortunately, the ....

....A insert A XS L . inssort : #Y# type . mode inssort L K. sortNil : inssort nil nil . sortCons : inssort (X XS L inssort XS K insert X K L . In its current implementation, Twelf is equipped with termination and reduction checking (as found by Walther [35]) based on the subterm ordering (cf. Pientka [29] In the given natural specification, insertion sort is not recognized as terminating in Twelf. The function insert is strictly increasing the size of its input and thus no preserver. However, our proposed type system can e#ortlessly keep track of ....

C. Walther. Argument-Bounded Algorithms as a Basis for Automated Termination Proofs. In E. L. Lusk and R. A. Overbeek, editors, 9th International Conference on Automated Deduction, volume 310 of Lecture Notes in Computer Science, pages 602--621. Springer-Verlag, 1988.

....we proceed with stating a lemma: 8x; y : nat 9z : nat ( y = 0 :x y) y z) x Proving this lemma our system comes up with an algorithmic specification of minus. The system automatically recognizes that in case of :y = 0 :x = 0 the function minus is argument bounded on the first argument [Wa88] i.e. minus(x; y) is less than x according to a well founded ordering (i.e. the count ordering) Thus, the used scheme 8x; y : nat (y = 0 x y) Psi (x; y) y = 0 :x y) Psi (minus(x; y) y) Psi (x; y) 8x; y : nat Psi (x; y) is a sound induction axiom and the proof is ....

Walther, C. Argument-Bounded Algorithms as a Basis for Automated Termination Proofs. 9th CADE, LNCS310, Springer, 1988 This article was processed using the L a T E X macro package with LLNCS style

....on the automation of termination proofs has been done in the areas of term rewriting systems (for surveys see e.g. 11,27] and of logic programs (e.g. 24,25,28] in this paper we focus on functional programs. Up to now all methods for automated termination analysis of functional programs (e.g. [1,3,13,14,23,26,29,32]) aim to prove that a program terminates for each input. However, if the termination proof fails then these methods provide no means to find a (sub )domain where termination is provable. Therefore these methods cannot be used to analyze the termination behavior of partial functional programs, ....

C. Walther, Argument-Bounded Algorithms as a Basis for Automated Termination Proofs, in: Proc. 9th CADE, Lecture Notes in Computer Science, Vol. 310 (Springer, Berlin, 1988) 602-621.

....Constructor variables are convenient in the field 22 of inductive theorem proving for expressing important lemmas that do not hold for undefined terms 23 . The means for automatically showing termination of the functions of classic inductive theorem proving (cf. Boyer Moore[10] Walther[26]) also depend on the variables in the function definitions being bound to constructor terms only. This dependence, however, and the intended meaning of the variables at all, are usually hidden in the formalism and not made as explicit as in Avenhaus Becker[1] where it is shown that the restriction ....

Christoph Walther. Argument-Bounded Algorithms as a Basis for Automated Termination Proofs. 9 th CADE 1988, LNAI 310. Springer-Verlag, Berlin 1988.

....could, in theory, be proved so, although the problem is semi decidable and resource limitations would severely restrict what was possible in practice. This is the alternative approach, based on well founded orderings, that was rejected in x2. It has been adopted, for instance, by Walther, Walther 88] The termination proving part of the task is essentially that of proving termination of sets of (conditional) rewrite rules. There is a significant literature on this problem. See, for instance, Huet Oppen 80] for a survey of the known techniques. It is a semi decidable problem, but some ....

C. Walther. Argument-bounded algorithms as a basis for automated termination proofs. In R. Lusk and R. Overbeek, editors, 9th International Conference on Automated Deduction, pages 602--621. Springer-Verlag, 1988. Revised version to appear in AI Journal. 32

....condition, t is a term of sort s and represents the argument of a recursive call of the algorithm under consideration, and q is a term of sort s and represents the original input to the algorithm. For these kind of proof obligations Walther developed a special calculus, the Estimation Calculus, [14, 15]. It consists of a set of inference rules which represent properties of the underlying data types, especially the UFP, and the property of some functions to be argument bounded. A function f is argument bounded if there is an argument position p such that for all x 1 ; x n # s (f(x 1 ; ....

....the intuition behind the Estimation Calculus. In addition, for freely generated data types both notions coincide. Hence, for each data type we can assume a well founded order relation, the minimal size order. 3 The Estimation Calculus The original Estimation Calculus was developed by Walther [14, 15] in order to automate termination proofs over freely generated data types using the size order. In this section we present a generalization of the original calculus to enable termination proofs over non freely generated data types as well. As we illustrated before a termination proof of an ....

C. Walther. Argument-Bounded Algorithms as a Basis for Automated Termination Proofs. In CADE 9. Springer Verlag, 1988.

....proofs has been done in the areas of term rewriting systems (for surveys see e.g. Der87, Ste95] and of logic programs (e.g. UV88, Plu90, SD94] in this paper we focus on functional programs. Up to now all methods for automated termination analysis of functional programs (e.g. BM79, Wal88, Hol91, Wal94b, NN95, Gie95b, Gie95c] aim to prove that a program terminates for each input. However, if the termination proof fails then these methods provide no means to find a (sub )domain where termination is provable. Therefore these methods cannot be used to analyze the termination ....

C. Walther. Argument-Bounded Algorithms as a Basis for Automated Termination Proofs. In Proc. 9th CADE, LNCS 310, Argonne, IL, 1988.

....of our method have been developed, datei.tex; 13 02 1998; 10:59; p.9 10 JRGEN GIESL, CHRISTOPH WALTHER, JRGEN BRAUBURGER e.g. techniques for the simplification of difference predicates which ease the proof of termination hypotheses considerably. Further details and refinements can be found in (Walther, 1988; Walther, 1991; Walther, 1994b) Our method works for polymorphic types as well and an adaptation to non free data types can be found in (Sengler, 1996; Sengler, 1997) Based on our schema for argument bounded functions, McAllester and Arkoudas (1996) suggest a programming discipline such that ....

Walther, C.: 1988, `Argument--Bounded Algorithms as a Basis for Automated Termination Proofs'. In: Proc. CADE-9. Argonne, IL, pp. 602--621. LNCS 310.

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C. Walther. Argument-bounded algorithms as a basis for automated termination proofs. In Proceedings of the 9th International Conference on Automated Deduction, Lecture Notes in Computer Science 310, 1988. 28

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C. Walther. Argument-Bounded Algorithms as a Basis for Automated Termination Proofs. In E. L. Lusk and R. A. Overbeek, editors, 9th International Conference on Automated Deduction, volume 310 of Lecture Notes in Computer Science, pages 602-621. Springer, 1988.

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C. Walther. Argument-bounded algorithms as a basis for automated termination proofs. Proceedings of 9th International Conference on Automated Deduction, Argonne, Illinois, volume 310 of LNCS, 1988.

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