By Lutz Plümer (auth.)

Termination proofs represent an important a part of software verification. a lot learn approximately termination has been performed within the context of time period rewriting platforms. yet previously there has been little wish that termination proofs for nontrivial courses might be completed immediately. This publication offers a complete dialogue of the termination challenge within the context of common sense programming. even supposing good judgment courses pose detailed problems for termination proofs it seems that automation of this activity is on the market to a miles better measure than for courses in critical languages. a strategy for the automated derivation of termination proofs is gifted intimately. The dialogue of numerous nontrivial examples illustrates its variety of applicability. The technique relies at the proposal of declarative semantics, and therefore uses a huge characteristic of common sense programming.

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2 Terminating Queries We now make precise what is meant by a terminating query. The idea is that a terminating query is one which has a finite number of derivations, all of which are finite. 1 DEFINITION: Terminating Queries For a given program P we have a) Termn(P) = {Q~ HBp I V0, all computed proof trees for (P,Q0) are of depth < n} b) Term(P) = {Q~ HBp I V0, all computed proof trees for (P,Q0) are of a finitely bounded depth} Term(P) is called the set of terminating queries of P. • 45 TERMINATINGLOGICPROGRAMS Note that • Termn(P) G Terrnn+l(P), • Term(P) = [-Jn~N Termn(P), • Term(P) is dependent on the computation rule, • Term(P) is closed under substitution.

Sm) >* f(h ..... tin), where s >* ti for i= I ..... m, the tuples (sl ..... sin) and (t~ ..... t,n) are compared lexicographically instead of comparing them as multisets. Lexicographic comparison can be made from left to right, from right to left or in any fixed order. The specific way of comparing arguments is referred to as the status of a function symbol [BAC88]. An example where this modification of RPO is useful is associativity: (V) (X,Y),Z >* X, ( V . Z). Whereas the multiset {(X • Y), Z} is not bigger than {(Y • Z), X}, lexicographic comparison of the corresponding tuples is possible: (VI) ((X • Y), Z) >x (X,(Y • Z)) since X • Y >* X.

Tn) Ix = ~ I ti Iz i=l otherwise. PROGRAM PROPERTIES AND TRANSFORMATIONS 31 This norm counts the number of occurrences of reflexive constructors for x in a given term (which is not necessarily of type x). Since the type norm is induced by function symbol weights it is linear. (f(a),nil)). We have I[a,f(a)] Ii = 3 for the 1-norm, I[a,f(a)] In I[a,f(a)] Ix = 5 = 2 for the n-norm, for the type norm and x = type list(any). Another norm, which is interesting for historical reasons, is the Knuth-Bendix-norm.

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