Certification of nontermination proofs using strategies and nonlooping derivations

© 2014 Springer International Publishing Switzerland. The development of sophisticated termination criteria for term rewrite systems has led to powerful and complex tools that produce (non)termination proofs automatically. While many techniques to establish termination have already been formalized—thereby allowing to certify such proofs—this is not the case for nontermination. In particular, the proof checker CeTA was so far limited to (innermost) loops. In this paper we present an Isabelle/HOL formalization of an extended repertoire of nontermination techniques. First, we formalized techniques for nonlooping nontermination. Second, the available strategies include (an extended version of) forbidden patterns, which cover in particular outermost and context-sensitive rewriting. Finally, a mechanism to support partial nontermination proofs further extends the applicability of our proof checker

Context-sensitive Rewriting

[EN] The appropriate selection of the arguments of functions that can be evaluated in function calls is often useful to improve efficiency, speed, termination behavior, and so on. This is essential, e.g., in the conditional if - then - else operator. We can specify this by associating a set mu( f) of indices of evaluable arguments to each function symbol f. With mu(if-then-else) = {1}, only the Boolean argument b in calls if b then e else e' is evaluated. In the realm of term rewriting, this is called context-sensitive rewriting. It has been proven useful to improve the termination behavior of rewriting computations while it is still able to compute (or approximate) canonical forms like head-normal forms, (infinite) values, and (infinite) normal forms by requiring a few reasonable conditions. 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