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Degrees of undecidability in term rewriting
 Proceedings of Computer 30 Logic (CSL09), volume 5771 of Lecture Notes in Computer Science
, 2009
"... Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy cl ..."
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Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy, where also quantification over function variables is allowed. In this paper we consider properties of first order term rewriting systems and classify them in this hierarchy. Most of the standard properties are Π 0 2complete, that is, of the same level as uniform halting of Turing machines. In this paper we show two exceptions. Weak confluence is Σ 0 1complete, and therefore essentially easier than ground weak confluence which is Π 0 2complete. The most surprising result is on dependency pair problems: we prove this to be Π 1 1complete, which means that this property exceeds the arithmetical hierarchy and is essentially analytic. A minor variant, dependency pair problems with minimality flag, turns out be Π 0 2complete again, just like the original termination problem for which dependency pair analysis was developed. 1
OrderSorted Dependency Pairs
, 2008
"... Types (or sorts) are pervasive in computer science and in rewritingbased programming languages, which often support subtypes (subsorts) and subtype polymorphism. Programs in these languages can be modeled as ordersorted term rewriting systems (OSTRSs). Often, termination of such programs heavily d ..."
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Types (or sorts) are pervasive in computer science and in rewritingbased programming languages, which often support subtypes (subsorts) and subtype polymorphism. Programs in these languages can be modeled as ordersorted term rewriting systems (OSTRSs). Often, termination of such programs heavily depends on sort information. But few techniques are currently available for proving termination of OSTRSs; and they often fail for interesting OSTRSs. In this paper we generalize the dependency pairs approach to prove termination of OSTRSs. Preliminary experiments suggest that this technique can succeed where existing ones fail, yielding easier and simpler termination proofs.
Termination of Narrowing via Termination of Rewriting
"... Narrowing extends rewriting with logic capabilities by allowing logic variables in terms and by replacing matching with unification. Narrowing has been widely used in different contexts, ranging from theorem proving (e.g., protocol verification) to language design (e.g., it forms the basis of funct ..."
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Narrowing extends rewriting with logic capabilities by allowing logic variables in terms and by replacing matching with unification. Narrowing has been widely used in different contexts, ranging from theorem proving (e.g., protocol verification) to language design (e.g., it forms the basis of functional logic languages). Surprisingly, the termination of narrowing has been mostly overlooked. In this work, we present a novel approach for analyzing the termination of narrowing in leftlinear constructor systems—a widely accepted class of systems—that allows us to reuse existing methods in the literature on termination of rewriting.
Termination of Lazy Rewriting Revisited
, 2007
"... Lazy rewriting is a proper restriction of term rewriting that dynamically restricts the reduction of certain arguments of functions in order to obtain termination. In contrast to contextsensitive rewriting, reductions at such argument positions are not completely forbidden but delayed. Based on the ..."
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Lazy rewriting is a proper restriction of term rewriting that dynamically restricts the reduction of certain arguments of functions in order to obtain termination. In contrast to contextsensitive rewriting, reductions at such argument positions are not completely forbidden but delayed. Based on the observation that the only existing (nontrivial) approach to prove termination of such lazy rewrite systems is flawed, we develop a modified approach for transforming lazy rewrite systems into contextsensitive ones that is sound and complete with respect to termination. First experimental results with this transformation based technique are encouraging.
Termination Analysis by Dependency Pairs and Inductive Theorem Proving
, 2009
"... Current techniques and tools for automated termination analysis of term rewrite systems (TRSs) are already very powerful. However, they fail for algorithms whose termination is essentially due to an inductive argument. Therefore, we show how to couple the dependency pair method for TRS termination ..."
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Current techniques and tools for automated termination analysis of term rewrite systems (TRSs) are already very powerful. However, they fail for algorithms whose termination is essentially due to an inductive argument. Therefore, we show how to couple the dependency pair method for TRS termination with inductive theorem proving. As confirmed by the implementation of our new approach in the tool AProVE, now TRS termination techniques are also successful on this important class of algorithms.
Goaldirected and Relative Dependency Pairs for Proving the Termination of Narrowing ⋆
"... Abstract. In this work, we first consider a goaloriented extension of the dependency pair framework for proving termination w.r.t. a given set of initial terms. Then, we introduce a new result for proving relative termination in terms of a dependency pair problem. Both contributions put together al ..."
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Abstract. In this work, we first consider a goaloriented extension of the dependency pair framework for proving termination w.r.t. a given set of initial terms. Then, we introduce a new result for proving relative termination in terms of a dependency pair problem. Both contributions put together allow us to define a simple and powerful approach to analyzing the termination of narrowing, an extension of rewriting that replaces matching with unification in order to deal with logic variables. Our approach could also be useful in other contexts where considering termination w.r.t. a given set of terms is also natural (e.g., proving the termination of functional programs). 1
Automated Termination Analysis for Programs with SecondOrder Recursion
"... Abstract. Many algorithms on data structures such as terms (finitely branching trees) are naturally implemented by secondorder recursion: A firstorder procedure f passes itself as an argument to a secondorder procedure like map, every, foldl, foldr, etc. to recursively apply f to the direct subte ..."
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Abstract. Many algorithms on data structures such as terms (finitely branching trees) are naturally implemented by secondorder recursion: A firstorder procedure f passes itself as an argument to a secondorder procedure like map, every, foldl, foldr, etc. to recursively apply f to the direct subterms of a term. We present a method for automated termination analysis of such procedures. It extends the approach of argumentbounded functions (i) by inspecting type components and (ii) by adding a facility to take care of secondorder recursion. Our method has been implemented and automatically solves the examples considered in the literature. This improves the state of the art of inductive theorem provers, which (without our approach) require user interaction even for termination proofs of simple secondorder recursive procedures. 1
Modular Termination of Basic Narrowing and Equational Unification
, 2009
"... Basic narrowing is a restricted form of narrowing which constrains narrowing steps to a set of unblocked (or basic) positions. In this work, we study the modularity of termination of basic narrowing in hierarchical combinations of TRSs, which provides new algorithmic criteria to prove termination of ..."
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Basic narrowing is a restricted form of narrowing which constrains narrowing steps to a set of unblocked (or basic) positions. In this work, we study the modularity of termination of basic narrowing in hierarchical combinations of TRSs, which provides new algorithmic criteria to prove termination of basic narrowing. Basic narrowing has a number of important applications including equational unification in canonical theories. Another application is analyzing termination of narrowing by checking the termination of basic narrowing, as done in pioneering work by Hullot. As a particularly interesting application, we consider solving equations modulo a theory that is given by a TRS, and then distill a number of modularity results for the decidability of equational unification via the modularity of basic narrowing (completeness and) termination.
Shallow Dependency Pairs
"... Abstract. We show how the dependency pair approach, commonly used to modularize termination proofs of rewrite systems, can be adapted to establish termination of recursive functions in a system like Isabelle/HOL or Coq. It turns out that all that is required are two simple lemmas about wellfoundedne ..."
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Abstract. We show how the dependency pair approach, commonly used to modularize termination proofs of rewrite systems, can be adapted to establish termination of recursive functions in a system like Isabelle/HOL or Coq. It turns out that all that is required are two simple lemmas about wellfoundedness. 1