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A SATBased Approach to Size Change Termination with Global Ranking Functions
"... Abstract. We describe a new approach to proving termination with size change graphs. This is the first decision procedure for size change termination (SCT) which makes direct use of global ranking functions. It handles a welldefined and significant subset of SCT instances, designed to be amenable t ..."
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Abstract. We describe a new approach to proving termination with size change graphs. This is the first decision procedure for size change termination (SCT) which makes direct use of global ranking functions. It handles a welldefined and significant subset of SCT instances, designed to be amenable to a SATbased solution. We have implemented the approach using a stateoftheart Boolean satisfaction solver. Experimentation indicates that the approach is a viable alternative to the complete SCT decision procedure based on closure computation and local ranking functions. Our approach has the extra benefit of producing an explicit witness to prove termination in the form of a global ranking function. 1
Complexity analysis by rewriting
"... Abstract. In this paper we introduce a restrictive version of the multiset path order, called polynomial path order. This recursive path order induces polynomial bounds on the maximal number of innermost rewrite steps. This result opens the way to automatically verify for a given program, written ..."
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Abstract. In this paper we introduce a restrictive version of the multiset path order, called polynomial path order. This recursive path order induces polynomial bounds on the maximal number of innermost rewrite steps. This result opens the way to automatically verify for a given program, written in an eager functional programming language, that the maximal number of evaluation steps starting from any function call is polynomial in the input size. To test the feasibility of our approach we have implemented this technique and compare its applicability to existing methods. 1
A3PAT, an Approach for Certified Automated Termination Proofs
 In ACM SIGPLAN Workshop on Partial Evaluation and Program Manipulation (PEPM 10
, 2010
"... Software engineering, automated reasoning, rulebased programming or specifications often use rewriting systems for which termination, among other properties, may have to be ensured. This paper presents the approach developed in Project A3PAT to discover and moreover certify, with full automation, t ..."
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Software engineering, automated reasoning, rulebased programming or specifications often use rewriting systems for which termination, among other properties, may have to be ensured. This paper presents the approach developed in Project A3PAT to discover and moreover certify, with full automation, termination proofs for term rewriting systems. It consists of two developments: the COCCINELLE library formalises numerous rewriting techniques and termination criteria for the COQ proof assistant; the CiME3 rewriting tool translates termination proofs (discovered by itself or other tools) into traces that are certified by COQ assisted by COCCINELLE. The abstraction level of our formalisation allowed us to weaken premises of some theorems known in the literature, thus yielding new termination criteria, such as an extension of the powerful subterm criterion (for which we propose the first full COQ formalisation). Techniques employed in CiME3 also improve on previous works on formalisation and analysis of dependency graphs. Categories and Subject Descriptors F.3.1 [Logics and Meaning
KBO Orientability
 J AUTOM REASONING
, 2009
"... This article presents three new approaches to prove termination of rewrite systems with the KnuthBendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo)propositional logic or linear arithmetic and the resulting formula is tested for satisfia ..."
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Cited by 8 (3 self)
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This article presents three new approaches to prove termination of rewrite systems with the KnuthBendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo)propositional logic or linear arithmetic and the resulting formula is tested for satisfiability using dedicated solvers. Any satisfying assignment represents a weight function and a precedence such that the induced KnuthBendix order orients the rules of the encoded rewrite system from left to right. This means that in contrast to the dedicated methods our approach does not directly solve the problem but transforms it to equivalent formulations for which sophisticated backends exist. In order to make all approaches complete we present a method to compute upper bounds on the weights. Furthermore, our encodings take dependency pairs into account to increase the applicability of the order.
Proving Termination by Dependency Pairs and Inductive Theorem Proving
 JOURNAL OF AUTOMATED REASONING
"... 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 termination of T ..."
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Cited by 6 (3 self)
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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 termination of TRSs 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.
Polytool: Polynomial interpretations as a basis for termination analysis of logic programs
, 2009
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Argument Filterings and Usable Rules for Simply Typed Dependency Pairs (extended abstract)
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On the formalization of termination techniques based on multiset orderings
"... Multiset orderings are a key ingredient in certain termination techniques like the recursive path ordering and a variant of sizechange termination. In order to integrate these techniques in a certifier for termination proofs, we have added them to the Isabelle Formalization of Rewriting. To this en ..."
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Cited by 4 (2 self)
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Multiset orderings are a key ingredient in certain termination techniques like the recursive path ordering and a variant of sizechange termination. In order to integrate these techniques in a certifier for termination proofs, we have added them to the Isabelle Formalization of Rewriting. To this end, it was required to extend the existing formalization on multiset orderings towards a generalized multiset ordering. Afterwards, the soundness proofs of both techniques have been established, although only after fixing some definitions. Concerning efficiency, it is known that the search for suitable parameters for both techniques is NPhard. We show that checking the correct application of the techniques—where all parameters are provided—is also NPhard, since the problem of deciding the generalized multiset ordering is NPhard.
Lazy Abstraction for SizeChange Termination
, 2010
"... Sizechange termination is a widely used means of proving termination where source programs are first abstracted to sizechange graphs which are then analyzed to determine if they satisfy the sizechange termination property. Here, the choice of the abstraction is crucial to the success of the meth ..."
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Sizechange termination is a widely used means of proving termination where source programs are first abstracted to sizechange graphs which are then analyzed to determine if they satisfy the sizechange termination property. Here, the choice of the abstraction is crucial to the success of the method, and it is an open problem how to choose an abstraction such that no critical loss of precision occurs. This paper shows how to couple the search for a suitable abstraction and the test for sizechange termination via an encoding to a single SAT instance. In this way, the problem of choosing the right abstraction is solved en passant by a SAT solver. We show that for the setting of term rewriting, the integration of this approach into the dependency pair framework works smoothly and gives rise to a new class of sizechange reduction pairs. We implemented sizechange reduction pairs in the termination prover AProVE and evaluated their usefulness in extensive experiments.
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