Orderings for term-rewriting systems

Dershowitz, Nachum

doi:10.1109/sfcs.1979.32

Cited by 147 publications

(202 citation statements)

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“…. , σ n )), we construct the peeling order S by using the recursive path order > rpo [5] with the argument filtering method [1] over first-order term rewriting systems. We take the argument filtering function by π(tp n ) = n, π(→) = [1,2], and π(c) = [1, .…”

Section: Example 38mentioning

confidence: 99%

Static Dependency Pair Method in Rewriting Systems for Functional Programs with Product, Algebraic Data, and ML-Polymorphic Types

Kusakari

2013

IEICE Trans. Inf. & Syst.

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SUMMARYFor simply-typed term rewriting systems (STRSs) and higher-order rewrite systems (HRSs)à la Nipkow, we proposed a method for proving termination, namely the static dependency pair method. The method combines the dependency pair method introduced for first-order rewrite systems with the notion of strong computability introduced for typed λ-calculi. This method analyzes a static recursive structure based on definition dependency. By solving suitable constraints generated by the analysis, we can prove termination. In this paper, we extend the method to rewriting systems for functional programs (RFPs) with product, algebraic data, and ML-polymorphic types. Although the type system in STRSs contains only product and simple types and the type system in HRSs contains only simple types, our RFPs allow product types, type constructors (algebraic data types), and type variables (ML-polymorphic types). Hence, our RFPs are more representative of existing functional programs than STRSs and HRSs. Therefore, our result makes a large contribution to applying theoretical rewriting techniques to actual problems, that is, to proving the termination of existing functional programs. key words: rewriting systems for functional programs, termination, static dependency pair method IntroductionVarious extensions of term rewriting systems (TRSs) [24] for handling higher-order functions have been proposed [10], [12], [14], [19], [20]. Simply-typed term rewriting systems (STRSs) introduced by Kusakari [14], and higher-order rewrite systems (HRSs) introduced by Nipkow [19] are two such extensions. In this paper, we introduce rewriting systems for functional programs (RFPs), which is an extension of TRSs with product, algebraic data, and ML-polymorphic types. For example, the typical higher-order function foldl can be represented by the following RFP R foldl :Here we suppose that the function foldl has the type:in which α and β are type variables, and list is a type constructor.The static dependency pair method is a powerful )) xsTo prove the non-loopingness of static recursion components, the notions of subterm criterion and reduction pair have been proposed. The subterm criterion was introduced on TRSs [9], and slightly improved by extending the subterms permitted by the criterion on STRSs [16], and extended on HRSs [18]. Reduction pairs [15] are an abstraction of weak-reduction order [1]. By using the subterm criterion, we can prove the non-loopingness of the above static recursion component from the following fact:cons(x, xs) sub xs (xs is a subterm of cons (x, xs))By recapitulating such a termination proof by the static dependency pair method, we obtain the following claim:The function foldl is explicitly recursively defined on the third argument. Hence, the function foldl is well-defined (terminating).This claim is an assertion of the static dependency pair method, and it may be very natural reasoning. However, it is quite difficult to verify the claim because its reduction may be affected by unanticipated behaviors of functio...

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Section: Example 38mentioning

confidence: 99%

Static Dependency Pair Method in Rewriting Systems for Functional Programs with Product, Algebraic Data, and ML-Polymorphic Types

Kusakari

2013

IEICE Trans. Inf. & Syst.

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“…As the precedence is well-founded, so is the RPO [22], therefore > is wellfounded. Furthermore, since a RPO is a simplification ordering, subproofs are indeed smaller according to >.…”

Section: Applicationmentioning

confidence: 97%

Embedding Deduction Modulo into a Prover

Burel¹

2010

Computer Science Logic

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Abstract. Deduction modulo consists in presenting a theory through rewrite rules to support automatic and interactive proof search. It induces proof search methods based on narrowing, such as the polarized resolution modulo. We show how to combine this method with more traditional ordering restrictions. Interestingly, no compatibility between the rewriting and the ordering is requested to ensure completeness. We also show that some simplification rules, such as strict subsumption eliminations and demodulations, preserve completeness. For this purpose, we use a new framework based on a proof ordering. These results show that polarized resolution modulo can be integrated into existing provers, where these restrictions and simplifications are present. We also discuss how this integration can actually be done by diverting the main algorithm of state-of-the-art provers.Whatever their applications, proofs are rarely searched for without context: mathematical proofs rely on set theory, or Euclidean geometry, or arithmetic, etc.; proofs of program correctness are done using e.g. pointer arithmetic and/or theories defining data structures (chained lists, trees, . . . ); concerning security, theories are used for instance to model properties of encryption algorithms. It is therefore essential to have theoretical foundations and practical methods that handle theories conveniently and efficiently. For this purpose, there are two directions: to develop methods that are really specific to a particular theory; or to develop a generic framework that can handle all theories. The first option is appealing for efficiency reasons: for instance, combining a SAT solver with the Simplex method leads to very powerful SMT solvers for linear arithmetic. However, developing methods for new theories is hard. Even the combination of such specific methods is not trivial, although there have been a lot of interesting results in that direction in the recent years. In this paper, we are more interested in the second option: having a generic way to handle theories efficiently. A naive way to do so would be to use an axiomatization of the theory, but in general, this approach would be really inefficient for automated proving. Somehow, we need to present the theory so as to take advantage of its properties.A first idea is to use the consistency of the theory. When proving a goal in a consistent theory by refutation, resolving the clauses of the theory is useless, since it will not bring out a contradiction. This idea defines the set-of-support

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“…Define the TRS R = {l → c | l → r ∈ R} over the signature F. Clearly → R ⊆ → s . Consider a precedence (i.e., a well-founded proper order on F) > with f > c for every function symbol f ∈ F different from c. The TRS R is compatible with the induced recursive path order > rpo ( [3]) and thus terminating. Since R s and R have the same normal forms, it follows that R s is weakly normalizing.…”

Section: Suppose That Q /mentioning

confidence: 99%

On the Modularity of Deciding Call-by-Need

Durand¹,

Middeldorp²

2001

Lecture Notes in Computer Science

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Abstract. In a recent paper we introduced a new framework for the study of call by need computations. Using elementary tree automata techniques and ground tree transducers we obtained simple decidability proofs for a hierarchy of classes of rewrite systems that are much larger than earlier classes defined using the complicated sequentiality concept. In this paper we study the modularity of membership in the new hierarchy. Surprisingly, it turns out that none of the classes in the hierarchy is preserved under signature extension. By imposing various conditions we recover the preservation under signature extension. By imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructor-sharing combinations.

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Orderings for term-rewriting systems

Cited by 147 publications

References 6 publications

Static Dependency Pair Method in Rewriting Systems for Functional Programs with Product, Algebraic Data, and ML-Polymorphic Types

Static Dependency Pair Method in Rewriting Systems for Functional Programs with Product, Algebraic Data, and ML-Polymorphic Types

Embedding Deduction Modulo into a Prover

On the Modularity of Deciding Call-by-Need

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