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Dependency pairs for rewriting with builtin numbers and semantic data structures
, 2007
"... Abstract. This paper defines an expressive class of constrained equational rewrite systems that supports the use of semantic data structures (e.g., sets or multisets) and contains builtin numbers, thus extending our previous work presented at CADE 2007 [6]. These rewrite systems, which are based on ..."
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Abstract. This paper defines an expressive class of constrained equational rewrite systems that supports the use of semantic data structures (e.g., sets or multisets) and contains builtin numbers, thus extending our previous work presented at CADE 2007 [6]. These rewrite systems, which are based on normalized rewriting on constructor terms, allow the specification of algorithms in a natural and elegant way. Builtin numbers are helpful for this since numbers are a primitive data type in every programming language. We develop a dependency pair framework for these rewrite systems, resulting in a flexible and powerful method for showing termination that can be automated effectively. Various powerful techniques are developed within this framework, including a subterm criterion and reduction pairs that need to consider only subsets of the rules and equations. It is wellknown from the dependency pair framework for ordinary rewriting that these techniques are often crucial for a successful automatic termination proof. Termination of a large collection of examples can be established using the presented techniques. 1
Operational termination of conditional rewriting with builtin numbers and semantic data structures
, 2007
"... Abstract. Rewrite systems on free data structures have limited expressive power since semantic data structures like sets or multisets cannot be modeled elegantly. In this work we define a class of rewrite systems that allows the use of semantic data structures. Additionally, builtin natural numbers ..."
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Abstract. Rewrite systems on free data structures have limited expressive power since semantic data structures like sets or multisets cannot be modeled elegantly. In this work we define a class of rewrite systems that allows the use of semantic data structures. Additionally, builtin natural numbers, including (dis)equality, ordering, and divisibility constraints, are supported. The rewrite mechanism is a combination of normalized equational rewriting with evaluation of conditions and validity checking of instantiated constraints. The framework is highly expressive and allows modeling of algorithms in a natural way. Termination is one of the most important properties of conditional normalized equational rewriting. For this it is not sufficient to only show wellfoundedness of the rewrite relation, but it also has to be ensured that evaluation of the conditions does not loop. The notion of operational termination is a way to capture these properties. In this work we show that it is possible to transform a conditional constrained equational rewrite system into an unconditional one such that termination of the latter implies operational termination of the former. Methods for showing termination of unconditional constrained equational rewrite system are presented in a companion paper. 1
Termination of ContextSensitive Rewriting with BuiltIn Numbers and Collection Data Structures ⋆
"... Abstract. Contextsensitive rewriting is a restriction of rewriting that can be used to elegantly model declarative specification and programming languages such as Maude. Furthermore, it can be used to model lazy evaluation in functional languages such as Haskell. Building upon our previous work on ..."
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Abstract. Contextsensitive rewriting is a restriction of rewriting that can be used to elegantly model declarative specification and programming languages such as Maude. Furthermore, it can be used to model lazy evaluation in functional languages such as Haskell. Building upon our previous work on an expressive and elegant class of rewrite systems (called CERSs) that contains builtin numbers and supports the use of collection data structures such as sets or multisets, we consider contextsensitive rewriting with CERSs in this paper. This integration results in a natural way for specifying algorithms in the rewriting framework. In order to prove termination of this kind of rewriting automatically, we develop a dependency pair framework for contextsensitive rewriting with CERSs, resulting in a flexible termination method that can be automated effectively. Several powerful termination techniques are developed within this framework. An implementation in the termination prover AProVE has been successfully evaluated on a large collection of examples. 1
References
"... This erratum corrects the proof of Lemma 12. For the definition of ∼E∪PA, recall that PA is the following set of equations given in the beginning of Section 2: PA = {x + (y + z) ≈ (x + y) + z, x + y ≈ y + x, x + 0 ≈ x} We use the standard definition of relation composition, i.e., we let x(P ◦ Q)z i ..."
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This erratum corrects the proof of Lemma 12. For the definition of ∼E∪PA, recall that PA is the following set of equations given in the beginning of Section 2: PA = {x + (y + z) ≈ (x + y) + z, x + y ≈ y + x, x + 0 ≈ x} We use the standard definition of relation composition, i.e., we let x(P ◦ Q)z iff there is a y such that xPy and yQz. Lemma 12. Let (R, S, E) be a CCES. Then ∼E∪PA ◦ S → PA‖E\R ⊆ S → PA‖E\R ◦ ∼E∪PA. Furthermore, the S → PA‖E\R steps are performed using the same conditional constrained rewrite rule and PAbased substitution. Proof. Assume s ′ ∼E∪PA s S → PA‖E\R t. We then need to show that s ′ S → PA‖E\R t ′ ∼E∪PA t for some t ′. From s S → PA‖E\R t we get s = C[f(u ∗)] for some context C and some f ∈ D(R), where f(u ∗ ) <ε