Commutation, Transformation, and Termination

  title={Commutation, Transformation, and Termination},
  author={Leo Bachmair and Nachum Dershowitz},
In this paper we study the use of commutation properties for proving termination of rewrite systems. Commutation properties may be used to prove termination of a combined system R∪S by proving termination of R and S separately. We present termination methods for ordinary and for equational rewrite systems. Commutation is also important for transformation techniques. We outline the application of transforms—mappings from terms to terms—to termination in general, and describe various specific… 
Commutativity of Term Rewriting Systems∗
Using the concept of critical pairs between two term rewriting systems, a sufficient condition for commutativity is proposed and a new sufficient condition is offered for the Church-Rosser property of left-linearterm rewriting systems.
Proving Termination of Associative Commutative Rewriting Systems by Rewriting
This paper proposes a special reduction ordering for proving termination of Associative Commutative rewriting systems, based on a transformation of the terms by a rewriting system with rules similar to distributivity, and shows cases where this ordering fails.
Termination Modulo Equations by Abstract Commutation with an Application to Iteration
Full-Commutation and Fair-Termination in Equational (and Combined) Term-Rewriting Systems
This paper defines the notion of fairness in equational term-rewriting systems, where a derivation step is a composition of the equality generated by a (finite) set of equations with one step rewriting using a set of rules.
Incremental Termination Proofs and the Length of Derivations
It is shown how an incremental termination proof for a term rewriting system T can be used to derive upper bounds on the length of derivations in T, and how these results can be applied to yield (sharp) low-degree polynomial complexity bounds.
Non-looping rewriting
In this paper we present a number of necessary conditions for the existence of loops, i.e. reductions of the form t !R c[t ]. We investigate which of the known termination preserving transformation
Remarks on isomorphisms of simple inductive types
On Lazy Commutation
The notion of a constricting sequence is developed, which can be applied to generic path orderings used in termination proofs and combinatorial commutation properties for reordering a sequence of two kinds of steps.
Termination of Associative-Commutative Rewriting by Dependency Pairs
It is shown how this criterion for termination of rewriting can be generalized to rewriting modulo associativity and commutativity, and how one can build weak AC-compatible reduction orderings which may be used in this criterion.
Isomorphisms of simple inductive types through extensional rewriting
  • D. Chemouil
  • Mathematics
    Mathematical Structures in Computer Science
  • 2005
The notion of a faithful copy of an inductive type and a corresponding conversion relation that also preserves the good properties of the calculus are defined.


Associative-Commutative Rewriting
Methods for proving termination of associative-commutative systems are described, incorporating a set of rules for Boolean algebra that provides a refutationally-complete theorem prover and a new programming paradigm.
Proofs by induction in equational theories with constructors
  • G. Huet, J. Hullot
  • Mathematics, Computer Science
    21st Annual Symposium on Foundations of Computer Science (sfcs 1980)
  • 1980
Computing with Rewrite Systems
Termination of Linear Rewriting Systems (Preliminary Version)
Limitations, such as right-linearity, on the form of rules in a term-rewriting system are shown to restrict the class of derivations that must be considered when determining whether or not the system
On Proving Uniform Termination and Restricted Termination of Rewriting Systems
A new method of proving uniform termination is proposed, which shows that no cycles can occur if the rewriting relation is globally finite and nontermination can occur only if there are cycles.
Conflunt reductions: Abstract properties and applications to term rewriting systems
  • G. Huet
  • Computer Science
    18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
  • 1977
This paper gives new results, and presents old ones in a unified formalism, concerning Church-Rosser theorems for rewriting systems, and shows how these results yield efficient methods for the mechanization of equational theories.
Termination of a Set of Rules Modulo a Set of Equations
It is shown here that termination of the rewriting relation and E-termination are the same whenever the used rewriting relation is E-commuting, a property inspired from Peterson and Stickel’s E-compatibility property.
Orderings for term-rewriting systems
  • N. Dershowitz
  • Computer Science, Mathematics
    20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
  • 1979