# Higher-order critical pairs

@article{Nipkow1991HigherorderCP, title={Higher-order critical pairs}, author={Tobias Nipkow}, journal={[1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science}, year={1991}, pages={342-349} }

A subclass of lambda -terms, called patterns, which have unification properties resembling those of first-order terms, is introduced. Higher-order rewrite systems are defined to be rewrite systems over lambda -terms whose left-hand sides are patterns: this guarantees that the rewrite relation is easily computable. The notion of critical pair is generalized to higher-order rewrite systems, and the analog of the critical pair lemma is proved. The restricted nature of patterns is instrumental in…

## 313 Citations

### Equivalence of Reductions in Higher-Order Rewriting

- Mathematics, Computer Science
- 2008

The main result of this dissertation is that for local, orthogonal HRSs the three notions of equivalence coincide, and the proof that H RSs enjoy finite family developments, meaning that in infinite reductions, there is no bound on the number of steps which were involved in creating a given symbol.

### Orthogonal Higher-Order Rewrite Systems are Confluent

- MathematicsTLCA
- 1993

The results about higher-order critical pairs and the confluence of OHRSs provide a firm foundation for the further study of higher-order rewrite systems. It should now be interesting to lift more…

### Confluence in Non-Left-Linear Untyped Higher-Order Rewrite Theories

- Computer SciencePPDP
- 2021

We develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of critical pairs for higher-order rewrite rules extending beta-reduction on pure…

### From Higher-Order to First-Order Rewriting ( Extended Abstract )

- Computer Science
- 2001

A characterization of the class of higher-order rewriting systems which can be encoded by first- order rewriting modulo an empty theory (that is, E = ∅) is obtained, which includes of course the λ-calculus.

### On Dependency Pair Method for Proving Termination of Higher-Order Rewrite Systems

- MathematicsIEICE Trans. Inf. Syst.
- 2005

A new notion of dependency forest is introduced that characterize infinite reductions and infinite R-chains, and it is shown that the termination property of higher-order rewrite systems R can be checked by showing the non-existence of an infiniteR-chain, if R is strongly linear or non-nested.

### Higher-Order Rewriting: Framework, Confluence and Termination

- Computer ScienceProcesses, Terms and Cycles
- 2005

Higher-order rewriting is considered, that is, rewriting higher-order functional expressions at higher-types, which is a useful generalization of first- order rewriting.

### From Higher-Order to First-Order Rewriting

- Computer ScienceRTA
- 2001

A characterization of the class of higher-order rewriting systems which can be encoded by first- order rewriting modulo an empty theory (that is, Ɛ = θ), which includes of course the λ-calculus.

### Polymorphic higher-order recursive path orderings

- MathematicsJACM
- 2007

A family of recursive path orderings for terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols is defined, which can be generated from two given well-founded orderings, on the function symbols and on the type constructors.

### Higher Order Conditional Rewriting and Narrowing

- Computer ScienceCCL
- 1994

This paper first shows that the satisfactory way to combine higher order terms and unconditional rewriting can be carried over to conditional higher order rewrite systems, and then studies narrowing using higher order rewriting systems.

### Unification of Higher-Order patterns in a Simply Typed Lambda-Calculus with Finite Products and terminal Type

- Computer Science, MathematicsRTA
- 1996

This work develops a higher-order unification algorithm for a restricted class of simply typed lambda terms with function space and product type constructors that terminates on each input and produces a most general unifier if one exists.

## References

SHOWING 1-10 OF 15 REFERENCES

### Combining algebra and higher-order types

- Computer Science, Mathematics[1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science
- 1988

The author shows that provability in the higher-order equational proof system obtained by adding the simply typed beta and eta axions to some many-sorted algebraic proof system is effectively reducible to Provability in that algebraicProof system.

### Conflunt reductions: Abstract properties and applications to term rewriting systems

- Computer Science18th 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.

### Rewrite Systems

- Computer ScienceHandbook of Theoretical Computer Science, Volume B: Formal Models and Sematics
- 1990

### Higher-Order Unification Revisited: Complete Sets of Transformations

- Computer ScienceJ. Symb. Comput.
- 1989

### A Logic Programming Language with Lambda-Abstraction, Function Variables, and Simple Unification

- Computer ScienceJ. Log. Comput.
- 1991

This paper presents a logic programming language, called Lλ, that also contains both function variables and λ-abstractions, but certain restriction are placed on occurrences of function variables, so that an implementation of Lλ does not need to implement full higher-order unification.

### Isabelle: The Next 700 Theorem Provers

- Computer ScienceArXiv
- 1993

A thorough history of Isabelle is given, beginning with its origins in the LCF system, and an account of how logics are represented is presented, illustrated using classical logic.

### A Formulation of the Simple Theory of Types

- MathematicsJ. Symb. Log.
- 1940

A formulation of the simple theory oftypes which incorporates certain features of the calculus of λ-conversion into the theory of types and is offered as being of interest on this basis.

### Simple Word Problems in Universal Algebras

- Mathematics
- 1983

An algorithm is described which is capable of solving certain word problems: i.e. of deciding whether or not two words composed of variables and operators can be proved equal as a consequence of a…

### Canonical Equational Proofs

- Computer Science, MathematicsProgress in Theoretical Computer Science
- 1991

Equational Proofs, a guide to proving consistency and ground reducibility in Equational Proofing, 1.1.