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Completion for rewriting modulo a congruence
[chapter]

1987
*
Lecture Notes in Computer Science
*

We present

doi:10.1007/3-540-17220-3_17
fatcat:spyymf3dknewxnswap7kk2tjjq
*completion*methods*for**rewriting**modulo**a**congruence*, generalizing previous methods by Peterson and Stickel (1981) and Jouannaud and Kirchner (1986) . ... We formalize our methods as equational inference systems and describe techniques*for*reasoning about such systems. ... Various methods have been proposed*for**rewriting**modulo**a**congruence*. ...##
###
Completion for rewriting modulo a congruence

1989
*
Theoretical Computer Science
*

*Completion*

*modulo*

*a*

*congruence*is

*a*method

*for*constructing

*a*presentation of an equational theory as

*a*

*rewrite*system that defines unique normal forms with respect to the

*congruence*. ... We formulate this

*completion*method as an equational inference system and present techniques

*for*proving the correctness of procedures based on the inference system. ...

*Completion*

*for*

*rewriting*

*modulo*

*a*

*congruence*was first shown to be practically feasible by Peterson and Stickel [24] ,

*for*associative-commutative

*congruences*. ...

##
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Automated Theorem Proving in First-Order Logic Modulo: On the Difference between Type Theory and Set Theory
[chapter]

2000
*
Lecture Notes in Computer Science
*

Resolution

doi:10.1007/3-540-46508-1_1
fatcat:pddj5d3slzbcfbor7f63unxyki
*modulo*is*a*first-order theorem proving method that can be applied both to first-order presentations of simple type theory (also called higher-order logic) and to set theory. ... Indeed, the reduction process is deterministic because the*rewrite*system is confluent. Cut elimination and*completeness*Resolution*modulo*is not*complete**for*all*congruences*. ...*Completeness*Another difference is that, as type theory verifies the cut elimination property, resolution*modulo*this*congruence*is*complete*, while it is incomplete*modulo*the*congruence*of set theory. ...##
###
Order-Sorted Rewriting and Congruence Closure
[chapter]

2016
*
Lecture Notes in Computer Science
*

The satisfiability problems

doi:10.1007/978-3-662-49630-5_29
fatcat:nyojilw2ijdh3ngamfnuzdt3oa
*for*the theories of: (i) order-sorted uninterpreted function symbols, and (ii) of such symbols*modulo**a*subset Δ of associative-commutative ones are reduced to the unsorted ... New results on order-sorted*rewriting*are needed to achieve this reduction. ... I thank Maria Paola Bonacina*for*suggested improvements. ...##
###
Deduction modulo theory
[article]

2015
*
arXiv
*
pre-print

This paper is

arXiv:1501.06523v1
fatcat:6g6jl4gthzahdmg55eegcg54yi
*a*survey on Deduction*modulo*theory ...*A**congruence*≡ is said to be valid in*a*model when*A*≡ B implies*A*φ = B φ*for*all valuations φ, and*a*soundness and*completeness*theorem can be proved using standard methods. ... is performed*modulo*this*congruence*. ...##
###
Semantic A-translations and Super-Consistency Entail Classical Cut Elimination
[chapter]

2013
*
Lecture Notes in Computer Science
*

Deduction

doi:10.1007/978-3-642-45221-5_28
fatcat:nm25kwa6cncejoy6l7szv76siu
*Modulo*is*a*formalism that aims at separating computation from reasoning in proofs by making inferences*modulo*some*congruence*. ... We show that if*a*theory R defined by*a**rewrite*system is super-consistent, the classical sequent calculus*modulo*R enjoys the cut elimination property, which was an open question. ...*A**rewrite*system R (*a**congruence*≡) in deduction*modulo*is super-consistent if it has aB-valued model*for*all full, ordered and*complete*pseudo-Heyting algebraB. ...##
###
Semantic A-translation and Super-consistency entail Classical Cut Elimination
[article]

2014
*
arXiv
*
pre-print

We show that if

arXiv:1401.0998v1
fatcat:npcwlc3fbvflpa7hgc7oghy4ly
*a*theory R defined by*a**rewrite*system is super-consistent, the classical sequent calculus*modulo*R enjoys the cut elimination property, which was an open question. ...*For*such theories it was already known that proofs strongly normalize in natural deduction*modulo*R, and that cut elimination holds in the intuitionistic sequent calculus*modulo*R. ...*A**rewrite*system R (*a**congruence*≡) in deduction*modulo*is super-consistent if it has aB-valued model*for*all full, ordered and*complete*pseudo-Heyting algebraB. ...##
###
Schematization of infinite sets of rewrite rules generated by divergent completion processes

1989
*
Theoretical Computer Science
*

Infinite sets of

doi:10.1016/0304-3975(89)90007-8
fatcat:qivrf2xlw5et3i57yw447ofi2u
*rewrite*rules may be generated*for*example by*completion*of termrewriting systems or by*a*narrowing process*for*solving equations in equational theories. ... This is*a*severe limitation to the practical use of these processes. We propose in this paper*a*notion of schematization*for*an infinite set of*rewrite*rules. ... Remy and to the referees*for*their valuable comments. I am grateful to M. Schmidt-Schauss*for*pointing out*a*problem with the associativity and idempotency axioms in*a*previous version of this paper. ...##
###
Proof Search and Proof Check for Equational and Inductive Theorems
[chapter]

2003
*
Lecture Notes in Computer Science
*

Different concepts, especially

doi:10.1007/978-3-540-45085-6_26
fatcat:jy44cmtpb5e4llgyvpu4yv2khe
*rewriting*calculus and deduction*modulo*, contribute to define and to relate proof search, proof representation and proof check. ... In practice, we instantiate the theoretical study on the Coq proof assistant and the ELAN*rewriting*based system, focusing first on equational and then on inductive proofs. ... Many thanks to the Protheo team*for*stimulating discussions on many subjects developed in this paper, and in particular to Jürgen Stuber. ...##
###
Narrowing Based Inductive Proof Search
[chapter]

2013
*
Lecture Notes in Computer Science
*

We present in this paper

doi:10.1007/978-3-642-37651-1_9
fatcat:vb57hiivgjdlhfhhfklaw2fixy
*a*narrowing-based proof search method*for*inductive theorems. ... It has the specificity to be grounded on deduction*modulo*and to yield*a*direct translation from*a*successful proof search derivation to*a*proof in the sequent calculus. ... The reader can refer to [7] and to [8]*for**a*detailed exposition. In deduction*modulo*, terms but also propositions can be identified*modulo**a**congruence*. ...##
###
Deduction versus Computation: The Case of Induction
[chapter]

2002
*
Lecture Notes in Computer Science
*

We extend slightly the original version of the deduction

doi:10.1007/3-540-45470-5_3
fatcat:4bjblblahbhkrhbmrrmxx2ivsu
*modulo*framework and we provide modularity properties*for*it. ... We show how this applies to*a*uniform understanding of the so called induction by*rewriting*method and how this relates directly to the general use of an induction principle. ... It is thus not surprising to have our framework based on deduction*modulo*. This presentation of first-order logic relies on the sequent calculus*modulo**a**congruence*defined on terms and propositions. ...##
###
Experimenting with Deduction Modulo
[chapter]

2011
*
Lecture Notes in Computer Science
*

*For*the integration of

*rewriting*, we also compare several implementation techniques, based

*for*instance on discrimination trees or on compilation. ... Deduction

*modulo*is

*a*generic framework to describe proofs in

*a*theory better than using raw axioms. This is done by presenting the theory through rules

*rewriting*terms and propositions. ... It consists in presenting

*a*theory as

*a*

*congruence*over propositions, and in applying the inference rules of deductive systems

*modulo*this

*congruence*. ...

##
###
Page 6630 of Mathematical Reviews Vol. , Issue 88m
[page]

1988
*
Mathematical Reviews
*

process of

*rewriting**modulo*R, the word problem*for*({*a*, b}; Ro) can be solved in time O(n?) ... Jantzen [“Thue*congruences*and*complete*string-*rewriting*systems”, Habilitationsschrift, Univ. Hamburg, Hamburg, 1986; per bibl.; see also Inform. Process. ...##
###
Cut Admissibility by Saturation
[chapter]

2014
*
Lecture Notes in Computer Science
*

Deduction

doi:10.1007/978-3-319-08918-8_9
fatcat:iac7kwgrqzecdojd2rdjeumjqy
*modulo*is*a*framework in which theories are integrated into proof systems such as natural deduction or sequent calculus by presenting them using*rewriting*rules. ... This work relies on*a*view of proposition*rewriting*rules as oriented clauses, like term*rewriting*rules can be seen as oriented equations. ... In Deduction*Modulo*1 ,*a*theory is represented by*a**congruence*over formulae, and proofs are searched*for**modulo*this*congruence*. ...##
###
The problems of cyclic equality and conjugacy for finite complete rewriting systems

1986
*
Theoretical Computer Science
*

Thus,

doi:10.1016/0304-3975(86)90131-3
fatcat:hnim3jomwzbmtpvx5pz6geni6a
*for**a*finite*rewriting*system, while the property of being*complete*is sufficient to guarantee the decidability of the word problem, it is not sufficient*for*the decidability of the conjugacy problem ... This does not hold*for*the problem of cyclic equality as is shown here by presenting*a*finite, length-reducing, and*complete**rewriting*system with an undecidable problem of cyclic equality. ... CYCLIC Cyclic equality is undecidable*for*finite length-reducing*complete**rewriting*systems Here we are going to present*a*finite length-reducing*complete**rewriting*system T1 with the property that the ...
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