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Schematization of infinite sets of rewrite rules. Application to the divergence of completion processes [chapter]

Hélène Kirchner
1987 Lecture Notes in Computer Science  
We determine here precise conditions that have to be checked in order to get a sound and complete schematization of the infinite set of rules.  ...  The practical interest of a completion procedure is limited by the fact that it can generate infinite sets of rewrite rules.  ...  But whenever it is possible to find a set of meta-rules that is a complete schematization for this infinite term rewriting system, a complete set of solutions for an equation in this theory can be found  ... 
doi:10.1007/3-540-17220-3_16 fatcat:qig6ekcs75bapacy5cjqsxepfe

Schematization of infinite sets of rewrite rules generated by divergent completion processes

Hélène Kirchner
1989 Theoretical Computer Science  
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.  ...  Infinite sets of rewrite rules may be generated for example by completion of termrewriting systems or by a narrowing process for solving equations in equational theories.  ...  Schmidt-Schauss for pointing out a problem with the associativity and idempotency axioms in a previous version of this paper.  ... 
doi:10.1016/0304-3975(89)90007-8 fatcat:qivrf2xlw5et3i57yw447ofi2u

Author index volume 67 (1989)

1989 Theoretical Computer Science  
Stefinekova, Nondeterministic multicounter machines and complementation (No/e) , H., Schematization of infinite sets of rewrite rules generated by divergent completion processes (2,3) 303-332 Klop, J.W  ...  Choppy Techniques and Applications (2,3) 141 Madlener, K. and F. Otto, About the descriptive power of certain classes of finite string-rewriting systems Marongiu, G. and S.  ... 
doi:10.1016/0304-3975(89)90008-x fatcat:cvlxybreoba4lhoj44kb7u5zgy

NARROWER: a new algorithm for unification and its application to Logic Programming [chapter]

Pierre Rety, Claude Kirchner, Hélène Kirchner, Pierre Lescanne
1985 Lecture Notes in Computer Science  
Kirchner: "Schematization of infinite sets of rewrite rules. Application to the divergence of completion processes", Proc.  ...  Kirchner: "Completion of a set of rules modulo a set of equations" Proc. 11th ACM Symposium on Principles of Programming Languages Salt Lake City (Utah, USA), 1984.  ... 
doi:10.1007/3-540-15976-2_7 fatcat:xaxsw22i4jh2dgzl7nascq4eei

A Proof of Weak Termination Providing the Right Way to Terminate [chapter]

Olivier Fissore, Isabelle Gnaedig, Hélène Kirchner
2005 Lecture Notes in Computer Science  
of subterms, and narrowing, schematizing rewriting steps.  ...  For that, we observe proof trees built from the rewrite system, schematizing the innermost rewriting tree of any ground term, and generated with two mechanisms: abstraction, schematizing normalization  ...  A formal description with a complete set of inference rules for describing the subtree cut process, and proofs of theorems are given in [11] .  ... 
doi:10.1007/978-3-540-31862-0_26 fatcat:xhv2g5ilvbamjgx6idnfj6nit4

System Presentation -- CARIBOO

Olivier Fissore, Isabelle Gnaedig, Hélène Kirchner
2002 Proceedings of the 4th ACM SIGPLAN international conference on Principles and practice of declarative programming - PPDP '02  
We describe Cariboo, the implementation of an inductive process recently proposed to prove termination of rewriting under strategies on ground term algebras.  ...  On many examples, Cariboo even enables to succeed without considering any constraint at all ; the process is then completely automatic. Cariboo offers a graphical view of the proof process.  ...  Moreover, the process can diverge if we have an infinite application of Abstract and Narrow.  ... 
doi:10.1145/571157.571164 dblp:conf/ppdp/FissoreGK02 fatcat:z2vfukg7yzc6zf3snm6ubcskaq

Outermost Ground Termination

Olivier Fissore, Isabelle Gnaedig, Hélène Kirchner
2004 Electronical Notes in Theoretical Computer Science  
We propose a semi-automatic inductive process to prove termination of outermost rewriting on ground term algebras.  ...  Evaluated in a lazy manner, inf (0) is reduced to cons(0, inf(s(0))), and then, since application of the first rule fails, the sub-expression inf (s(0)) has to be evaluated before considering the whole  ...  Moreover, the narrowing process itself had to be adapted to suitably schematize outermost rewriting on ground terms.  ... 
doi:10.1016/s1571-0661(05)82535-6 fatcat:s576puihdfbmpof5xgjtk2mi3u

A rule-based system for automatic decidability and combinability

E. Tushkanova, A. Giorgetti, C. Ringeissen, O. Kouchnarenko
2015 Science of Computer Programming  
To reason on the superposition calculus, a schematic superposition calculus has been developed to build the schematic form of the saturations allowing to automatically prove decidability of single theories  ...  This paper presents a rule-based logical framework and a tool implementing a complete many-sorted schematic superposition calculus for arbitrary theories.  ...  This process continues to generate deeper and deeper schematic clauses so that the calculus will diverge.  ... 
doi:10.1016/j.scico.2014.02.005 fatcat:zakpk5jbgvfjbcrc2ddcl73dwi

Induction for positive almost sure termination

Isabelle Gnaedig
2007 Proceedings of the 9th ACM SIGPLAN international conference on Principles and practice of declarative programming - PPDP '07  
We extend to the probabilistic case a technique we proposed for termination of usual rewriting under strategies.  ...  The proof is developed in generating proof trees, modelizing rewriting trees, in alternatively applying abstraction steps, expressing the application of the induction hypothesis, and narrowing steps, simulating  ...  Acknowledgments We would like to thank Olivier Bournez for fruitful discussions on termination of probabilistic rewriting and on this paper, and for giving us motivating examples.  ... 
doi:10.1145/1273920.1273943 dblp:conf/ppdp/Gnaedig07 fatcat:3rdtzrr5g5g4jpcu57475etdaq

Page 3189 of Mathematical Reviews Vol. , Issue 88f [page]

1988 Mathematical Reviews  
. 156-167); Zhenyu Qian, Structured con- textual rewriting (pp. 168-179); Héléne Kirchner, Schematization of infinite sets of rewrite rules.  ...  Application to the divergence of completion processes (pp. 180-191); Leo Bachmair and Nachum Dershowitz, Completion for rewriting modulo a congruence (pp. 192-203); Hans-Jirgen Birckert, Alexander Herold  ... 

Automatic decidability and combinability

Christopher Lynch, Silvio Ranise, Christophe Ringeissen, Duc-Khanh Tran
2011 Information and Computation  
While being able to cope only with quantifier-free formulae is not an obstacle for some verification applications, it may become a serious limitation in the verification of complex data structures.  ...  When proving the validity of a formula, general-purpose ATPs have only one way to consider the interpretations allowed by a background theory T : add as a premise to the formula a conjunction of the axioms  ...  Acknowledgments The authors would like to thank the referee for his comments and suggestions that helped to improve the clarity of the paper.  ... 
doi:10.1016/j.ic.2011.03.005 fatcat:43hij62ivrc3plqi2seqpzq73m

Deciding observational congruence of finite-state CCS expressions by rewriting

P. Inverardi, M. Nesi
1995 Theoretical Computer Science  
The rewriting relation is shown to be complete with respect to the axiomatization by proving that every structured fair derivation computes a term that denotes an rz-normal process graph.  ...  Starting from the complete axiomatization of observational congruence for this subset of CCS, a non-terminating rewriting relation has been defined.  ...  Acknowledgements We would like to thank the anonymous referees and Giovanni Mandorino for their very helpful comments on previous versions of this paper.  ... 
doi:10.1016/0304-3975(94)00074-s fatcat:aayhbmidq5dazkqs4rz3h5sfye

Termination and completion modulo associativity, commutativity and identity

Jean-Pierre Jouannaud, Claude Marché
1992 Theoretical Computer Science  
We build up on this idea in two ways: by giving a complete set of rules for completion module these axioms; by showing how to build appropriate orderings for proving termination of constrained rewriting  ...  In 1989, Baird, Peterson and Wilkerson introduced the notion of constrained rewriting, to avoid the problem of nontermination inherent to the use of identities.  ...  We thank Leo Bachmair who pointed out an error well hidden in a previous version of this work.  ... 
doi:10.1016/0304-3975(92)90165-c fatcat:icqheynbenbbhbayskklsaptde

Computing constructor forms with non terminating rewrite programs

Isabelle Gnaedig, Hélène Kirchner
2006 Proceedings of the 8th ACM SIGPLAN symposium on Principles and practice of declarative programming - PPDP '06  
In the context of the study of rule-based programming, we focus in this paper on the property of C-reducibility, expressing that every term reduces to a constructor term on at least one of its rewriting  ...  This property implies completeness of function definitions, and enables to stop evaluations of a program on a constructor form, even if the program is not terminating.  ...  The subtree cut process is similar to the one defined for weak termination, for which a formal description with a complete set of inference rules is given in [16] .  ... 
doi:10.1145/1140335.1140351 dblp:conf/ppdp/GnaedigK06 fatcat:pdgpkwdwubcxplpiqyjmih2jwa

Termination of rewriting strategies: a generic approach [article]

Isabelle Gnaedig, Helene Kirchner
2005 arXiv   pre-print
Abstraction constraints can be used to control the narrowing mechanism, well known to easily diverge. The generic method is then instantiated for the innermost, outermost and local strategies.  ...  The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof.  ...  for the interest he took in our approach, and for his advice on the manuscript of this paper.  ... 
arXiv:cs/0507064v1 fatcat:i2wno2wclrex7ohochjbqsd4lq
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