Automated Mathematical Induction

Bouhoula, Adel; Kounalis, Emmanuel; Rusinowitch, Michaël

doi:10.1093/logcom/5.5.631

Cited by 42 publications

(30 citation statements)

References 31 publications

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“…a rewrite system R, we will consider a simplified version of the Spike inference system [5,3], represented in Fig. 1, that implements this particular case of Noetherian induction in order to perform implicit induction proofs.…”

Section: Noetherian Induction For Implicit Induction Proofsmentioning

confidence: 99%

“…if e is subsumed by some conjecture from H ∪ L. presented in [3] and is different from the 'test set' method from [5]. Finally, the current conjecture is added to the set of premises.…”

Section: Noetherian Induction For Implicit Induction Proofsmentioning

confidence: 99%

“…We are interested in certifying properties about conditional specifications using automated proof techniques like those employed by Spike [5,19,3], a rewrite-based implicit induction proof system. The implementation of Spike's inference system is spread over thousands of lines of OCaml [12] code.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Automated Certification of Implicit Induction Proofs

Stratulat¹,

Demange²

2011

Certified Programs and Proofs

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Abstract. Theorem proving is crucial for the formal validation of properties about user specifications. With the help of the Coq proof assistant, we show how to certify properties about conditional specifications that are proved using automated proof techniques like those employed by the Spike prover, a rewrite-based implicit induction proof system. The certification methodology is based on a new representation of the implicit induction proofs for which the underlying induction principle is an instance of Noetherian induction governed by an induction ordering over equalities. We propose improvements of the certification process and show that the certification time is reasonable even for industrial-size applications. As a case study, we automatically prove and certify more than 40% of the lemmas needed for the validation of a conformance algorithm for the ABR protocol.

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Section: Noetherian Induction For Implicit Induction Proofsmentioning

confidence: 99%

“…if e is subsumed by some conjecture from H ∪ L. presented in [3] and is different from the 'test set' method from [5]. Finally, the current conjecture is added to the set of premises.…”

Section: Noetherian Induction For Implicit Induction Proofsmentioning

confidence: 99%

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Automated Certification of Implicit Induction Proofs

Stratulat¹,

Demange²

2011

Certified Programs and Proofs

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show abstract

“…This algorithm is used to synchronize a collaborative text object, shared by two or more users. There are two editing operations: Ins(p, c, pr) to insert a character c at position p and Del(p, pr) to delete a character at position p. Operations Ins and Del are extended with another parameter pr 1 . This one represents a priority scheme that is used to solve a conflict occurring when two concurrent insert operations were originally intended to insert different characters at the same position.…”

Section: Convergence Problemsmentioning

confidence: 99%

“…(iii) In the same way, op 1 is received on site 2 and must be transformed according to op 2 . T (Ins(1, x), Del (1)) is called and returns Ins(0, x). Condition C 1 is violated.…”

Section: Violation Of Cmentioning

confidence: 99%

Towards Synchronizing Linear Collaborative Objects with Operational Transformation

Imine

Molli

Oster

et al. 2005

Formal Techniques for Networked and Distributed Systems - FORTE 2005

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Abstract.A collaborative object represents a data type (such as a text document or a filesystem) designed to be shared by multiple geographically separated users. Data replication is a technology to improve performance and availability of data in distributed systems. Indeed, each user has a local copy of the shared objects, upon which he may perform updates. Locally executed updates are then transmitted to the other users. This replication potentially leads, however, to divergent (i.e. different) copies. In this respect, Operational Transformation (OT) algorithms are applied for achieving convergence of all copies, i.e. all users view the same objects. Using these algorithms users can apply the same set of updates but possibly in different orders since the convergence should be ensured in all cases. However, achieving convergence with the OT approach is still a critical and challenging issue. In this paper, we address an open convergence problem when the shared data has a linear structure such as list, text, ordered XML tree, etc. We analyze the source of this problem and we propose a generic solution with its formal correctness.

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Test Sets for the Universal and Existential Closure of Regular Tree Languages

Hofbauer¹,

Huber²

2002

Information and Computation

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Finite test sets are a useful tool for deciding the membership problem for the universal closure of a given tree language, that is, for deciding whether a term has all its ground instances in the given language. A uniform test set for the universal closure must serve the following purpose: In order to decide membership of a term, it is sufficient to check whether all its test set instances belong to the underlying language. A possible application, and our main motivation, is ground reducibility, an essential concept for many approaches to inductive reasoning. Ground reducibility modulo some rewrite system is membership in the universal closure of the set of reducible ground terms. Here, test sets always exist, and several algorithmic approaches are known. The resulting sets, however, are often unnecessarily large. In this paper we consider regular languages and linear closure operators. We prove that universal as well as existential closure, defined analogously, preserve regularity. By relating test sets to tree automata and to appropriate congruence relations, we show how to characterize, how to compute, and how to minimize ground and non-ground test sets. In particular, optimal solutions now replace previous ad hoc approximations for the ground reducibility problem.

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Automated Mathematical Induction

Cited by 42 publications

References 31 publications

Automated Certification of Implicit Induction Proofs

Automated Certification of Implicit Induction Proofs

Towards Synchronizing Linear Collaborative Objects with Operational Transformation

Test Sets for the Universal and Existential Closure of Regular Tree Languages

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