A variant of Herbrand's theorem for formulas in prenex form

Lyaletskii, A. V.

doi:10.1007/bf01307049

Cited by 4 publications

(5 citation statements)

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“…To improve Kanger's Fachievements_, another notion of admissibility was suggested in [2] which later gave rise to appearance of computer-oriented sequent calculi having higher efficiency of logical inference search than K (see, for example,…”

Section: General Remarksmentioning

confidence: 99%

“…The introduced notion of admissible substitutions is closely connected with the notion of admissibility used in [2,3].…”

Section: New Notion Of Admissibilitymentioning

confidence: 99%

“…The nearest purpose of our consideration is to Fincorporate_ the notion of admissibility of [2] into the calculus GL in such a way that can give a possibility for the construction of its modification mGL leading to sequent forms of the Herbrand theorem.…”

Section: General Remarksmentioning

confidence: 99%

“…For performing the transformation, let us use the convolution algorithm from [2] for the construction of a word w in the same way that was used in the proof of Proposition 2. As a result, the following word w can be generated: 1 a 1 1 x 2 1 b 2 1 x 4 1 b 4 1 x 3 1 b 3 1 x 1 1 b 1 1 x 1 a 2 1 a 3 1 u 1 1 a 1 u 2 1 y 2 1 u 3 1 y 3 1 z 3 1 u 4 1 y 4 1 z 4 giving the order of quantifier rules applications that determine an order of propositional rules applications, which leads to the construction of a proof tree in GL.…”

Section: Proof Tree In Glmentioning

confidence: 99%

“…The form A does not need the skolemization of F, and it only uses constants, functional symbols, and certain quantifier variables from F. But its formulation requires checking a large number of quantifier sequences constructed from Fschemes_ (in the terminology of [1]) in order to find at least one sequence satisfying a certain condition that guarantees the validity of F. To optimize the item-by-item examination of all the possible sequences, in [2] there was proposed the original notion of admissible substitutions that permits to detect existence of such a sequence by means of analyzing both a selected substitution of terms for variables and the quantifier structure of F. Note that in [2], a Herbrand theorem form was given for formulas in prefix forms. Later, it was generalized to the case of arbitrary formulas [3].…”

Section: Introductionmentioning

confidence: 99%

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Sequent forms of Herbrand theorem and their applications

Lyaletski

2006

Ann Math Artif Intell

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New sequent forms * of the famous Herbrand theorem are proved for first-order classical logic without equality. These forms use the original notion of an admissible substitution and a certain modification of the Herbrand universe, which is constructed from constants, special variables, and functional symbols occurring only in the signature of an initial theory. Other well-known forms of the Herbrand theorem are obtained as special cases of the sequent ones. Besides, the sequent forms give an approach to the construction and theoretical investigation of computer-oriented calculi for efficient logical inference search in the signature of an initial theory. In a comparably simple way, they provide us with some technique for proving the completeness and soundness of the calculi.

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Section: General Remarksmentioning

confidence: 99%

“…The introduced notion of admissible substitutions is closely connected with the notion of admissibility used in [2,3].…”

Section: New Notion Of Admissibilitymentioning

confidence: 99%

Section: General Remarksmentioning

confidence: 99%

Section: Proof Tree In Glmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sequent forms of Herbrand theorem and their applications

Lyaletski

2006

Ann Math Artif Intell

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Evidence Algorithm and System for Automated Deduction: A Retrospective View

Lyaletski

Verchinine

2010

Lecture Notes in Computer Science

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A research project aimed at the development of an automated theorem proving system was started in Kiev (Ukraine) in early 1960s. The mastermind of the project, Academician V.Glushkov, baptized it "Evidence Algorithm", EA 3 . The work on the project lasted, off and on, more than 40 years. In the framework of the project, the Russian and English versions of the System for Automated Deduction, SAD, were constructed. They may be already seen as powerful theoremproving assistants. The paper 4 gives a retrospective view to the whole history of the development of the EA and SAD. Theoretical and practical results obtained on the long way are systematized. No comparison with similar projects is made.

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Evidence Algorithm and Inference Search in First-Order Logics

Lyaletski

2015

J Autom Reasoning

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A variant of Herbrand's theorem for formulas in prenex form

Cited by 4 publications

References 2 publications

Sequent forms of Herbrand theorem and their applications

Sequent forms of Herbrand theorem and their applications

Evidence Algorithm and System for Automated Deduction: A Retrospective View

Evidence Algorithm and Inference Search in First-Order Logics

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