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Cutelimination and proofsearch for biintuitionistic logic using nested sequents
, 2008
"... We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant cal ..."
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Cited by 15 (4 self)
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We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cutelimination proof. We then present the derived calculus, and then present a proofsearch strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for biintuitionistic logic. As far as we know, our new calculus is the first sequent calculus for biintuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cutelimination proof, and which can be used naturally for backwards proofsearch.
Combining Derivations and Refutations for Cutfree Completeness in BiIntuitionistic Logic
, 2008
"... Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree se ..."
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Cited by 7 (0 self)
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Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cutfree ” sequent calculus has recently been shown to fail cutelimination. We present a new cutfree sequent calculus for biintuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. 1
Logical Foundations of Eval/Quote Mechanisms, and the Modal Logic S4
 IN PRESS S15708683(05)000431/FLA AID:71 Vol.•••(•••) [DTD5] P.12 (112) JAL:m1a v 1.40 Prn:15/07/2005; 8:08 jal71 by:SL p. 12 12 N. Alechina, D. Shkatov / Journal of Applied Logic
, 1997
"... Starting from the idea that cut elimination is the precise meaning of program execution, we design two languages of constructions for the minimal logic S4, yielding calculi with idealized versions of Lisp's eval and quote. The first, the S4 calculus, is based on Bierman and De Paiva's ..."
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Starting from the idea that cut elimination is the precise meaning of program execution, we design two languages of constructions for the minimal logic S4, yielding calculi with idealized versions of Lisp's eval and quote. The first, the S4 calculus, is based on Bierman and De Paiva's proposal, and has all desirable logical properties, except for its nonoperational flavor. The second, the evQcalculus, is more complicated, but has a clear operational meaning: it is a tower of interpreters in the style of Lisp's reflexive tower. Remarkably, this language was developed from purely logical principles, but nonetheless provides some operational insight into eval/quote mechanisms. 1 Introduction Let's consider two dual questions. The first is: is there a proofsasprograms, formulasas types correspondence for the modal logic S4? There is one between minimal and intuitionistic logics and  calculi [How80], and also for classical logic [Gri90] or linear logic [Abr93], so why not S4? A...
Bisimulation and Propositional Intuitionistic Logic
 PROCEEDINGS OF THE 8TH INTERNATIONAL CONFERENCE ON CONCURRENCY THEORY
, 1996
"... The BrouwerHeytingKolmogorov interpretation of intuitionistic logic suggests that p oe q can be interpreted as a computation that given a proof of p constructs a proof of q. Dually, we show that every finite canonical model of q contains a finite canonical model of p. If q and p are interderivabl ..."
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The BrouwerHeytingKolmogorov interpretation of intuitionistic logic suggests that p oe q can be interpreted as a computation that given a proof of p constructs a proof of q. Dually, we show that every finite canonical model of q contains a finite canonical model of p. If q and p are interderivable, their canonical models contain each other. Using this insight, we are able to characterize validity in a Kripke structure in terms of bisimilarity. Theorem 1 Let K be a finite Kripke structure for propositional intuitionistic logic, then two worlds in K are bisimilar if and only if they satisfy the same set of formulas. This theorem lifts to structures in the following manner. Theorem 2 Two finite Kripke structures K and K 0 are bisimilar if and only if they have the same set of valid formulas. We then generalize these results to a variety of infinite structures; finite principal filter structures and saturated structures.
Formal Proof: Reconciling Correctness and Understanding
, 2009
"... Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical th ..."
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Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no penandpaper proof can match, but an essential component of mathematics — the insight and understanding — seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding.
Combining Derivations and Refutations for Cutfree Completeness in Biintuitionistic Logic
"... Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ‘cutfree ’ sequent calculus has recently been shown to fail cutelimination. We present a new cutfree se ..."
Abstract
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Biintuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent ‘cutfree ’ sequent calculus has recently been shown to fail cutelimination. We present a new cutfree sequent calculus for biintuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose.
ALGORITHMIC INFORMATION THEORY Third Printing
"... Press as the rst volume in the series Cambridge ..."
Completeness vs. Incompleteness
"... Metamathematical results on formally undecidable propositions: ..."