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15
Cutfree Display Calculi for Nominal Tense Logics
 Conference on Tableaux Calculi and Related Methods (TABLEAUX
, 1998
"... . We define cutfree display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Krac ..."
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. We define cutfree display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Kracht's results. The rules of the display calculus ffiMNTL for MNTL mimic those of the display calculus ffiMTL 6= for MTL 6= . Since ffiMNTL does not satisfy Belnap's condition (C8), we extend Wansing's strong normalisation theorem to get a similar theorem for any extension of ffiMNTL by addition of structural rules satisfying Belnap's conditions (C2)(C7). Finally, we show a weak Sahlqviststyle theorem for extensions of MNTL, and by Kracht's techniques, deduce that these Sahlqvist extensions of ffiMNTL also admit cutfree display calculi. 1 Introduction Background: The addition of names (also called nominals) to modal logics has been investigated recently with different motivations; see...
Light Affine Set Theory: A Naive Set Theory of Polynomial Time
, 2004
"... In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal ju ..."
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In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal justification of the claim is not provided sufficiently. Moreover, the syntax is quite complicated in that it is based on a nontraditional hybrid sequent calculus which is required for formulating LLL. In this paper, we consider a naive set theory based on Intuitionistic Light Affine Logic (ILAL), a simplification of LLL introduced by [1], and call it Light Affine Set Theory (LAST). The simplicity of LAST allows us to rigorously verify its polytime character. In particular, we prove that a function over {0, 1} ∗ is computable in polynomial time if and only if it is provably total in LAST.
Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems
 INFORMATION AND COMPUTATION
"... We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the calculus due ..."
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We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the calculus due to Bergstra and Klop, and others. To unify our results with those in the literature, we introduce Contextsensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal CCERSs. We then prove (1) perpetuality (in a specific context) of a redex whose contraction does not erase potentially infinite arguments, which are possibly finite (i.e., strongly normalizable) arguments that may become infinite after a number of outside steps, and (2) perpetuality (in every con...
A Solver for QBFs in Negation Normal Form
"... Various problems in artificial intelligence can be solved by translating them into a quantified boolean formula (QBF) and evaluating the resulting encoding. In this approach, a QBF solver is used as a black box in a rapid implementation of a more general reasoning system. Most of the current solvers ..."
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Various problems in artificial intelligence can be solved by translating them into a quantified boolean formula (QBF) and evaluating the resulting encoding. In this approach, a QBF solver is used as a black box in a rapid implementation of a more general reasoning system. Most of the current solvers for QBFs require formulas in prenex conjunctive normal form as input, which makes a further translation necessary, since the encodings are usually not in a specific normal form. This additional step increases the number of variables in the formula or disrupts the formula’s structure. Moreover, the most important part of this transformation, prenexing, is not deterministic. In this paper, we focus on an alternative way to process QBFs without these drawbacks and describe a solver, qpro, which is able to handle arbitrary formulas. To this end, we extend algorithms for QBFs to the nonnormal form case and compare qpro with the leading normal form provers on several problems from the area of artificial intelligence. We prove properties of the algorithms generalized to nonclausal form by using a novel approach based on a sequentstyle formulation of the calculus. 1.
Notes on the Simply Typed Lambda Calculus
, 1998
"... Contents 1 Deduction 11 1.1 Inference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.2 Adding extra axioms . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.3 Semantics for Inference System ..."
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Cited by 5 (0 self)
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Contents 1 Deduction 11 1.1 Inference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.1 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.2 Adding extra axioms . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.3 Semantics for Inference Systems . . . . . . . . . . . . . . . . 12 1.1.4 Formal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.5 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Intuitionistic Implication . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 A Hilbertstyle formal system, H . . . . . . . . . . . . . . . . 14 1.2.2 Natural Deduction . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.3 Sequent Formulation, ND, of Natural Deduction . . . . . . . 17 1.2.4 Normal ND treeproofs . . . . . . . . . . . . . . . . . . . . . 17 1.2.5 Sequent Calculus SC . . . . . . . . . . . .
A TERM ASSIGNMENT FOR DUAL INTUITIONISTIC LOGIC.
"... Abstract. We study the prooftheory of coHeyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a singleassumption multipleconclusions Natural Deduction system NJ � � for this logic: unlike the bestknown treatm ..."
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Abstract. We study the prooftheory of coHeyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a singleassumption multipleconclusions Natural Deduction system NJ � � for this logic: unlike the bestknown treatments of multipleconclusion systems (e.g., Parigot’s λ−µ calculus, or Urban and Bierman’s termcalculus) here the termassignment is distributed to all conclusions, and exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The present construction can be regarded as the construction of a free coCartesian
Revisiting cutelimination: One difficult proof is really a proof
 RTA 2008
, 2008
"... Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of termrewriting systems. The first author used such a logical relation argument to establish strong normalising for a cutelimination procedure in classical logic. ..."
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Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of termrewriting systems. The first author used such a logical relation argument to establish strong normalising for a cutelimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulties in this proof arise from a quite subtle substitution operation. We have formalised this proof in the theorem prover Isabelle/HOL using the Nominal Datatype Package, closely following the first authors PhD. In the process, we identified and resolved a gap in one central lemma and a number of smaller problems in others. We also needed to make one informal definition rigorous. We thus show that the original proof is indeed a proof and that present automated proving technology is adequate for formalising such difficult proofs.
Towards Hilbert's 24th Problem: Combinatorial Proof Invariants
, 2006
"... Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for ..."
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Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand’s theorem.
A Sequent Calculus for Signed Interval Logic
, 2001
"... We propose and discuss a complete sequent calculus formulation for Signed Interval Logic (SIL) with the chief purpose of improving proof support for SIL in practice. The main theoretical result is a simple characterization of the limit between decidability and undecidability of quantifierfree SIL. ..."
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We propose and discuss a complete sequent calculus formulation for Signed Interval Logic (SIL) with the chief purpose of improving proof support for SIL in practice. The main theoretical result is a simple characterization of the limit between decidability and undecidability of quantifierfree SIL. We present a mechanization of SIL in the generic proof assistant Isabelle and consider techniques for automated reasoning. Many of the results and ideas of this report are also applicable to traditional (nonsigned) interval logic and, hence, to Duration Calculus. 1
Cut Formulas for Kalmar Elementary Functions
, 2000
"... We show that for every Kalmar elementary function we can nd appropriate cutformulas and using them we nd proofs of length linear in the arguments of the function in predicate logic of the well denedness of the function. The usual proof of cut elimination shows the converse. This gives a new charact ..."
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We show that for every Kalmar elementary function we can nd appropriate cutformulas and using them we nd proofs of length linear in the arguments of the function in predicate logic of the well denedness of the function. The usual proof of cut elimination shows the converse. This gives a new characterization of the Kalmar elementary functions as the functions which we can feasibly prove to be well dened in predicate logic using cut. 1 Main result We consider classical predicate logic with equality. The language consists of a unary predicate N , the constant 0, the unary successor function s and possibly a number of function symbols. As axioms we have the elementary successor axioms: 0 : N 8x : N : sx : N and a number of axioms dening the function symbols not involving the predicate N like: +0y = y +sxy = s + xy ?0y = 0 ?sxy = +y ? xy Email: herman.jervell@ilf.uio.no. Phone: +4722856650. Fax: +4722856919. y Email: wenhui.zhang@hrp.no. These equational axioms are r...