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202
FirstOrder Logic of Proofs
, 2011
"... The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capa ..."
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The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capable of realizing firstorder modal logic S4 and, therefore, the firstorder intuitionistic logic HPC. FOLP enjoys a natural provability interpretation; this provides a semantics of explicit proofs for firstorder S4 and HPC compliant with BrouwerHeytingKolmogorov requirements. FOLP opens the door to a general theory of firstorder justification.
A generalization of the linzhao theorem
 Annals of Mathematics and Artificial Intelligence
, 2006
"... The theorem on loop formulas due to Fangzhen Lin and Yuting Zhao shows how to turn a logic program into a propositional formula that describes the program’s stable models. In this paper we simplify and generalize the statement of this theorem. The simplification is achieved by modifying the definiti ..."
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Cited by 27 (8 self)
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The theorem on loop formulas due to Fangzhen Lin and Yuting Zhao shows how to turn a logic program into a propositional formula that describes the program’s stable models. In this paper we simplify and generalize the statement of this theorem. The simplification is achieved by modifying the definition of a loop in such a way that a program is turned into the corresponding propositional formula by adding loop formulas directly to the conjunction of its rules, without the intermediate step of forming the program’s completion. The generalization makes the idea of a loop formula applicable to stable models in the sense of a very general definition that covers disjunctive programs, programs with nested expressions, and more. 1
A systematic proof theory for several modal logics
 Advances in Modal Logic, volume 5 of King’s College Publications
, 2005
"... abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence betw ..."
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Cited by 26 (1 self)
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abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a HilbertLewis style and conditions on the accessibility relation on frames. By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is adhoc. While we can formulate several modal logics in the sequent calculus that enjoy cutelimination, their formalisation arises through systembysystem fine tuning to ensure that the cutelimination holds, and the correspondence to the axioms of the HilbertLewis systems becomes opaque. This paper introduces a systematic presentation for the systems K, D, M, S4, and S5 in the calculus of structures, a structural proof theory that employs deep inference. Because of this, we are able to axiomatise the modal logics in a manner directly analogous to the HilbertLewis axiomatisation. We show that the calculus possesses a cutelimination property directly analogous to cutelimination for the sequent calculus for these systems, and we discuss the extension to several other modal logics. 1
Orderenriched categorical models of the classical sequent calculus
 LECTURE AT INTERNATIONAL CENTRE FOR MATHEMATICAL SCIENCES, WORKSHOP ON PROOF THEORY AND ALGORITHMS
, 2003
"... It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contra ..."
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It is wellknown that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the wellknown categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cutreduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish nondeterministic choices of cutelimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.
A system of interaction and structure II: the need for deep inference
 Logical Methods in Computer Science
, 2006
"... Vol. 2 (2:4) 2006, pp. 1–24 ..."
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Comparing Control Constructs by Doublebarrelled CPS
 Higherorder and Symbolic Computation
, 2002
"... We investigate callbyvalue continuationpassing style transforms that pass two continuations. Altering a single variable in the translation of #abstraction gives rise to di#erent control operators: firstclass continuations; dynamic control; and (depending on a further choice of a variable) eithe ..."
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We investigate callbyvalue continuationpassing style transforms that pass two continuations. Altering a single variable in the translation of #abstraction gives rise to di#erent control operators: firstclass continuations; dynamic control; and (depending on a further choice of a variable) either the return statement of C; or Landin's Joperator. In each case there is an associated simple typing. For those constructs that allow upward continuations, the typing is classical, for the others it remains intuitionistic, giving a clean distinction independent of syntactic details. Moreover, those constructs that make the typing classical in the source of the CPS transform break the linearity of continuation use in the target.
The maximality of the typed lambda calculus and of cartesian closed categories
 Publ. Inst. Math. (N.S
"... From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here ..."
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From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson.
Fibring Labelled Deduction Systems
 Journal of Logic and Computation
, 2002
"... We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial ..."
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Cited by 16 (9 self)
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We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial characterization of fibring of models. Based on this, we establish the conditions under which our systems are sound and complete with respect to the general semantics for the corresponding logics, and establish requirements on logics and systems so that completeness is preserved by both forms of fibring.
A Deterministic Terminating Sequent Calculus for GödelDummett logic
, 1999
"... We give a short prooftheoretic treatment of a terminating contractionfree calculus G4LC for the zeroorder GödelDummett logic LC. This calculus is a slight variant of a calculus given by Avellone et al, who show its completeness by modeltheoretic techniques. In our calculus, all the rules of G4 ..."
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Cited by 16 (0 self)
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We give a short prooftheoretic treatment of a terminating contractionfree calculus G4LC for the zeroorder GödelDummett logic LC. This calculus is a slight variant of a calculus given by Avellone et al, who show its completeness by modeltheoretic techniques. In our calculus, all the rules of G4LC are invertible, thus allowing a deterministic proofsearch procedure.