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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
Expansion nets: proofnets for propositional classical logic
 IN PROCEEDINGS OF THE 17TH INTERNATIONAL CONFERENCE ON LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE, AND REASONING, LPAR’10
, 2010
"... We give a calculus of proofnets for classical propositional logic. These nets improve on a proposal due to Robinson by validating the associativity and commutativity of contraction, and provide canonical representants for classical sequent proofs modulo natural equivalences. We present the relation ..."
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Cited by 4 (1 self)
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We give a calculus of proofnets for classical propositional logic. These nets improve on a proposal due to Robinson by validating the associativity and commutativity of contraction, and provide canonical representants for classical sequent proofs modulo natural equivalences. We present the relationship between sequent proofs and proofnets as an annotated sequent calculus, deriving formulae decorated with expansion/deletion trees. We then see a subcalculus, expansion nets, which in addition to these good properties has a polynomialtime correctness criterion.
Proof nets for Herbrand’s Theorem
"... This paper explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence ” of a sequent calculus proof. We we see how to view a calculus of abstract Herbrand proofs (“Herbrand nets”) as an analytic proof system with syntactic cute ..."
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Cited by 3 (1 self)
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This paper explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence ” of a sequent calculus proof. We we see how to view a calculus of abstract Herbrand proofs (“Herbrand nets”) as an analytic proof system with syntactic cutelimination. Herbrand nets can also be seen as a natural generalization of Miller’s expansion tree proofs to a setting including cut. We demonstrate sequentialization of Herbrand nets into a sequent calculus LKH; each net corresponds to an equivalence class of LKH proofs under natural proof transformations. A surprising property of our cutreduction algorithm is that it is nonconfluent, despite not supporting the usual examples of nonconfluent reduction in classical logic.
The Frame Problem and the Semantics of Classical Proofs
"... We outline the logic of current approaches to the socalled “frame problem ” (that is, the problem of predicting change in the physical world by using logical inference), and we show that these approaches are not completely extensional since none of them is closed under uniform substitution. The und ..."
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We outline the logic of current approaches to the socalled “frame problem ” (that is, the problem of predicting change in the physical world by using logical inference), and we show that these approaches are not completely extensional since none of them is closed under uniform substitution. The underlying difficulty is something known, in the philosophical community, as Goodman’s “new riddle of induction ” or the “Grue paradox”. Although it seems, from the philosophical discussion, that this paradox cannot be solved in purely a priori terms and that a solution will require some form of realworld data, it nevertheless remains obscure both what the logical form of this realworld data might be, and also how such data actually interacts with logical deduction. We show, using work of McCain and Turner, that this data can be captured using the semantics of classical proofs developed by Bellin, Hyland and Robinson, and, consequently, that the appropriate arena for solutions of the frame problem lies in proof theory. We also give a very explicit model for the categorical semantics of classical proof theory using techniques derived from work on the frame problem.
Canonical proof nets for classical logic
"... Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proofnet, they are in essence the same proof. Providing a convincing proofnet counterpart to proofs in the classical sequent calculus is thus an ..."
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Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proofnet, they are in essence the same proof. Providing a convincing proofnet counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cutelimination procedure which preserves correctness. Previous attempts to give proofnetlike objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK ∗ in this paper, is a novel onesided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a selfcontained extended version of [23]) , we give a full proof of (c) for expansion nets with respect to LK ∗, and in addition give a cutelimination procedure internal to expansion nets – this makes expansion nets the first notion of proofnet for classical logic satisfying all four criteria. 1
Categories and Subject Descriptors: F4.1 [Mathematical logic and formal languages]: Mathematical
"... This paper explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence ” of a sequent calculus proof. We see how to view a calculus of abstract Herbrand proofs (“Herbrand nets”) as an analytic proof system with syntactic cutelim ..."
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This paper explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence ” of a sequent calculus proof. We see how to view a calculus of abstract Herbrand proofs (“Herbrand nets”) as an analytic proof system with syntactic cutelimination. Herbrand nets can also be seen as a natural generalization of Miller’s expansion tree proofs to a setting including cut. We demonstrate sequentialization of Herbrand nets into a sequent calculus LKH; each net corresponds to an equivalence class of LKH proofs under natural proof transformations. A surprising property of our cutreduction algorithm is that it is nonconfluent, despite not supporting the usual examples of nonconfluent reduction in classical logic.
concurrency and
"... IOS Press On the πcalculus and cointuitionistic logic. Notes on logic for ..."
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IOS Press On the πcalculus and cointuitionistic logic. Notes on logic for
An Institutional View on Categorical Logic
"... We introduce a generic notion of categorical propositional logic and provide a construction of a preorderenriched institution out of such a logic, following the CurryHowardTait paradigm. The logics are specified as theories of a metalogic within the logical framework LF such that institution com ..."
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We introduce a generic notion of categorical propositional logic and provide a construction of a preorderenriched institution out of such a logic, following the CurryHowardTait paradigm. The logics are specified as theories of a metalogic within the logical framework LF such that institution comorphisms are obtained from theory morphisms of the metalogic. We prove several logicindependent results including soundness and completeness theorems and instantiate our framework with a number of examples: classical, intuitionistic, linear and modal propositional logic. We dedicate this work to the memory of our dear friend and colleague Joseph Goguen who passed away during its preparation. 1
HIGHER DIMENSIONAL ALGEBRAS VIA COLORED PROPS
, 809
"... Abstract. Starting from any unital colored PROP P, we define a category P(P) of shapes called Ppropertopes. Presheaves on P(P) are called Ppropertopic sets. For 0 ≤ n ≤ ∞ we define and study ntime categorified Palgebras as Ppropertopic sets with some lifting properties. Taking appropriate PROPs ..."
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Abstract. Starting from any unital colored PROP P, we define a category P(P) of shapes called Ppropertopes. Presheaves on P(P) are called Ppropertopic sets. For 0 ≤ n ≤ ∞ we define and study ntime categorified Palgebras as Ppropertopic sets with some lifting properties. Taking appropriate PROPs P, we obtain higher categorical versions of polycategories, 2fold monoidal categories, topological quantum field theories, and so on.