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16
Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λCalculi
, 1992
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A typed foundation for directional logic programming
 In Proc. Workshop on Extensions to Logic Programming
, 1992
"... Abstract. A long standing problem in logic programming is how to impose directionality on programs in a safe fashion. The benefits of directionality include freedom from explicit sequential control, the ability to reason about algorithmic properties of programs (such as termination, complexity and d ..."
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Abstract. A long standing problem in logic programming is how to impose directionality on programs in a safe fashion. The benefits of directionality include freedom from explicit sequential control, the ability to reason about algorithmic properties of programs (such as termination, complexity and deadlockfreedom) and controlling concurrency. By using Girard’s linear logic, we are able to devise a type system that combines types and modes into a unified framework, and enables one to express directionality declaratively. The rich power of the type system allows outputs to be embedded in inputs and vice versa. Type checking guarantees that values have unique producers, but multiple consumers are still possible. From a theoretical point of view, this work provides a “logic programming interpretation ” of (the proofs of) linear logic, adding to the concurrency and functional programming interpretations that are already known. It also brings logic programming into the broader world of typed languages and typesaspropositions paradigm, enriching it with static scoping and higherorder features.
Turning Cycles into Spirals
, 1999
"... Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14 ..."
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Cited by 6 (3 self)
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Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14]. There is a link between the presence of cut formulas with nested quantifiers and the nonelementary expansion needed to prove a theorem without the help of such formulas. If one considers the graph defined by tracing the flow of occurrences of formulas (in the sense of [2]) for proofs allowing a nonelementary compression, one Preprint submitted to Elsevier Preprint 7 November 1997 finds that such graphs contain cycles [5] or almost cyclic structures[6]. These cycles codify in a small space (i.e. a proof with a small number of lines) all the information which is present in the proof once cuts on formulas wit
Adding modalities to MTL and its extensions
 Proceedings of the Linz Symposium
, 2005
"... Abstract. Monoidal tnorm logic MTL and related fuzzy logics are extended with various modalities distinguished by the axiom (A ∨ B) → (A ∨ B). Such modalities include Linear logiclike exponentials, the globalization (or Delta) operator, and truth stressers like “very true”. Extensions of MTL with ..."
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Abstract. Monoidal tnorm logic MTL and related fuzzy logics are extended with various modalities distinguished by the axiom (A ∨ B) → (A ∨ B). Such modalities include Linear logiclike exponentials, the globalization (or Delta) operator, and truth stressers like “very true”. Extensions of MTL with modalities are presented here via axiomatizations, hypersequent calculi, and algebraic semantics, and related to standard algebras based on tnorms. Embeddings of logics, decidability, and the finite embedding property are also investigated. 1
Uniqueness of Normal Proofs in Implicational Intuitionistic Logic
 Journal of Logic, Language and Information
, 1999
"... . A minimal theorem in a logic L is an Ltheorem which is not a nontrivial substitution instance of another Ltheorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has be ..."
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. A minimal theorem in a logic L is an Ltheorem which is not a nontrivial substitution instance of another Ltheorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has been known to be partially positive and generally negative. It is shown here that a minimal implicational theorem A in intuitionistic logic has a unique finormal proof in NJ whenever A is provable without nonprime contraction. The nonprime contraction rule in NJ is the implication introduction rule whose cancelled assumption differs from a propositional variable and appears more than once in the proof. Our result improves the known partial positive solutions to Komori's problem. Also, we present another simple example of a minimal implicational theorem in intuitionistic logic which does not have a unique fijnormal proof in NJ. Key words: natural deduction, uniqueness of normal proofs, coh...
A Clausal Approach to Proof Analysis in SecondOrder Logic
 In Symposium on Logical Foundations of Computer Science (LFCS 2009), Lecture Notes in Computer Science
, 2009
"... Abstract. This work defines an extension CERES 2 of the firstorder cutelimination method CERES to the subclass of sequent calculus proofs in secondorder logic using quantifierfree comprehension. This extension is motivated by the fact that cutelimination can be used as a tool to extract informa ..."
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Abstract. This work defines an extension CERES 2 of the firstorder cutelimination method CERES to the subclass of sequent calculus proofs in secondorder logic using quantifierfree comprehension. This extension is motivated by the fact that cutelimination can be used as a tool to extract information from real mathematical proofs, and often a crucial part of such proofs is the definition of sets by formulas. This is expressed by the comprehension axiom scheme, which is representable in secondorder logic. At the core of CERES 2 lies the production of a set of clauses CL(ϕ) from a proof ϕ that is always unsatisfiable. From a resolution refutation γ of CL(ϕ), a proof without essential cuts can be constructed. The main theoretical obstacle in the extension of CERES to secondorder logic is the construction of this proof from γ. This issue is solved for the subclass considered in this paper. Moreover, we discuss the problems that have to be solved to extend CERES 2 to the complete class of secondorder proofs. Finally, the method is applied to a simple mathematical proof that involves induction and comprehension and the resulting proof is analyzed. 1
Some Pitfalls of LKtoLJ Translations and How to Avoid Them
 Proc CADE14, LNCS 1249
, 1997
"... . In this paper, we investigate translations from a classical cutfree sequent calculus LK into an intuitionistic cutfree sequent calculus LJ. Translations known from the literature rest on permutations of inferences in classical proofs. We show that, for some classes of firstorder formulae, all m ..."
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. In this paper, we investigate translations from a classical cutfree sequent calculus LK into an intuitionistic cutfree sequent calculus LJ. Translations known from the literature rest on permutations of inferences in classical proofs. We show that, for some classes of firstorder formulae, all minimal LJproofs are nonelementary but there exist short LKproofs of the same formula. Similar results are obtained even if some fragments of intuitionistic firstorder logic are considered. The size of the corresponding minimal search spaces for LK and LJ are also nonelementarily related. We show that we can overcome these difficulties by extending LJ with an analytic cut rule. 1 Introduction Characterizing classes of formulae for which classical derivability implies intuitionistic derivability was one topic in the Leningrad group around Maslov in the sixties. Such classes are called (complete) Glivenko classes which were extensively characterized by Orevkov [7]. More recently, people ar...
Automation of HigherOrder Logic
 THE HANDBOOK OF THE HISTORY OF LOGIC, EDS. D. GABBAY & J. WOODS; VOLUME 9: LOGIC AND COMPUTATION, EDITOR JÖRG SIEKMANN
, 2014
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Defunctionalizing Focusing Proofs (Or, How Twelf Learned To Stop Worrying And Love The Ωrule)
"... Abstract. In previous work, the author gave a higherorder analysis of focusing proofs (in the sense of Andreoli’s search strategy), with a role for infinitary rules very similar in structure to Buchholz’s Ωrule. Among other benefits, this “patternbased ” description of focusing simplifies the cut ..."
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Abstract. In previous work, the author gave a higherorder analysis of focusing proofs (in the sense of Andreoli’s search strategy), with a role for infinitary rules very similar in structure to Buchholz’s Ωrule. Among other benefits, this “patternbased ” description of focusing simplifies the cutelimination procedure, allowing cuts to be eliminated in a connectivegeneric way. However, interpreted literally, it is problematic as a representation technique for proofs, because of the difficulty of inspecting and/or exhaustively searching over these infinite objects. In the spirit of infinitary proof theory, this paper explores a view of patternbased focusing proofs as façons de parler, describing how to compile them down to firstorder derivations through defunctionalization, Reynolds ’ program transformation. Our main result is a representation of patternbased focusing in the Twelf logical framework, whose core type theory is too weak to directly encode infinitary rules—although this weakness directly enables socalled “higherorder abstract syntax ” encodings. By applying the systematic defunctionalization transform, not only do we retain the benefits of the higherorder focusing analysis, but we can also take advantage of HOAS within Twelf, ultimately arriving at a proof representation with surprisingly little bureaucracy. 1