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16
Inductive Data Type Systems
 THEORETICAL COMPUTER SCIENCE
, 1997
"... In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, w ..."
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Cited by 44 (10 self)
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In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, which generalizes the usual recursor definitions for natural numbers and similar “basic inductive types”. This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called “strictly positive”, and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.
Proof Search in the Intuitionistic Sequent Calculus
 11th International Conference on Automated Deduction
, 1991
"... The use of Herbrand functions (more popularly known as Skolemization) plays an important role in classical theorem proving and logic programming. We define a notion of Herbrand functions for the full intuitionistic predicate calculus. The definition is based on the view that the prooftheoretic role ..."
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Cited by 42 (1 self)
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The use of Herbrand functions (more popularly known as Skolemization) plays an important role in classical theorem proving and logic programming. We define a notion of Herbrand functions for the full intuitionistic predicate calculus. The definition is based on the view that the prooftheoretic role of Herbrand functions (to replace universal quantifiers), and of unification (to find instances corresponding to existential quantifiers), is to ensure that the eigenvariable conditions on a sequent proof are respected. The propositional impermutabilities that arise in the intuitionistic but not the classical sequent calculus motivate a generalization of the classical notion of Herbrand functions. Proof search using generalized Herbrand functions also provides a framework for generalizing logic programming to subsets of intuitionistic logic that are larger than Horn clauses. The search procedure described here has been implemented and observed to work effectively in practice. The generaliza...
Types as graphs: Continuations in type logical grammar
, 2005
"... Using the programminglanguage concept of CONTINUATIONS, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of insitu quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as c ..."
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Cited by 11 (8 self)
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Using the programminglanguage concept of CONTINUATIONS, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of insitu quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, typelogical way to model evaluation order and side effects in natural language. We illustrate with an improved account of quantificational binding, weak crossover, whquestions, superiority, and polarity licensing.
Higman's Lemma in Type Theory
 PROCEEDINGS OF THE 1996 WORKSHOP ON TYPES FOR PROOFS AND PROGRAMS
, 1997
"... This thesis is about exploring the possibilities of a limited version of MartinLöf's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type t ..."
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Cited by 5 (0 self)
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This thesis is about exploring the possibilities of a limited version of MartinLöf's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type theory itself is studied and one in which it is used to prove Higman's lemma. In the first paper, A Lambda Calculus Model of MartinLöf's Theory of Types with Explicit Substitution, we present the formal calculus in complete detail. It consists of MartinLof's logical framework with explicit substitution extended with some inductively defined sets, also given in complete detail. These inductively defined sets are precisely those we need in the second paper of this thesis for the formal proof of Higman's lemma. The limitations of the formalism come from the fact that we do not introduce universes. It is known that for other versions of type theory, the absence of universes implies the impossib...
A Deconstruction of Nondeterministic Classical Cut Elimination
 TLCA'01, LNCS 2044, 268282
"... This paper shows how a symmetric and nondeterministic cut elimination procedure for a classical sequent calculus can be faithfully simulated using a nondeterministic choice operator to combine different `doublenegation' translations of each cut. The resulting interpretation of classical proof ..."
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Cited by 4 (0 self)
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This paper shows how a symmetric and nondeterministic cut elimination procedure for a classical sequent calculus can be faithfully simulated using a nondeterministic choice operator to combine different `doublenegation' translations of each cut. The resulting interpretation of classical proofs in a calculus with nondeterministic choice leads to a simple proof of termination for cut elimination.
Classical And Constructive Hierarchies In Extended Intuitionistic Analysis
"... This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(#) on Baire space with t ..."
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Cited by 4 (3 self)
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This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(#) on Baire space with the property that every constructive partial functional defined on {# : R(#)} is continuous there. The domains of continuity for coincide with the stable relations (those equivalent in to their double negations), while every relation R(#) is equivalent in #) for some stable A(#, #) (which belongs to the classical analytical hierarchy). The logic of is intuitionistic. The axioms of include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems.
Decidability Extracted: Synthesizing ``CorrectbyConstruction'' Decision Procedures from Constructive Proofs
, 1998
"... The topic of this thesis is the extraction of efficient and readable programs from formal constructive proofs of decidability. The proof methods employed to generate the efficient code are new and result in clean and readable Nuprl extracts for two nontrivial programs. They are based on the use of ..."
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Cited by 3 (0 self)
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The topic of this thesis is the extraction of efficient and readable programs from formal constructive proofs of decidability. The proof methods employed to generate the efficient code are new and result in clean and readable Nuprl extracts for two nontrivial programs. They are based on the use of Nuprl's set type and techniques for extracting efficient programs from induction principles. The constructive formal theories required to express the decidability theorems are of independent interest. They formally circumscribe the mathematical knowledge needed to understand the derived algorithms. The formal theories express concepts that are taught at the senior college level. The decidability proofs themselves, depending on this material, are of interest and are presented in some detail. The proof of decidability of classical propositional logic is relative to a semantics based on Kleene's strong threevalued logic. The constructive proof of intuitionistic decidability presented here is the first machine formalization of this proof. The exposition reveals aspects of the Nuprl tactic collection relevant to the creation of readable proofs; clear extracts and efficient code are illustrated in the discussion of the proofs.
A generalization of conservativity theorem for classical versus intuitionistic arithmetic
 Mathematical Logic Quarterly
, 2004
"... A basic result in Intuitionism is Π0 2Conservativity. Take any proof p in Classical Arithmetic of some Π0 2statement (some arithmetical statement ∀x.∃y.P (x, y), with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we genera ..."
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Cited by 1 (0 self)
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A basic result in Intuitionism is Π0 2Conservativity. Take any proof p in Classical Arithmetic of some Π0 2statement (some arithmetical statement ∀x.∃y.P (x, y), with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we generalized this result: any classical proof p of an arithmetical statement ∀x.∃y.P (x, y), with P of degree k, may be effectively turned into some proof of the same statement, using Excluded Middle only over degree k formulas. When k = 0, we get the original conservativity result as particular case. This result was a byproduct of a semantical construction. J. Avigad, of Carnegie Mellon University, found a short, direct syntactical derivation of the same result, using H. Friedman’s Atranslation. His proof is included here with his permission.
Interpolation via translations
"... A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties betwe ..."
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A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a KolmogorovGentzenGödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a KiriyamaOno style translation with features of a KolmogorovGentzenGödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics. AMS Classification: 03C40, 03F03, 03B22
New dimensions on translations between logics
"... Abstract. After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: conservative translations, transfers and contextual translations. Though independent, such approaches are here compared and assess ..."
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Abstract. After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: conservative translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another.