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The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
Programming Metalogics with a Fixpoint Type
, 1992
"... A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category th ..."
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Cited by 12 (6 self)
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A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category theory and treats recursion in a new way. The notion of a category with fixpoint object is defined. Corresponding to this categorical structure there are type theoretic equational rules which will be present in all of the metalogics considered. These rules define the fixpoint type which will allow the interpretation of recursive declarations. With these core notions FIX categories are defined. These are the categorical equivalent of an equational logic which can be viewed as a very basic programming metalogic. Recursion is treated both syntactically and categorically. The expressive power of the equational logic is increased by embedding it in an intuitionistic predicate calculus, giving rise to the FIX logic. This contains propositions about the evaluation of computations to values and an induction principle which is derived from the definition of a fixpoint object as an initial algebra. The categorical structure which accompanies the FIX logic is defined, called a FIX hyperdoctrine, and certain existence and disjunction properties of FIX are stated. A particular FIX hyperdoctrine is constructed and used in the proof of the same properties. PCFstyle languages are translated into the FIX logic and computational adequacy reaulta are proved. Two languages are studied: Both are similar to PCF except one has call by value recursive function declararations and the other higher order conditionals. ...
Hierarchies of Decidable Extensions of Bounded Quantification
 IN 22ND ACM SYMP. ON PRINCIPLES OF PROGRAMMING LANGUAGES
, 1994
"... The system F , the wellknown secondorder polymorphic typed calculus with subtyping and bounded universal type quantification [CW85, BL90, CG92, Pie92, CMMS94], appears to be undecidable [Pie92] because of undecidability of its subtyping component. Attempts were made to obtain decidable type sys ..."
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Cited by 7 (5 self)
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The system F , the wellknown secondorder polymorphic typed calculus with subtyping and bounded universal type quantification [CW85, BL90, CG92, Pie92, CMMS94], appears to be undecidable [Pie92] because of undecidability of its subtyping component. Attempts were made to obtain decidable type systems with subtyping by weakening F [CP94, KS92], and also by reinforcing or extending it [Vor94a, Vor94b, Vor95]. However, for the moment, these extensions lack the important prooftheoretic minimum type property, which holds for F and guarantees that each typable term has the minimum type, being a subtype of any other type of the term in the same context [CG92, Vor94c]. As a preparation step to introducing the extensions of F with the minimum type property and the decidable term typing relation (which we do in [Vor94e]), we define and study here the hierarchies of decidable extensions of the F subtyping relation. We demonstrate conditions providing that each theory in a hierarchy: 1. ext...
Herbrand Methods in Sequent Calculi: Unification in LL
 Proc. of the Joint International Conference and Symposium on Logic Programming
, 1992
"... We propose a reformulation of quantifiers rules in sequent calculi which allows to replace blind existential instantiation with unification, thereby reducing nondeterminism and complexity in proofsearch. Our method, based on some ideas underlying the proof of Herbrand theorem for classical logic, m ..."
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We propose a reformulation of quantifiers rules in sequent calculi which allows to replace blind existential instantiation with unification, thereby reducing nondeterminism and complexity in proofsearch. Our method, based on some ideas underlying the proof of Herbrand theorem for classical logic, may be applied to any "reasonable" nonclassical sequent calculus, but here we focus on sequent calculus for linear logic, in view of an application to linear logic programming. We prove that the new linear proofsystem which we propose, the so called system LLH, is equivalent to standard linear sequent calculus LL. 1 Introduction A result in classical logic which has been widely exploited in logic programming is Herbrand theorem. Several versions of this result are present in the literature; we recall here one of them (see [13]). Herbrand Theorem Let F be a prenex formula of the form 9w8x9y8zA[w; x; y; z] with A quantifierfree. F is provable in predicate calculus if and only if a disjun...
A Linear Logic Approach to Consistency Preserving Updates
, 1996
"... The aim of this paper is to propose linear logic as a proof system allowing to perform updates of databases containing incomplete information. In our approach, a database is specified by facts, deduction rules (among which default rules) and update constraints. Updates will always preserve consisten ..."
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The aim of this paper is to propose linear logic as a proof system allowing to perform updates of databases containing incomplete information. In our approach, a database is specified by facts, deduction rules (among which default rules) and update constraints. Updates will always preserve consistency, i.e. any update of a "consistent" database will produce a new base which is "consistent". The calculus of the "static semantics" of a database DB corresponds to the construction of a proof in a logical theory Th(DB) associated to the database. Similarly, the calculus of the "update semantics" of a database DB w.r.t. the insertion of a literal L, is the construction of a proof in Th(DB). Key words : Deductive Databases, Incomplete Information, Linear Logic, Updates. R'esum'e Ce papier propose la logique lin'eaire comme un syst`eme de preuve permettant de mettre `a jour des bases de donn'ees qui contiennent des informations incompl`etes. Selon notre approche, une base de donn'ees est spe...
Inductive, projective, and retractive types
, 1993
"... We give an analysis of classes of recursive types by presenting two extensions of the simplytyped lambda calculus. The first language only allows recursive types with builtin principles of wellfounded induction, while the second allows more general recursive types which permit nonterminating com ..."
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We give an analysis of classes of recursive types by presenting two extensions of the simplytyped lambda calculus. The first language only allows recursive types with builtin principles of wellfounded induction, while the second allows more general recursive types which permit nonterminating computations. We discuss the expressive power of the languages, examine the properties of reductionbased operational semantics for them, and give examples of their use in expressing iteration over large ordinals and in simulating both callbyname and callbyvalue versions of the untyped lambda calculus. The motivations for this work come from category theoretic models. 1
aspert i&s. unibo. it
"... We analyze the inherent complexity of implementing L&y’s notion of optimal evaluation for the &calculus, where similar redexes are contracted in one step via socalled parallel /%rekction. optimal evaluation IQSS finally realized by Lamping, who introduced a beautiful graph reduction technol ..."
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We analyze the inherent complexity of implementing L&y’s notion of optimal evaluation for the &calculus, where similar redexes are contracted in one step via socalled parallel /%rekction. optimal evaluation IQSS finally realized by Lamping, who introduced a beautiful graph reduction technology for sharing evaluation. contexts dual to the sharing of values. His pioneering insights have been modified and improved in subsequent implementations of optimal reduction. We prove that the cost of parallel Preduction is not bounded by any ICalm&elementary recursive function. Not merely do we establish that the parallel