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87
Indexed InductionRecursion
, 2001
"... We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in ..."
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Cited by 44 (16 self)
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We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductiverecursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.
Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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Cited by 38 (11 self)
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
On Bunched Predicate Logic
 Proceedings of the IEEE Symposium on Logic in Computer Science
, 1999
"... We present the logic of bunched implications, BI, in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication, and may be viewe ..."
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Cited by 29 (17 self)
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We present the logic of bunched implications, BI, in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication, and may be viewed as a merging of intuitionistic logic and multiplicative, intuitionistic linear logic. The predicate version of BI includes, in addition to usual additive quantifiers, multiplicative (or intensional) quantifiers 8new and 9new , which arise from observing restrictions on structural rules on the level of terms as well as propositions. Moreover, these restrictions naturally allow the distinction between additive predication and multiplicative predication for each propositional connective. We provide a natural deduction system, a sequent calculus, a Kripke semantics and a BHK semantics for BI. We mention computational interpretations, based on locality and sharing, at both the propositiona...
A probabilistic language based upon sampling functions
 In Conference Record of the 32nd Annual ACM Symposium on Principles of Programming Languages
, 2005
"... As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages which treat probability distributions as primitive datatypes. Most probabilistic languages, however, focus only on discrete distributions and have limited expressive p ..."
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Cited by 26 (1 self)
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As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages which treat probability distributions as primitive datatypes. Most probabilistic languages, however, focus only on discrete distributions and have limited expressive power. This paper presents a probabilistic language, called λ○, whose expressive power is beyond discrete distributions. Rich expressiveness of λ ○ is due to its use of sampling functions, i.e., mappings from the unit interval (0.0, 1.0] to probability domains, in specifying probability distributions. As such, λ ○ enables programmers to formally express and reason about sampling methods developed in simulation theory. The use of λ ○ is demonstrated with three applications in robotics: robot localization, people tracking, and robotic mapping. All experiments have been carried out with real robots.
An Authorization Logic with Explicit Time
, 2008
"... We present an authorization logic that permits reasoning with explicit time. Following a prooftheoretic approach, we study the metatheory of the logic, including cut elimination. We also demonstrate formal connections to proofcarrying authorization’s existing approach for handling time and commen ..."
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Cited by 26 (8 self)
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We present an authorization logic that permits reasoning with explicit time. Following a prooftheoretic approach, we study the metatheory of the logic, including cut elimination. We also demonstrate formal connections to proofcarrying authorization’s existing approach for handling time and comment on the enforceability of our logic in the same framework. Finally, we illustrate the expressiveness of the logic through examples, including those with complex interactions between time, authorization, and mutable state.
Extensions and Applications of Higherorder Unification
, 1990
"... ... unification problems. Then, in this framework, we develop a new unification algorithm for acalculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (objectlanguages). The ric ..."
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Cited by 25 (1 self)
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... unification problems. Then, in this framework, we develop a new unification algorithm for acalculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (objectlanguages). The rich structure of a typedcalculus,asopposedtotraditional,rst generalideaistouseacalculusasametalanguageforrepresentingvariousotherlanguages thelattercase,thealgorithmisincomplete,thoughstillquiteusefulinpractice. Thelastpartofthedissertationprovidesexamplesoftheusefulnessofthealgorithms.The algorithmrstfordependentproduct()types,andsecondforimplicitpolymorphism.In involvessignicantcomplicationsnotarisingHuet'scorrespondingalgorithmforthesimply orderabstractsyntaxtrees,allowsustoexpressrules,e.g.,programtransformationand typedcalculus,primarilybecauseitmustdealwithilltypedterms.Wethenextendthis Wecanthenuseunicationinthemetalanguagetomechanizeapplicationoftheserules.
Deliverables: A Categorical Approach to Program Development in Type Theory
, 1992
"... This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's ..."
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Cited by 24 (1 self)
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This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higherorder primitive recursion and higherorder intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigmatypes of the calculus enable us to achieve this. There are many similarities with the subset interpretation of MartinLof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...
A Relevant Analysis of Natural Deduction
 Journal of Logic and Computation
, 1999
"... Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and ..."
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Cited by 23 (7 self)
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Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lLcalculus; the representation mechanism is judgementsastypes, developed for relevant logics. The lLcalculus type theory is a firstorder dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required prooftheoretic metatheory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I calculus and ML with references. We describe the CurryHowardde Bruijn correspondence of the lLcalculus with a s...