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An Effective Theory of Type Refinements
, 2002
"... We develop an explicit two level system that allows programmers to reason about the behavior of effectful programs. The first level is an ordinary MLstyle type system, which confers standard properties on program behavior. The second level is a conservative extension of the first that uses a logic ..."
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Cited by 62 (5 self)
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We develop an explicit two level system that allows programmers to reason about the behavior of effectful programs. The first level is an ordinary MLstyle type system, which confers standard properties on program behavior. The second level is a conservative extension of the first that uses a logic of type refinements to check more precise properties of program behavior. Our logic is a fragment of intuitionistic linear logic, which gives programmers the ability to reason locally about changes of program state. We provide a generic resource semantics for our logic as well as a sound, decidable, syntactic refinementchecking system. We also prove that refinements give rise to an optimization principle for programs. Finally, we illustrate the power of our system through a number of examples.
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
Ordered Linear Logic and Applications
, 2001
"... This work is dedicated to my parents. Acknowledgments Firstly, and foremost, I would like to thank my principal advisor, Frank Pfenning, for his patience with me, and for teaching me most of what I know about logic and type theory. I would also like to acknowledge some useful discussions with Kevin ..."
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Cited by 36 (0 self)
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This work is dedicated to my parents. Acknowledgments Firstly, and foremost, I would like to thank my principal advisor, Frank Pfenning, for his patience with me, and for teaching me most of what I know about logic and type theory. I would also like to acknowledge some useful discussions with Kevin Watkins which led me to simplify some of this work. Finally, I would like to thank my other advisor, John Reynolds, for all his kindness and support over the last five years. Abstract This thesis introduces a new logical system, ordered linear logic, which combines reasoning with unrestricted, linear, and ordered hypotheses. The logic conservatively extends (intuitionistic) linear logic, which contains both unrestricted and linear hypotheses, with a notion of ordered hypotheses. Ordered hypotheses must be used exactly once, subject to the order in which they were assumed (i.e., their order cannot be changed during the course of a derivation). This ordering constraint allows for logical representations of simple data structures such as stacks and queues. We construct ordered linear logic in the style of MartinL"of from the basic notion of a hypothetical judgement. We then show normalization for the system by constructing a sequent calculus presentation and proving cutelimination of the sequent system.
Relating Natural Deduction and Sequent Calculus for Intuitionistic NonCommutative Linear Logic
, 1999
"... We present a sequent calculus for intuitionistic noncommutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cutfree derivations, and arbitrary natura ..."
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Cited by 27 (14 self)
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We present a sequent calculus for intuitionistic noncommutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cutfree derivations, and arbitrary natural deductions to sequent derivations with cut. This gives us a syntactic proof of normalization for a rich system of noncommutative natural deduction and its associated calculus. INCLL conservatively extends linear logic with means to express sequencing, which has applications in functional programming, logical frameworks, logic programming, and natural language parsing. 1 Introduction Linear logic [11] has been described as a logic of state because it views linear hypotheses as resources which may be consumed in the course of a deduction. It thereby significantly extends the expressive power of both classical and intuitionistic logics, yet it does not offer means to express sequencing. Th...
Hybridizing a logical framework
 In International Workshop on Hybrid Logic 2006 (HyLo 2006), Electronic Notes in Computer Science
, 2006
"... The logical framework LF is a constructive type theory of dependent functions that can elegantly encode many other logical systems. Prior work has studied the benefits of extending it to the linear logical framework LLF, for the incorporation linear logic features into the type theory affords good r ..."
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Cited by 20 (1 self)
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The logical framework LF is a constructive type theory of dependent functions that can elegantly encode many other logical systems. Prior work has studied the benefits of extending it to the linear logical framework LLF, for the incorporation linear logic features into the type theory affords good representations of state change. We describe and argue for the usefulness of an extension of LF by features inspired by hybrid logic, which has several benefits. For one, it shows how linear logic features can be decomposed into primitive operations manipulating abstract resource labels. More importantly, it makes it possible to realize a metalogical framework capable of reasoning about stateful deductive systems encoded in the style familiar from prior work with LLF, taking advantage of familiar methodologies used for metatheoretic reasoning in LF.Acknowledgments From the very first computer science course I took at CMU, Frank Pfenning has been an exceptional teacher and mentor. For his patience, breadth of knowledge, and mathematical good taste I am extremely thankful. No less do I owe to the other two major contributors to my programming languages
ConnectionBased Proof Construction in Linear Logic
 14 th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 1249
, 1997
"... Abstract. We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrixbased proof search procedure for this fragment and a procedure which translates the machinefound proofs back into the usual sequent calculus for linea ..."
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Cited by 13 (7 self)
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Abstract. We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrixbased proof search procedure for this fragment and a procedure which translates the machinefound proofs back into the usual sequent calculus for linear logic. Both procedures are straightforward extensions of methods which originally were developed for a uniform treatment of classical, intuitionistic and modal logics. They can be extended to further fragments of linear logic once a matrix characterization has been found. 1
Linear Logic Programming with an Ordered Context
 2nd International Conference on Principles and Practice of Declarative Programming (PPDP 2000
, 2000
"... We begin with a review of ordered linear logic (OLL), a refinement of intuitionistic linear logic with an inherent notion of order. We then develop a logic programming interpretation for OLL in two steps: (1) we give a system of ordered uniform derivations which is sound and complete with respect to ..."
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Cited by 12 (0 self)
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We begin with a review of ordered linear logic (OLL), a refinement of intuitionistic linear logic with an inherent notion of order. We then develop a logic programming interpretation for OLL in two steps: (1) we give a system of ordered uniform derivations which is sound and complete with respect to OLL, and (2) we present a model of resource consumption which removes nondeterminism from ordered resource allocation during search for uniform derivations. We also illustrate the expressive power of the resulting ordered linear logic programming language with several example programs.
Extension of WAM for a linear logic programming language
, 1996
"... This paper describes an extension of the WAM (Warren Abstract Machine) for a logic programming language called LLP which is based on intuitionisic linear logic. LLP is a subset of Lolli and includes additive and multiplicative conjunction, linear implication in a goal, exponential (!), and the const ..."
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Cited by 11 (6 self)
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This paper describes an extension of the WAM (Warren Abstract Machine) for a logic programming language called LLP which is based on intuitionisic linear logic. LLP is a subset of Lolli and includes additive and multiplicative conjunction, linear implication in a goal, exponential (!), and the constant 1. The extension of the WAM is mainly for efficient resource management: especially for resource lookup and deletion. In our design, only one table is maintained to keep resources during the execution. Lookingup of a resource is done through a hash table. Deletion of a resource is done by just "marking" the entry in the table. Our prototype compiler produces 25 times faster code compared with a compiled Prolog program which represents resources by a list structure.
ProofTheoretic Foundation of Compilation in Logic Programming Languages
 Proceedings of the Joint International Conference and Symposium on Logic Programming (JICSLPâ€™98
"... Commercial implementations of logic programming languages are engineered Appeared in the Proceedings of the 1998 Joint International Conference and Symposium on Logic Programming  JICSLP'98 (J. Jaffar editor), pp ????, MIT Press, Manchester, UK, 1619 June 1998. around a compiler based on War ..."
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Cited by 11 (1 self)
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Commercial implementations of logic programming languages are engineered Appeared in the Proceedings of the 1998 Joint International Conference and Symposium on Logic Programming  JICSLP'98 (J. Jaffar editor), pp ????, MIT Press, Manchester, UK, 1619 June 1998. around a compiler based on Warren's Abstract Machine (WAM) or a variant of it. In spite of various correctness proofs, the logical machinery relating the prooftheoretic specification of a logic programming language and its compiled form is still poorly understood. In this paper, we propose a logicindependent definition of compilation for logic programming languages. We apply this methodology to derive the first cut of a compiler and the corresponding abstract machine for the language of hereditary Harrop formulas and then for its linear refinement. 1 Introduction Compiled logic programs run over an order of magnitude faster than their interpreted source and constitute therefore a key step to combining the advantages of...
Forum as a Logic Programming Language: Preliminary Report
 Proceedings of the Linear Logic '96 Meeting
, 1996
"... When Miller introduced Forum he called it a specification logic, rather than a logic programming language. In this paper we outline those features that create problems in attempting to implement an interpreter for the language, and describe solutions to those problems. We show how techniques used in ..."
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Cited by 10 (1 self)
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When Miller introduced Forum he called it a specification logic, rather than a logic programming language. In this paper we outline those features that create problems in attempting to implement an interpreter for the language, and describe solutions to those problems. We show how techniques used in the implementation of Lolli can be extended naturally to Forum. Finally, we show two Forum programs in order to demonstrate some of the paradigms that arise in using the language. 1 Introduction Forum, a fragment of Linear Logic introduced by Dale Miller in 1994, is distinguished by two key features. First, it is complete for all of Linear Logic, in the sense that Linear Logic operators that are not part of Forum can be mapped to Forum by a provabilitypreserving translation. Second, a form of goaldirected proof search (as characterized by uniform proofs) is complete. Historically, logics for which uniform proofs are complete, such as Horn Clauses, Hereditary Harrop Formulas, and Linear H...