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132
A new approach to abstract syntax with variable binding
 Formal Aspects of Computing
, 2002
"... Abstract. The permutation model of set theory with atoms (FMsets), devised by Fraenkel and Mostowski in the 1930s, supports notions of ‘nameabstraction ’ and ‘fresh name ’ that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variablebinding op ..."
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Cited by 286 (64 self)
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Abstract. The permutation model of set theory with atoms (FMsets), devised by Fraenkel and Mostowski in the 1930s, supports notions of ‘nameabstraction ’ and ‘fresh name ’ that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variablebinding operations. Inductively defined FMsets involving the nameabstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntaxmanipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science. 1.
A Linear Logical Framework
, 1996
"... We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. ..."
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Cited by 238 (49 self)
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We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of MiniML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cutelimination. 1 Introduction A logical framework is a formal system desig...
Nominal Logic: A First Order Theory of Names and Binding
 Information and Computation
, 2001
"... This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal L ..."
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Cited by 218 (15 self)
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This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of firstorder manysorted logic with equality containing primitives for renaming via nameswapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...
A Judgmental Reconstruction of Modal Logic
 Mathematical Structures in Computer Science
, 1999
"... this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deductio ..."
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Cited by 193 (47 self)
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this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic [FM97] and find that it is already contained in modal logic, using the decomposition of the lax modality fl A as
A New Approach to Abstract Syntax Involving Binders
 In 14th Annual Symposium on Logic in Computer Science
, 1999
"... Syntax Involving Binders Murdoch Gabbay Cambridge University DPMMS Cambridge CB2 1SB, UK M.J.Gabbay@cantab.com Andrew Pitts Cambridge University Computer Laboratory Cambridge CB2 3QG, UK ap@cl.cam.ac.uk Abstract The FraenkelMostowski permutation model of set theory with atoms (FMsets) ..."
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Cited by 176 (20 self)
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Syntax Involving Binders Murdoch Gabbay Cambridge University DPMMS Cambridge CB2 1SB, UK M.J.Gabbay@cantab.com Andrew Pitts Cambridge University Computer Laboratory Cambridge CB2 3QG, UK ap@cl.cam.ac.uk Abstract The FraenkelMostowski permutation model of set theory with atoms (FMsets) can serve as the semantic basis of metalogics for specifying and reasoning about formal systems involving name binding, ffconversion, capture avoiding substitution, and so on. We show that in FMset theory one can express statements quantifying over `fresh' names and we use this to give a novel settheoretic interpretation of name abstraction. Inductively defined FMsets involving this nameabstraction set former (together with cartesian product and disjoint union) can correctly encode objectlevel syntax modulo ffconversion. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated n...
Reasoning with higherorder abstract syntax in a logical framework
, 2008
"... Logical frameworks based on intuitionistic or linear logics with highertype quantification have been successfully used to give highlevel, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natu ..."
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Cited by 103 (26 self)
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Logical frameworks based on intuitionistic or linear logics with highertype quantification have been successfully used to give highlevel, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natural to consider proving properties about the specified systems in the framework: for example, given the specification of evaluation for a functional programming language, prove that the language is deterministic or that evaluation preserves types. One challenge in developing a framework for such reasoning is that higherorder abstract syntax (HOAS), an elegant and declarative treatment of objectlevel abstraction and substitution, is difficult to treat in proofs involving induction. In this paper, we present a metalogic that can be used to reason about judgments coded using HOAS; this metalogic is an extension of a simple intuitionistic logic that admits higherorder quantification over simply typed λterms (key ingredients for HOAS) as well as induction and a notion of definition. The latter concept of definition is a prooftheoretic device that allows certain theories to be treated as “closed ” or as defining fixed points. We explore the difficulties of formal metatheoretic analysis of HOAS encodings by considering encodings of intuitionistic and linear logics, and formally derive the admissibility of cut for important subsets of these logics. We then propose an approach to avoid the apparent tradeoff between the benefits of higherorder abstract syntax and the ability to analyze the resulting encodings. We illustrate this approach through examples involving the simple functional and imperative programming languages PCF and PCF:=. We formally derive such properties as unicity of typing, subject reduction, determinacy of evaluation, and the equivalence of transition semantics and natural semantics presentations of evaluation.
A syntactic approach to foundational proofcarrying code
 In Seventeenth IEEE Symposium on Logic in Computer Science
, 2002
"... ProofCarrying Code (PCC) is a general framework for verifying the safety properties of machinelanguage programs. PCC proofs are usually written in a logic extended with languagespecific typing rules. In Foundational ProofCarrying Code (FPCC), on the other hand, proofs are constructed and verifie ..."
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Cited by 96 (19 self)
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ProofCarrying Code (PCC) is a general framework for verifying the safety properties of machinelanguage programs. PCC proofs are usually written in a logic extended with languagespecific typing rules. In Foundational ProofCarrying Code (FPCC), on the other hand, proofs are constructed and verified using strictly the foundations of mathematical logic, with no typespecific axioms. FPCC is more flexible and secure because it is not tied to any particular type system and it has a smaller trusted base. Foundational proofs, however, are much harder to construct. Previous efforts on FPCC all required building sophisticated semantic models for types. In this paper, we present a syntactic approach to FPCC that avoids the difficulties of previous work. Under our new scheme, the foundational proof for a typed machine program simply consists of the typing derivation plus the formalized syntactic soundness proof for the underlying type system. We give a translation from a typed assembly language into FPCC and demonstrate the advantages of our new system via an implementation in the Coq proof assistant. 1.
Automating the Meta Theory of Deductive Systems
, 2000
"... not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a metalogical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, a ..."
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Cited by 88 (16 self)
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not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a metalogical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, and experimental results related to the areas of programming languages, type theory, and logics. Design: The metalogical framework extends the logical framework LF [HHP93] by a metalogic M + 2. This design is novel and unique since it allows higherorder encodings of deductive systems and induction principles to coexist. On the one hand, higherorder representation techniques lead to concise and direct encodings of programming languages and logic calculi. Inductive de nitions on the other hand allow the formalization of properties about deductive systems, such as the proof that an operational semantics preserves types or the proof that a logic is is a proof calculus whose proof terms are recursive functions that may be consistent.M +
Flexible Type Analysis
 In 1999 ACM International Conference on Functional Programming
, 1999
"... Runtime type dispatch enables a variety of advanced optimization techniques for polymorphic languages, including tagfree garbage collection, unboxed function arguments, and flattened data structures. However, modern typepreserving compilers transform types between stages of compilation, making ty ..."
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Cited by 83 (22 self)
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Runtime type dispatch enables a variety of advanced optimization techniques for polymorphic languages, including tagfree garbage collection, unboxed function arguments, and flattened data structures. However, modern typepreserving compilers transform types between stages of compilation, making type dispatch prohibitively complex at low levels of typed compilation. It is crucial therefore for type analysis at these low levels to refer to the types of previous stages. Unfortunately, no current intermediate language supports this facility. To fill this gap, we present the language LX, which provides a rich language of type constructors supporting type analysis (possibly of previousstage types) as a programming idiom. This language is quite flexible, supporting a variety of other applications such as analysis of quantified types, analysis with incomplete type information, and type classes. We also show that LX is compatible with a typeerasure semantics. 1 Introduction Typedirected co...