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Nominal logic programming
, 2006
"... Nominal logic is an extension of firstorder logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, αequivalence). This article investigates logic programming based on nominal logic. This technique is especial ..."
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Cited by 23 (8 self)
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Nominal logic is an extension of firstorder logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, αequivalence). This article investigates logic programming based on nominal logic. This technique is especially wellsuited for prototyping type systems, proof theories, operational semantics rules, and other formal systems in which bound names are present. In many cases, nominal logic programs are essentially literal translations of “paper” specifications. As such, nominal logic programming provides an executable specification language for prototyping, communicating, and experimenting with formal systems. We describe some typical nominal logic programs, and develop the modeltheoretic, prooftheoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via two examples.
Metaprogramming through typeful code representation
 In Proceedings of the Eighth ACM SIGPLAN International Conference on Functional Programming
, 2003
"... By allowing the programmer to write code that can generate code at runtime, metaprogramming offers a powerful approach to program construction. For instance, metaprogramming can often be employed to enhance program efficiency and facilitate the construction of generic programs. However, metaprog ..."
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Cited by 22 (4 self)
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By allowing the programmer to write code that can generate code at runtime, metaprogramming offers a powerful approach to program construction. For instance, metaprogramming can often be employed to enhance program efficiency and facilitate the construction of generic programs. However, metaprogramming, especially in an untyped setting, is notoriously errorprone. In this paper, we aim at making metaprogramming less errorprone by providing a type system to facilitate the construction of correct metaprograms. We first introduce some code constructors for constructing typeful code representation in which program variables are replaced with deBruijn indices, and then formally demonstrate how such typeful code representation can be used to support metaprogramming. The main contribution of the paper lies in recognition and then formalization of a novel approach to typed metaprogramming that is practical, general and flexible.
A Sequent Calculus for Nominal Logic
 In Proc. LICS’04
, 2004
"... Nominal logic is a theory of names and binding based on the primitive concepts of freshness and swapping, with a selfdual N  (or "new")quantifier, originally presented as a Hilbertstyle axiom system extending firstorder logic. We present a sequent calculus for nominal logic called Fresh Logic, ..."
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Cited by 22 (10 self)
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Nominal logic is a theory of names and binding based on the primitive concepts of freshness and swapping, with a selfdual N  (or "new")quantifier, originally presented as a Hilbertstyle axiom system extending firstorder logic. We present a sequent calculus for nominal logic called Fresh Logic, or FL, admitting cutelimination. We use FL to provide a prooftheoretic foundation for nominal logic programming and show how to interpret $FOL^{\Delta abla}$, another logic with a selfdual quantifier, within FL.
Focusing on binding and computation
 In IEEE Symposium on Logic in Computer Science
, 2008
"... Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the CurryHoward interpretation of a focused ..."
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Cited by 21 (6 self)
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Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the CurryHoward interpretation of a focused sequent calculus with two kinds of implication, of opposite polarity. The representational arrow extends systems of definitional reflection with a notion of scoped inference rules, which are used to represent binding. On the other hand, the usual computational arrow classifies recursive functions defined by patternmatching. Unlike many previous approaches, both kinds of implication are connectives in a single logic, which serves as a rich logical framework capable of representing inference rules that mix binding and computation. 1
The Complexity of Equivariant Unification
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004), volume 3142 of LNCS
"... Nominal logic is a firstorder theory of names and binding based on a primitive operation of swapping rather than substitution. Urban, Pitts, and Gabbay have developed a nominal unification algorithm that unifies terms up to nominal equality. However, because of nominal logic's equivariance principl ..."
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Cited by 21 (7 self)
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Nominal logic is a firstorder theory of names and binding based on a primitive operation of swapping rather than substitution. Urban, Pitts, and Gabbay have developed a nominal unification algorithm that unifies terms up to nominal equality. However, because of nominal logic's equivariance principle, atomic formulas can be provably equivalent without being provably equal as terms, so resolution using nominal unification is sound but incomplete. For complete resolution, a more general form of unification called equivariant unification, or "unification up to a permutation" is required. Similarly, for rewrite rules expressed in nominal logic, a more general form of matching called equivariant matching is necessary. In this paper, we study the complexity of the decision problem for equivariant unification and matching. We show that these problems are NPcomplete in general. However, when one of the terms is essentially firstorder, equivariant and nominal unification coincide. This shows that equivariant unification can be performed efficiently in many interesting common cases: for example, anypurely firstorder logic program or rewrite system can be run efficiently on nominal terms.
Spatial Logics for Bigraphs
 In Proceedings of ICALP’05, volume 3580 of LNCS
, 2005
"... Abstract. Bigraphs are emerging as an interesting model for concurrent calculi, like CCS, picalculus, and Petri nets. Bigraphs are built orthogonally on two structures: a hierarchical place graph for locations and a link (hyper)graph for connections. With the aim of describing bigraphical structur ..."
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Cited by 21 (2 self)
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Abstract. Bigraphs are emerging as an interesting model for concurrent calculi, like CCS, picalculus, and Petri nets. Bigraphs are built orthogonally on two structures: a hierarchical place graph for locations and a link (hyper)graph for connections. With the aim of describing bigraphical structures, we introduce a general framework for logics whose terms represent arrows in monoidal categories. We then instantiate the framework to bigraphical structures and obtain a logic that is a natural composition of a place graph logic and a link graph logic. We explore the concepts of separation and sharing in these logics and we prove that they generalise some known spatial logics for trees, graphs and tree contexts. 1
A Simpler Proof Theory for Nominal Logic
 In FOSSACS 2005, number 3441 in LNCS
, 2005
"... Nominal logic is a variant of firstorder logic which provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the newquantifier, which quantifies over fresh names (names not appearing in any values considered so far). Previous attempts have been made ..."
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Cited by 20 (10 self)
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Nominal logic is a variant of firstorder logic which provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the newquantifier, which quantifies over fresh names (names not appearing in any values considered so far). Previous attempts have been made to develop convenient rules for reasoning with the newquantifier, but we argue that none of these attempts is completely satisfactory. In this paper we develop a new sequent calculus for nominal logic in which the rules for the newquantifier are much simpler than in previous attempts. We also prove several structural and metatheoretic properties, including cutelimination, consistency, and conservativity with respect to Pitts' axiomatization of nominal logic; these proofs are considerably simpler for our system. 1
Barendregt’s variable convention in rule inductions
 In Proc. of the 21th International Conference on Automated Deduction (CADE), volume 4603 of LNAI
, 2007
"... Abstract. Inductive definitions and rule inductions are two fundamental reasoning tools in logic and computer science. When inductive definitions involve binders, then Barendregt's variable convention is nearly always employed (explicitly or implicitly) in order to obtain simple proofs. Using this c ..."
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Cited by 20 (8 self)
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Abstract. Inductive definitions and rule inductions are two fundamental reasoning tools in logic and computer science. When inductive definitions involve binders, then Barendregt's variable convention is nearly always employed (explicitly or implicitly) in order to obtain simple proofs. Using this convention, one does not consider truly arbitrary bound names, as required by the rule induction principle, but rather bound names about which various freshness assumptions are made. Unfortunately, neither Barendregt nor others give a formal justification for the variable convention, which makes it hard to formalise such proofs. In this paper we identify conditions an inductive definition has to satisfy so that a form of the variable convention can be built into the rule induction principle. In practice this means we come quite close to the informal reasoning of "pencilandpaper " proofs, while remaining completely formal. Our conditions also reveal circumstances in which Barendregt's variable convention is not applicable, and can even lead to faulty reasoning. 1 Introduction In informal proofs about languages that feature bound variables, one often assumes (explicitly or implicitly) a rather convenient convention about those bound variables. Barendregt's statement of the convention is: Variable Convention: If M1; : : : ; Mn occur in a certain mathematical context (e.g. definition, proof), then in these terms all bound variables are chosen to be different from the free variables. [2, Page 26]
Nominal rewriting
 Information and Computation
"... Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting ma ..."
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Cited by 19 (7 self)
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Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the objectlanguage (atoms) and of the metalanguage (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced firstorder character, since substitution of terms for variables is not captureavoiding. We show how good properties of firstorder rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem
Modal proofs as distributed programs
 13th European Symposium on Programming
, 2003
"... We develop a new foundation for distributed programming languages by defining an intuitionistic, modal logic and then interpreting the modal proofs as distributed programs. More specifically, the proof terms for the various modalities have computational interpretations as remote procedure calls, com ..."
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Cited by 18 (1 self)
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We develop a new foundation for distributed programming languages by defining an intuitionistic, modal logic and then interpreting the modal proofs as distributed programs. More specifically, the proof terms for the various modalities have computational interpretations as remote procedure calls, commands to broadcast computations to all nodes in the network, commands to use portable code, and finally, commands to invoke computational agents that can find their own way to safe places in the network where they can execute. We prove some simple metatheoretic results about our logic as well as a safety theorem that demonstrates that the deductive rules act as a sound type system for a distributed programming language. 1