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22
Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 219 (28 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
Programming with proofs and explicit contexts
 In Symposium on Principles and Practice of Declarative Programming, 2008. François Pottier and Nadji
"... This paper explores a new point in the design space of functional programming: functional programming with dependentlytyped higherorder data structures described in the logical framework LF. This allows us to program with proofs as higherorder data. We present a decidable bidirectional type syste ..."
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Cited by 30 (10 self)
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This paper explores a new point in the design space of functional programming: functional programming with dependentlytyped higherorder data structures described in the logical framework LF. This allows us to program with proofs as higherorder data. We present a decidable bidirectional type system that distinguishes between dependentlytyped data and computations. To support reasoning about open data, our foundation makes contexts explicit. This provides us with a concise characterization of open data, which is crucial to elegantly describe proofs. In addition, we present an operational semantics for this language based on higherorder pattern matching for dependently typed objects. Based on this development, we prove progress and preservation.
Induction and coinduction in sequent calculus
 Postproceedings of TYPES 2003, number 3085 in LNCS
, 2003
"... Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than sett ..."
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Cited by 28 (8 self)
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Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than settheoretic) notion of definition [13, 20, 25, 51]. Definitions are akin to (stratified) logic programs, where the left and right rules for defined atoms allow one to view theories as “closed ” or defining fixed points. The use of definitions makes it possible to reason intensionally about syntax, in particular enforcing free equality via unification. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and coinductively about properties of computational system making full use of higherorder abstract syntax. Consistency is guaranteed via cutelimination, where we give the first, to our knowledge, cutelimination procedure in the presence of general inductive and coinductive definitions. 1
Case analysis of higherorder data
"... Abstract. We discuss coverage checking for data that is dependently typed and is defined using higherorder abstract syntax. Unlike previous work on coverage checking that required objects to be closed, we consider open data objects, i.e. objects that may depend on some context. Our work may therefo ..."
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Cited by 16 (14 self)
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Abstract. We discuss coverage checking for data that is dependently typed and is defined using higherorder abstract syntax. Unlike previous work on coverage checking that required objects to be closed, we consider open data objects, i.e. objects that may depend on some context. Our work may therefore provide insights into coverage checking in Twelf, and serve as a basis for coverage checking in functional languages such as Delphin and Beluga. More generally, our work is a foundation for proofs by case analysis in systems that reason about higherorder abstract syntax. 1
Nominal System T
, 2010
"... This paper introduces a new recursion principle for inductive data modulo ..."
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Cited by 14 (1 self)
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This paper introduces a new recursion principle for inductive data modulo
Structural Recursion with Locally Scoped Names
"... This paper introduces a new recursion principle for inductively defined data modulo αequivalence of bound names that makes use of Oderskystyle local names when recursing over bound names. It is formulated in simply typed λcalculus extended with names that can be restricted to a lexical scope, tes ..."
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Cited by 13 (2 self)
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This paper introduces a new recursion principle for inductively defined data modulo αequivalence of bound names that makes use of Oderskystyle local names when recursing over bound names. It is formulated in simply typed λcalculus extended with names that can be restricted to a lexical scope, tested for equality, explicitly swapped and abstracted. The new recursion principle is motivated by the nominal sets notion of “αstructural recursion”, whose use of names and associated freshness sideconditions in recursive definitions formalizes common practice with binders. The new calculus has a simple interpretation in nominal sets equipped with name restriction operations. It is shown to adequately represent αstructural recursion while avoiding the need to verify freshness sideconditions in definitions and computations. The paper is a revised and expanded version of (Pitts, 2010). 1
Nominal Abstraction
, 2009
"... Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logicbased reasoning about such descriptions: the interpretation of atomic judgments through recursive ..."
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Cited by 11 (8 self)
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Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logicbased reasoning about such descriptions: the interpretation of atomic judgments through recursive definitions and an encoding of binding constructs via generic judgments. However, logics encompassing these two features do not currently allow for the definition of relations that embody dynamic aspects related to binding, a capability needed in many reasoning tasks. We propose a new relation between terms called nominal abstraction as a means for overcoming this deficiency. We incorporate nominal abstraction into a rich logic also including definitions, generic quantification, induction, and coinduction that we then prove to be consistent. We present examples to show that this logic can provide elegant treatments of binding contexts that appear in many proofs, such as those establishing properties of typing calculi and of arbitrarily cascading substitutions that play a role in reducibility arguments.
A metaprogramming approach to realizing dependently typed logic programming
 In ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP
, 2010
"... Dependently typed lambda calculi such as the Logical Framework (LF) can encode relationships between terms in types and can naturally capture correspondences between formulas and their proofs. Such calculi can also be given a logic programming interpretation: the Twelf system is based on such an int ..."
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Cited by 7 (5 self)
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Dependently typed lambda calculi such as the Logical Framework (LF) can encode relationships between terms in types and can naturally capture correspondences between formulas and their proofs. Such calculi can also be given a logic programming interpretation: the Twelf system is based on such an interpretation of LF. We consider here whether a conventional logic programming language can provide the benefits of a Twelflike system for encoding type and proofandformula dependencies. In particular, we present a simple mapping from LF specifications to a set of formulas in the higherorder hereditary Harrop (hohh) language, that relates derivations and proofsearch between the two frameworks. We then show that this encoding can be improved by exploiting knowledge of the wellformedness of the original LF specifications to elide much redundant typechecking information. The resulting logic program has a structure that closely resembles the original specification, thereby allowing LF specifications to be viewed as hohh metaprograms. Using the Teyjus implementation of λProlog, we show that our translation provides an efficient means for executing LF specifications, complementing the ability that the Twelf system provides for reasoning about them. Keywords logical frameworks, dependently typed lambda calculi, higherorder logic programming, translation 1.
Relating nominal and higherorder abstract syntax specifications
 in: Proceedings of the 2010 Symposium on Principles and Practice of Declarative Programming, ACM
"... Nominal abstract syntax and higherorder abstract syntax provide a means for describing binding structure which is higherlevel than traditional techniques. These approaches have spawned two different communities which have developed along similar lines but with subtle differences that make them dif ..."
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Cited by 5 (2 self)
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Nominal abstract syntax and higherorder abstract syntax provide a means for describing binding structure which is higherlevel than traditional techniques. These approaches have spawned two different communities which have developed along similar lines but with subtle differences that make them difficult to relate. The nominal abstract syntax community has devices like names, freshness, nameabstractions with variable capture, and thequantifier, whereas the higherorder abstract syntax community has devices like λbinders, λconversion, raising, and the ∇quantifier. This paper aims to unify these communities and provide a concrete correspondence between their different devices. In particular, we develop a semanticspreserving translation from αProlog, a nominal abstract syntax based logic programming language, to G − , a higherorder abstract syntax based logic programming language. We also discuss higherorder judgments, a common and powerful tool for specifications with higherorder abstract syntax, and we show how these can be incorporated into G −. This establishes G − as a language with the power of higherorder abstract syntax, the finegrained variable control of nominal specifications, and the desirable properties of higherorder judgments.
Contributions to the Theory of Syntax with Bindings and to Process Algebra
, 2010
"... We develop a theory of syntax with bindings, focusing on: methodological issues concerning the convenient representation of syntax; techniques for recursive definitions and inductive reasoning. Our approach consists of a combination of FOAS (FirstOrder Abstract Syntax) and HOAS (HigherOrder Abst ..."
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Cited by 5 (4 self)
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We develop a theory of syntax with bindings, focusing on: methodological issues concerning the convenient representation of syntax; techniques for recursive definitions and inductive reasoning. Our approach consists of a combination of FOAS (FirstOrder Abstract Syntax) and HOAS (HigherOrder Abstract Syntax) and tries to take advantage of the best of both worlds. The connection between FOAS and HOAS follows some general patterns and is presented as a (formally certified) statement of adequacy. We also develop a general technique for proving bisimilarity in process algebra Our technique, presented as a formal proof system, is applicable to a wide range of process algebras. The proof system is incremental, in that it allows building incrementally an a priori unknown bisimulation, and patternbased, in that it works on equalities of process patterns (i.e., universally quantified equations of process terms containing process variables), thus taking advantage of equational reasoning in a “circular ” manner, inside coinductive proof loops. All the work presented here has been formalized in the Isabelle theorem prover. The formalization is performed in a general setting: arbitrary manysorted syntax with bindings and arbitrary SOSspecified process algebra in de Simone format. The usefulness of our techniques is illustrated by several formalized case studies: a development of callbyname and callbyvalue λcalculus with constants, including ChurchRosser theorems, connection with de Bruijn representation, connection with other Isabelle formalizations, HOAS representation, and contituationpassingstyle (CPS) transformation; a proof in HOAS of strong normalization for the polymorphic secondorder λcalculus (a.k.a. System F). We also indicate the outline and some details of the formal development. ii to Leili R. Marleene iii