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115
An axiomatic approach to metareasoning on nominal algebras in HOAS
 Leeuwen (Eds.), 28th International Colloquium on Automata, Languages and Programming, ICALP 2001
, 2001
"... We present a logical framework # for reasoning on a very general class of languages featuring binding operators, called nominal algebras, presented in higherorder abstract syntax (HOAS). # is based on an axiomatic syntactic standpoint and it consists of a simple types theory a la Church extended wi ..."
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Cited by 19 (1 self)
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We present a logical framework # for reasoning on a very general class of languages featuring binding operators, called nominal algebras, presented in higherorder abstract syntax (HOAS). # is based on an axiomatic syntactic standpoint and it consists of a simple types theory a la Church extended with a set of axioms called the Theory of Contexts, recursion operators and induction principles. This framework is rather expressive and, most notably, the axioms of the Theory of Contexts allow for a smooth reasoning of schemata in HOAS. An advantage of this framework is that it requires a very low mathematical and logical overhead. Some case studies and comparison with related work are briefly discussed.
A recursion combinator for nominal datatypes implemented in Isabelle/HOL
 IN PROC. OF THE 3RD INTERNATIONAL JOINT CONFERENCE ON AUTOMATED REASONING (IJCAR), VOLUME 4130 OF LNAI
, 2006
"... The nominal datatype package implements an infrastructure in Isabelle/HOL for defining languages involving binders and for reasoning conveniently about alphaequivalence classes. Pitts stated some general conditions under which functions over alphaequivalence classes can be defined by a form of str ..."
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Cited by 18 (8 self)
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The nominal datatype package implements an infrastructure in Isabelle/HOL for defining languages involving binders and for reasoning conveniently about alphaequivalence classes. Pitts stated some general conditions under which functions over alphaequivalence classes can be defined by a form of structural recursion and gave a clever proof for the existence of a primitiverecursion combinator. We give a version of this proof that works directly over nominal datatypes and does not rely upon auxiliary constructions. We further introduce proving tools and a heuristic that made the automation of our proof tractable. This automation is an essential prerequisite for the nominal datatype package to become useful.
Scrap your Nameplate  Functional Pearl
"... Recent research has shown how boilerplate code, or repetitive code for traversing datatypes, can be eliminated using generic programming techniques already available within some implementations of Haskell. One particularly intractable kind of boilerplate is nameplate, or code having to do with names ..."
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Cited by 17 (5 self)
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Recent research has shown how boilerplate code, or repetitive code for traversing datatypes, can be eliminated using generic programming techniques already available within some implementations of Haskell. One particularly intractable kind of boilerplate is nameplate, or code having to do with names, namebinding, and fresh name generation. One reason for the difficulty is that operations on data structures involving names, as usually implemented, are not regular instances of standard map, fold , or zip operations. However, in nominal abstract syntax, an alternative treatment of names and binding based on swapping, operations such as #equivalence, captureavoiding substitution, and free variable set functions are much betterbehaved.
A Complete Characterization of Observational Equivalence in Polymorphic λCalculus with General References
, 2009
"... We give a (sound and complete) characterization of observational equivalence in full polymorphic λcalculus with existential types and firstclass, higherorder references. Our method is syntactic and elementary in the sense that it only employs simple structures such as relations on terms. It is ne ..."
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Cited by 16 (2 self)
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We give a (sound and complete) characterization of observational equivalence in full polymorphic λcalculus with existential types and firstclass, higherorder references. Our method is syntactic and elementary in the sense that it only employs simple structures such as relations on terms. It is nevertheless powerful enough to prove many interesting equivalences that can and cannot be proved by previous approaches, including the latest work by Ahmed, Dreyer and Rossberg (to appear in POPL 2009). 1.
Syntax for free: Representing syntax with binding using parametricity
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2009
"... We show that, in a parametric model of polymorphism, the type ∀α.((α → α) → α) → (α → α → α) → α is isomorphic to closed de Bruijn terms. That is, the type of closed higherorder abstract syntax terms is isomorphic to a concrete representation. To demonstrate the proof we have constructed a mode ..."
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Cited by 14 (5 self)
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We show that, in a parametric model of polymorphism, the type ∀α.((α → α) → α) → (α → α → α) → α is isomorphic to closed de Bruijn terms. That is, the type of closed higherorder abstract syntax terms is isomorphic to a concrete representation. To demonstrate the proof we have constructed a model of parametric polymorphism inside the Coq proof assistant. The proof of the theorem requires parametricity over Kripke relations. We also investigate some variants of this representation.
Recursive function definition for types with binders
 In Proc. TPHOLS 2004
, 2004
"... Abstract. This work describes the proof and uses of a theorem allowing definition of recursive functions over the type of λcalculus terms, where terms with bound variables are identified up to αequivalence. The theorem embodies what is effectively a principle of primitive recursion, and the analog ..."
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Cited by 13 (0 self)
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Abstract. This work describes the proof and uses of a theorem allowing definition of recursive functions over the type of λcalculus terms, where terms with bound variables are identified up to αequivalence. The theorem embodies what is effectively a principle of primitive recursion, and the analogues of this theorem for other types with binders are clear. The theorem’s sideconditions require that the putative definition be wellbehaved with respect to fresh name generation and name permutation. A number of examples over the type of λcalculus terms illustrate the use of the new principle. 1
A unifying model of variables and names
 Foundations of Software Science and Computational Structures, vol. 3441, Lect. Notes in Comp. Sci
, 2005
"... Abstract. We investigate a category theoretic model where both “variables” and “names”, usually viewed as separate notions, are particular cases of the more general notion of distinction. The key aspect of this model is to consider functors over the category of irreflexive, symmetric finite relation ..."
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Cited by 13 (3 self)
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Abstract. We investigate a category theoretic model where both “variables” and “names”, usually viewed as separate notions, are particular cases of the more general notion of distinction. The key aspect of this model is to consider functors over the category of irreflexive, symmetric finite relations. The models previously proposed for the notions of “variables ” and “names ” embed faithfully in the new one, and initial algebra/final coalgebra constructions can be transferred from the formers to the latter. Moreover, the new model admits a definition of distinctionaware simultaneous substitutions. As a substantial application example, we give the first semantic interpretation of MillerTiu’s FOλ ∇ logic. 1
Fine Control of Demand in Haskell
 In 6th International Conference on the Mathematics of Program Construction, Dagstuhl
, 2002
"... Functional languages have the Acalculus at their core, but then depart fi'om this fu'm foundation by including features that alter their default evaluation order. The resulting mixed evaluationpartly lazy and partly strictcomplicates the formal semantics of these languages. ..."
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Cited by 13 (6 self)
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Functional languages have the Acalculus at their core, but then depart fi'om this fu'm foundation by including features that alter their default evaluation order. The resulting mixed evaluationpartly lazy and partly strictcomplicates the formal semantics of these languages.
Logical reasoning for higherorder functions with local state
 In Foundations of Software Science and Computation Structure
"... ABSTRACT. We introduce an extension of Hoare logic for callbyvalue higherorder functions with MLlike local reference generation. Local references may be generated dynamically and exported outside their scope, may store higherorder functions and may be used to construct complex mutable data stru ..."
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Cited by 13 (4 self)
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ABSTRACT. We introduce an extension of Hoare logic for callbyvalue higherorder functions with MLlike local reference generation. Local references may be generated dynamically and exported outside their scope, may store higherorder functions and may be used to construct complex mutable data structures. This primitive is captured logically using a predicate asserting reachability of a reference name from a possibly higherorder datum and quantifiers over hidden references. We explore the logic’s descriptive and reasoning power with nontrivial programming examples combining higherorder procedures and dynamically generated local state. Axioms for reachability and local invariant play a central role for reasoning about the examples.
Consistency of the Theory of Contexts
, 2001
"... The Theory of Contexts is a typetheoretic axiomatization which has been recently proposed by some of the authors for giving a metalogical account of the fundamental notions of variable and context as they appear in Higher Order Abstract Syntax. In this paper, we prove that this theory is consistent ..."
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Cited by 12 (3 self)
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The Theory of Contexts is a typetheoretic axiomatization which has been recently proposed by some of the authors for giving a metalogical account of the fundamental notions of variable and context as they appear in Higher Order Abstract Syntax. In this paper, we prove that this theory is consistent by building a model based on functor categories. By means of a suitable notion of forcing, we prove that this model validates Classical Higher Order Logic, the Theory of Contexts, and also (parametrised) structural induction and recursion principles over contexts. The approach we present in full detail should be useful also for reasoning on other models based on functor categories. Moreover, the construction could be adopted, and possibly generalized, also for validating other theories of names and binders. Contents 1 The object language 4 2 The metalanguage (Framework System #) 6 2.1 Syntax 6 2.2 Typing and logical judgements 7 2.3 Adequacy of the encoding 8 2.4 Remarks on the design of # 9 3 Categorytheoretic preliminaries 11 4.1 The ambient categories 4.2 Interpreting types 16 4.3 Interpreting environments 18 4.4 Interpreting the typing judgement of terms 19 4.5 Interpreting logical judgements 21 is a model of # 22 5.1 Forcing 22 5.2 Characterisation of Leibniz equality 23 models logical axioms and rules 26 models the Theory of Contexts 27 6 Recursion 28 6.1 Firstorder recursion 28 6.2 Higherorder recursion 31 7 Induction 33 7.1 Firstorder induction 34 7.2 Higherorder induction 37 8 Connections with tripos theory 38 9 Related work 41 9.1 Semantics based on functor categories 41 9.2 Logics for nominal calculi 44 10 Conclusions 45 A Proofs 46 A.1 Proof of Proposition 4.2 46 A.2 Proof of Proposition 4.3 47 A.3 Proof of Theorem 5.1 48 A.4 Proof of...