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15
Nominal Unification
 Theoretical Computer Science
, 2003
"... We present a generalisation of firstorder unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms #equivalent, i.e. equal up to renaming bound names. For the a ..."
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We present a generalisation of firstorder unification to the practically important case of equations between terms involving binding operations. A substitution of terms for variables solves such an equation if it makes the equated terms #equivalent, i.e. equal up to renaming bound names. For the applications we have in mind, we must consider the simple, textual form of substitution in which names occurring in terms may be captured within the scope of binders upon substitution. We are able to take a `nominal' approach to binding in which bound entities are explicitly named (rather than using nameless, de Bruijnstyle representations) and yet get a version of this form of substitution that respects #equivalence and possesses good algorithmic properties. We achieve this by adapting an existing idea and introducing a key new idea. The existing idea is terms involving explicit substitutions of names for names, except that here we only use explicit permutations (bijective substitutions). The key new idea is that the unification algorithm should solve not only equational problems, but also problems about the freshness of names for terms. There is a simple generalisation of the classical firstorder unification algorithm to this setting which retains the latter's pleasant properties: unification problems involving #equivalence and freshness are decidable; and solvable problems possess most general solutions.
Alphastructural recursion and induction
 Journal of the ACM
, 2006
"... The nominal approach to abstract syntax deals with the issues of bound names and αequivalence by considering constructions and properties that are invariant with respect to permuting names. The use of permutations gives rise to an attractively simple formalisation of common, but often technically i ..."
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Cited by 48 (6 self)
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The nominal approach to abstract syntax deals with the issues of bound names and αequivalence by considering constructions and properties that are invariant with respect to permuting names. The use of permutations gives rise to an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax modulo αequivalence. At the heart of this approach is the notion of finitely supported mathematical objects. This paper explains the idea in as concrete a way as possible and gives a new derivation within higherorder logic of principles of αstructural recursion and induction for αequivalence classes from the ordinary versions of these principles for abstract syntax trees.
Abstract syntax and variable binding (extended abstract
 In Proc. 14 th LICS
, 1999
"... Abstract We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with both a (binding) algebra and a substitution structure compatible with each other. The syntax generated by the signature is the ..."
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Cited by 23 (0 self)
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Abstract We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with both a (binding) algebra and a substitution structure compatible with each other. The syntax generated by the signature is the initial model. This gives a notion of initial algebra semantics encompassing the traditional one; besides compositionality, it automatically verifies the semantic substitution lemma.
From Action Calculi to Linear Logic
, 1998
"... . Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to ..."
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Cited by 19 (7 self)
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. Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give a sound translation of our type theory in the (type theory of) intuitionistic linear logic, corresponding to the relation between Benton's models of linear logic and models of action calculi. The conservativity of the syntactic translation is proved by a modelembedding construction using the Yoneda lemma. Finally, we briefly discuss how these techniques can also be used to give conservative translations between various extensions of action calculi. 1 Introduction Action calculi arose directly from the ßcalculus [MPW92]. They were introduced by Milner [Mil96], to provide a uniform notation for capturing many calculi of interaction such as the ßcalculus, the calculus, models of distribut...
Simple Situation Theory and its Graphical Representation
 DYANA Report R2.1.C
, 1991
"... ion If we have box B containing free parameter symbols, so that it represents some parametric object b(X; Y; Z; : : :) (possibly with a restriction) we indicate that some of its parameters have been abstracted over by putting the corresponding parameter symbols in a small box at the top right of th ..."
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Cited by 17 (4 self)
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ion If we have box B containing free parameter symbols, so that it represents some parametric object b(X; Y; Z; : : :) (possibly with a restriction) we indicate that some of its parameters have been abstracted over by putting the corresponding parameter symbols in a small box at the top right of the box B representing b. Thus for example, we can bind the parameter symbols X and Z by creating the box: (7) X;Z b(X; Y; Z; : : :) If we want to apply this abstract to some objects, say those denoted by a; c, we would write (8) X;Z b(X; Y; Z; : : :) [a; c] According to the semantics we will give, this will denote the same object as (9) b(a; Y; c; : : :) In certain cases, an abstract like this may denote the sort of thing one can predicate of objects. In this case, we will provide ourselves with a variant notation, to show that the abstract can be used as a predicate. The variant is: (10) X;Z b(X; Y; Z; : : :) Our notation will allow us ways to indicate predication that is different from...
A Cutfree Sequent Calculus for Elementary Situated Reasoning
, 1991
"... A rstorder language is interpreted in the following way: terms are regarded as referring to situations and the truth of formulae is relativized to a situation. The language is then extended to include formulae of the form t : (where t is a term and is a formula) meaning that is true in the s ..."
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Cited by 10 (3 self)
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A rstorder language is interpreted in the following way: terms are regarded as referring to situations and the truth of formulae is relativized to a situation. The language is then extended to include formulae of the form t : (where t is a term and is a formula) meaning that is true in the situation referred to by t. Gentzen's sequent calculus for classical rstorder logic is extended with rules which capture this interpretation. Variants of the calculus and extensions of the language are discussed and the Cut rule is shown to be eliminable from some of the proposed calculi. Situation theory has been concerned with a range of issues centring around the partiality, context dependency and intensional structure of information. In formalizing situation theory one must focus on a specic aspect of the whole package  there is too much uncertainty and equivocation about the connections between the various parts. A dominant approach in recent years has been to focus on build...
Alphastructural recursion and induction (Extended Abstract)
 THEOREM PROVING IN HIGHER ORDER LOGICS, 18TH INTERNATIONAL CONFERENCE, TPHOLS 2005, OXFORD UK, AUGUST 2005, PROCEEDINGS, VOLUME 3603 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... There is growing evidence for the usefulness of name permutations when dealing with syntax involving names and namebinding. In particular they facilitate an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax tr ..."
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Cited by 6 (2 self)
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There is growing evidence for the usefulness of name permutations when dealing with syntax involving names and namebinding. In particular they facilitate an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax trees modulo αequivalence. At the heart of this formalisation is the notion of finitely supported mathematical objects. This paper explains the idea in as concrete a way as possible and gives a new derivation within higherorder logic of principles of αstructural recursion and induction for αequivalence classes from the ordinary versions of these principles for abstract syntax trees.
Language Support for Program Generation: Reasoning, Implementation, and Applications
, 2001
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Substitution in nonwellfounded . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 82 NO. 1 (2003)
, 2003
"... Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable bin ..."
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Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. In the nonwellfounded case, the fundamental operation of substitution is more beneficially defined in terms of primitive corecursion than coiteration.