A standard monad of continuations, when constructed with domains in the world of FM-sets [4], is shown to provide a model of dynamic allocation of fresh names that is both simple and useful. In particular, it is used to prove that the powerful facilities for manipulating fresh names and binding operations provided by the “Fresh ” series of metalanguages [15,17,18] respect α-equivalence of object-level languages up to meta-level contextual equivalence.

Abstract. Higher-order abstract syntax (HOAS) refers to the technique of representing variables of an object-language using variables of a meta-language. The standard first-order alternatives force the programmer to deal with superficial concerns such as substitutions, whose implementation is often routine, tedious, and error-prone. In this paper, we describe the underlying calculus of Delphin. Delphin is a fully implemented functional-programming language supporting reasoning over higher-order encodings and dependent types, while maintaining the benefits of HOAS. More specifically, just as representations utilizing HOAS free the programmer from concerns of handling explicit contexts and substitutions, our system permits programming over such encodings without making these constructs explicit, leading to concise and elegant programs. To this end our system distinguishes bindings of variables intended for instantiation from those that will remain uninstantiated, utilizing a variation of Miller and Tiu’s ∇-quantifier [1]. 1

3 A complete example

Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K.

Concurrent computation can be given an abstract mathematical treatment very similar to that provided for sequential computation by domain theory and denotational semantics of Scott and Strachey.

Nominal logic is a theory of names and binding based on the primitive concepts of freshness and swapping, with a self-dual N - (or "new")-quantifier, originally presented as a Hilbert-style axiom system extending first-order logic. We present a sequent calculus for nominal logic called Fresh Logic, or FL, admitting cut-elimination. We use FL to provide a proof-theoretic foundation for nominal logic programming and show how to interpret $FOL^{\Delta abla}$, another logic with a self-dual quantifier, within FL.

We present a meta-logic that contains a new quantifier (for encoding "generic judgment") and inference rules for reasoning within fixed points of a given specification. We then specify the operational semantics and bisimulation relations for the finite π-calculus within this meta-logic. Since we

Nominal logic is a variant of first-order logic which provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the new-quantifier, 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 new-quantifier, 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 cut-elimination, consistency, and conservativity with respect to Pitts' axiomatization of nominal logic; these proofs are considerably simpler for our system. 1

Nominal logic is a first-order 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 NP-complete in general. However, when one of the terms is essentially first-order, equivariant and nominal unification coincide. This shows that equivariant unification can be performed efficiently in many interesting common cases: for example, anypurely first-order logic program or rewrite system can be run efficiently on nominal terms.

We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on Fraenkel-Mostowski (FM) set theory how to address this through first-class names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier N , name-binding, and unique choice of fresh names. The Schanuel topos --- the category underlying FM set theory --- is an instance of this axiomatisation.