this paper, we do this by adding the #-quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the type o, and for all types # not containing o, # is a constant of type (# o) o. The expression # #x.B is ACM Transactions on Computational Logic, Vol. V, No. N, October 2003. 4 usually abbreviated as simply # x.B or as if the type information is either simple to infer or not important

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.

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 Curry-Howard 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 pattern-matching. 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

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

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 co-induction. These proof principles are based on a proof theoretic (rather than set-theoretic) 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 co-inductively about properties of computational system making full use of higher-order abstract syntax. Consistency is guaranteed via cut-elimination, where we give the first, to our knowledge, cut-elimination procedure in the presence of general inductive and co-inductive definitions. 1

We construct a logical framework supporting datatypes that mix binding and computation, implemented as a universe in the dependently typed programming language Agda 2. We represent binding pronominally, using well-scoped de Bruijn indices, so that types can be used to reason about the scoping of variables. We equip our universe with datatype-generic implementations of weakening, substitution, exchange, contraction, and subordination-based strengthening, so that programmers need not reimplement these operations for each individual language they define. In our mixed, pronominal setting, weakening and substitution hold only under some conditions on types, but we show that these conditions can be discharged automatically in many cases. Finally, we program a variety of standard difficult test cases from the literature, such as normalization-by-evaluation for the untyped λ-calculus, demonstrating that we can express detailed invariants about variable usage in a program’s type while still writing clean and clear code.

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Abstract. Model checking for transition systems specified in π-calculus has been a difficult problem due to the infinite-branching nature of input prefix, name-restriction and scope extrusion. We propose here an approach to model checking for π-calculus by encoding it into a logic which supports reasoning about bindings and fixed points. This logic, called F Oλ ∆ ∇ , is a conservative extension of Church’s Simple Theory of Types with a “generic ” quantifier. By encoding judgments about transitions in pi-calculus into this logic, various conditions on the scoping of names and restrictions on name instantiations are captured naturally by the quantification theory of the logic. Moreover, standard implementation techniques for (higher-order) logic programming are applicable for implementing proof search for this logic, as illustrated in a prototype implementation discussed in this paper. The use of logic variables and eigenvariables in the implementation allows for exploring the state space of processes in a symbolic way. Compositionality of properties of the transitions is a simple consequence of the meta theory of the logic (i.e., cut elimination). We illustrate the benefits of specifying systems in this logic by studying several specifications of modal logics for pi-calculus. These specifications are also executable directly in the prototype implementation of F Oλ ∆ ∇. 1