Linear lambda terms as invariants of rooted trivalent maps

The main aim of the article is to give a simple and conceptual account for the correspondence (originally described by Bodini, Gardy, and Jacquot) between $\alpha$-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the "easy" direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.Comment: accepted author manuscript, posted six months after publicatio

Generalizations of the Recursion Theorem

We consider two generalizations of the recursion theorem, namely Visser's ADN theorem and Arslanov's completeness criterion, and we prove a joint generalization of these theorems

Bounding normalization time through intersection types

Non-idempotent intersection types are used in order to give a bound of the length of the normalization beta-reduction sequence of a lambda term: namely, the bound is expressed as a function of the size of the term.Comment: In Proceedings ITRS 2012, arXiv:1307.784

Strong Normalization through Intersection Types and Memory

AbstractWe characterize β-strongly normalizing λ-terms by means of a non-idempotent intersection type system. More precisely, we first define a memory calculus K together with a non-idempotent intersection type system K, and we show that a K-term t is typable in K if and only if t is K-strongly normalizing. We then show that β-strong normalization is equivalent to K-strong normalization. We conclude since λ-terms are strictly included in K-terms

Gems of Corrado B\"ohm

The main scientific heritage of Corrado B\"ohm consists of ideas about computing, concerning concrete algorithms, as well as models of computability. The following will be presented. 1. A compiler that can compile itself. 2. Structured programming, eliminating the 'goto' statement. 3. Functional programming and an early implementation. 4. Separability in {\lambda}-calculus. 5. Compiling combinators without parsing. 6. Self-evaluation in {\lambda}-calculus

Characterising strongly normalising intuitionistic terms

This paper gives a characterisation, via intersection types, of the strongly normalising proof-terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary lambdal-calculus. The completeness of the typing system is obtained from subject expansion at root position. Next we use our result to analyze the characterisation of strong normalisability for three classes of intuitionistic terms: ordinary lambda-terms, LambdaJ-terms (lambda-terms with generalised application), and lambdax-terms (lambda-terms with explicit substitution). We explain via our system why the type systems iin the natural deduction format for LambdaJ and lambdax known from the literature contain extra, exceptional rules for typing generalised application or substitution; and we show a new characterisation of the beta-strongly normalising l-terms, as a corollary to a PSN-result, relating the lambda-calculus and the intuitionistic sequent calculus. Finally, we obtain variants of our characterisation by restricting the set of assignable types to sub-classes of intersection types, notably strict types. In addition, the known characterisation of the beta-strongly normalising lambda-terms in terms of assignment of strict types follows as an easy corollary of our results.Fundação para a Ciência e Tecnologi