NIDUS IDEARUM. Scilogs, IV: vinculum vinculorum

Welcome into my scientific lab! My lab[oratory] is a virtual facility with non-controlled conditions in which I mostly perform scientific meditation and chats: a nest of ideas (nidus idearum, in Latin). I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings ground , opinion , expectation ), combining the welly of both science and informal (via internet) talks (in English, French, and Romanian). * In this fourth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring mostly to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. Feel free to budge in or just use the scilogs as open source for your own ideas

A correspondence between rooted planar maps and normal planar lambda terms

A rooted planar map is a connected graph embedded in the 2-sphere, with one edge marked and assigned an orientation. A term of the pure lambda calculus is said to be linear if every variable is used exactly once, normal if it contains no beta-redexes, and planar if it is linear and the use of variables moreover follows a deterministic stack discipline. We begin by showing that the sequence counting normal planar lambda terms by a natural notion of size coincides with the sequence (originally computed by Tutte) counting rooted planar maps by number of edges. Next, we explain how to apply the machinery of string diagrams to derive a graphical language for normal planar lambda terms, extracted from the semantics of linear lambda calculus in symmetric monoidal closed categories equipped with a linear reflexive object or a linear reflexive pair. Finally, our main result is a size-preserving bijection between rooted planar maps and normal planar lambda terms, which we establish by explaining how Tutte decomposition of rooted planar maps (into vertex maps, maps with an isthmic root, and maps with a non-isthmic root) may be naturally replayed in linear lambda calculus, as certain surgeries on the string diagrams of normal planar lambda terms.Comment: Corrected title field in metadat

On the Number of Variables in Special Classes of Random Lambda-Terms

We investigate the number of variables in two special subclasses of lambda-terms that are restricted by a bound of the number of abstractions between a variable and its binding lambda, and by a bound of the nesting levels of abstractions, respectively. These restrictions are on the one hand very natural from a practical point of view, and on the other hand they simplify the counting problem compared to that of unrestricted lambda-terms in such a way that the common methods of analytic combinatorics are applicable. We will show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to C_1 n and C_2 n, respectively, where the constants C_1 and C_2 depend on the bound that has been imposed. So far we just derived closed formulas for the constants in case of the class of lambda-terms with a bounded number of abstractions between each variable and its binding lambda. However, for the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of nesting levels of abstractions, we investigate the number of variables in the different abstraction levels and thereby exhibit very interesting results concerning the distribution of the variables within those lambda-terms

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Unary profile of lambda terms with restricted De Bruijn indices

In this paper we present an average-case analysis of closed lambda terms with restricted values of De Bruijn indices in the model where each occurrence of a variable contributes one to the size. Given a fixed integer k, a lambda term in which all De Bruijn indices are bounded by k has the following shape: It starts with k De Bruijn levels, forming the so-called hat of the term, to which some number of k-colored Motzkin trees are attached. By means of analytic combinatorics, we show that the size of this hat is constant on average and that the average number of De Bruijn levels of k-colored Motzkin trees of size n is asymptotically Θ(√ n). Combining these two facts, we conclude that the maximal non-empty De Bruijn level in a lambda term with restrictions on De Bruijn indices and of size n is, on average, also of order √ n. On this basis, we provide the average unary profile of such lambda terms

On type-directed genmeration of lambda-terms

We describe a Prolog-based combined lambda term generator and type-inferrer for closed well-typed terms of a given size, in de Bruijn notation. By taking advantage of Prolog's unique bidirectional execution model and sound unification algorithm, our generator can build "customized" closed terms of a given type. This relational view of terms and their types enables the discovery of interesting patterns about frequently used type expressions occurring in well-typed functional programs. Our study uncovers the most "popular" types that govern function applications among a about a million small-sized lambda terms and hints toward practical uses to combinatorial software testing. It also shows the effectiveness of Prolog as a meta-language for modeling properties of lambda terms and their types. KEYWORDS: lambda calculus, de Bruijn notation, type inference, generation of lambda terms, logic programming as a meta-language