Corrado Böhm

Automatic synthesis of typed Λ-programs on term algebras☆

The notion of iteratively defined functions from and to heterogeneous term algebras is introduced as the solution of a finite set of equations of a special shape. Such a notion has remarkable consequences: (1) Choosing the second-order typed lamdda-calculus (Λ for short) as a programming language enables one to represent algebra elements and iterative functions by automatic uniform synthesis paradigms, using neither conditional nor recursive constructs. (2) A completeness theorem for Λ-terms with type of degree at most two and a companion corollary for Λ-programs have been proved. (3) A new congruence relation for the last-mentioned Λ-terms which is stronger than Λ-convertibility is introduced and proved to have the meaning of a Λ-program equivalence. Moreover, an extension of the paradigms to the synthesis of functions of higher complexity is considered and exemplified. All the concepts are explained and motivated by examples over integers, list- and tree-structures.

Logic and Computers

Shortly after the appearance of the first computers, that is computing machines with stored programs, there arose the problem of how to communicate efficiently with them, especially how to simplify the writing of coded instructions. Zuse (1949) was probably the first to suggest that machines should be employed to facilitate this work using logical propositional methods. Shortly after, the author of this article (1951–52) devised a ‘language’. Its phrases were to have a double meaning: for mathematicians they signified the description of some algorithm and for the computer they signified a list of instructions to be obeyed in order to implement that algorithm. Moreover the computer was to take upon itself the task of translating these phrases into sequences of instructions written in its own code (Bohm, 1954).

A discrimination algorithm inside λ-β-calculus

Abstract A finite set { F 1 ,…, F n } of λ-terms is said to be discriminable if, given n arbitrary λ-terms X 1 ,…, X n , there exists a λ-term Δ such that: ΔF i ⩾ X i for 1 ⩽ i ⩽ n . In the present paper each finite set of normal combinators which are pairwise non α-η-convertible is proved to be discriminable. Moreover a discrimination algorithm is given.

λ-Terms as total or partial functions on normal forms

In this paper the set of λ -terras is split into 2ω+1 disjoint classes Ph (−ω≤h≤ω). This classification takes into account the meaning of a λ-term F as function on normal forms, and more precisely: 1 iff when auccessively applied to any number of normal forms it gives a λ-term without normal form 2 (0<h<ω) iff when successively applied to h-1 arbitrary normal forms it gives a λ-term without normal form, but there exist h normal forms X1,...,Xh such that FX1...Xh possesses normal form 3 (0≤h>ω) iff when successively applied to h arbitrary normal forms it gives a λ-term which possesses normal form, but there exist h+1 normal forms X1,...,Xh+1 such that FX1...Xh+1 possesses no normal form 4 iff when successively applied to any number of normal forms it gives a λ-term which possesses normal form.

The Ant-Lion Paradigm for Strong Normalization

We look for strongly normalizable (SN) solutions of fixed point equations of the ?-calculus. We propose a paradigmatic way (the ant-lion paradigm) to solve some meaningful sample problems: (a) the adequacy theorem for numeral systems of the ?-K-calculus is extended to the ?-I-calculus and to the ?-?K-calculus (obtained by extending the ?-I-calculus by a restricted K as a ?-rule); (b) the fixed point equation X = FX is solved for F specialized into a deed; (c) numeral systems are classified by examples with respect to the key notions of the paper.

Subduing Self-Application

Recursive equational schemes, defining total functions on natural numbers, are embedded into a combinatory algebra producing an equation system having the following shape:

Surjectivity for Finite Sets of Combinators by Weak Reduction

f(0) = b,f(s(x)) = H(a,f,x).

Characterizing X-separability and one-side invertibility in lambda - beta - Omega -calculus

Two methods are described to derive the combinator representing f by means of a generalized morphism, avoiding the use of fixed point combinators and preserving strong normalizability, i.e. the same feature warranted by most type disciplines.