The strong relevance logics

The tautology p - q - p is not a theorem of the various relevance logics (see Anderson and Belnap [1]) because q is not considered to be relevant in the derivation of final p. We can take this lack of relevance to mean simply that p-q-p could have been proved without q and its -, i.e., p-p. By the same criterion we could say that in ((p-p) -q) -q p-p is not relevant. In general we will say that any theorem A of an implicational logic is strongly relevant if there is no subpart B ! which can be removed from A, leaving the rest still a theorem of the same logic. Such a subpart B - is said to be superfluous

Classical versions of BCI, BCK and BCIW logics

The question is, is there a formula X, independent of B,C,K1, I and W that creates distinct subclassical logics BCIX,BCKX and BCIWX, while BCKWX is the full classical implicational logic TV

On binary reflected Gray codes and functions

AbstractThe binary reflected Gray code function b is defined as follows: If m is a nonnegative integer, then b(m) is the integer obtained when initial zeros are omitted from the binary reflected Gray code of m.This paper examines this Gray code function and its inverse and gives simple algorithms to generate both. It also simplifies Conder's result that the jth letter of the kth word of the binary reflected Gray code of length n is 2n-2n-j-1⌊2n-2n-j-1-k/2⌋mod2by replacing the binomial coefficient by k-12n-j+1+12

Intersection type systems and logics related to the Meyer-Routley system B+

Some, but not all, closed terms of the lambda calculus have types; these types are exactly the theorems of intuitionistic implicational logic. An extension of these simple (→) types to intersection (or →∧) types allows all closed lambda terms to have types. The corresponding →∧ logic, related to the Meyer–Routley minimal logic B+ (without ∨), is weaker than the →∧ fragment of intuitionistic logic. In this paper we provide an introduction to the above work and also determine the →∧ logics that correspond to certain interesting subsystems of the full →∧ type theory

Horadam functions and powers of irrationals

This paper generalizes a result of Gerdemann to show (with slight variations in some special cases) that, for any real number m and Horadam function Hn(A, B, P,Q), mHn(A, B, P, Q) = i=h k∑(t,Hn+i(A, B, P, Q)), for two consecutive values of n, if and only if, m= i=h k∑ (tiai)= i=h k∑ (t ibi) where a =(P+(P2-4Q) 1/2)/2 and b = (P-(P2-4Q) 1/2)/2. (Horadam functions are defined by: H 0(A, B, P, Q) = A, H1(A,B,P,Q) = B, Hn+1(A, B, P,Q) = PHn(A,B, P,Q)-QHn-1(A, B, P,Q).) Further generalizations to the solutions of arbitrary linear recurrence relations are also considered

Some improvements to Turner\u27s algorithm for bracket abstraction

A computer handles A-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm-the (fab) algorithm of Curry (see Curry and Feys [6])-is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases

A classification of intersection type systems

The first system of intersection types. Coppo and Dezani [3], extended simple types to include intersections and added intersection introduction and elimination rules ((ΛI ) and (ΛE) ) to the type assignment system. The major advantage of these new types was that they were invariant under β-equality, later work by Barendregt, Coppo and Dezani [1], extended this to include an (η) rule which gave types invariant under βη-reduction. Urzyczyn proved in [6] that for both these systems it is undecidable whether a given intersection type is empty. Kurata and Takahashi however have shown in [5] that this emptiness problem is decidable for the sytem including (η). but without (ΛI). The aim of this paper is to classify intersection type systems lacking some of (ΛI), (ΛE) and (η), into equivalence classes according to their strength in typing λ-terms and also according to their strength in possessing inhabitants. This classification is used in a later paper to extend the above (un)decidability results to two of the five inhabitation-equivalence classes. This later paper also shows that the systems in two more of these classes have decidable inhabitation problems and develops algorithms to find such inhabitants

On self matching in ⌊nα⌋

For an arbitrary real number with convergents p0 q0 , p1 q1 , p2 q2 , . . ., b(n+qi)c−bnc is equal to pi, and so is independent of n, except at a small specified number of values of n. For fixed n, this relation holds for all or for all except a finite number of values of i

Expedited Broda-Damas bracket abstraction

A bracket abstraction algorithm is a means of translating λ-terms into combinators. Broda and Damas, in [1], introduce a new, rather natural set of combinators and a new form of bracket abstraction which introduces at most one combinator for each λ-abstraction. This leads to particularly compact combinatory terms. A disadvantage of their abstraction process is that it includes the whole Schonfinkel [4] algorithm plus two mappings which convert the Schonfinkel abstract into the new abstract. This paper shows how the new abstraction can be done more directly, in fact, using only 2n - 1 algorithm steps if there are n occurrences of the variable to be abstracted in the term. Some properties of the Broda-Damas combinators are also considered