Investigation into Combinatory systems with dual combinators

Combinatory logic is known to be related to substructural logics. Algebraic considerations of the latter, in particular, algebraic considerations of two distinct implications (→, ←), led to the introduction of dual combinators in Dunn & Meyer 1997. Dual combinators are "mirror images" of the usual combinators and as such do not constitute an interesting subject of investigation by themselves. However, when combined with the usual combinators (e.g., in order to recover associativity in a sequent calculus), the whole system exhibits new features. A dual combinatory system with weak equality typically lacks the Church-Rosser property, and in general it is inconsistent. In many subsystems terms "unexpectedly" turn out to be weakly equivalent. The paper is a preliminary attempt to investigate some of these issues, as well as, briefly compare function application in symmetric A-calculus (cf. Barbanera & Berardi 1996) and dual combinatory logic.