Towards Böhm trees for lambda - value : the operational and proof - theoretical machinery †

The pure lambda calculus has a well-established ‘standard theory’ in which the notion of solvability characterises the operational relevance of terms. Solvable terms, defined as solutions to a beta-equation, have a ‘syntactic’ characterisation as terms with head normal form. Unsolvable terms are irrelevant and can be beta-equated without affecting consistency. The derived notions of sensibility and Böhm trees connect the consistent theory with models and with a representation of approximate normal forms. The lambda-value calculus is the calculus that corresponds to a strict functional programming language whose operational semantics is defined by the SECD machine. The beta-equational definition of solvability has been duly adapted to the pure lambda-value calculus, but the syntactic characterisation (value head normal forms and the ahead machine) involves beta reduction and not beta-value reduction. The v-unsolvables terms cannot be equated without affecting consistency, and some v-normal forms are v-unsolvable and have to be considered irrelevant. This has been ignored in the context of weak reduction (not going under lambda, an ingredient of call-by-value reduction as specified by the SECD machine) because of the existence of initial models in this scenario. However, that does not answer in full the question of v-solvability nor provides a consistent ‘standard theory’ for pure lambda-value. The problem lies in the emphasis on operational equivalence of closed terms according to SECD. When considering open terms, different v-normal forms with stuck subterms which are operationally equivalent may differ on the scope at which a stuck term pops up. A notion of solvability should take into account this distinction which reflects ‘preserving confluence by preserving potential divergence’ intrinsic in the by-value mechanism. We introduce the syntactic notion of quasi-v-solvability, which shows that beta-value reduction is appropriate for solvability and restores the validity of v-normal forms and