Semi-constructive formal systems and axiomatization of abstract data types

Extracting information from intermediate semiconstructive HA-systems – extended abstract

“…Ferrari,C. Fiorentini,and P. Miglioli 590 Therefore, if T ⊕ L contains a proof π of a formula ∀x∃!yA(x, y) (respectively, a formula of the kind ∀x(B(x) ∨ ¬B(x))), then the whole system T ⊕ Cl can be used to compute the function (respectively, the predicate) associated with such a formula (Miglioli et al 1989;Miglioli et al 1994). If the system T ⊕ L does not satisfy further properties, the algorithm to compute the function (the predicate) is highly inefficient since it does not use the 'local' information contained in the proof π (the proof π is only used to guarantee the termination of the algorithm).…”

“…Formally, a system T ⊕ L, where T is a first order theory (the mathematical part) and L is a super-intuitionistic logic (the deductive apparatus) is semiconstructive if it satisfies the weak disjunction property (if a closed wff A ∨ B belongs to T ⊕ L, then either A or B belongs to the corresponding classical theory T ⊕ Cl) and the weak explicit definability property (if a closed wff ∃xA(x) belongs to T ⊕ L, then A(t) belongs to the corresponding classical theory T ⊕ Cl for some closed term t). The notion of semiconstructive system is relevant in the context of the authors' approach to program synthesis, formal verification and Abstract Data Types specification (Miglioli and Ornaghi 1981;Miglioli et al 1989;Miglioli et al 1994;Avellone et al 1999;Benini 1999). Indeed, if T is a theory completely formalizing an Abstract Data Type, according to the characterization of Abstract Data Types based on the notion of isoinitial model (Miglioli et al 1994), the addition of T to a semiconstructive deductive apparatus L gives rise to a recursively axiomatizable and semiconstructive system T ⊕ L.…”