Existential quantifiers in abstract data types

Algebraic definition of a functional programming language and its semantic models

“…by eliminating ail occurrences of the function subst and by local renaming of all identifiers such that the local identifiers in fl and F2 appear in some fixed linear order, i. e. MAP> tl = tl, According to this définition ail functions with range map f are simple term-constructor functions apart from the function sub J , which can always be eliminated. The values of the function occurs 7 are uniquely determined, since occurs is defined sufficiently complete.…”

“…So it remains to define the function apply 7 apply^/ 1 , xi, . .., x B ) f / if 3^eP^DATA : MAPN apply(/ s xi, .…”

“…In place of that we propose two other extensions of the basic theory: -an "algebraic" one using partial algebras and homomorphisms for partial algebras instead of total ones; -a "logical" one using explicitly existential quantifiers in the axioms {cf Broy et al [7]). …”

Initial and Final Algebra Semantics for Data Type Specifications: Two Characterization Theorems

“…Moreover, in the first paper to explicitly formulate final algebra semantics [23], Wand argues that it is indeed the denotational semantics of Guttag's theory of specifications.) Mathematically exact declarations in favor of the far less well-understood final algebra semantics can be seen in Hornung and Raulefs [13], Kamin [14], Kapur and Srivas [15], the Munich Group [8], [24] and Wand [23].…”

“…PROJ (X) = PROJ (PROJ (X)), Successor (4) SUCC (X) = SUCC (PROJ (X)), Addition (5) ADD (X, 0) = PROJ (X), (6) ADD (X, SUCC ( Y)) = SUCC (ADD (X, Y)), (7) ADD (X, Y) = ADD (PROJ (X), PROJ (Y)) Multiplication (8) MULT (X, 0) = 0, (9) MULT (X, SUCC ( Y)) = ADD (MULT (X, Y), X), (10) MULT (X, Y) = MULT (PROJ (X), PROJ (Y)).…”