Z. Luo. Program specification and data type refinement in type theory. Math. Struct. in Comp. Sci., 3:333--363, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....one to derive in the logic that two concrete data types are equal if and only if there exists a simulation relation between their operational parts. At first order, this in turn corresponds to a notion of observational equivalence. Thus, the type theoretic formalism of refinement due to Luo [20] automatically gives a notion in the logic of specification refinement up to observational equivalence; a key issue in program development. In [11] a formal connection is shown at first order between an account of algebraic specification refinement due to Sannella and Tarlecki [37, 36] and the ....

....(Fu) is derivable. We write SP # F SP # for this fact. The notion of constructor in Def. 3 is based on the notion of parameterised program [9] Given a program P that is a realisation of SP # , the instantiation F (P ) is then a realisation of SP . Constructors correspond to refinement maps in [20] and derived signature morphisms in [15] It is evident that the refinement 7 relation of Def. 3 is transitive, i.e. for F # F # def = #u:Sig SP ## .F (F # u) SP # F SP # and SP # # F # SP ## # SP # F#F # SP ## If T[X ] is first order, we get a string of interesting results in the ....

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Z. Luo. Program specification and data type refinement in type theory. Math. Struct. in Comp. Sci., 3:333--363, 1993.

....their implementation parts. Together with the fact that at first order, equality at existential type is derivably equivalent to a notion of observational equivalence, this formalises the semantic proof principle of Mitchell [25] This lifts the type theoretic formalism of refinement due to Luo [22] to a notion in the logic of specification refinement up to observational equivalence; a key issue in program development. In this paper, we discuss the above type theoretic notion of specification refinement in more generality, i.e. we treat data types whose operations may be higher order and ....

....x:top(x:push(x; s) x d We reserve T[X ] for the function profile part of abstract data types 9X:T[X ] For brevity, in this paper we do not consider parameterised specifications and so assume X to be the only free type variable in T[X ] The notion of specification of Def. 1 resembles that of [22]. However, as we are about to see, the important difference is that here equality of data type inhabitants is inherently behavioural, and implementation is up to observational equivalence. In analogy to the meta level notion in [25] we define observational equivalence in terms of observable ....

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Z. Luo. Program specification and data type refinement in type theory. Math. Struct. in Comp. Sci., 3:333--363, 1993.

....into type theory to show a correspondence between two notions of refinement. But this importation is also interesting in its own right, and our results complement those of [25] in that we consider also partial congruences. Other work linking algebraic specification and type theory includes [17] encoding constructor implementations in ECC, 26] expressing module algebra axioms in ECC, 23] encoding behavioural equalities in UTT, 2] treating the specification language ASL , 35] using Nuprl as a specification language, and [34] promoting dependent types in specification. Only [25] ....

....existential types arise from algebraic specifications as in Def. 1. We do not consider free type variables in existential types since this corresponds to parameterised algebraic specifications, which is outside this paper s scope. The type theory specification of Def. 1 is essentially that of [17]. The important difference is that with parametricity, equality of data type inhabitants is inherently behavioural, so implementation is up to observational equivalence. In algebraic specification we said that two Sigma algebras A and B are observationally equivalent w.r.t. Obs and In iff for ....

Z. Luo. Program specification and data type refinement in type theory. Math. Struct. in Comp. Sci., 3:333--363, 1993.

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