Deepak Kapur. Towards a theory for abstract data types. Technical Report MIT-LCS-TR237, MIT, June 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....relational structures do. Composition is written in diagrammatic order f ; g for g(f( For a binary relation function OE, OE denotes its inverse = fhy; xi : hx; yi 2 OEg. 2 Multialgebras Our interest in relational structures originates from earlier studies of multialgebras [4, 5, 10, 13, 17, 21, 27, 30, 32] which provide means for modeling nondeterminism in the context of algebraic specification of abstract data types [13, 17 19, 31 33] Multilagebras can be viewed as relational structures with a specific composition of relations of arbitrary arities. According to definition 1.2, ....

....its inverse = fhy; xi : hx; yi 2 OEg. 2 Multialgebras Our interest in relational structures originates from earlier studies of multialgebras [4, 5, 10, 13, 17, 21, 27, 30, 32] which provide means for modeling nondeterminism in the context of algebraic specification of abstract data types [13, 17 19, 31 33] Multilagebras can be viewed as relational structures with a specific composition of relations of arbitrary arities. According to definition 1.2, relations are viewed as set valued functions where the last, n th argument corresponds to an element of the result set obtained by ....

Kapur, D. Towards a Theory of Abstract Data Types. Ph. D. thesis, Laboratory for CS, MIT. (1980).

....and reasoning about static collections of values, functions, and relations. Usually a collection of interdependent immutable abstract data types is identified with the mathematical notion of a heterogeneous algebra, and algebras are described either axiomatically, as in [Guttag78, Goguen78, Kapur80] or via set.theoretic constructions, as in [Abria180] Specification of mutable abstract data types, on the other hand, can be thought of as the problem of describing and reasoning about the dynamic behavior of a collection of objects that can be manipulated using a limited set of procedures ....

Kapur, D., "Towards a Theory for Abstract Data Types," MIT/LCS/TR.237, May, 1980.

....resulting from evaluating the program. This does not hold for nondeterministic programs. The input to such a program does not uniquely determine the resultant output. Since a nondeterministic program cannot be modeled as an inputoutput function, the set approach [Hesselink 1988, Kaplan 1988, Kapur 1980, Maibaum 1977, Subrahmanyam 1981] generalizes the standard algebraic notion of programs as functions, and regards nondeterministic programs as functions in a multialgebra. A program then denotes a function which instead of returning single values returns sets of values. For a particular ....

....according to the set semantics of equality the result will always be 5, because the term 1 denotes one and the same set valued constant the equality 1 =#. 0, 1 is interpreted as a test of whether the set 0, 1 equals the set 0, 1 , which is always true [Hesselink 1988, Kaplan 1988, Kapur 1980, Maibaum 1977, Subrahmanyam 1981] Thus the set semantic interpretation of equality implies the possibility of two terms returning the same result, but it does not guarantee that the same results actually will be produced. This leads to particular difficulties in treating properly such notions ....

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Kapur, D., Towards a theory of abstract data types, Ph.D. thesis, Laboratory for CS, MIT, 1980.

....an empirical investigation of the consequences of applying the theory of initiality to the new domain of nondeterminism. The presentation is rather informal. We will present a simple, yet illustrative, example and work it out in different formalisms. The models we will consider are multialgebras [3,4,7], computations [15] and unified algebras [11,12] In each case we will give a short description of the language used. We will have in mind only initial models, and the interpretation of a ground term t in such a model will be denoted H(t) or t. The example itself was used in [11] but has ....

Kapur, D., Towards a theory of abstract data types, Ph.D. thesis, Laboratory for CS, MIT, 1980.

....superscript : a plural variable will be denoted by x , the set of plural variables in a term t by V [t] a specification with plural arguments SP , the corresponding extension of the language L by L etc. 3. A note on the intuitive interpretation Multialgebraic semantics [10, 13] interprets specifications in some form of power structures where the (nondeterministic) operations correspond to set valued functions. This means that a (ground) term is interpreted as a set of possibilities it denotes the set of possible results of the corresponding operation. We, on the ....

....terms as applications of the respective operations. Every two syntactic occurrences of a term t will refer to possibly distinct applications of t. For nondeterministic terms this means that they may denote two distinct values. Typically, equality is interpreted in a multialgebra as set equality [13, 23, 12]. For instance, the formula a t=s means that the sets corresponding to all possible results of the operations t and s are equal. This gives a model which is mathematically plausible, but which does not correspond to our operational intuition. The (set) equality a t=s does not guarantee that the ....

Kapur, D., Towards a theory of abstract data types, Ph.D. thesis, Laboratory for CS, MIT, 1980.

....resulting from evaluating the program. 2 This does not hold for nondeterministic programs. The input to such a program does not uniquely determine the resultant output. Since a nondeterministic program cannot be modeled as an inputoutput function, the set approach [Hesselink 1988, Kaplan 1988, Kapur 1980, Maibaum 1977, Subrahmanyam 1981] generalizes the standard algebraic notion of programs as functions, and regards nondeterministic programs as functions in a multialgebra. A program then denotes a function which instead of returning single values returns sets of values. For a particular ....

....the set semantics of equality the result will always be 5, because the term #. 0, 1 denotes one and the same set valued constant the equality #. 0, 1 =#. 0, 1 is interpreted as a test of whether the set 0, 1 equals the set 0, 1 , which is always true [Hesselink 1988, Kaplan 1988, Kapur 1980, Maibaum 1977, Subrahmanyam 1981] Thus the set semantic interpretation of equality implies the possibility of two terms returning the same result, but it does not guarantee that the same results actually will be produced. This leads to particular difficulties in treating properly such notions ....

[Article contains additional citation context not shown here]

Kapur, D., Towards a theory of abstract data types, Ph.D. thesis, Laboratory for CS, MIT, 1980.

....last, n th argument of an n ary relation corresponds to an element of the result set obtained by applying the function to the first n Gamma 1 arguments. This view appears in [19] was elaborated in [11, 12] then in [20] and re emerged in recent years in the algebraic approaches to nondeterminism [13, 9, 1, 26, 27]. This view is based on the simple observation that any (setvalued) operation f : A 1 Theta : Theta An P(A) determines a relation R f A 1 Theta : Theta An Theta A and vice versa, via the isomorphism: A 1 Theta : Theta An P(A) A 1 Theta : Theta An [A Bool] A 1 ....

Kapur, D. Towards a Theory of Abstract Data Types. Ph. D. thesis, Laboratory for CS, MIT. (1980).

....argument of an n ary relation corresponds to an element of the result set obtained by applying the set valued function to the first n Gamma 1 arguments. This view appears in [24] was elaborated in [15, 16] then in [25] and re emerged in recent years in the algebraic approaches to nondeterminism [17, 14, 1, 28, 29]. It is based on the observation that any (set valued) operation f : A 1 Theta : Theta An P(A) determines a relation R f A 1 Theta : Theta An Theta A and vice versa, via the isomorphism: A 1 Theta : Theta An P(A) A 1 Theta : Theta An [A Bool] A 1 Theta : ....

Kapur, D. Towards a Theory of Abstract Data Types. Ph. D. thesis, Laboratory for CS, MIT. (1980).

....state. Its other arguments are hypothetical; they are not objects that actually exist in the object universe. The diamond rule is stronger than necessary because it requires equality between abstract values. We need only the weaker notion of observable equivalence (e.g. see Kapur s definition[20]) since values that are distinct may not be observably different if the supertype s set of methods (in particular, observers) is too weak to let us perceive the difference. In practice, 5 Thanks to Ian Maung for pointing out this problem and inspiring this example. 6 An alternative solution to ....

Deepak Kapur. Towards a theory of abstract data types. Technical Report 237, MIT LCS, June 1980. Ph.D. thesis.

..... All variables occurring in a sequent are implicitly universally quantified over the whole sequent. For a specification SP= L(SP) is the restriction of L to W ( V . The semantics of L specifications uses multistructures. Our definitions are very similar to those used by other authors [8,10,12,20] except for the notion of equality. Also, we provide the means to interpret the occurrences of terms in L as applications of (possibly nondeterministic) operations rather than, as it is usually the case, as the sets of possible results. Definition 3.1 (Multistructures) Let SP be an L ....

Kapur, D., Towards a theory of abstract data types, Ph.D. thesis, Laboratory for CS, MIT, 1980.

....include changes in value to state objects, observable changes in the store (new objects that appear and old objects that disappear) objects returned by the operation, and signaled exceptions or errors. Another way to think about observable behavior is to think about observable equivalence [15, 11, 10]. Ask Can I distinguish between these two things where things might be states, individual entities in a state, traces of a process, or behavior sets of a process, depending on what you are specifying. If the answer is yes, then there must be way to tell them apart (perhaps by using unique ....

Deepak Kapur. Towards a theory of abstract data types. Technical Report 237, MIT LCS, June 1980. Ph.D. Thesis.

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Deepak Kapur. Towards a theory for abstract data types. Technical Report MIT-LCS-TR237, MIT, June 1980.

No context found.

Deepak Kapur. Towards a Theory of Abstract Data Types. Technical Report 237, MIT LCS, June, 1980. Ph.D. thesis.

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Deepak Kapur. Towards a theory for abstract data types. Technical Report MIT-LCS-TR-237, MIT, June 1980.

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