More, T.: Axioms and Theorems for a Theory of Arrays. IBM J. Res. Dev. 17(2) (1973) 135--175  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this important database research area. 2. Related Work Arrays have been defined as an operational data structure in many instances, ranging from low level programming language definitions to related work on query languages for arrays. A notable e#ort to formalize arrays is the Theory of Arrays [5], which contains many axioms and theorems with their formal proof. As mentioned above, 1] identified the failure of most DBMS systems to support ordered data collections natively. The authors hypothesize that the mismatch in domains between database systems based on unordered sets, and ....

....is [f( Function f may apply the operators defined on the base type in the database layer to values indexed in previously defined arrays, the index values themselves, as well as constant values. A simple example follows: The matrix: 3 4 5 6 7 8 [x 3 y x 3, y 3] # [ 0, 1, 2] [3, 4, 5], 6, 7, 8] Although the array comprehension syntax is related to set comprehension syntax (the foundation of database query languages like SQL) its semantics are fundamentally di#erent. A set comprehension D C 1 , C 2 , C n (easily recognized in the SQL variant SELECT FROM D ....

T. More jr. Axioms and theorems for a theory of arrays. IBM Journal of Research and Development, 17(2):135--157, March 1973.

....given parallel architectures. They thus fit very well in a functional context and it is not hard to imagine an implementation strategy for data fields where certain operations on data fields are translated into skeletal code. There is some work on the formal modelling of arrays. The Array Theory [50] of Trenchard More, Jr. is an attempt to define an axiomatic theory of APL arrays. A virtue of this theory is the consistent handling of singularities, like empty arrays. This model is highly APL specific and it is not easily generalized to other indexed structures. Another approach to arrays is ....

T. More, Jr. Axioms and theorems for a theory of arrays. IBM Journal of Research and Development, 17(2):135--175, Mar. 1973.

....Maier and Vance [21] who propose syntax similar to our tabulation construct. Constructs of the same flavor are used in a number of functional language that provide support for arrays, see [25, 1, 14, 10] and also in APL [15] General techniques for reasoning about arrays were studied in [26]. The view of arrays as functions was also explored by [13] in the context of parallel computations. A number of proposals for object oriented query languages include arrays, see [22, 9] The ODMG proposal includes one dimensional arrays with a limited set of supported operations [8] Vandenberg ....

T. More. Axioms and theorems for a theory of arrays. IBM J. Res. and Development 17 (1973), 135--175.

....Constructs of the same flavor are used in a number of functional languages that provide support for arrays, see [1, 10, 15, 26] and also in APL [16] The view of arrays as functions was also explored by [14] in the context of parallel computations. An early axiomatization for arrays was given in [27]. A number of proposals for object oriented query languages include arrays, see [9, 23] The ODMG proposal [8] includes one dimensional arrays with operations for creating, inserting, updating, subscripting and resizing. Their arrays seem to support in place updates, since they can have holes and ....

T. More. Axioms and theorems for a theory of arrays. IBM J. Res. and Development 17 (1973), 135--175.

....array which they then apply to produce a new array reference. There are times when we need direct knowledge of these things. We define for the purpose (but do not propose as a standard) the following primitives: Sh returns the shape of an array, which is defined in More s array theory [Mor 73] to be a vector of the same length as the dimensionality of the array, containing in each element the size of the corresponding dimension of the array. Loc returns the offset of an array element within its container, also in the form of a vector. Suppose A is a 3 x 4 x 5 array and shapeA = Sh(A) ....

Trenchard More. Axioms and Theorems for a Theory of Arrays. In IBM Journal of Research and Development 17(2):135-175, March 1973.

....parallel array operations. A more abstract recent proposal for basing parallel computation on functional operators over lists appears in [28] Third, we can modify SVP to include some features for arrays, and some kinds of nested arrays. More has developed an extensive theory of nested arrays in [20, 21], and has shown how it generalizes and improves the (non nested) model of arrays used in APL. Others have gotten very interesting results with recursive arrays. The classic paper [36] explores the results of storing square arrays as recursive quadtrees (which Wise calls 2 d ary trees, or ....

T. More, Axioms and Theorems for A Theory of Arrays, IBM J. Res. Devel. 17:2, 195--175, 1973.

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More, T.: Axioms and Theorems for a Theory of Arrays. IBM J. Res. Dev. 17(2) (1973) 135--175

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T. More. Axioms and theorems for a theory of arrays. IBM J. Res. and Development 17 (1973), 135--175.

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