Proofs for ‘Verifying Spatial Properties of Array Computations’

This technical report provides accompanying proofs for the paper: Verifying Spatial Properties of Array Computations. We show three properties of the lattice model of the stencil specification language. 1) that it is sound with respect to the equational theory of region specifications; 2) that is it sound with respect to the theory of region approximation; 3) that the inference algorithm is sound. We further derive useful identities on the specification language and properties of Union Normal Form—the data structure used to implement the model. Core definitions from the paper are restated here to make the overall report more self contained. Numbering for definitions, lemmas, propositions, theorems, and sections is aligned to the paper [1] for easy cross-reference. Definition 1 (Induction variables). An integer variable is an induction variable if it is the control variable of a “for” loop (do in Fortran), incremented by 1 per iteration. A variable is interpreted as an induction variable only within the scope of the loop body. Throughout, syntactic variables i, j, k range over induction variables. Definition 2 (Array subscripts and indices). An array subscript, denoted a(ē), is an expression which indicates the element of an N -dimensional array a at the N -dimensional index ē, specified by a comma-separated sequence of integer expressions e1, . . . , eN . We also refer to each one of these ei as an index. An index ei is called relative if the expression involves an induction variable. An absolute index is an integer expression which is constant with respect to the enclosing loop, i.e., it does not involve an induction variable. Definition 3 (Origin). For an array subscript a(ē), the index ē is called an origin index, if all e ∈ ē are naked induction variable expressions, e.g., a(i, j). We refer to this as the “origin” since the indices are offset by 0 in each dimension relative to the induction variables. Definition 4 (Neighbourhood and affine index). For an array subscript a(ē), an index e ∈ ē is a neighbourhood index (or neighbour) if e is of the form e ≡ i, e ≡ i + c, or e ≡ i c, where c is an integer constant and i is an induction variable. That is, a neighbourhood index is a constant offset/translation of an induction variable. (The relation ≡ here identifies terms up-to commutativity of + and the inverse relationship of + and − e.g., (-b) + i ≡ i b). An affine index is an index expression of the form a*i + b, where a and b are constants.