By Markus Müller-Olm (auth.)

Program research is anxious with thoughts that immediately ascertain run-time homes of given courses sooner than run-time. it's used for validation on the way to make sure that courses serve their meant goal and in extra processing for effective execution comparable to in optimizing compilers. optimum application research offers a warrantly concerning the precision of the computed results.

This monograph, a revised model of the author's habilitation thesis, focusses on optimum circulate research of sequential and parallel courses. It experiences algorithmic houses of assorted types of the well known constant-propagation challenge. so one can come to grips with the variations thought of, it combines recommendations from assorted components akin to linear algebra, computable ring idea, summary interpretation, application verification, complexity thought, and so on. mix of thoughts is the main to extra growth in computerized research and constant-propagation permits us to demonstrate this element in a theoretical study.

After a normal evaluation, the monograph comprises 3 primarily self-contained elements that may be learn independently of one another. those components learn: a hierarchy of constants in sequential courses, inherent limits of movement research of parallel courses, and the way to beat those limits via forsaking a vintage atomic execution assumption.

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**Extra resources for Variations on Constants: Flow Analysis of Sequential and Parallel Programs**

**Example text**

0) and b = (1). In the concrete algorithm the elements of D are represented in this way by a matrix A and a vector b but this further representation step is suppressed in this section. We show, however, that all the needed operations on aﬃne spaces can eﬃciently be performed on their representation by a matrix and a vector. Conceptually, it is simpler to consider the aﬃne spaces themselves as representations because they are ordered. On pairs (A, b) we have only the pre-order induced by their interpretation as aﬃne spaces: (A, b) ≤ (A , b ) :⇔ {x | Ax = b} ⊆ {x | A x = b } .

The most important reason for this assumption is that it ensures that we can validly compute on representations without losing precision: if we precisely mirror the equations characterizing weakest preconditions on representations, the largest solution of the resulting equation system on representations characterizes the representation of the weakest precondition by the following wellknown lemma. It appears in the literature (for the dual situation of least ﬁxpoints) under the name Transfer Lemma [4] or μ-Fusion Rule [49].

Xn ] by standard ﬁxpoint iteration. 2 Zeros As mentioned, we represent assertions by the zeros of ideals in our algorithm. A state σ is called a zero of polynomial p if pσ = 0; we denote the set of zeros of polynomial p by Z(p). More generally, for a subset B ⊆ Z[x1 , . . , xn ], Z(B) = {σ | ∀p ∈ B : pσ = 0}. For later use some facts concerning zeros are collected in the following lemma, in particular of the relationship of ideal operations with operations on their zeros. 1. Suppose B, B are sets of polynomials, q is a polynomial, I, I are ideals, and I is a set of ideals in Z[x1 , .