O'Keefe, R. A., Finite fixed-point problems, in: J.-L. Lassez, editor, Logic Programming, Proceedings of the Fourth International Conference (1987), pp. 729--743.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....time and space complexity. 9 Damian 5 Related work A number of authors describe algorithms for computing least fixed points as solutions to program analyses using chaotic iteration, which are also adapted to compute approximation using widenings or narrowings [6] O Keefe s bottomup algorithm [14] has inspired a significant number of articles, where the convergence speed is improved using refined strategies on choosing the next iteration, or exploiting locality properties of the specifications [1,11,15] Such algorithms have also been applied to languages with dynamic control flow. Chen, ....

O'Keefe, R. A., Finite fixed-point problems, in: J.-L. Lassez, editor, Logic Programming, Proceedings of the Fourth International Conference (1987), pp. 729--743.

.... h i h i h h i h i i h i h i Dynamic neededness information functional disregard redynamic per equation f x ; x E f ; f ; x ; x j m x OE ; OE f ; f x ; x :E ; x ; x :E E f ; f ; x ; x j f ; f x ; x m [11] A more general treatment of the idea can be found in O Keefe [8]. For an overview of dataflow analyses that can be formalised as an equation system (1) see Marlowe and Ryder [12] In this paper we investigate how a neededness analysis can be used to optimise the fixpoint computation when the fixpoint is a tuple of functions rather than primitive values. Thus ....

....may safely be omitted, since the applications will not change the current fixpoint approximation. The method or basic idea described in the section is very simple. In particular, Lemma 4 is a straightforward extension to functionals of ideas underlying algorithms proposed by Kildall [11] O Keefe [8], and others. The neededness function is defined in terms of the set of subexpressions of having as the outermost function symbol, the recursive calls to : is a subexpression of Observe that is at most the number of occurrences of in . The size of the set may be ....

R.A. O'Keefe. Finite fixed-point problems. , 1987, pp. 729--743.

.... [13] Kildall [44] Tenenbaum [77] and others [36, 42, 16, 67, 43, 74] to specify and implement global program analysis problems, are important to program verification [16, 17, 20, 23] arise in complexity theory [79, 39, 40, 33, 35, 59] and are used to support high level program transformations [2, 29, 7, 49, 62, 53, 56, 11, 54, 70, 60, 81, 51]. We are further encouraged by the following facts: Any set generated by inductive definitions can also be defined as the least fixed point of a monotone function [1] Without a fixed point operator, a first order language on finite structures cannot express transitive closure [2] The ....

O'Keefe, R.. Finite Fixed-Point Problems. Logic Programming, 1987, pp. 729-743. Proc. 4th Intl. Conf. on Logic Prog..

....as a generalization of computing the values of a function from its recursive definition. This leads to a top down approach in contrast to the more usual approach inspired by the Kleene s sequence. We now compare our algorithm with a representative bottom up algorithm proposed by O Keefe [24]. O Keefe s algorithm solves finite sets of equations of the form x i = expr i (1 i n) where x 1 ; x n are distinct variables, ranging on lattices of finite depth T 1 ; T n , and expr 1 ; expr n are well typed monotone expressions possibly involving x 1 ; x n . ....

....entry and procedure exit. The abstract semantics can be expressed as system of 2n equations with 2n variables where n is the number of procedures. Since this system is small, the naive bottom up method based on the Kleene s sequence can be used as done in Mellish s algorithm. O Keefe s paper [24] was motivated by Mellish s work and provides a faster method. The framework of Nilsson has static but finer granularity: all program points are considered (clause entry and exit, and any point between two calls) This results in a system of m equations with m variables where m is the number of ....

R.A. O'Keefe. Finite Fixed-Point Problems. In J-L. Lassez, editor, Fourth International Conference on Logic Programming, pages 729--743, Melbourne, Australia, 1987.

....functions consists of processing each function at each fixpoint iteration until no changes occur. More precisely, the iteration process is complete when the abstract values have been propagated through every control flow path and the abstract descriptions of the functions have stabilized. O Keefe [O Keefe 1987], Hall and Kennedy [Hall and Kennedy 1992] and others noticed that this strategy causes needless recomputations to be performed. Indeed, a given abstract function can be reanalyzed even though the changes that triggered a new iteration do not a#ect it. One solution to remedy this situation ....

....programs. Bondorf and Jrgensen [Bondorf and Jorgensen 1994] implemented this approach in the partial evaluator Similix [Bondorf 1991] The analysis is monovariant however, and it is unclear at this time whether or not the analysis can be made polyvariant while preserving its e#ciency. O Keefe [O Keefe 1987] gave an e#cient fixpoint algorithm that is similar to Henglein s in that terms are rewritten to an intermediate form better suited for analysis. His algorithm is able to find solutions to sets of equations with arbitrary lattices of finite height. The algorithm is not parameterized with respect ....

O'Keefe, R. A. 1987. Finite fixed-point problems. In J.-L. Lassez (Ed.), Proceedings of the Fourth International Conference on Logic Programming, Melbourne, Australia, pp. 729-- 743. MIT Press, Cambridge, Massachusetts.

....in Prolog compilers to be competitive with procedural languages and the declarative nature of the language which makes it more amenable to static analysis. Considerable progress has been realised in this area in terms of the frameworks (e.g. 1, 3, 2, 7, 20, 21, 23, 30] the algorithms (e.g. [2, 6, 15, 16, 26]) the abstract domains (e.g. 4, 14, 25] and the implementations (e.g. 13, 18, 12, 29] An abstract domain which has raised much interest in recent years is the domain Prop proposed by Marriott and Sondergaard [22] The domain is intended to compute groundness information in Prolog programs. ....

R.A. O'Keefe. Finite Fixed-Point Problems. In J-L. Lassez, editor, Fourth International Conference on Logic Programming, pages 729--743, Melbourne, Australia, 1987.

....natural and important improvement in comparison to Jacobian or Gauss Seidel iteration is to note that a function only needs to be recomputed when one of its arguments has changed. Each function must be computed at least once as it may yield a non bottom result even if all its arguments are bottom [83, 108]. 6.2.1 Strongly connected components In order to optimize the algorithm for calculation of the fixpoint we analyse the fixpoint problem and try to find an order in which to compute the domain functions which is guaranteed to reduce the number of domain function applications. x 1 = f 1 (x 8 ; x ....

R. A. O'Keefe. Finite Fixed-Point Problems. In Logic Programming: Proceedings of the Fourth International Conference, MIT Press, 1987.

.... EXAMPLES We have implemented the abstract interpretation framework and the polymorphic type analysis described above in SWI Prolog on a Sun SPARC IV station running SunOS operating system. The abstract interpretation framework is implemented using O Keefe s least fixedpoint algorithm [41]. Both the abstract interpretation framework and the polymorphic type analysis are implemented as meta interpreters using ground representations for program variables and derived type parameters. The following examples present results of the polymorphic type analysis on some Prolog programs. 1 is ....

R. A. O'Keefe. Finite fixed-point problems. In J.-L. Lassez, editor, Proceedings of the fourth International Conference on Logic programming, volume 2, pages 729--743. The MIT Press, 1987.

....of continuous functions over complete lattices, which are classically computed by iterative computations starting from either the smallest element or the largest element of the lattice. Efficient computation of extremal fixed points of functions over lattices of finite height is a classical topic [17, 19]. Unfortunately, abstract interpretation also has to deal with lattices of infinite or very large height. For instance, when the values of the integer variables of a program are coded on n bits, the lattice of intervals, which is used to compute the maximum range of these variables, is of height ....

....quadratic, finding this set would be by far too costly. Hence, two distinct problems have to be solved: ffl Determine a good iteration strategy, that is, an order in which to apply the equations. ffl Determine a good set of widening points W. The first problem has been addressed by many authors [6, 16, 17, 19] but, to our knowledge, no algorithm exists for finding good sets of widening points, and authors who mention widening operators use them everywhere or improperly [18] In the next section, we introduce the notion of weak topological ordering of a directed graph and we show that this notion yields ....

R.A. O'Keefe: "Finite fixed-point problems", in J.-L. Lassez editor, Proc. of the Fourth International Conference of Logic Programming, MIT Press (1987) 729--743

....efficient calculation of fixpoints comprises an efficient representation and detection of fixpoints. Our approach to the problem is to use heuristics derived from the program to avoid non productive computations. The two approaches can be combined to result in a more efficient solution. O Keefe [8] presents an algorithm for a more general class of fixpoint problems. When applied to find the least fixpoint of Phi P;ff , O Keefe s algorithm corresponds to a variation of our algorithm 3.3 with a selection function that always selects the most recently activated clause from the agenda. This is ....

R. A. O'Keefe. Finite fixed-point problems. In J.-L. Lassez, editor, Proceedings of the fourth International Conference on Logic programming, volume 2, pages 729--743. The MIT Press, 1987.

....6 Implementation and Examples We have implemented the abstract interpretation framework and the polymorphic groundness analysis in SWI Prolog on a Sun SPARC IV station running SunOS operating system. The abstract interpretation framework is implemented using O Keefe s least fixed point algorithm [40]. Both the abstract interpretation framework and the polymorphic type analysis are implemented as meta interpreters using ground representations for program variables and mode parameters. 6.1 Examples The following examples present the results of the polymorphic groundness analysis on some ....

R. A. O'Keefe. Finite fixed-point problems. In J.-L. Lassez, editor, Proceedings of the fourth International Conference on Logic programming, volume 2, pages 729--

....to the roots, propagating the results and then iterating the process. The processing order of the components can be determined by a topological sort on the c graph. The entire process necessary to obtain the scheduling order can be seen as a static version of the algorithm presented in [31]. So, we have turned the problem of solving an I system into the problem of solving a root, that is: solving a subsystem of the initial system that corresponds to a maximal strongly connected component without a predecessor. In a root, we make a distinction between two kinds of nodes: the V nodes ....

R. O'Keefe. Finite fixed point problems. In Proc. Of the Int. Conf. on Logic Programming - ICLP'87, Melbourne, 1987. MIT Press.

....functions consists of processing each function at each fixpoint iteration until no changes occur. More precisely, the iteration process is complete when the abstract values have been propagated through every control flow path and the abstract descriptions of the functions have stabilized. O Keefe [O Keefe 1987], Hall and Kennedy [Hall and Kennedy 1992] and others noticed that this strategy causes needless recomputations to be performed. Indeed, a given abstract function can be reanalyzed even though the changes that triggered a new iteration do not affect it. One solution to remedy this situation ....

....programs. Bondorf and J rgensen [Bondorf and J rgensen 1993] implemented this approach in the partial evaluator Similix [Bondorf 1991] The analysis is monovariant however, and it is unclear at this time whether or not the analysis can be made polyvariant while preserving its efficiency. O Keefe [O Keefe 1987] gave an efficient fixpoint algorithm that is similar to Henglein s in that terms are rewritten to an intermediate form better suited for analysis. His algorithm is able to find solutions to sets of equations with arbitrary lattices of finite height. The algorithm is not parameterized with respect ....

O'Keefe, R. A. 1987. Finite fixed-point problems. In J.-L. Lassez (Ed.), Proceedings of the Fourth International Conference on Logic Programming, Melbourne, Australia, pp. 729-- 743. MIT Press.

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