M.H. Srensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J.W. Lloyd, editor, Logic Programming: Proceedings of the 1995 International Symposium, pages 465--479. MIT Press, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(e.g. generalization algorithms) that repeatedly traverse metacode will suffer from a significant computational degrading effect as can be seen from this example. The area of online program transformation (e.g. partial deduction, supercompilation) has seen a recent flurry of activity, e.g. [2, 5, 19, 21, 22, 26, 27]. Recently well quasi orders in general, and homeomorphic embedding in particular, 3 have gained popularity to ensure termination of program analysis, specialization and other program transformation techniques because well quasi orders are strictly more powerful [18] than a large class of ....

.... the sets [ cval n 1 ] n and [ cval n 2 ] n , but it should be as specific as possible (to minimize the loss of information) These concepts can be formalized by extending familiar concepts of mostspecific generalization, as applied previously to program specialization in e.g. [22], to level index constructors and elevated metavariables. 4.1 Elevation and Generalization Not only the positioning of metavariables by their degree is important in a hierarchy of metasystems (cf. Example 6) but also the domain specified by the elevation index is crucial. Elevation can be seen ....

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M.H. Srensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J.W. Lloyd, editor, Logic Programming: Proceedings of the 1995 International Symposium, pages 465--479. MIT Press, 1995.

....to (d3) IR[let x = e in e 0 ] let x = e in IR[e 0 ] Of course, too much generalisation can prevent interesting transformation from occurring. Too little, and it can be hard to ensure termination. The question of when and where to generalise is beyond the scope of this paper. See e.g. [36,31,32] [36] 7.3 Adding Laws In principle, any strong improvement laws e 0 s e 1 , can be added, and correctness follows providing that the law is domain independent of the definitions thus transformed. As a simple example, we can add properties about the list concatenate function 42 append such ....

M H Srensen and R Gluck. An algorithm for generalization in positive supercompolation. In International Logic Programming Symposium (ILPS'95), Portland, Oregon, 1995. MIT Press.

....approaches to ensure local termination exist. The methods in [6, 50, 49, 46] are based on well founded orders, inspired by their usefulness in the context of static termination analysis (see e.g. 13, 10] Instead of well founded ones, well quasi orders can be used [4, 58] Homeomorphic embedding [63, 40] on selected atoms has recently gained popularity as the basis for such an order. These techniques ensure termination, while at the same time allowing unfolding related to the structural aspect of the program and goal to be partially deduced, by for instance allowing the consumption of relevant ....

M. H. Srensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, Proceedings of ILPS'95, the International Logic Programming Symposium, pages 465--479, Portland, USA, December 1995. MIT Press.

....is guaranteed and results are satisfactory This problem has been tackled from two (until now) largely separate angles: the so called off line versus on line approaches. Partial evaluation of functional programs [8, 31] has mainly stressed the former, while supercompilation of functional [67, 68, 65] and partial deduction of logic programs [21, 64, 4, 6, 52, 54] have concentrated on on line control. Some exceptions are [69, 56, 38, 33] Basically, within the off line approach, an analysis phase (by hand and or automatically) prior to the specialisation proper, provides annotations that ....

....e 2 ; in V there are i j such that e i V e j . We also say that V is a well quasi order (wqo) on V . An interesting wqo is the homeomorphic embedding relation . It has been adapted from [16, 17] where it is used in the context of term rewriting systems, for use in supercompilation in [65]. Its usefulness as a stop criterion for partial evaluation is also discussed and advocated in [50] Some complexity results can be found in [66] also summarised in [50] Recall that expressions are formulated using the alphabet A P which we implicitly assume underlying the programs and queries ....

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M. Srensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. Lloyd, editor, Proceedings of ILPS'95, the International Logic Programming Symposium, pages 465--479, Portland, USA, December 1995. MIT Press.

....the essential ingredients of the partial deduction method we use in this paper. Experiments were conducted using the ecce automatic partial deduction system [26] 2.1 Local control As a starting point for local control, we use the one proposed in Section 4. 2 of [17] partially imported by [34, 24]. The following definitions are taken from that paper. Definition 2.1 Given an SLD tree . Let G = A 1 : A n be a goal in , Am the selected atom in G, A A 0 1 : A 0 k a clause of P such that is an mgu of Am and A. Then in (A 1 : A 0 1 : A 0 k : A n ....

M. H. Srensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J.W. Lloyd, editor, Proceedings ILPS'95, pages 465--479, Portland, Oregon, December 1995. MIT Press.

....will be required. This analysis phase provides annotations which then guide the specialisation phase proper, often to the point of making it almost trivial. Partial evaluation of functional programs ( 13, 31] has mainly stressed offline approaches, while supercompilation of functional ([70, 67]) and partial deduction of logic programs ( 21, 64, 6, 9, 54, 55, 39, 48, 49, 34] have concentrated on on line control. Some exceptions are [57, 42, 38] The main reason for using the off line approach is to achieve effective selfapplication ( 32] But the off line approach is in general also ....

M. H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, Proceedings of ILPS'95, the International Logic Programming Symposium, pages 465--479, Portland, USA, December 1995. MIT Press.

....transformers. The insight that various transformers are very similar has led to the exchange of many ideas between researchers working on different transformers, especially techniques to ensure termination. Variations of one technique, used to ensure termination of positive supercompilation [35], have been adopted in partial deduction [23] conjunctive partial deduction [16] Turchin s supercompiler [41] and partial evaluation of functional logic programs [1] While the technique is fairly easily transported between different settings, a separate correctness proof has been given in each ....

.... deduction [16] compiling control [10] loop absorption [31] partial evaluation of functionallogic languages [1] unfold fold transformation of functional programs [11] unfold fold transformation of logic programs [38] tupling [4, 29] supercompilation [39, 40] positive supercompilation [18, 35], generalized partial computation [15] deforestation [42] and online partial evaluation of functional programs [43, 33, 21] Although offline transformers (i.e. transformers making use of analyses prior to the transformation to make changes in the program ensuring termination) may fit into the ....

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M.H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J.W. Lloyd, editor, Logic Programming: Proceedings of the 1995 International Symposium, pages 465--479. MIT Press, 1995.

....analysis technology do not directly contribute to improving the control of partial deduction and the quality of specialised code produced by partial deduction systems. The work of this chapter aims to bridge this gap. Moreover, the control described in [9, 36, 35] as well as the more recent [43, 29] are inherently online, meaning that they are much slower than offline approaches and that they are not based on a global analysis of the program s behaviour which enables control decisions to be taken before the actual specialisation phase itself. Offline approaches to local control of partial ....

....Embedding It turns out that, for the online approach, well founded orders as used in Sections 6.1 and 6.2 are sometimes too rigid or (conceptually) too complex. Recently, well quasi orders have therefore gained popularity to ensure online termination of program manipulation techniques ([5, 42, 43, 21, 24, 31, 1, 26, 45, 32]) Definition 1 (well quasi order) An ordered set S( is called well quasi ordered iff for any infinite sequence e 1 ; e 2 ; of elements of S, there exist elements e i and e j with i j such that e i e j . 2 The additional power of well quasi orders stems from the fact that ....

M. H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, Proceedings of ILPS'95, the International Logic Programming Symposium, pages 465--479, Portland, USA, December 1995. MIT Press.

....in the next section. However, it has been felt by several researchers that well founded orders are sometimes too rigid or (conceptually) too complex in an online setting. Recently, well quasi orders have therefore gained popularity to ensure online termination of program manipulation techniques [6, 69, 72, 46, 47, 21, 35, 2, 37, 1, 76, 1, 13]. Unfortunately, this move to well quasi orders has never been formally justified nor has the relation to well founded approaches been investigated. We strive to do so in the first part of this paper and will actually prove that a rather simple well quasi approach the homeomorphic embedding ....

....6 we will actually develop wbr s which are not wqo s. An interesting wqo is the homeomorphic embedding relation Theta, which derives from results by Higman [27] and Kruskal [36] It has been used in the context of term rewriting systems in [14, 15] and adapted for use in supercompilation [75] in [72]. Its usefulness as a stop criterion for partial evaluation is also discussed and advocated in [52] Some complexity results can be found in [74] and [24] also summarised in [52] The following is the definition from [72] which adapts the pure homeomorphic embedding from [15] by adding a ....

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M. H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, Proceedings of ILPS'95, the International Logic Programming Symposium, pages 465--479, Portland, USA, December 1995. MIT Press.

....data structures with a narrowing based partial evaluator. Thus, we prove that the partial evaluator can achieve in a principled, non ad hoc way, the same transformation effect as some (postunfolding) techniques that eliminate intermediate functions in the driving approach to program specialization [14, 36, 37]. The preference for deterministic computations is a good heuristic to avoid code explosion which is also comparable to the determinism based criterium of [12] that explores maximal deterministic paths. The structure of the paper is as follows. Basic definitions are given in Section 2. Section 3 ....

....Figure 1 for the sequence of terms: q append append x y z append x y . Note that append has been abbreviated to a in the picture. Then we get the following residual program : append append nil y z append y z append append x x y z x append append x y z append nil z z append y y z y append y z [37] (6 ) 1 1 The embedding relation is a well quasi ordering of the set (6 ) for finite 6 [2] that is, any infinite sequence of terms with a finite number of operators is self embedding, i.e. there are numbers with and . In order to avoid an infinite sequence of diverging calls, we ....

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M.H. Sørensen and R. Gluck. An Algorithm of Generalization in Positive Supercompilation. In J.W. Lloyd, editor, , pages 465--479. The MIT Press, Cambridge, MA, 1995.

....PD) However, a number of concepts have been generalized for dealing with features such as nested function calls, eager and lazy evaluation strategies and the standard optimization based on deterministically reducing functions. Control issues are managed by using standard techniques as in [33, 38]. 4] distinguishes a local level of control and a global level of control. At the local level, finite) narrowing trees for (nested) function calls are constructed. At the global level, the calls extracted from the leaves of the local trees are considered for the next iteration of the algorithm, ....

....tree for sorted bits(x : xs) 1 x and the call sorted bits(x : xs) 1 x. The lazy narrowing tree depicted 1 in Fig. 1 is built up by using the nonembedding unfolding rule of [3] which expands derivations while new redexes are not greater (with the homeomorphic embedding ordering, see e.g. [4, 38]) than previous, comparable redexes in the branch (i.e. redexes with the same outermost function symbol) From this tree, we can identify two main weaknesses of the plain NPE algorithm: The rightmost branch stops because the leftmost redex sorted bits(x 0 : xs 0 ) of the leaf embeds the ....

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M.H. Sørensen and R. Gluck. An Algorithm of Generalization in Positive Supercompilation. In J.W. Lloyd, editor, Proc. of ILPS'95, pages 465--479. The MIT Press, Cambridge, MA, 1995.

....trees and, what is of even greater significance, how to stop constructing them. The unfolding rule can be set up by typing the command unfold. Currently, there are three options: ffl embgoal: the nonembedding unfolding rule, which uses the homeomorphic embedding ordering (see, e.g. AFV98b, SG95] to stop the construction of the narrowing trees; ffl embredex: this unfolding rule uses a dynamic narrowing selection strategy to determine the set of positions to be unfolded by a don t care selection within a conjunction. It implements the dependency clash test of [AAF 98] using ....

M.H. Sørensen and R. Gluck. An Algorithm of Generalization in Positive Supercompilation. In J.W. Lloyd, editor, Proc. of ILPS'95, pages 465--479. The MIT Press, Cambridge, MA, 1995.

....the interactions that may occur while evaluating these subgoals. For functional and imperative programs, various transformation methods have been proposed in the literature, which can be classified under this category, e.g. finite differencing [40] deforestation [59] and super compilation [56, 51]. Loop merging, in Section 2.1.3, also called loop fusion by Debray [14] is one of the transformation techniques that was proposed for improving programs that were written in compositional style. This technique transforms the program for a relation that is defined as the composition of two ....

M.H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In: J. Lloyd (ed), Proc. of ISLP'95, pp. 465--479. The MIT Press, 1995. BIBLIOGRAPHY 139

....which is analogous to unfolding. The driving process generates a tree of configurations, or a graph if we identify nodes with similar configurations. This graph is called graph of states and transitions. A similar model is the partial process tree used in the positive supercompilation technique [47, 48]. There is, however, a difference between unfolding a la Burstall and Darlington and driving a la Turchin: by unfolding we replace an expression which matches the left hand side of an equation by the corresponding instance of the right hand side, whereas by driving a sort of unification process, ....

....necessary to construct a symbolic computation model which is more informative than the one based on the m dags used in Section 2. This more informative model is based on the construction of a set of trees which can be obtained by an extension of the positive supercompilation technique described in [47]. The extension is motivated by the fact that in positive supercompilation it is not possible to directly exploit the interactions among different function calls because they belong to different branches of the process trees. We will not give here the formal rules for the construction of this more ....

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M. H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, Proceedings of the 1995 International Logic Programming Symposium (ILPS '95), pages 465--479. MIT Press, 1995.

.... on (S rensen, Gluck and Jones, 1994) and (S rensen, 1994a) It belongs to a line of work which aims at a better understanding of supercompilation and its relation to other program transformers (Gluck and Klimov, 1993; Jones, 1994; Gluck and J rgensen, 1994; Gluck and S rensen, 1994; Nielsen and S rensen, 1995). The remainder of the paper is organized as follows. In Section 2 the object language is presented. In Section 3 we introduce the KMP test which allows us to assess the amount of information that a program transformer maintains. In Section 4 we present some preliminaries which are used in Section ....

....be present in its left side. 3 We have made a number of syntactic choices. First, we use case expressions rather than functions defined by patterns, following (Wadler, 1990) The advantage of case expression is that the self contained syntax is easier to analyze. Moreover, the 2 Recent work (S rensen and Gluck, 1995) develops generalization for the positive supercompiler; this is beyond the scope of this paper. 3 We use the shorthand notation (x : xs) and [ for the constructors Cons x xs and N il, respectively. For x1 : x2 : xn : we write [x1 ; x2 ; xn ] or sometimes x1x2 : xn ....

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Sørensen, M.H. and Gluck, R. 1995. An algorithm of generalization in positive supercompilation.

....with the ancestor atoms from which the current atom descends (via resolution) However, in an on line setting, well founded orders are sometimes too rigid or too complex. Recently, well quasi orders have therefore gained popularity to ensure on line termination of program manipulation techniques [3, 41, 43, 26, 27, 13, 18, 1, 20, 46]. Indeed, as we will see below, well quasi orders are often much more flexible than well founded orders in an on line context. We start examining them in the next section. 2 Well quasi orders and homeomorphic embedding From now on, we suppose familiarity with basic notions in logic programming ....

....1 V v 2 and v 1 2 V v 2 , is also a wqo on V . An interesting wqo is the homeomorphic embedding relation Theta, which derives from results by Higman [15] and Kruskal [19] It has been used in the context of term rewriting systems in [6, 7] and adapted for use in supercompilation ( 45] in [43]. Its usefulness as a stop criterion for partial evaluation is also discussed and advocated in [31] Some complexity results can be found in [44] and [14] also summarised in [31] The following is the definition from [43] which adapts the pure homeomorphic embedding from [7] by adding a ....

[Article contains additional citation context not shown here]

M. H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, Proceedings of ILPS'95, the International Logic Programming Symposium, pages 465--479, Portland, USA, December 1995. MIT Press.

....1 ; xn ) T rue) i : 0; repeat i 1 = extend( i ; P 0 ) ff ; i = i 1; until i = i Gamma1 return i P 0 : the program obtained from extracting the set of statements of i . The ff operator generalises the m tree, using the generalisation and folding procedures of [SG95, GS96], extended to analyse the constraints when performing these generalisations. The global and local control of the algorithm is beyond the scope of this abstract; this will be discussed in the seminar. 3 Example: Match We have implemented the partial evaluator for Escher (in Godel) and an initial ....

M. H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, International Logic Programming Symposium, page to appear. MIT Press, 1995.

....using existing Prolog compilers. The transformational elimination of intermediate data structures via function composition, is an old idea (see, for instance, 2] However, the design of effective transformation algorithms, such as the deforestation algorithm [21] and positive supercompilation [16], which perform this elimination in an automatic way, needs further study. Our transformation technique also realizes what in a functional setting is called the tupling strategy [2, 11] that is, it may be used to avoid multiple traversals of data structures by defining new predicates which ....

M. H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, Proceedings of the 1995 International Logic Programming Symposium (ILPS '95), pages 465--479. MIT Press, 1995.

....of big step semantics derivations and its use of both inherited (downwards flowing) and synthesized (upwards flowing) attributes. However, the latter characteristics do not suit on line partial evaluation 1 algorithms that are oriented towards a term rewriting (small step) framework, e.g. [10, 12, 13, 20, 27, 28, 31, 33]. And simulations of two dimensional big step derivations by term rewriting on one dimensional representations destroy the beauty and many of the advantages of the big step framework [15, 23] In this paper, we develop an on line partial evaluator that lets a big step semantics be evaluated in ....

M. H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J.W. Lloyd, editor, Logic Programming: Proceedings of the 1995 International Sympo sium, pages 465--479. MIT Press, 1995.

....The analysis phase provides annotations which then guide the control aspect of the proper specialisation phase, often to the point of making it completely trivial. Partial evaluation of functional programs [5, 17] has mainly stressed off line approaches, while supercompilation of functional [52, 51, 14] and partial deduction of logic programs [11, 50, 1, 2, 40, 42, 24, 32] have concentrated on on line control. The main reason for using the off line approach is to make specialisation more amenable to effective self application [18] On line methods, however, usually obtain better specialisation, ....

....e 2 ; in V there are i j such that e i V e j . An interesting wqo is the homeomorphic embedding relation Theta, which derives from results by Higman [16] and Kruskal [22] It has been used in the context of term rewriting systems in [8] and adapted for use in supercompilation ( 52] in [51]. Its usefulness as a stop criterion for partial evaluation is also discussed and advocated in [37] Some complexity results are summarised in[37] The following is the definition from [51] which adapts the pure homeomorphic embedding from [8] by adding a rudimentary treatment of variables. ....

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M. H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. W. Lloyd, editor, Proceedings of ILPS'95, the International Logic Programming Symposium, pages 465--479, Portland, USA, December 1995. MIT Press.

.... incorporates part of the unfold fold technology [7, 39, 30] and bears some relationship to automated methods proposed in [33, 34, 35] It also approaches more closely techniques for the specialisation and transformation of functional programs, such as deforestation [42] and supercompilation [40, 36]. Especially the latter constituted, together with unfold fold transformations, a source of inspiration for the conception and design of conjunctive partial deduction. In the present paper, we endeavour to put conjunctive partial deduction on trial. We use a large set of small and medium size ....

....termination, tend not to worsen a program, but are often somewhat too conservative. ffl well founded measures [8, 28] Imposing some (essentially) well founded order on selected atoms guarantees termination, but, on its own, can lead to overly eager unfolding. ffl homeomorphic embedding [36, 24] Instead of well founded ones, well quasi orders can be used [37, 2] Homeomorphic embedding on selected atoms has recently gained popularity as the basis for such an order. At the global control level, closedness [26] is ensured and the degree of polyvariance is decided: For which atoms should ....

M.H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J. Lloyd, editor, Proceedings ILPS'95, pages 465--479, Portland, Oregon, December 1995, MIT Press.

....exists an extensive literature on this topic in classical partial deduction, see e.g. 2, 4, 25] We propose a method which is sophisticated enough to usually give good results for the kind of transformations we have in mind. The following homeomorphic embedding relation Theta is adapted from [30, 20]. As usual, e 1 OE e 2 denotes that e 2 is a strict instance of e 1 . Definition18. strict homeomorphic embedding) Let X;Y range over variables, f over functors, and p over predicates. Define Theta on terms and atoms: X Theta Y s Theta f(t1 ; tn) s Theta t i for some i f(s1 ; ....

....is present in the notion of minimal foldable upper portion of an unfolding tree. Turchin s supercompiler [32, 14] can also do deforestation. Supercompilation performs driving (normal order unfolding and unification based information propagation) and generalisation (a form of abstraction) [33, 30]. Tree structures are used to record the history of configurations [12] The connection between driving and classical partial deduction was established in [13] Recently, a transformation scheme has been proposed for functional logic languages based on an automatic unfolding algorithm that builds ....

M.H. Sørensen, R. Gluck. An algorithm of generalization in positive supercompilation. In [23], 465--479.

....more historical details, see Turchin s personal account [95] Despite these remarkable contributions, supercompilation has not found recognition outside a small circle of experts. This paper gives a gentle introduction to the principles of supercompilation in terms of a positive supercompiler [34, 73, 74, 75, 76] comprising two components, driving (Sect. 2) and generalization (Sect. 3) The supercompiler is compared to related program transformers (Sect. 4) and put into the larger perspective of metacomputation (Sect. 5) We give references to the literature throughout the text, which can hopefully be ....

....be present in its left side. Figure 2 shows the function a for appending two lists, using the short notation [ and (x : xs) for the list constructors nil and cons xxs. a [ ys 4 = ys a (x : xs) ys 4 = x : a xs ys Fig. 2. Example program append. Our language contained case expressions in [76, 74], because the connection between positive supercompilation and deforestation is clearest for case expressions. However, g functions [21] lead to a simpler presentation of generalization. 2 Driving Driving takes a term and a program and constructs a possibly infinite process tree, representing all ....

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M.H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J.W. Lloyd, editor, Logic Programming: Proceedings of the 1995 International Symposium, pp. 465--479. MIT Press, 1995.

....Sometimes, splitting a goal into subparts is still necessary to guarantee termination, but, in general, it is avoided when possible. The technique approaches more closely techniques for the specialisation and transformation of functional programs, such as deforestation [77] and supercompilation [75, 72]. Especially the latter constituted, together with unfold fold transformations, a source of inspiration for the conception and design of conjunctive partial deduction. 4 The present paper offers an up to date, comprehensive, and uniform presentation of conjunctive partial deduction as developed ....

....heuristic, apart from not guaranteeing termination, are often somewhat too conservative. ffl well founded measures [11, 58] Imposing some (essentially) well founded order on selected atoms guarantees termination, but, on its own, can lead to overly eager unfolding. 19 ffl homeomorphic embedding [72, 50] Instead of well founded ones, well quasi orders can be used [67, 4] Homeomorphic embedding on selected atoms has recently gained popularity as the basis for such an order. Global Control At the global control level, closedness [55] is ensured and the degree of polyvariance is decided: For which ....

[Article contains additional citation context not shown here]

M.H. Sørensen, R. Gluck. An algorithm of generalization in positive supercompilation. In J. Lloyd (ed.), Proceedings ILPS'95, 465--479, MIT Press, 1995.

....that introducing a technique to ensure termination of deforestation requires some motivation, since there are several transformers with similar aims and effects as deforestation with known techniques for ensuring termination. Most notably, termination techniques exist for positive supercompilation [60], partial evaluation [30] and partial deduction [34] The techniques for positive supercompilation and partial deduction are online; that is, the efforts taken to ensure termination are carried out during the transformation process. In contrast, the technique for deforestation described in this ....

M.H. Sørensen and R. Gluck. An algorithm of generalization in positive supercompilation. In J.W. Lloyd, editor, Logic Programming: Proceedings of the 1995 International Symposium, pages 465--479. MIT Press, 1995.

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