B. Lang. Complete evaluation of Horn clauses: An automata theoretic approach. Rapport de Recherche 913, Institut National de Recherche en Informatique et en Automatique, Rocquencourt, France, November 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Research (NWO) under grant 00 62 518 A first attempt to solve this problem is to use predictive LR (PLR) parsing. PLR parsing allows simulta neous processing of a common prefix ct, provided that the left hand sides of the rules are the same. However, in case we have e.g. the rules A ct 3x and B where again o is not the empty string but now A 5k B, then PLR parsing will not improve the efficiency. We therefore go one step further and discuss extended LR (ELR) and common prefix (CP) parsing, which are algorithms capable of simultaneous processing of all common prefixes. ELR and CP ....

....the left hand side Ohs) A is an element from N and the right hand side (rhs) c is an element from V , where V denotes (NUT) P can also be seen as a relation on N x V . We use symbols A, B, C . to range over N, symbols a, b, to range over T, symbols X, Y, Z to range over V, symbols c, 3, to range over V , and v, w, x, to range over T . We let e denote the empty string. The notation of rules A el, A a2, with the same lhs is often simplified to A A rule of the form A e is called an epsilon rule. We assume grammars do not have epsilon rules unless stated ....

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B. Lang. Complete evaluation of Horn clauses: An automata theoretic approach. Rapport de Recherche 913, Institut National de Recherche en Informatique et en Automatique, Rocquencourt, France, November 1988.

....from the same defects as the parser on which it is based. This phenomenon has prompted the development of other, more sophisticated proof procedures for logic programs. In spite of the resemblance between logic programs and context free grammars, there are few proof procedures based on parsers [8, 17, 18, 23]. We find this contrast puzzling because logic programs can naturally represent context free grammars, e.g. using well known difference list techniques. The difference list representation of a context free grammar associates a production: A B 1 B 2 : Bn with the clause: a(X 0 ; Xn ) b 1 ....

Bernard Lang. Complete evaluation of Horn clauses: An automata theoretic approach. Technical Report 913, INRIA, November 1988.

.... this kind of situation where a small step operational semantics has been favored [11,46] The small step operational semantics of recursion is most naturally modeled by pushdown automata (PDAs) Related concepts have successfully been used for imperative languages [49,33,2,4] as well as for Prolog [39,16,17]. In order to derive an analysis engine, this framework proceeds in three stages. In the first stage, the concrete operational semantics is simulated by an abstract operational semantics [46] Ideally, while preserving the control structure, we would only abstract data and the operations on ....

....or Edge) as subscripts. Thus, we will write for instance Exit r d instead of Exit r d and Comb e (d 1 ; d 2 ) instead of Comb e (d 1 ; d 2 ) 4 The Operational Semantics: Pushdown Automata For the (small step) operational semantics of a program we rely on pushdown automata (PDAs for short) [49,33,39,16,37]. There are several ways how such pushdown automata can be defined. Lang [39] for instance follows the tradition in parsing theory. He views intermediate configurations as sequences of stack symbols and the set of transitions as left rewrite system. For a recent presentation, see also [17] ....

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Bernard Lang. Complete Evaluation of Horn Clauses: an Automata Theoretic Approach. Technical Report 913, INRIA, 1988.

....Prolog implementations in order to decrease the number of possible rules for the next move. It should also be possible to use the finite state control in advance or parallel to the filtering by unification. The idea to enrich automata by means of unification is also present in the work of B. Lang [15] and of Y. Shemesh and N. Francez [7] Lang considers logic push down automata (The name logic finite automaton was chosen on this background. He has, however, quite distinct motivations. Lang shows how unificationbased push down automata may be used for a complete evaluation of Horn clauses, ....

B.Lang, "Complete Evaluation of Horn Clauses: an Automata Theoretic Approach," Research Note, INRIA, Les Chesnay, France 1988.

....which is considered to represent configurations in the transducer. To be more precise, S 1 uses only the top, while S 2 uses also the previous element. 3 Essentially, automata that store atoms and substitutions on their stack, and use unification to apply transitions. They are due to Lang [17], which obtains an exponential reduction in complexity over the traditional resolution methods. 132 M. Vilares M. A. Alonso Correctness and completeness of S 2 is directly derived from S T since in the worst case transitions in S T depend on the first two elements in the stack. This is ....

....EXPAND for token 64, at possition #[14] Non determinism 1 at state 472. Fork of branch (1) in branches: 1) 1) 1) 2) EXPAND for token 74, at possition #[15] EXPAND for token 3, at possition #[16] Unification state 426, symbol 225, in branch (1) EXPAND for token 30, at possition #[17] EXPAND for token 5, at possition #[18] EXPAND for token 0, at possition #[19] EXPAND for token 0, at possition #[20] Figure 6: Analyzing a non deterministic Pascal program In comparison with the method applied by Ice, previously described, this approach seems to be less general. In effect: ....

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B. Lang, "Complete evaluation of horn clauses, an automata theoretic approach", Tech. Rep. 913, INRIA, Rocquencourt, France, 1988.

....for PDA computations in different domains. In context free parsing, we find the CockeKasami Younger algorithm [1] Earley s algorithm [5,7] chart parsing [6] graph structured stacks [18] In logic programming, among the main references, we find the Earley resolution [13] OLDT [17] LPDA [8], magic set [2] In abstract interpretation we have [9] Although based on the same principles, all these algorithms are expressed in specific ways and each of them has been proved correct independently. As a consequence it is rather difficult to compare them, and to identify clearly their ....

....the items most general with respect to the instance relation. An item is added to the item table only if it is not an instance of an already tabulated item. Furthermore, all the instances of the newly added item may then be removed from the table, though this may be more costly than profitable [8]. 13 t 2 t 2 t 2 t 2 Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma . ffl3 ffl2 9 ffl2 ffl1 8 ffl1 ffl0 2 ffl0 1 0ffl 5 1ffl 6 2ffl 7 1ffl ffl0 4 2ffl ffl1 3 3ffl ....

Bernard Lang. Complete evaluation of Horn clauses: an automata theoretic approach. Technical Report 913, INRIA, Rocquencourt, France, Nov. 1988.

....Push Down Automata We present in this section an example of Subsumption oriented Push Down Automata which works on Herbrand domains. They are used to evaluate Horn Clause Programs by encoding various resolution mechanisms (OLD, Bottom Up, and Earley Deduction) They were first introduced in [Lan88] with a slightly different presentation than the one used in this paper for simplicity reasons. The reader can find in Appendix A an example of an evaluation of a tiny logic program by an automaton. We assume the reader is familiar with the notions of free algebra T ( Sigma; V ) subsumption ....

Bernard Lang. Complete evaluation of Horn clauses: an automata theoretic approach. Technical Report 913, INRIA, Rocquencourt, France, nov 1988. to appear in Int. Journal of Foundations of Computer Science.

....(by cross product) the answers to G 1 and G 2 . However, non input models mainly include models developed to handle the incompleteness and termination problems of PROLOG. We may cite the tabular models[TS86, Vie87] the magic set like models[BMSU86] and our Dynamic Programming (DP) model[Lan88, BVdlC92]. Tabular models use a goal indexed look up table to store computed answers. When a goal is created which matches an existing entry, the present, as well as the future answers attached to the entry will be reused instead of starting a new computation. The match test may be an identity test (for ....

....to the new goal. Magic set models achieve a bottom up evaluation of transformed programs where prediction (i.e. top down) information has been added to restrict fact propagation. Our DP interpreter DyALog is relatively close to magic set models 2 . We use Logical Push Down Automata[Lan88] to describe non deterministic stack based computations on Herbrand domains. We are able to simulate SLD, Bottom Up, or Magic Set resolution mechanisms, but also more exotic ones. The DP interpretation of these LPDA breaks stack computations into sub computations compactly represented by ....

Bernard Lang. Complete evaluation of Horn clauses: an automata theoretic approach. Technical Report 913, INRIA, Rocquencourt, France, nov 1988. to appear in Int. Journal of Foundations of Computer Science.

....anonymous variables and the marked lines denote the lines to be extracted (in order) from the stack resulting of the unification. 2 2. 3 Logic Push Down Automata Logic Push Down Automata [LPDA] extend the Push Down Automata used in Context Free parsing by handling arguments via unification [Lan88]. They are designed to work on atom stacks and use unification for transition application. The interesting points of the LPDA are first their ability to describe stack based resolution strategies and secondly the existence of tabulationbased evaluators for them. The LPDAs use three kinds of ....

Bernard Lang. Complete evaluation of Horn clauses: an automata theoretic approach. Technical Report 913, INRIA, Rocquencourt, France, nov 1988.

....another interesting property of DyALog is offered by the possibility of describing and simulating different resolution strategies. 2. 1 A stack automata formalism Resolution strategies (OLD, bottom up, Earley deduction, can be encoded by non deterministic Logic Push Down Automata [LPDA][Lan88]. More precisely, during the compilation pass, an input program (including a query) is given to a compilation scheme (associated to a resolution strategy) which returns an LPDA A ; the transitions of A denote the valid computational steps allowed by the chosen strategy. We have three kinds of ....

Bernard Lang. Complete evaluation of Horn clauses: an automata theoretic approach. Technical Report 913, INRIA, Rocquencourt, France, nov 1988. to appear in Int. Journal of Foundations of Computer Science.

....recovery, as is shown in Fig. 1. These are: ffl Total recovery. The idea is to detect when the parsing process becomes independent of the modification. 7 essentially, automata that store atoms and substitutions on their stack, and use unification to apply transitions. They are due to Lang [19], which obtains an exponential reduction in complexity over the traditional resolution methods. ffl Partial recovery. In this case, recovery applies to all trees of the shared forest corresponding to a part of the remaining input. To be more exact, we distinguish three cases in relation to this, ....

B. Lang, "Complete evaluation of Horn Clauses, an automata theoretic approach", Tech. Rep. 913, INRIA, Rocquencourt, France, 1988.

....computation to a useful part of the search space. We propose a third approach which, we believe, generalizes and explains Magic Set transformations. Rather than mixing into a single logic program, both data and resolution mechanisms, we introduce a distinct formalism of Logic Push Down Automata [Lan88a, Lan91] which we believe to be an appropriate operational formalism to express logic programs computations. The key idea is that the operational proof tree exploration strategy (e.g. bottom up, top down, predictive, is naturally expressed, as for context1 free grammars (CFG) by a non deterministic ....

....where bindings are stored while refuting the head of fl k . Multiple definitions of variable tuples X k:i may be given, but to simplify this presentation 2 , we use X k:i = X k = Var(fl k ) Var(A k:0 ) Var(A k:1 ) Var(A k:n k ) 1 These predicates were introduced in [Lan88b, Lan88a], and are similar to the supplementary predicates of the Magic Set construction [BR87] They are the logic counterpart of the dotted rules of Earley s algorithm [Ear70] which are the crucial step for achieving the O(n 3 ) CF parsing complexity. 2 In section 4, we keep in each tuple only the ....

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B. Lang. Complete evaluation of Horn clauses: an automata theoretic approach. Technical Report 913, INRIA, Rocquencourt, France, nov 1988.

.... E E 0 , iff there is a unifier s of AA 0 and BD , and a unifier s 0 of A 0 s and E. This produces the item Css 0 E 0 s 0 . 3. 2 Top down compilation of DC programs into LPDAs Given a DC program, many different compilation schemata may be used to build a corresponding LPDA [Lan 88c]. We give here a very simple and unoptimized top down construction. The DC program to be compiled is composed of a set of clauses fl k : A k;0 : A k;1 ; A k;n k , where each A k;i is a logical literal. The query is assumed to be the head literal A 0;0 of the first clause fl 0 . The ....

....Prolog interpretation of the original DC program. 3. 3 A very simple example The following example has been produced with a prototype implementation realized by Eric Villemonte de la Clergerie and Alain Zanchetta [VilZ 88] This example, as well as the top down construction above, are taken from [Lan 88c]. The definite clause program to be executed is given in figure 11. Note that a search for all solutions in a backtrack evaluator would not terminate. 7 If k = k 0 then we rename the variable in t k 0 since the transition corresponds to the use of two distinct variants of the clause fl k . ....

Lang, B. 1988 Complete Evaluation of Horn Clauses: an Automata Theoretic Approach. INRIA Research Report 913. To appear in the International Journal of Foundations of Computer Science.

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