K. Marriott and H. Sndergaard. Precise and Efficient Groundness Analysis for Logic Programs. ACM Letters on Programming Languages and Systems 2(1-- 4):181--196, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....structure of constraint systems can be weakened to include more analyses, as discussed in Section 6. 5.1. An Example: Rigidity Analysis A number of researchers have considered abstract interpretation techniques for the analysis of ground dependences for pure logic programs (see, for examples, [5, 21, 40, 58, 59]) this notion can be generalized to that of rigidity with respect to size measures, or norms , for terms. Intuitively, a norm is a function from the set of terms to the set of natural numbers such that the norm of a term depends only on the its principal functor and (some of) its subterms. In ....

....is a special case of rigidity under the selection of the norm size, since Vrel size (t) iff t is ground. Proposition 5.5. Vrel S is a morphism of term systems. Marriott and Sndergaard have proposed an elegant domain, named Prop, to represent ground dependences among arguments in atoms ([21, 56, 58, 59]) This domain can be expressed as an instance of our framework using the algebra of propositional formulae with disjunction. Let P rop = P rop V ; true; false ; 9X ; t) t ) X V ;t;t 0 2V [f;g be the algebra of possibly existentially quantified disjunctions of formulae, defined on ....

K. Marriott and H. Sndergaard. Precise and Efficient Groundness Analysis for Logic Programs. ACM Letters on Programming Languages and Systems 2(1-- 4):181--196, 1993.

....X is in all of the models and if Y is in a model (becomes ground) then so is Z. Notice that = fX=ag is not described by as fX=a; Y=ag is a further instantiation of and fX; Y g is not a model of . P os satisfies the algebraic properties required of an abstract domain [9] See also [23] for more details. A simple way of implementing a P os based groundness analysis is described in [7] and is illustrated in Figure 1. For the purpose of this paper it is sufficient to understand that the problem of analyzing the concrete program (on the left part of Figure 1) is reduced to the ....

....the problem of analyzing the concrete program (on the left part of Figure 1) is reduced to the problem of computing the concrete semantics of the abstract program (in the middle and on the right) The result is given at the bottom of the figure. For additional details of why this is so, refer to [7, 10, 23]. The least model of the abstract program (e.g. computed using metainterpreters such as those described in [5, 7] is interpreted as representing the propositional formula x 1 x 2 and (x 1 x 2 ) x 3 for the atoms rotate(X 1 ; X 2 ) and append(X 1 ; X 2 ; X 3 ) respectively. This illustrates a ....

K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1-4):181--196, 1993.

....1. However, this time, no calls to unfold funnyapp have to be evaluated during specialisation. 3.3 Automation To weave the step by step analysis sketched above in a single analysis, a special purpose tool has to be built. We implemented a system based on the abstract domain POS, also called PROP [24]. It describes the state of the program variables by means of positive boolean formulas, i.e. formulas built from ; and . Its most popular use is for groundness analysis. In that case, the formula X expresses that the program variable X is (definitely) bound to a ground term, X Y expresses ....

K. Marriott and H. Sndergaard. Precise and Efficient Groundness Analysis for Logic Programs. ACM Letters on Progr. Lang. and Syst., 2(1--4):181--196, 1993.

....X is in all of the models and if Y is in a model (becomes ground) then so is Z. Notice that = fX=ag is not described by as fX=a; Y=ag is a further instantiation of and fX; Y g is not a model of . P os satisfies the algebraic properties required of an abstract domain [9] See also [23] for more details. A simple way of implementing a P os based groundness analysis is described in [7] and is illustrated in Figure 1. For the purpose of this paper it is sufficient to understand that the problem of analyzing the concrete program (on the left part of Figure 1) is reduced to the ....

....the problem of analyzing the concrete program (on the left part of Figure 1) is reduced to the problem of computing the concrete semantics of the abstract program (in the middle and on the right) The result is given at the bottom of the figure. For additional details of why this is so, refer to [7, 10, 23]. The least model of the abstract program (e.g. computed using metainterpreters such as those described in [5, 7] is interpreted as representing the propositional formula x 1 x 2 and (x 1 x 2 ) x 3 for the atoms rotate(X 1 ; X 2 ) and append(X 1 ; X 2 ; X 3 ) respectively. This illustrates a ....

K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1-4):181--196, 1993.

....instantiation information with respect to a given symbolic norm. Size relations are obtained as described in Section 5. Instantiation information is obtained by performing any standard groundness analysis on the abstract program, such as that based on the Pos domain of positive Boolean functions ([35,11,9]) The operations on this domain consist of the standard operation on both domains. We denote by mgu ff the abstract most general unifier over this combined domain. Equivalence of syntactic objects is denoted by . The analysis is for any initial atomic goal which is described by an initial goal ....

K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1-- 4):181--196, 1993.

....structure of constraint systems can be weakened to include more analyses, as discussed in Section 6. 5.1. An Example: Rigidity Analysis A number of researchers have considered abstract interpretation techniques for the analysis of ground dependences for pure logic programs (see, for examples, [5, 21, 40, 58, 59]) this notion can be generalized to that of rigidity with respect to size measures, or norms , for terms. Intuitively, a norm is a function from the set of terms to the set of natural numbers such that the norm of a term depends only on the its principal functor and (some of) its subterms. In ....

....is a special case of rigidity under the selection of the norm size, since Vrel size (t) iff t is ground. Proposition 5.5. Vrel S is a morphism of term systems. Marriott and Sndergaard have proposed an elegant domain, named Prop, to represent ground dependences among arguments in atoms ([21, 56, 58, 59]) This 27 domain can be expressed as an instance of our framework using the algebra of propositional formulae with disjunction. Let P rop = P rop V ; true; false; 9X ; t) t 0 ) X V ;t;t 0 2V [f;g be the algebra of possibly existentially quantified disjunctions of formulae, defined on ....

K. Marriott and H. Sndergaard. Precise and Efficient Groundness Analysis for Logic Programs. ACM Letters on Programming Languages and Systems 2(1-- 4):181--196, 1993.

....D abs , rather than the concrete domain of computation D conc . A concretization function conc : D abs Gamma D conc maps each abstract domain element to the concrete domain element it describes. Abstract interpretation of Horn programs has been studied by various researchers (see, for example, [4, 9, 10, 25, 26, 27, 29]) Given the structural relationships between the abstract and concrete domains, the notion of Omega in D abs can be derived without much trouble from the notion of instance in D conc . For example, one plausible definition is the following: given elements s and t in D abs , s is an instance of ....

....= ffi k (mgi(t 1 ; t 2 ) where mgi(t 1 ; t 2 ) is the most general instance of t 1 and t 2 (in the usual first order sense) if one exists, otherwise; and ffi k ( ffl The normalization operator ] is essentially the identity function: S ] S n f g. Example 6. 2 (Groundness Analysis [10, 11, 26, 27, 23]) First, consider a very simple groundness analysis that uses the special constant g to represent terms known to be definitely ground, and the constant any to represent the set of all terms of the language [11, 23] For notational convenience, define the predicate is ground as follows: Given a ....

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K. Marriott and H. Sndergaard, "Precise and Efficient Groundness Analysis of Logic Programs ", ACM Letters on Programming Languages and Systems vol. 2 nos. 1--4, March--Dec. 1993, pp. 181--196.

....Many analyses for logic programs use the lattice of positive Boolean functions ordered by implication (P os) to express dependencies between program variables. One of the best known applications of this type of analysis involves reasoning about groundness dependencies in logic programs (e.g. [10]) For example, in this context the formula x (y z) is interpreted to describe a program state in which x is definitely bound to a ground term and there exists a grounding dependency such that whenever z becomes bound to a ground term then so does y. For a predicate p=n (of arity n) a program ....

K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Lett. Program. Lang. Syst., 2(4):181--196, 1993.

....behavior. Formal justification of program analyses is reduced to proving conditions on the relation between data and data descriptions and on the elementary operations defined on the data descriptions. The domain Prop of propositional formulae is proposed in [30, 14] and further discussed in [31] as a means to describe substitutions and as a basis for groundness analysis defined in terms of abstract interpretation. A similar domain of dependency formulae is introduced in [16] For example, the formula X Y describes any substitution which binds both X and Y to ground terms. Likewise, X ....

K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, ACM-LOPLAS, 2(1--4):181---196, March--December 1993.

....which characterize the set of possible directed types of a given program and hence generalize the notion of directional types. The key idea is to express the dependencies between the types of a predicates arguments similar to the way groundness dependencies are expressed in the P os domain (e.g. [15, 36]) Type dependencies are first mentioned in [31] and are reminiscent of the implication types of [39] As a typical example, consider the following directed type for the well known append relation which specifies that the result of concatenating two lists of elements of type is a list of ....

....when they involve an abstract unification algorithm which is idempotent and commutative and a least upper bound function which is additive. It is interesting to note that our definitions satisfy these properties almost trivially and hence our abstract domain is condensing in the terminology of [36]. Idempotence, means that performing the unification algorithm twice does not give more information. In other words, the unification algorithm makes use of all of the information in the type descriptions in one shot . Commutativity means that the order in which we solve ACI equations does not ....

K. Marriott and H. Sndergaard. Precise and Efficient Groundness Analysis for Logic Programs. ACM Letters on Programming Languages and Systems (LOPLAS), 2(4):181--196, March 1993.

....relation between X and Y is established by unifying c 2 with anc(X ; Y ) in the body of c 1 . Thus, this relation propagates the groundness from Y (which is derived from R) to X . In order to solve this problem, a new abstract domain, called Def , has been introduced in the beginning of 90 (cf. [47]) Def consists of definite propositional formulae. For instance, an implicational formula X : g Y : g represents the relational information that X is ground iff Y is ground. By using Def , we get a more precise groundness analysis, namely hanc(X ; Y ) X : g Y : gi for c 1 and hanc(X ; Y ) ....

....a separated abstract domain. Applications. In the field of groundness logic program analysis, we show that the well known abstract domain of definite propositional formulae Def , introduced by Dart in [24] for studying groundness in deductive databases, and then used by Marriott and Sndergaard in [47] for grounddependency analysis of logic programs, is isomorphic to the reduced relative power having the simpler Jones and Sndergaard s domain for (nonrelational) plain groundness analysis (denoted Gr , cf. 43] as base and exponent. Our reduced relative power is here necessary to handle a ....

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K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Lett. Program. Lang. Syst., 2(1-4):181--196, 1993.

.... domains to compare the expressive power of some well known abstract domains for ground dependency analysis of logic languages, like Def , Pos and their common disjunctive completion IP(Def ) cf. Armstrong et al. 1997; Cortesi et al. 1996; Fil e and Ranzato 1998; Giacobazzi and Ranzato 1998; Marriott and Sndergaard 1993]) More in detail, considering a standard bottom up least fixpoint semantics of reference like those of Barbuti et al. 1993] Codish et al. 1994] and Marriott et al. 1994] we prove that the least complete and fully complete extensions of Def with respect to the disjunctive completion IP(Def ....

.... analyses formulated in terms of concrete context domains and context functions (basically additive functions) of Hughes [1988] This is also the case of abstract and concrete unification and projection in ground dependency analysis of logic languages [Armstrong et al. 1997; Cortesi et al. 1996; Marriott and Sndergaard 1993], and of dependency analysis for imperative languages of Nielson and Nielson [1992] Moreover, in the field of comparative semantics for logic programs, all the operators for building SLD trees of Comini and Levi [1994] are additive. Nielson and Nielson 1992] thoroughly discuss this issue of ....

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Marriott, K. and Sndergaard, H. 1993. Precise and efficient groundness analysis for logic programs.

....considering typical abstract domains for grounddependency and aliasing analysis in logic programming. The fragment L of a first order assertion language introduced in [19] actually, a slight extension of this) is used. Logical descriptions of various abstract domains are given: Def [10] and Pos [21, 22] for grounddependency analysis; Sharing [15] and ASub [26] for aliasing analysis. Maximal factorizations for these domains are obtained by inspecting the structure of the assertions in the abstract domains, and they are used for analyzing and comparing the abstract domains. Moreover, we study the ....

....composite domains that use Sharing and ASub, called equations systems, are investigated. We deal with the disjunctive completion of these domains in the last subsection. 9 4. 1 Def in Logical Form The abstract domain Def was introduced by Marriott and Sndergaard for grounddependency analysis in [22], based on previous work by Dart ( 10] on groundness analysis in deductive databases. We show that Def can be factorized into two reduced domains, describing groundness and covering, respectively. First, we recall the definition of Def . Def is the largest class of positive boolean functions ....

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K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1--4):181--196, 1993.

....of various abstract domains. 15] proposed propositional formulas to represent groundness relations. Many authors followed this approach [4,2] and contributed to develop and study the domains Def and Pos, while others focused on the abstract operations or slightly different characterizations [16,14]. All these authors share the same approach: They construct abstract domains independently from the property to be analyzed (i.e. from the semantics or concrete domain) and then prove some properties of the abstraction. Their attention is focused entirely on the representation of formulas in the ....

K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1-4):181--196, 1993.

....closed. An important and simple type, which distinguishes whether a term contains variables or not, is groundness [1, 5, 6] The usual domain for groundness analysis, Pos , features some desirable properties: simplicity, human readability, effectivity, usefulness. Moreover, it has been shown [13, 15] that Pos is condensing and is the most precise domain for groundness analysis that does not consider the name Part of this work was supported by EPSRC grant GR M05645. Part of this work was done while Fausto Spoto was visiting the School of Computer Studies of the University of Leeds, ....

K. Marriott and H. Sndergaard. Precise and Efficient Groundness Analysis for Logic Programs. ACM Letters on Programming Languages and Systems, 2(1-- 4):181--196, 1993.

....lattice of possibly occurring transformers D D, F f can also be viewed as system of equations over F . In this case, ordinary fixpoint methods may be applied, see [37,36] or [28 30] for instances of this idea for imperative languages. For logic languages, this approach has been suggested in [32,40]. Prerequisite always is that every (occurring) function f 2 F is succinctly representable and that the necessary operations on F , especially composition, t and equality, are efficiently computable. By fixpoint induction, we prove: Theorem 4 For all p 2 Proc, v 2 N p and d 2 D, 1) F ( R ....

Kim Marriott and Harald Sndergaard. Precise and Efficient Groundness Analysis for Logic Programs. ACM Letters on Programming Languages and Systems (LOPLAS), 2:181--196, 1993.

....of various abstract domains. 23] proposed propositional formulas to represent groundness relations. Many authors followed this approach [5,2] and contributed to develop and study the domains Def and Pos, while others focused on the abstract operations or slightly different characterizations [24,22]. All these authors share the same approach: They construct abstract domains independently from the property to be analyzed and then prove some properties of the abstraction. Their attention is entirely focused on the representation of formulas in the abstract domains. This forces to always work ....

....the two domains. Moreover, this is not only a result of differentiation, but a complete characterization of Pos since it turns out that Pos is the most abstract domain which satisfies this equality, which could be taken as an alternative definition of Pos. 9 Reachability analysis Some authors (cf. [5,24,2]) consider a slightly different definition of the domain Pos, by including also the empty set of substitutions as least element. Given a domain A, we denote by A the domain A [ f ; g, which is always a Moore family. For V ar = f x; y g, the domain Pos and Pos are depicted below. ffl ffl ffl ffl ....

K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(14) :181--196, 1993.

....1 Introduction The most popular technique for analyzing logic programs is currently abstract interpretation. Abstract interpretation is a program analysis technique that works by interpreting programs using an abstraction of the programs actual data structures [4] The abstract domain P os [5] uses the positive Boolean functions, i.e. functions that evaluate to true when all inputs are true. P os can be used to express Boolean properties of program variables where the property of one variable may depend on that property of other variables. For example, in the compilation of logic ....

Kim Marriott and Harald Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1--4):181--196, 1993.

....which can be efficiently represented by their maximal elements. Abstract unification in #JL is defined on such representations and its soundness is proven. It is shown that #JL represents sharing and definite groundness, but no ground dependencies. Therefore, we add the groundness domain POS [20, 1, 27] to #JL yielding the new sharing domain #JL POS. ffl We implemented and JL and #JL POS with the help of the Prolog analyzer generator GENA [14] The generated analyzers were run on a large set of benchmark programs. The new domain #JL POS is much more efficient than JL. Even more important, it ....

....does not model ground dependencies between variables. Hence, #JL may detect less ground variables than JL. Since no ground variable can share with any other variable, #JL may detect much more sharing pairs than JL. In order to compensate this loss of precision, we add the groundness domain POS [20, 1, 27] to #JL. Although groundness analysis with POS has to solve a theoretically intractable problem, implementations of POS are amazingly fast in practice. The reduced product [9] of #JL and POS is denoted #JL POS. Abstract domain POS will never infer less ground variables than DEF. As a ....

K. Marriott and H. Søndergaard. Precise and Efficient Groundness Analysis for Logic Programs. ACM Letters on Programming Languages and Systems, 2:181--196, 1993.

....for set sharing in logical form. 1 Introduction Of the abstract domains used in abstract interpretation of logic programs, the two that have received the most attention are Pos and Sharing. The former, originally introduced by Marriott and S ndergaard [13] and more formally presented in [4, 14, 5] consists of the class of positive Boolean functions. The Pos domain is most commonly applied to the analysis of groundness dependencies for logic programs. The Sharing domain, introduced by Jacobs and Langen [9, 11, 10] consists of sets of sets of variables and is applied to the analysis of ....

K. Marriott and H. Søndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1--4):181--196, 1993.

....of possibly occurring transformers D D, F f (M) can also be viewed as system of equations over F . In this case, ordinary fixpoint methods may be applied, see [26, 25] or [20 22] for instances of this idea for imperative languages. For logic languages, this approach has been suggested in [23, 28]. Prerequisite always is that every (occurring) function f 2 F is succinctly representable and that the necessary operations on F , especially composition, t and equality, are efficiently computable. Example 6. Let us consider the pda from Example 1. Then constraint system F f (M) is given ....

Kim Marriott and Harald Søndergaard. Precise and Efficient Groundness Analysis for Logic Programs. ACM Letters on Programming Languages and Systems (LOPLAS), 2:181--196, 1993.

....B ff JG 0 in PK [ by Point 1 of Theorem 7.7 ] B ff JA in PK e Theta B ff JG 0 in PK [ by Point 2 of Theorem 7.7 ] B ff JG in PK: 7. 1 The Observable for Groundness Analysis of Computed Answers We show now how to obtain Groundness analysis of computed answers for pure logic programs [3, 38, 16] by applying our scheme. In order to define the abstract domain we have to do several small steps. We will use propositional formulas to represent the groundness dependencies of variables. In particular, we will use the domain POS [3] of positive propositional formulas classes modulo logical ....

K. Marriott and H. Søndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1--4):181--196, 1993.

....sharing analysis. 1 Introduction Of the abstract domains used in abstract interpretation of logic programs, two that have received considerable attention are Pos and Sharing. The former, originally introduced by Marriott and Sndergaard [20] consists of the class of positive Boolean functions [1, 9, 11, 21]. The Pos domain is most commonly applied to the analysis of groundness dependencies for logic programs. The Sharing domain, introduced by Jacobs and Langen [16, 17, 18] consists of sets of sets of variables and is applied to the analysis of possible sharing amongst sets of program variables. The ....

....(u (x y) w (x z) append(Y,V,Z) g fl w (u v) For example, the annotation at point b fl states that u is ground at this point, and the annotation at c fl says that furthermore, v will become ground when w does, and vice versa. For details, see for example Marriott and Sndergaard [21]. At a first sight there is nothing much in common between the two analyses. However, consider what happens if we read the sharing annotations as sets of models and ask which Boolean functions possess exactly those sets of models. The following annotations simply rephrase the sharing annotations ....

K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1--4):181--196, 1993.

....a list of numbers. This behavior is captured by the function xs ys . One consequence which can be read out of this formula is: whenever quicksort succeeds given one of its arguments is ground, the other argument has been made ground. This information can be obtained automatically as follows [25]. As a first step we translate the program to its Clark completion [10] Since we will need to manipulate rather complex formulas involving predicate and variable names, we deviate from Prolog conventions and use lower case for variables, and nil and : for list construction. This yields q(xs ; ....

....with its first two arguments ground, information that a compiler can utilize. This way of computing call patterns is slightly simpler than traditional abstract interpretation based methods. It relies on condensation [22] For the positive functions it is as precise as more traditional approaches [25]. 2.2 Finiteness Analysis Finiteness analysis is one of the most important dataflow analyses for deductive databases as it is used to identify possibly non terminating queries. In a finiteness analysis, the description x y for a predicate p(x ; y) is read as for any finite assignment of values ....

K. Marriott and H. Søndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems 2 (1--4): 181--196, 1993.

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K. Marriott and H. Sndergaard. Precise and efficient groundness analysis for logic programs. ACM Letters on Programming Languages and Systems, 2(1--4):181--196, 1993.

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