G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proceedings of the Conference and Symposium on Logic Programming, pages 1649--1665, Seattle, 1988. ALP, IEEE, The MIT Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the syntactic operators use clauses which are renamed apart. Note that, according to previous remark, unf denotes also a binary (semantic) operator on interpretations. The unfolding operator is of interest as it can be applied to formalize both top down and bottom up semantics for logic programs [21]. Below we show the bottom up construction of the semantics for the open programs. For a set of predicate symbols Omega we denote by the set fp( x ) Gammap ( x ) 2 C j p g. Definition 2.4 (Fixpoint semantics [5] The fixpoint semantics of an Omega open program P is given by the ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

.... = Clause) Programs and interpretations have then the same structure (i.e. sets of clauses) The concrete semantics is formalized in terms of unfolding of clauses unf : Int Theta Int Int , specifying every possible way to unfold each literal in each clause of P 1 once using clauses in P 2 (see [28]) unf (P 1 ; P 2 ) h Gamma oe fi fi fi fi fi fi fi c = h Gamma oe [ g 1 ; g n 2 P 1 hh i Gamma oe i [ b i i i=1 c P 2 = oe (oe i fg i = h i g) H j= oe = unf is a binary associative operator (see [15] on ....

....h i g) H j= oe = unf is a binary associative operator (see [15] on Int , which is additive on its first argument and continuous on its second one. This operator is of interest as it can be applied to formalize both top down and bottom up semantics for logic programs (cf. [28]) A bottom up semantics for open logic programs is defined in [5] in terms of iterated unfolding that is, repeatedly unfolding the clauses in a program until further unfolding produces no change. The fixpoint semantics of a program P is given by the function F : Int Int , defined as F(P) ....

[Article contains additional citation context not shown here]

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....operator. However, this equivalence is not fully achieved, since the standard operational model of a CLP program does not mirror its observable properties, i.e. the real program behaviour, given in terms of a set of answer constraints, is not reflected in the minimal model. As argued in [7], the key issue consists in a suitable choice of the notion of model. In [4] we can find a reconstruction of the semantics of CLP, obtained by defining various model notions, each corresponding to a specific operationally observable property, as firstly defined in [3] for the pure logic ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

.... = Clause) Programs and interpretations have then the same structure (i.e. sets of clauses) The concrete semantics is formalized in terms of unfolding of clauses unf : Int Theta Int Int , specifying every possible way to unfold each literal in each clause of P 1 once using clauses in P 2 (see [27]) unf (P 1 ; P 2 ) 8 : h Gamma oe 0 [ b 1 : Delta Delta Delta : b n fi fi fi fi fi fi fi fi fi fi c = h Gamma oe [ g 1 ; g n 2 P 1 hh i Gamma oe i [ b i i n i=1 c P 2 oe 0 = oe n i=1 (oe i fg i = h i g) H j= ....

....j= oe 0 9 = unf is a binary associative operator (see [14] on Int , which is additive on its first argument and continuous on its second one. This operator is of interest as it can be applied to formalize both top down and bottom up semantics for logic programs (cf. [27]) A bottom up semantics for open logic programs is defined in [5] in terms 1 The predicate disjoint restriction has the advantage that abducible information (for open(P) is basic [26] i.e. it is not defined in terms of other abducible information. For a set of abducible predicates, we can ....

[Article contains additional citation context not shown here]

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....the positive completion of P , compl(P ) is even(y) y = 0) 9x(y = s(s(x) even(x) Therefore the first unfolding of :even(s(0) is : s(0) 0) 9x(s(0) s(s(x) even(x) which is true even with respect to the empty program. These ideas are closely related to those used by Levi et al. [Lev88, GL92], who define the unfolding semantics of a program via the limit of the process of unfolding the program. These ideas allow us to sidestep the problem of asymmetry of success and failure in the least model and fixpoint semantics by viewing the process of unfolding as central. 1.3 Organization ....

....Kunen [Kun87] do. But one of the main goals of this paper has been to give a logical and operational semantics which are sound and complete with respect to each other. 6. 3 Unfolding Semantics I have already noted the similarity between this work and that of Levi s group on unfolding semantics ([Lev88, GL92] and many other papers) Whereas Levi et al. unfold a program in order to find all the atoms entailed by it, the approach of this paper takes the positive Clark completion of the program and unfolds a goal in order to characterize its truth value. Both lines of work have a basis in Tamaki and ....

Giorgio Levi. Models, unfolding rules and fixpoint semantics. In Proceedings of the 1988 International Conference and Symposium on Logic Programming, pages 1649--1665, Seattle, Washington, 1988.

....i; hh 1 ; h n i) 9 = Intuitively, the unfolding operator yields every possible way to unfold each literal in each clause of P 1 once using clauses in P 2 . This operator is of interest as it can be applied to formalize both top down and bottom up semantics for logic programs [19]. The following formalizes a bottom up semantics for open logic programs in terms of iterated unfolding that is, repeatedly unfolding the clauses in a program until further unfolding produces no change: Definition 3.3 [fixpoint semantics [4] The fixpoint semantics of a program P is given ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pp. 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....Prolog. The approach considered in this section is based on the observation that both top down and bottom up semantics of logic programs can be expressed in terms of program unfolding. In particular, the non ground TP semantics of [11] can be viewed as deriving the facts from the unfoldings of P [17]. While the first two enhancements described below follow the usual bottom up approach; the third is technically simpler when described by viewing unfolding as a top down process. The resulting semantics, while top down, maintains the structure and advantages of the bottom up approach mentioned ....

....is possible because both programs and interpretations have the same structure. Switching P and I in the TP definition corresponds to switch the construction of proof trees from bottom up to top down. The resulting immediate consequence operator corresponds to the unfolding semantics of the program [17]. An alternative approach to solve this problem is in [5] However that solution is rather complex and more difficult to abstract for the purposes of abstract interpretation. Definition17. oracle fixpoint semantics) Let P be a logic program, f an oracle for P and I a clausal interpretation. T f ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....In order to specify the oracle from within the semantics we enhance the semantic objects to capture also partial computations and derivation paths. Our approach is based on the observation that both top down and bottom up semantics of logic programs can be expressed in terms of program unfolding [Levi 88] Section 4.1 introduces a top down unfolding semantics for logic programs with oracles. Later, in Section 4.3, a bottom up unfolding semantics is introduced by allowing the same oracle to be applied in the process of unfolding, in a reversed way . In Sections 5.1 and 5.2 we consider two ....

....G and G 0 G 00 . 4.3 Bottom up oracle semantics Top down constructions typically unfold clauses from an interpretation (a set of partial computations) with program clauses. In contrast, bottom up constructions involve the unfolding of program clauses with clauses from an interpretation ( Levi 88] The two approaches correspond to top down and bottom up constructions of the computation trees. A bottom up oracle semantics is derived by switching the role of the program and the interpretation in Definition 4.5. In this case we are able to specify directly a semantics for successful ....

G. Levi. Models, unfolding rules and fixpoint semantics. In Robert A. Kowalski and Kenneth A. Bowen, editors, Proceedings of the Fifth International Conference and Symposium on Logic Programming, pages 1649--1665, Seattle, 1988. The MIT Press.

.... (Clause) Programs and interpretations have then the same structure (i.e. sets of clauses) The concrete semantics is formalized in terms of unfolding of clauses unf : Int Theta Int Int , specifying every possible way to unfold each literal in each clause of P 1 once using clauses in P 2 (see [19]) 1 The predicate disjoint restriction has the advantage that abducible information (for open(P) is basic [18] i.e. it is not defined in terms of other abducible information. For a set of abducible predicates, we can always transform P (as suggested in [18] in such a way that it does not ....

....observation. In the concrete semantics, for a program P and goal G = oe [ b, G has a successful derivation in P iff, by extending G to a clause: ans Gamma G with a new atom ans such that var(ans) var( b) there exists ans Gamma oe 0 [ true 2 unf (fans Gamma Gg; F(P) e.g. see [19]) In this case we say that oe 0 is the answer constraint. In the following, for any set of variables V we denote ans(V ) a flat and linear atom such that var(ans(V ) V . To simplify the notation, for any syntactic object s , we denote ans(s) the atom ans(var(s) and we will omit V when it ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....In order to specify the oracle from within the semantics we enhance the semantic objects to capture also partial computations and derivation paths. Our approach is based on the observation that both top down and bottom up semantics of logic programs can be expressed in terms of program unfolding [Levi 88] Section 4.1 introduces a top down unfolding semantics for logic programs with oracles. Later, in Section 4.3, a bottom up unfolding semantics is introduced by allowing the same oracle to be applied in the process of unfolding, in a reversed way . In Sections 5.1 and 5.2 we consider two ....

....G and G 0 G 00 . 4.3 Bottom up oracle semantics Top down constructions typically unfold clauses from an interpretation (a set of partial computations) with program clauses. In contrast, bottom up constructions involve the unfolding of program clauses with clauses from an interpretation ( Levi 88] The two approaches correspond to top down and bottom up constructions of the computation trees. A bottom up oracle semantics is derived by switching the role of the program and the interpretation in Definition 4.5. In this case we are able to specify directly a semantics for successful ....

G. Levi. Models, unfolding rules and fixpoint semantics. In Robert A. Kowalski and Kenneth A. Bowen, editors, Proceedings of the Fifth International Conference and Symposium on Logic Programming, pages 1649--1665, Seattle, 1988. The MIT Press.

....the syntactic operators use clauses which are renamed apart. Note that, according to previous remark, unf denotes also a binary (semantic) operator on interpretations. The unfolding operator is of interest as it can be applied to formalize both top down and bottom up semantics for logic programs [21]. Below we show the bottom up construction of the semantics for the open programs. For a set of predicate symbols Omega we denote by id Omega the set fp( x ) Gammap ( x ) 2 C j p 2 Omega g. Definition 2.4 (Fixpoint semantics [5] The fixpoint semantics of an Omega open program P is given ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

.... (Clause) Programs and interpretations have then the same structure (i.e. sets of clauses) The concrete semantics is formalized in terms of unfolding of clauses unf : Int Theta Int Int , specifying every possible way to unfold each literal in each clause of P 1 once using clauses in P 2 (see [28]) unf (P 1 ; P 2 ) 8 : h Gamma oe 0 [ b 1 : Delta Delta Delta : b n fi fi fi fi fi fi fi fi fi c = h Gamma oe [ g 1 ; g n 2 P 1 hh i Gamma oe i [ b i i n i=1 c P 2 oe 0 = oe n i=1 (oe i fg i = h i g) H j= oe 0 9 = ....

....i = h i g) H j= oe 0 9 = unf is a binary associative operator (see [15] on Int , which is additive on its first argument and continuous on its second one. This operator is of interest as it can be applied to formalize both top down and bottom up semantics for logic programs (cf. [28]) A bottom up semantics for open logic programs is defined in [5] in terms of iterated unfolding that is, repeatedly unfolding the clauses in a program until further unfolding produces no change. The fixpoint semantics of a program P is given by the function F : Int Int , defined as F(P) ....

[Article contains additional citation context not shown here]

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....semantics of concurrent logic programming languages. A central issue in our semantic treatment is the representation of a failure. A failure could be represented as a sequence of ask tell annotated constraints. Such a strategy is followed in earlier work on related languages, for example in [Sar85,Lev88,GL90] GCLS88] GMS89] etc. However, ask tell sequences are far too concrete in this setting. They store too much information which must then be abstracted from (usually via complex closure conditions) since they encode the precise path followed by a process to arrive at a resting point. ....

Giorgio Levi. Models, unfolding rules and fixpoint semantics. In Proceeedings of the Fifth International Conference and Symposium on Logic Programming, Seattle, pages 1649--1665, August 1988.

....i; hh 1 ; h n i) 9 = Intuitively, the unfolding operator yields every possible way to unfold each literal in each clause of P 1 once using clauses in P 2 . This operator is of interest as it can be applied to formalize both top down and bottom up semantics for logic programs [19]. The following formalizes a bottom up semantics for open logic programs in terms of iterated unfolding that is, repeatedly unfolding the clauses in a program until further unfolding produces no change: Definition 3.3 [fixpoint semantics [4] The fixpoint semantics of a program P is given ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pp. 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....htrue : y = 0i 2: The success set of P with respect to T is fp(0) p(s(0) p(s(s(0) g: 3.3 Declarative semantics We now define a declarative semantics for programs which is open [8] with respect to the interpretation of ask constraints. It is defined in terms of unfoldings of clauses [18] which contain uninterpreted ask conditions. This approach facilitates separation of termination from computation when applied to modelling Prolog control. Unfoldings of clauses with ask and tell constraints result in sequences of constraints which we term constrained atoms. Definition 3.7 ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....0 P htrue : y = 0i 2: The success set of P with respect to T is fp(0) p(s(0) p(s(s(0) g: 4.3 Declarative Semantics We now define a declarative semantics for programs which is open with respect to the interpretation of ask constraints. It is defined in terms of unfoldings of clauses [21] which contain uninterpreted ask conditions. This approach facilitates separation of termination from computation when applied to modelling Prolog control. Unfoldings of clauses with ask and tell constraints result in sequences of constraints [18] which we term constrained atoms. 10 Definition ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....3. 1 Unfolding semantics and equivalence results The equivalence between the operational semantics of an isa hierarchy HP and the fixpoint semantics of the corresponding HP can be proved in a concise and elegant way by introducing the intermediate notion of unfolding semantics U(P ) [18, 19, 8]. The unfolding semantics is obtained as the limit of the top down unfolding process. Definition 3.11 Let P be a h Sigma; Delta; Thetai program. Then we define the collection of cs interpretations P 1 = P Pn 1 = unf Pn (P [ Id Open(P ) The unfolding semantics U(P ) of the program P is ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....htrue : y = 0i 2: The success set of P with respect to T is fp(0) p(s(0) p(s(s(0) g: 3.3 Declarative semantics We now define a declarative semantics for programs which is open [8] with respect to the interpretation of ask constraints. It is defined in terms of unfoldings of clauses [18] which contain uninterpreted ask conditions. This approach facilitates separation of termination from computation when applied to modelling Prolog control. Unfoldings of clauses with ask and tell constraints result in sequences of constraints which we term constrained atoms. Definition 3.7 ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....of terms respectively, while B will denote a (possibly empty) conjunction of atoms. For any set A, A denotes the set of finite sequences of elements of A. will denote concatenation of sequences and is the empty sequence. 2 The s semantics approach The aim of the s semantics approach [82, 56, 52, 58] is modeling the observable behaviors (possibly in a compositional way) for a variety of logic languages. The approach is based on the idea of choosing (equivalence classes of) sets of clauses as semantic domains. The denotations are then defined by syntactic objects, as in the case of Herbrand ....

.... can be computed both by a top down construction (a success set) and by a bottom up construction (the least fixpoint of suitable continuous immediate consequences operators on interpretations) The link between the top down and the bottom up constructions is given by an unfolding operator [82, 83]. The equivalence proofs can be stated in terms of simple properties of the unfolding and the immediate consequences operators [41] It is worth noting that the aim of the approach is not defining a new notion of model. We are simply unhappy with the traditional declarative semantics, because it ....

[Article contains additional citation context not shown here]

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....the CLP case the s semantics framework [11, 12] defined for pure logic programs. The usefulness of the s semantics has already been shown by several projects related to the semantics, the analysis, and the transformation of logic programs. These include the semantics of concurrent logic languages [22, 6, 9, 13], the semantics of partial computations [10] the abstract interpretation of pure logic programs[1, 3, 18] correctness of program transformation techniques [23, 2] semantics of logic programs with negation [27] and the definition of a non ground finite failure set semantics [24] The same kind ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

....) p(a; a) r(X) q(X) r(a) s(Y ) q(Y ) s(a) g F Omega (P ) T Omega P 3 3 1.4. 1 Unfolding semantics and equivalence results The equivalence between the operational and the fixpoint semantics can be proved by introducing the intermediate notion of unfolding semantics U Omega (P ) [22, 23, 9]. U Omega (P ) is obtained as the limit of the (top down) unfolding process. Since the unfolding rule preserves computed answers in a compositional way, U Omega (P ) is equivalent to the operational semantics O Omega (P ) The proof of this equivalence is straightforward since O Omega (P ) ....

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

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G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proceedings of the Conference and Symposium on Logic Programming, pages 1649--1665, Seattle, 1988. ALP, IEEE, The MIT Press.

No context found.

G. Levi. Models, Unfolding Rules and Fixpoint Semantics. In R. A. Kowalski and K. A. Bowen, editors, Proc. Fifth Int'l Conf. on Logic Programming, pages 1649--1665. The MIT Press, Cambridge, Mass., 1988.

No context found.

G. Levi. Models, unfolding rules and fixpoint semantics. In Robert A. Kowalski and Kenneth A. Bowen, editors, Proceedings of the Fifth International Conference and Symposium on Logic Programming, pages 1649--1665, Seatle, 1988. ALP, IEEE, The MIT Press.

No context found.

G. Levi. Models, unfolding rules and fixpoint semantics. In Robert A. Kowalski and Kenneth A. Bowen, editors, Proceedings of the Fifth International Conference and Symposium on Logic Programming, pages 1649--1665, Seatle, 1988. ALP, IEEE, The MIT Press.

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