F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In Proc. of the Symposium on Theoretical Aspects of Computer Science, volume 480 of Lecture Notes in Computer Science, pages 511--522, Hamburg, Germany, February 1991. Springer-Verlag.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 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 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 ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

....a CLP program is again a possibly infinite CLP program. The computed answer constraint can be obtained by executing the goal into this set of clauses. The operational semantics is reflected into a program transformation technique which preserves the intended meaning of the program. As stressed in [1] an unfolding transformation technique, appears as intermediary between procedural and fixpoint semantics. They show that it is possible to give a quasi algebraic characterization of the relationship between these two semantics, which allows to give more simple proofs than usual, in particular ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In Proc. STACS'91, 1991.

....Theorem 1. Let P 0 be a bottom up partial deduction of a program P , P 0 = A C P n 0 [ Res(A C P n 0 ) Then, if A C P n 0 is inside closed, M s (P 0 ) jPred(P ) M s (P ) thus P 0 is sound and complete w.r.t. P . In order to prove this theorem, we include a useful lemma from [6]. Lemma 1. Proposition II.1 from [6] Let A = fA i g 1in be a set of positive literals, B = fB i B i;1 ; B i;n i g 1in and C = fC i;j C i;j;1 ; C i;j;p i;j g 1in;1jn i be sets of definite clauses. Let us further assume that V ar(A) V ar(B) and V ar(C) are disjoint sets. ....

....partial deduction of a program P , P 0 = A C P n 0 [ Res(A C P n 0 ) Then, if A C P n 0 is inside closed, M s (P 0 ) jPred(P ) M s (P ) thus P 0 is sound and complete w.r.t. P . In order to prove this theorem, we include a useful lemma from [6] Lemma 1. Proposition II.1 from [6]) Let A = fA i g 1in be a set of positive literals, B = fB i B i;1 ; B i;n i g 1in and C = fC i;j C i;j;1 ; C i;j;p i;j g 1in;1jn i be sets of definite clauses. Let us further assume that V ar(A) V ar(B) and V ar(C) are disjoint sets. Then, there exist substitutions ff and fi ....

F. Denis and J.P. Delahaye. Unfolding, procedural and fixpoint semantics of logic programs. In 8th Annual Symposium on Theoretical Aspects of Computer Science, volume 480 of LNCS, pages 511--522. Springer, 1991.

....of the transformation. Theorem 1. Let P 0 be a bottom up partial deduction of a program P , P 0 = A C P n 0 [ Res(A C P n 0 ) Then, if A C P n 0 is inside closed, M s (P 0 ) M s (P ) thus P 0 is sound and complete w.r.t. P . Proof. Outline) Using Proposition II.1 from [5], we can prove by induction that for all i n 0 : M s (P i ) M s (P ) where P i = A C P i [ Res(A C P i) and thus M s (P 0 ) M s (P ) for every k; T C P k M s (P 0 ) and hence M s (P ) M s (P 0 ) A worked out proof can be found in the full version of this paper. ....

F. Denis and J.P. Delahaye. Unfolding, procedural and fixpoint semantics of logic programs. In 8th Annual Symposium on Theoretical Aspects of Computer Science, volume 480 of LNCS, pages 511--522. Springer, 1991.

....result can be obtained in a simpler way by using 4 of Proposition 4.8, Proposition 4.4 and the fact that P and Q are logically equivalent iff they have the same set of atomic logic consequences [23] Definition 5.27 Let P be a program. P is u closed iff unf P[Id Pi (P ) P . Lemma 5. 28 [14] Let P; Q; R be programs. Then unfR (unfQ (P ) unf unfR (Q) P ) Lemma 5.29 If P is an u closed program then, for any set of ground atoms Q, P ] s (Q) T s P id) Q) Proof Let us denote P [ Id Pi by P and Ground(unfQ (P ) by unfgQ (P ) By definition of T s P , T s P ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

....program. The compositional fixpoint semantics F Omega (P ) of P is defined as F Omega (P ) T Omega P . 7. 2 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 ) [35, 14] and following the lines of the similar proof in [7] U Omega (P ) is obtained as the limit of the (top down) unfolding process and is equivalent to the operational semantics O Omega (P ) The proof of this equivalence is straightforward since O Omega (P ) and U Omega (P ) are based on 27 ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

....for predicates defined in other modules that are lower in the program call graph. The concrete semantics is formalized in terms of unfolding of clauses. The unfolding operator unf specifies the result of unfolding clauses from an interpretation P 1 with clauses from an interpretation P 2 (cf. [16]) Definition 3.1 [unfolding] The unfolding operator unf : Int Theta Int Int is defined as unf (P 1 ; P 2 ) 8 : h oe 0 [ b 1 : Delta Delta Delta : bn fi fi fi fi fi fi fi fi fi c = h oe [ g 1 ; gn 2 P 1 ; hh i oe i [ b i i n ....

.... : gn 2 P a 1 ; hh i i [ b i i n i=1 c P a 2 0 = Omega n Omega i=1 ( i Omega ff A (g i = h i ) 9 = The following lemma states that abstract unfolding satisfies the basic properties of concrete unfoldings (for generic properties of concrete unfoldings see [16]) Lemma 4.3 [associativity, continuity and additivity] For any abstract constraint system A, unf A is associative, continuous in the second argument and additive in the first argument. In the following we assume the obvious extensions of the notions of modular logic programs, open ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

.... Depr (S) fl) Therefore, for a traceinterpretation I 2 (aT) ff(I ) x :ff S (I rfl S (x ) is a generic dependency in Depr (S) As observed in Section 5, a dependency x :ff S (I rfl S (x ) is the best correct approximation of the concrete operator x :I rx : aT) 7 Gamma (aT) As hinted in [25], it turns out that any function x :I rx is a TP like function. In our context, this observation is formalized by the next lemma, where we use the sequence abstraction ff to approximate finite traces by the pair of their initial and final states in a clause like form: ff : h(aT) i 7 Gamma ....

....the proofs of some results in Section 7. Proposition 7.3 The operator r is monotone, associative, left additive, right continuous, and ; is a left annihilator for r (i.e. 8Y 2 (aT) rY = Proof. We omit the proof of associativity of r, because it follows from a similar result proved in [25]. Moreover, by definition, it is immediate that r is monotone, left additive, right continuous, and ; is a left annihilator for r. Proposition 7.4 T s P = ff S ffi P ffi fl S . Proof. We prove a stronger relation between T s P and P , namely ff S ffi P = T s P ffi ff S . Let I 2 (aT) By ....

F. Denis and J.-P. Delahaye. Unfolding, procedural and fixpoint semantics of logic programs. In C. Choffrut and M. Jantzen, editors, Proc. of the 8th Internat. Symp. on Theoretical Aspects of Computer Science (STACS '91 ), volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

.... 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= 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 ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

.... 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= oe 0 9 = unf is a binary associative operator (see [11]) on Int , which is additive on its first argument and continuous on its second one. A bottom up semantics for open logic programs is defined in [3] in terms of iterated unfolding that is, repeatedly unfolding the clauses in a program until further unfolding produces no change. The fixpoint ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

....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 ....

.... 1 ; A k ) H 1 ; H k ) fl jvar(G) # jvar(G) The proof of the above result can be carried out by using a straightforward inductive argoment, since U(P ) is based on a top down definition which mimics a parallel SLD derivation (the proof is essentially the same of those given in [19, 8] for the case of standard programs) Then, the desired equivalence can be stated in terms of the following Theorem 3.14 Let HP be an isa hierarchy, HP be the corresponding h Sigma; Delta; Thetai program and G = A 1 ; A k be a goal with P red(G) Sigma [ Delta [ Theta) Then: ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

.... Gamma hb 1 ; b n i c 1 Gamma : c k Gamma b : hb k 1 ; b n i 9 = 2 The trace unfolding operator above satisfies the following basic algebraic properties. Proposition 5. 5 The operator r is left additive, i.e. i2I X i )rX = i2I (X i rX ) associative ([16]) and ; is a left annihilator for r, i.e. rX = Hence, h(T a P (SLD) ri is a semantic structure where the basic Assumption 3.6 is satisfied. In particular, Corollary 3.5 implies that if ( T a P (SLD) ff 1 ; D 1 ; fl 1 ) and ( T a P (SLD) ff 2 ; D 2 ; fl 2 ) are G.c. s, then ( T ....

....by unfolding traces in I . In particular, the elements in S r S (i.e. x :ff S (I rfl S (x ) are actually the best correct approximations for the concrete operators x :I rx : T a P (SLD) T a P (SLD) for I 2 (T a P (SLD) It is well known that x :I rx is a TP like function ([16]) Hence, the isomorphism above is based on the correspondence between TP like functions and clauses, the latter encoding the functional dependency between their bodies and heads. It is worth noticing that S r S corresponds to the well known compositional semantics of Bossi et al. introduced ....

F. Denis and J.-P. Delahaye. Unfolding, procedural and fixpoint semantics of logic programs. In Proc. STACS '91, LNCS 480, pp. 511--522, 1991.

.... 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 characterizes the logical properties only and we look for new notions of program denotation useful from the programming point of ....

....cases, given a interpretation I, the relation T P (I) unfP (I) holds. The proof of equivalence between U(P ) and F(P ) can be based on such a relation. In particular the equivalence immediately holds for those immediate consequences operators which are compatible with the unfolding rule [41]. The above relations suggest a methodology to obtain the immediate consequences operator by first defining the unfolding operator, which is easier to define because of its strong relation to the operational semantics. 2.3 Model theoretic semantics Let us first note that the original ....

[Article contains additional citation context not shown here]

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

.... 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 = 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 ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

....) 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 ) ....

.... q(X) r(a) s(Y ) q(Y ) s(a) g U Omega (P ) P 3 3 In order to prove the equivalence between the fixpoint and the unfolding semantics we need two lemmata. The first one states the associativity of the unfolding operator. The second lemma shows some properties of the unfolding. Lemma 23 [9] Let P; Q; W be programs. Then unfP (unfQ (W ) unf unfP (Q) W ) Lemma 24 [3] Let P be an Omega open program and let Pn be as in definition 22. Let W be a set of clauses and let us define unf 1 P (W ) unfP (W ) and, for n 1, unf n P (W ) unfP (unf n Gamma1 P (W ) Then for n 1 ....

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

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F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In Proc. of the Symposium on Theoretical Aspects of Computer Science, volume 480 of Lecture Notes in Computer Science, pages 511--522, Hamburg, Germany, February 1991. Springer-Verlag.

No context found.

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

No context found.

F. Denis and J.-P. Delahaye. Unfolding, Procedural and Fixpoint Semantics of Logic Programs. In C. Choffrut and M. Jantzen, editors, STACS 91, volume 480 of Lecture Notes in Computer Science, pages 511--522. Springer-Verlag, Berlin, 1991.

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