K. Kunen. Signed data dependencies in logic programs. Journal of Logic Programming, 7:978--992, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....So PROLOG (P; p(x) NO and PROLOG (P; p(a) ffl] YES. So the lifting property does not hold. ut Remark. It is known that there exist subclasses of the class of normal programs and goals, for which SLDNF resolution is sound and complete with respect to Clark s program completion. [Kunen89] contains such a result for allowed programs and inferences in Kleene s three valued logic. Plaza90, Plaza91] contains such a result for propositional normal programs, and inferences in classical logic. DM91] contains such a result for allowed programs and inferences in classical logic. ut From ....

K. Kunen, Signed Data Dependencies in Logic Programs, Journal of Logic Programming, 1989, vol. 7, no. 3, pp.231-247.

....equivalent under 3 valued logic. In addition to program rules from Comp(P ) for literal rewriting, we also have the following two types of rewrite rules: simplification rules to transform and simplify goals, and loop rules for handling loops. The notion of signed goals was introduced in [19] for a similar purpose. 10 We introduce these two types of rules in the next two subsections, following which goal rewrite systems are defined and their properties investigated. 4.1 Simplification rules The simplification subsystem is formulated with a mechanism of loop handling in mind, which ....

K. Kunen. Signed data dependencies in logic programs. J. Logic Programming, 7(3):231--245, 1989. 35

....Definition 6. Let # be a logic program with signature #.Asigning of # is a set atoms(#) such that (1) for any rule (1) from #, either m #S, S or m #S, 2) for any atom l does not appear in #. If a program has a signing, we say that it is signed [29,44]. The notion of signing for finite logic programs without classical negation was introduced by Kunen [29] who used it as a tool in his proof that, for a certain class of programs, two different semantics of logic programs coincide. Turner in [44] extends the definition to the class of logic ....

....(1) for any rule (1) from #, either m #S, S or m #S, 2) for any atom l does not appear in #. If a program has a signing, we say that it is signed [29,44] The notion of signing for finite logic programs without classical negation was introduced by Kunen [29], who used it as a tool in his proof that, for a certain class of programs, two different semantics of logic programs coincide. Turner in [44] extends the definition to the class of logic programs with two kinds of negation. Obviously, programs without negation as failure are signed with an empty ....

K. Kunen, Signed data dependencies in logic programs, J. Logic Programming 7(3) (1989) 231--245.

....extending it with specialised syntactic constructs to represent measures of uncertainty, preserving its unambiguous semantic characterisation and computability properties. Our initial language is the language of normal clauses with SLDNF as the inference rule (i.e. the language of pure PROLOG [Kun89]) which: this language is reviewed in chapter 2 1. is expressive enough to represent a significant portion of first order logic, 2. admits computationally tractable implementations, and 3. has a well defined formal semantics. We select three specific facets of uncertainty for our study, which ....

....B) we cannot guarantee that Pm1 P m ( A B) Hence we use Pm2 , since it is clear that m ( A B) A similar argument explains the use of P m2 instead of P m1 in the denominator of P m (AjB) 2. 2 Logic Programming with Negation The language presented here is defined after [Kun89]. The class of logic programs supported by this language is that of normal non cyclical programs which are strict with respect to queries and allowed (see definitions below) The symbols of the language are: ffl variables x; y; ffl constants a; b; ffl n ary predicates p; q , ffl ....

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K. Kunen. Signed Data Dependencies in Logic Programs. Journal of Logic Programming, 7:231--245, 1989.

....rules. A goal is a formula which may involve , and . A goal is also referred to as a goal formula. A goal resulted from a literal rewriting from another goal is called a derived goal. A goal with negation appearing only in front of atoms is said to be signed, a term introduced in [Kunen, 1989] for a similar purpose. For convenience, we assume that all goals are signed, which can be achieved easily by simple transformations using the following rules: for any formulas and , T . 0 . 0 Like a formula, a goal may be further ....

K. Kunen. Signed data dependencies in logic programs. J. Logic Programming, 7(3):231--245, 1989.

....Model Semantics is well known, and partly responsible for the non existence of goal directed stable model computation engines. A class of programs has been devised that is guaranteed to not su er from these non relevant inconsistencies. Programs in this class are called call consistent programs [8, 14]. All models computed by XNMR for queries over such programs correspond to total stable models of the original program. Even though XNMR may compute more models than there are stable models for a given program, it has been shown in [2] that all stable models of the program are always represented ....

Kenneth Kunen. Signed data dependencies in logic programs. The Journal of Logic Programming, 7(3):231-246, November 1989.

....Building a knowledge base: an example 13 1. for any rule (1) from , either jl 0 j; jl m j 2 S, jl m 1 j; jl n j 62 S or jl 0 j; jl m j 62 S, jl m 1 j; jl n j 2 S; 2. for any atom l 2 S, l does not appear in . If a program has a signing, we say that it is signed [28,44]. The notion of signing for finite logic programs without classical negation was introduced by Kunen [28] who used it as a tool in his proof that, for a certain class of program, two different semantics of logic programs coincide. Turner in [44] extends the definition to the class of logic ....

....j; jl n j 62 S or jl 0 j; jl m j 62 S, jl m 1 j; jl n j 2 S; 2. for any atom l 2 S, l does not appear in . If a program has a signing, we say that it is signed [28,44] The notion of signing for finite logic programs without classical negation was introduced by Kunen [28], who used it as a tool in his proof that, for a certain class of program, two different semantics of logic programs coincide. Turner in [44] extends the definition to the class of logic programs with two kinds of negation. Obviously, programs without negation as failure are signed with an empty ....

K. Kunen. Signed data dependencies in logic programs. Journal of Logic Programming, 7(3):231-245, 1989.

....is nite and all its leaves are marked failed. The SLDNF tree succeeds ( nitely fails) if the main tree does. To each success leaf of the main tree there corresponds a computed answer substitution for Q (and a computed answer Q ) de ned as expected. In this section we assume (following Kunen [Kun89] that the set of function symbols of each arity is in nite. The requirement about the arity can actually be abandoned. De nition 4.4 We say that a program P is complete for a query Q w.r.t. a speci cation spec = specS; specC) i (i) specS [ specC 0 j= Q 0 implies that some SLDNF tree ....

K. Kunen. Signed data dependencies in logic programs. Journal of Logic Programming, 7(3):231-245, 1989.

....are finite. 12 In the proof we will use multisets over 1 (the set of natural numbers with added ) Such multisets will be values of our level mapping for goals. We will use a multiset ordering obtained in a standard way from the usual ordering of 1. For definitions, see [AP93] or [Kun89]. Definition 7.3 (Multiset level mapping for goals) j fIgA j = supf jAoej : j= fIgoe g (where by j= fIgoe we mean that j= J oe for any element fJg of sequence fIg) jj ; fI 0 gA 1 fI 1 gA 2 Delta Delta Delta A n fI n g jj = bag( jfI 0 gA 1 j; jfI n Gamma1 gA n j ) where bag(x 1 ....

K. Kunen. Signed data dependencies in logic programs. J. Logic Programming, 7(3):231--245, November 1989.

....of the body. This condition is very stringent and excludes many common Prolog constructs. Both clauses in the definition of the standard member 2 predicate are not allowed. A third reason which makes it di#cult to apply the theory of logic programming is that the completeness result of Kunen [17] and its extension in [29] are for three valued logic and not for classical two valued logic. In this article we try to solve these problems. We introduce a new modified completion of a logic program, called the # completion of a logic program, which is tailored for the Prolog search strategy. The ....

K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231-- 245, 1989.

....the least fixed point of this operator can be # 1 1 complete and that the closure ordinal can be # CK 1 even for definite Horn clause programs. The finite stages of Fitting s operator, however, are decidable and correspond to what is computed by SLDNF resolution. This has been shown by Kunen in [15] for allowed logic programs and by the author in [21] for mode correct programs. The class of allowed programs is considered as too restrictive in general. The # Research supported by the Swiss National Science Foundation. This article has been written at the Department of Mathematics, Stanford ....

....contains most programs of practical interest, since a programmer has always modes in mind when he writes a program. Moreover, every allowed program is also mode correct. Even though the use of three valued logic can be eliminated in Fitting s operator (cf. e.g. 13] the completeness results of [15] and [21] cannot be applied to existing implementations of logic programming like, for example, Prolog. The reason is that these systems use special search strategies which depend on the order of clauses in the program and on the order of literals in the bodies of clauses. Apt and Pedreschi, ....

K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231--245, 1989.

....from the completed database (completion of a program) in predicate calculus then it has also a successful evaluation (a successful SLDNF refutation) A su#cient condition which ensures that the ranges of variables in negative literals are bounded is allowedness. And indeed, Kunen has proved in [14] that for allowed programs SLDNF resolution is complete with respect to the completion of programs in three valued logic. But the condition of allowedness is very stringent and excludes many common Prolog constructs. For instance, both clauses in the definition of the standard member 2 predictate ....

....In other words, a clause is S correct i# every variable of the clause occurs also in a positive literal of the body of the clause. A goal is S correct i# every variable of the goal occurs also in positive literal of the goal. In this way we obtain exactly the allowed programs and allowed goals of [14]. And again every goal is S closed. Example 2.10 Decomposable programs: This class of programs has been introduced in [19] Decomposable programs are interesting, since they have the cutproperty and are therefore negation complete [20] For a decomposable program we have the following canonical ....

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K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231--245, 1989.

....(rewrite) rules. A goal is a formula which may involve : and . A goal is also referred to as a goal formula. A goal resulted from a literal rewriting from another goal is called a derived goal. A goal with negation appearing only in front of atoms is said to be signed, a term introduced in [Kunen, 1989] for a similar purpose. For convenience, we assume that all goals are signed, which can be achieved easily by simple transformations using the following rules: for any formulas Phi and Psi, Phi Phi : Phi Psi) Phi : Psi : Phi Psi) Phi : Psi Like a formula, a goal may be ....

K. Kunen. Signed data dependencies in logic programs. J. Logic Programming, 7(3):231--245, 1989.

....literal of the body. This condition is very stringent and excludes many common Prolog constructs. Both clauses in the definition of the standard member 2 predicate are not allowed. A third reason which makes it di#cult to apply the theory of logic programming is that the completeness results of [13] and [19] are for three valued logic and not for classical logic. In this paper we try to solve all three problems. We introduce a new modified completion of a logic program, called the # completion, which is tailored for the Prolog search strategy, and we prove that Prologresolution is sound and ....

K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231--245, 1989.

....P and most general unifier # of A and B, # # #)# # F then # # F. F2) If # = ##A, A R #, and # is a renaming substitution for A then # # F. We do not require in (R3) and (F2) that the negative literal A must be ground. If we did then we would obtain exactly the definition of Kunen in [4] which is equivalent to the definition of SLDNF resolution in Lloyd [5] Our extension of SLDNF resolution is called ESLDNF resolution and it is also sound for the completion. This means that for any program P (1) if (L 1 , L n ) R # then comp(P ) #(L 1 # # . # L n #) 2) ....

K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231--245, 1989. 7

....which contains definite programs and allowed programs. Our third result is that SLDNF resolution is sound and complete for decomposable programs and admissible goals with respect to three valued models of the completion [Theorem 89, page 64] This theorem extends the completeness results of [4, 13, 18]. In the proof of the completeness theorem we use proof theoretic methods. If a goal is a three valued consequence of the completion then it is provable in the sequent calculus for the completion without axioms of the form #, A, A. The complexity of the proof can be bounded by a partial ....

....) fails. Otherwise, there exists a positive literal A in # such that every goal derived from #, R(t 1 , t n ) fails with rank less than the rank of #, R(t 1 , t n ) This case is treated like the previous one. # Chapter 3 A sequent calculus for the three valued completion In [18] Kunen proves that for allowed programs and allowed goals SLDNF resolution is sound and complete with respect to three valued models of the completion. We want to extend these results to larger classes of programs by proof theoretic methods. To do this we need a sequent calculus in which a sequent ....

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K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231--245, 1989.

....machine stops with input A after finitely many steps with answer no . A classical result states that i) negation as failure is sound for the Clark completion comp(P ) of a (normal) logic program [4] ii) there are some classes of programs for which SLDNF resolution is complete for comp(P ) [2, 9], iii) some results hold if the classical consequence relation is replaced by 3 valued logic, intuitionistic logic or linear logic [8, 9, 3] iv) some results hold if SLDNF Resolution is extended [13] The negation as failure rule is easily implemented but provides di#culties for theoretic ....

.... for the Clark completion comp(P ) of a (normal) logic program [4] ii) there are some classes of programs for which SLDNF resolution is complete for comp(P ) 2, 9] iii) some results hold if the classical consequence relation is replaced by 3 valued logic, intuitionistic logic or linear logic [8, 9, 3], iv) some results hold if SLDNF Resolution is extended [13] The negation as failure rule is easily implemented but provides di#culties for theoretic investigations. One di#culty are the nested negative calls. Consider the following PROLOG clauses, which are from a Tic Tac Toe game program. ....

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K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231--245, 1989.

....to the definition of the predicates in the program plus appropriate axioms for built in predicates. 1 Introduction There are several reasons that we have implemented an interactive theorem prover for the verification of pure Prolog programs. First of all, we wanted to show that results of [8,10,11] about the foundations of logic programming are not only of theoretical interest. In the spirit of Apt [1] we wanted to show that the results can be extended to a rather large subset of Prolog. Secondly, we believe that if computer programs become bigger and more complex, it will be inevitable ....

....list. 10 The inductive extension is related to Clark s completion and Kunen s threevalued completion in the following way. Let (FIX) be the collection of the fixedpoint axioms for the R s and R f relations. Then (CET) UNI) FIX) is equivalent to Kunen s three valued completion of [8]. Moreover, Clark s completion of [3] can be obtained from the three valued completion be adding the stronger totality axiom R s (t) # R f (t) Example 4. The simultaneous induction scheme is more natural for logic programs than structural induction on the Herbrand universe. Assume that the ....

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K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231--245, 1989.

....be chosen, and it succeeds and fails according to the rules above. The class of # programs contains the class of allowed programs. It also contains the class of definite programs. Therefore our completeness result extends the results of Shepherdson in [13] Cavedon and Lloyd in [2] and Kunen in [9]. In the program above, the # property means that if the goal good(s, t) succeeds with answer #, then the terms s and t have to be closed or the goal member(s, t) has to fail. Having this large class of programs for which ESLDNF resolution is complete in three valued logic the question is, ....

...., x n ) #= g(y 1 , y m ) if f #= g) 7) t #= x (if t is a term di#erent from x and x # var(t) The axiom x 1 = y 1 # . # x n = y n # r(x 1 , x n ) # r(y 1 , y n ) is not needed because it is derivable from comp(P ) Following Kunen in [8] and [9] we use the three valued logic of Kleene for the interpretation of comp(P ) In this logic the three truth values are t (true) f (false) and u (undefined) with the partial ordering defined by u t and u f . Then x # y is equivalent to the two statements (1) if x = t then y = t and (2) if x ....

K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231--245, 1989.

....like the class of decomposable programs. This class contains the definite programs and the allowed programs. We show that SLDNF resolution is sound and complete for decomposable programs with respect to three valued models of the completion. This theorem extends the completeness results of [3, 13, 18]. It is also possible to prove the completeness of SLDNF resolution for programs which have the cut property directly, using model theoretic methods (see [32] The 2 proof theoretic method, however, which we will use in this paper, gives more insight. The motivation to use proof theory for ....

....than or equal to n. The number k is the depth of nesting of negation as failure calls. In other words, ##, ## # R k,n or # R k,n # means the goal # returns answer # or the goal # succeeds with answer # ; # # F k,n means the goal # finitely fails . The following definition can be found in [18]. Since we want most general answers, we have to parametrize the sets R k,n by a set of protected variables V . The sets R[V ] k,n and F k,n are the least sets satisfying the following five closure conditions. R1) # R[V ] k,n #. The empty goal succeeds with answer the identity substitution. ....

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K. Kunen. Signed data dependencies in logic programs. J. of Logic Programming, 7(3):231--245, 1989.

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K. Kunen. Signed data dependencies in logic programs. Journal of Logic Programming, 7:978--992, 1989.

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Kenneth Kunen. Signed data dependencies in logic programs. Journal of Logic Programming, 7(3):231--245, 1989.

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K. Kunen. Signed data dependencies in logic programs. Journal of Logic Programming, 7(3):231-245, 1989.

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K. Kunen, Signed Data Dependencies in Logic Programs, Journal of Logic Programming, 1989, vol. 7, No 3, pp.231-247.

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K. Kunen, "Signed Data Dependencies in Logic Programs," Journal of Logic Programming, vol. 7, no. 3 (1989) pp.231-247.

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