J. C. Shepherdson. SLDNF-Resolution with Equality. Journal of Automated Reasoning, 8:297--306, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....language for Cognitive Robotics in which are combined a variety of challenging aspects of complex environments. Furnishing the calculus with a new axiomatic foundation, we have first of all overcome an important limitation of [65] caused by relying on the notion of unification completeness [62, 22]: Defining inequality of state terms as non unifiability wrt. AC1 did not permit any domain specific equalities like O#ce(Alice) R402 since this leads to a contradiction given that, e.g. the state terms InRoom(O#ce(Alice) and InRoom(R402 ) are not AC1 unifiable. The new, conceptually even ....

John C. Shepherdson. SLDNF-resolution with equality. Journal of Automated Reasoning, 8:297--306, 1992.

....by the equational logic programming approach [28, 23] augmented by specificity. We also illustrate the approach by examples motivated by the Broken Item and the Yale Shooting domain. Equational logic programs for computing change and specificity as well as their completion in the sense of [13] and [31, 50] are presented in Section 4. Section 5 focuses on models which interpret terms representing situations as multisets. In addition, we show how these models are related to the intended meaning of causality and specificity. Section 6 introduces SLDENF resolution as SLDNF resolution extended by a ....

....terms representing situations as multisets. In addition, we show how these models are related to the intended meaning of causality and specificity. Section 6 introduces SLDENF resolution as SLDNF resolution extended by a unification algorithm for equational theories. The soundness result of [50] for SLDENF resolution is recapitulated and a completeness result for SLDENF resolution is established, which can be applied to the equational logic programs specified in Section 4. In Section 7, we show how our approach can be used to perform backward reasoning, i.e. drawing conclusions about the ....

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J. C. Shepherdson. SLDNF-Resolution with Equality. Journal of Automated Reasoning, 8:297--306, 1992.

....language for Cognitive Robotics in which are combined a variety of challenging aspects of complex environments. Furnishing the calculus with a new axiomatic foundation, we have rst of all overcome an important limitation of [65] caused by relying on the notion of uni cation completeness [62, 22]: De ning inequality of state terms as non uni ability wrt. AC1 did not permit any domain speci c equalities like Oce(Alice) R402 since this leads to a contradiction given that, e.g. the state terms InRoom(Oce(Alice) and InRoom(R402 ) are not AC1 uni able. The new, conceptually even simpler ....

John C. Shepherdson. SLDNF-resolution with equality. Journal of Automated Reasoning, 8:297-306, 1992.

....2 ) then 8 x 2 4 t 1 = t 2 oe 2cUAC1 (t 1 ;t 2 ) 9 y: 3 5 where y denotes the variables which occur in = but not in x . The axioms of item 3, in conjunction with the standard uniqueness of names assumption in item 2, ensure that EUNA is unification complete [ Jaffar et al. 1984; Shepherdson, 1992 ] wrt. state terms and the equational theory AC1. The latter axiomatizes the arbitrary re arranging of the fluent terms that occur in a state term; hence, the following observation, which will be needed below, is a consequence of EUNA being AC1 unification complete: Observation 7 Let I; i be an ....

John C. Shepherdson. SLDNF-resolution with equality. Journal of Automated Reasoning, 8:297--306, 1992. 26

....language for Cognitive Robotics in which are combined a variety of challenging aspects of complex environments. Furnishing the calculus with a new axiomatic foundation, we have rst of all overcome an important limitation of [60] caused by relying on the notion of uni cation completeness [57, 21]: De ning inequality of state terms as non uni ability wrt. AC1 did not permit any domain speci c equalities like Oce(Alice) R402 since this contradicts the fact that, e.g. the state terms InRoom(Oce(Alice) and InRoom(R402 ) are not AC1uni able. The new, conceptually even simpler axiomatic ....

John C. Shepherdson. SLDNF-resolution with equality. Journal of Automated Reasoning, 8:297-306, 1992.

.... In order that the inequality of two state terms follows whenever they consist in different collections of fluent literals, an extension of the standard unique name assumption is needed, namely, the concept of unification completeness known from logic programming (see, e.g. Jaffar et al. 1984; Shepherdson, 1992; Thielscher, 1996 ] Let E be an equational theory, that is, a set of universally quantified equations. Two terms s and t are said to be E equal , written s =E t , iff s = t is entailed by E plus the standard axioms of equality (see (25) below) A substitution oe is called an E unifier of s ....

John C. Shepherdson. SLDNF-resolution with equality. Journal of Automated Reasoning, 8:297--306, 1992.

....unifiers cUAC1 (t 1 ; t 2 ) then 8 x 2 4 t 1 = t 2 oe 2cUAC1 (t 1 ;t 2 ) 9 y: 3 5 where y denotes the variables which occur in = but not in x . The axioms of item 3, in conjunction with the standard uniqueness of namesassumption in item 2, ensure that EUNA is unification complete [13, 19] wrt. state terms and the equational theory AC1. These axioms entail inequality of two state terms (or effect terms, resp. whenever these are composed of different fluent terms. The assertion that some fluent f holds (resp. does not hold) in some situation s is formalized as 9z: State(s) f ffi ....

John C. Shepherdson. SLDNF-resolution with equality. Journal of Automated Reasoning, 8:297--306, 1992. 192

....any interesting conclusions concerning the values of fluents in situations. The existing equational foundation of the Fluent Calculus, developed in [9] gives an answer to the two questions based on the equational theory of a commutative monoid along with the notion of unification completeness [12]. In the following section we show the limitations of this approach when it comes to incorporating domain specific equalities or the definition of functions among domain entities. In Section 3, a new and conceptually simpler equational foundation is developed, which is shown to overcome the ....

John C. Shepherdson. SLDNF-resolution with equality. Journal of Automated Reasoning, 8:297--306, 1992.

....the branches in the SLD tree either fail or instantiate the atom. This rule is then incorporated into a resolution method for general programs. The resulting method, called SLDNI resolution, was proved sound w.r.t. 2 valued semantics of program completion. Finally, let us mention here Shepherdson [160], where an extension of the SLDNF resolution with unification w.r.t. an equality is studied. 5.2 Prolog and its Variants Let us consider now Prolog. From the pure theoretical point of view it is an implementation of SLDNF resolution with the leftmost selection rule with the exception that the ....

J.C. Shepherdson. SLDNF resolution with equality. J. of Automated Reasoning, 8(2):297-- 306, 1992.

....set of clauses P = ELP D [ EPROP D [ f(8) 13)g . As we have negative literals in the body of some clauses, the adequate computational mechanism for P is SLDNF resolution where, due to our equational theory (AC1) standard unification is replaced by a theory unification procedure. Following [18], the semantics of our program is then given by its completion (cf. 3] P D ; AC1 ) where AC1 denotes a unification complete theory wrt. AC1 (see [13] or [18] The following theorem forms the basis of our soundness and completeness result regarding the completion of our constructed ....

....where, due to our equational theory (AC1) standard unification is replaced by a theory unification procedure. Following [18] the semantics of our program is then given by its completion (cf. 3] P D ; AC1 ) where AC1 denotes a unification complete theory wrt. AC1 (see [13] or [18]) The following theorem forms the basis of our soundness and completeness result regarding the completion of our constructed equational logic program. Theorem2. Let D be a domain description in AC determining a transition function Phi ; then,there exists a oe 0 such that the structure (oe 0 ; ....

J. C. Shepherdson. SLDNF-Resolution with Equality. Journal of Automated Reasoning, 8:297--306, 1992.

....we analyze the semantics of this program given by its completion, and in Subsection 5.3 we prove its soundness and completeness wrt the highlevel action semantics given by ANCC . Finally, in Subsection 5. 4 we discuss an adequate computation mechanism four our program, namely, SLDENF resolution [ Shepherdson, 1992; Thielscher, 1996a ] which is based on SLD resolution but with the standard unification procedure replaced by a special equality unification algorithm and negation as failure used to treat negative subgoals. Our translation allows automated reasoning about dynamic systems following the concepts ....

....construct a logic program corresponding to a domain description in ANCC . In Subsection 5. 2, we discuss the semantics of the resulting program by applying the standard completion procedure [ Clark, 1978 ] augmented by a special treatment of the underlying equational theory [ Jaffar et al. 1984; Shepherdson, 1992 ] In Subsection 5.3, we then prove soundness and completeness of the equational logic program (by taking the extended completion semantics) wrt the semantics of ANCC . Finally, in Subsection 5.4 we discuss the applicability of a special resolution variant, namely, SLDENF resolution [ ....

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John C. Shepherdson. SLDNF-resolution with equality. Journal of Automated Reasoning, 8:297--306, 1992.

....involves a certain kind of nonmonotnicity. Hence, its integration into our equational logic program (3) requires negative literals in the body of some clauses. To this end, the underlying refutation procedure, SLDE resolution, has to be extended by the (nonmonotonic) negation as failure principle [6, 43, 47]. ....

John C. Shepherdson. SLDNF-Resolution with Equality. Journal of Automated Reasoning, 8:297--306, 1992.

.... Since this logic program does not only require a special unification procedure but also contains negative literals, an adequate computation mechanism is SLDENF resolution, i.e. SLD resolution augmented by theory unification and the negation as failure principle to handle negative subgoals [ Shepherdson, 1992; Thielscher, 1995 ] In [ Holldobler and Thielscher, 1995 ] the application of this calculus to our ELP has been proved adequate wrt the semantics given by Definitions 1 3. 4 2 We use a Prolog like syntax, i.e. constants and predicates are in lower cases whereas variables are denoted by ....

J. C. Shepherdson. SLDNF-Resolution with Equality. J. of Autom. Reasoning, 8:297--306, 1992.

....failed SLDENF tree on the right hand side. Its two branches fail because st is not AC1 unifiable with on (s 1 ) ffi V nor with on (s 2 ) ffi V . This equational logic program contains negative literals in clause bodies, which requires an extended resolution principle, namely SLDENF resolution [ Shepherdson, 1992; Thielscher, 1996 ] i.e. SLDE resolution augmented by negation as failure. As usual, a negative literal as subgoal is solved by verifying that every derivation of the respective affirmative part fails, which in turn is shown by constructing a corresponding finitely failed derivation tree. ....

John C. Shepherdson. SLDNF-resolution with equality. Journal of Automated Reasoning, 8:297--306, 1992.

....world assumption applied to P . We are now able to derive negative information from P by deriving it from comp(P ) In fact, the following soundness and completeness result for definite 6 CET: Clark s Equational Theory. CET(LP ) axiomatizes the equality theory of all Herbrand(LP ) models. See [MMP88, She92] for the problem of equality and the underlying language. 3 ADDING DEFAULT NEGATION 23 programs P and definite queries Q = V i A i (consisting of only positive atoms) holds: Theorem 3.2 (COMP and Fair FF Trees) The following conditions are equivalent: ffl comp(P ) j= 8:Q ffl Every fair ....

John C. Shepherdson. SLDNF-Resolution with Equality. Journal of Automated Reasoning, 8, No. 2:297--306, 1992.

....of clauses P = ELP D [ EPROP D [ f(5:1) 5:6)g. As we have negative literals in the body of some clauses, the adequate computational mechanism for P is SLDENF Resolution where, due to our equational theory (AC1) standard unification is replaced by a theory unification procedure. Following [20, 24], the semantics of our program is then given by its completion (cf. 4] P D ; AC1 ) where AC1 denotes a unification complete theory wrt. AC1 [14, 20, 13, 24] The following theorem forms the basis of our soundness and completeness result regarding the completion of our constructed ....

....SLDENF Resolution where, due to our equational theory (AC1) standard unification is replaced by a theory unification procedure. Following [20, 24] the semantics of our program is then given by its completion (cf. 4] P D ; AC1 ) where AC1 denotes a unification complete theory wrt. AC1 [14, 20, 13, 24]. The following theorem forms the basis of our soundness and completeness result regarding the completion of our constructed equational logic program. Theorem 5.1 Let D be a domain description in AC determining a transition function Phi. Then, there exists some state oe 0 such that the ....

John C. Shepherdson. SLDNF-Resolution with Equality. Journal of Automated Reasoning 8:297--306, 1992.

....propositions in the same way, too. As we will have negated literals in the body of some clauses, the corresponding adequate computational mechanism is SLDENF Resolution where, due to the used equational theory (AC1) standard unification is replaced by a theory unification procedure. Following [21,19], the semantics of a translation PD of a domain description D is then given by its completion (cf. 6] P D ; AC1 ) where AC1 denotes the unification complete theory wrt. AC1 [21,11,12,19] The translation PD defines the predicate model such that (P D ; AC1 ) j= model(I) iff I ....

....equational theory (AC1) standard unification is replaced by a theory unification procedure. Following [21,19] the semantics of a translation PD of a domain description D is then given by its completion (cf. 6] P D ; AC1 ) where AC1 denotes the unification complete theory wrt. AC1 [21,11,12,19]. The translation PD defines the predicate model such that (P D ; AC1 ) j= model(I) iff I represents a model of D wrt. the semantics of AORC . The SLDENF Resolution is complete wrt. the program PD if I which represents an initial state does not contain any variables. Although, we do not ....

J. C. Shepherdson. SLDNF-Resolution with Equality. JAR, 8:297--306, 1992.

....our equational theory (AC1) Our equational program includes negative literals. Therefore, the adequate computation mechanism is SLDENF resolution, i.e. SLD resolution augmented by negation as failure along with an extended unification procedure which unifies wrt a concrete equational theory [30, 17, 32]. Moreover, we sometimes employ constructive negation [5] to avoid the problem of floundering. Semantics is defined by K. Clark s completion [6] along with a so called unification complete theory AC1 which allows to derive inequality of two terms whenever they are not unifiable [18, 30, 17] ....

....[30, 17, 32] Moreover, we sometimes employ constructive negation [5] to avoid the problem of floundering. Semantics is defined by K. Clark s completion [6] along with a so called unification complete theory AC1 which allows to derive inequality of two terms whenever they are not unifiable [18, 30, 17]. Let (P A ; AC1 ) denote the completion of our program depicted in Figure 1. In [17] it is argued that we can restrict ourselves to models of (P A ; AC1 ) where terms which are built up from the AC1 function ffi are interpreted as multisets. Let I denote such an interpretation. In ....

J. C. Shepherdson. SLDNF--Resolution with Equality. Journal of Automated Reasoning, 8:297--306, 1992.

....and change. We formalize our ideas by introducing an extension AORC of the Action Description Language AC [1] and give a sound and complete encoding of domain descriptions in AORC in terms of the Fluent Calculus [8] thereby enabling sound and complete automated inference using SLDENF Resolution [17, 15]. 1 Introduction In order to avoid domain descriptions of intractable length, approaches to the representation of concurrent actions usually describe the effects of concurrent actions by a combination of various partial descriptions. A crucial question is, how these partial descriptions are ....

....depend on whether they are related to fluents or resources, respectively. In Section 5, we briefly give a translation of domain descriptions in AORC into a corresponding variant FCORC of the Fluent Calculus, thereby enabling sound and complete reasoning about such domains via SLDENF Resolution [17, 15]. The Fluent Calculus basically models states by multisets. Similar to A , the execution of actions adds or removes elements from a particular multiset of atomic facts representing some state. Our results are summarized in Section 6. 2 AC We briefly repeat the concepts underlying AC . A domain ....

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John C. Shepherdson. SLDNF-Resolution with Equality. JAR, 8:297--306, 1992.

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J. C. Shepherdson. SLDNF-Resolution with Equality. Journal of Automated Reasoning, 8:297--306, 1992.

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J. C. Shepherdson. SLDNF-Resolution with Equality. J. of Autom. Reasoning, 8:297--306, 1992.

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J. C. Shepherdson. SLDNF--Resolution with Equality. Journal of Automated Reasoning, 8:297--306, 1992.

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Shepherdson, J. C., SLDNF-Resolution with Equality, Journal of Automated Reasoning 8:297-306 (1992).

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