S. Holldobler. Equational logic programming. In Proceedings of the Symposium on Logic Programming, Computer Society, Press of the IEEE, Washington, pages 335--346. IEEE, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with and with each other and . Let be a goal. If is not an empty constraint and the empty state then is the CLP( initial configuration. If is empty then . A terminal configuration has the form , where is an state and is the answer constraint. Let us consider the following CLP( program [30]: P = p1) p2) p3) p4) e1) e2) e3) e4) e5) e6) e7) knapsack states that items in the sublist of a given list can be packed into a knapsack such that the knapsack weights exactly f g b d d var G 2 3 3 3 3 3 H E H E H E H E ( 0 6 7 0 0 0 6 ....

S. Holldobler. Equational Logic Programming. In , pages 335--346. IEEE Computer Society Press, 1987.

....proves that this is possible and does not lead to intricacies. However, the connections with linear logic will be explored in future work. The concept of resource handling is closely related to the notions of action and change, which have recently been the subjects of many researches: see e.g. [8, 10, 11, 13, 14, 19, 20, 21, 22, 23, 25]. Our work differs from them as follows. The language AbstrAct [25] specifies the activities of systems by action rules characterizing agents capable of performing actions that operate on a global shared data space. A feature of AbstrAct is that it distinguishes between actions to induce state ....

....negation as failure for that purpose. The paper [8] has used normal logic programs with abduction. Our work differs from all of them by using a slight extension of Horn clause programs involving no negation and remaining in the (classical) mainstreams of logic programming. The work reported in [11, 13, 14] is the closest to ours. Rephrased in our terms, it proposes to describe each action by incorporating our pre and postconditions as predicate arguments and the non consumption problem by extending normal unification to cope with multisets. As an example (taken from [14] the clause action(P re; ....

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G. Grofie, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Tielscher. Equational Logic Programming, Actions, and Change. In K.R. Apt, editor, Proc. Joint International Conference and Symposium on Logic Programming, pages 177--191, Washington, USA, November 1992. The MIT Press.

....expression of the derivation. In general, there is no effective way to compute a standard derivation for any term in a regular term rewriting system. In the context of functional programs, however, each defining rule has the pattern based property (otherwise known as the constructor principle [Hol87]) Therefore, if the reduction strategy is based on call by need (which is also called lazy evaluation when infinite data structures are considered) it can be used to approximate standard derivations. That is, by reducing expressions in a pattern driven manner, a terminating derivation reducing ....

....regarded as an equational theory. A semidecidable solver, called pattern fitting, can be constructed as the transition system presented in Fig. 3.1 for solving rewriting constraints. This procedure is analogous to the the general E unification procedure based on a confluent term rewriting system [Hol87]. But it is considerably simpler than E unification Substitution: fx = rg [ G fx = rg [ aeG where ae = fx 7 rg Decomposition: fce = ctg [ G fe = tg [ G Elimination: fa = ag [ G G Failure: fc 1 e 1 = c 2 e 2 g [ G false c 1 6= c 2 Constrained Narrowing: ffe = csg [G 9X:fr = cs; e = ....

Steffen Holldobler. Equational logic programming. In Poceedings of the 4th Symposium on Logic Programming, San Francisco, pages 335--346. IEEE, 1987.

....3, 18] is widely recognized as a logic of concurrency, meaning that the proof theory underlying it faithfully represents some aspects of concurrent computations. Recently it has been pointed out that the same logic is suited for carefully modeling concepts of action and change in planning problems [10, 7]. In particular linear logic offers a very elegant solution to the so called frame problem [11] The foundation of logic programming languages by means of proof theory has been investigated by Miller et al. in the seminal work [16] where the concept of uniform proof is defined. In the case of ....

....inspired by the recent work of Masseron et al. 10] In that work the authors express planning problems in linear logic, and show that the frame problem [11] which is the major inconvenience when planning problems are expressed in classical logic) is brilliantly overcome. To this aim, see also [7], which establishes connections to Bibel s linear connection method. We shall show a formalization (somewhat dual of the one in [10] in Forum. In this way we have a logic programming language to reason about planning. The basic formalization we have will be enriched in the following section. The ....

Gerd Große, Steffen Holldobler, Josef Schneeberger, Ute Sigmund, and Michael Thielscher. Equational logic programming, actions, and change. In K. Apt, editor, Joint International Conference and Symposium on Logic Programming, pages 177--191. The MIT Press, 1992.

....10] is widely recognized as a logic of concurrency, meaning that the proof theory underlying it faithfully represents some aspects of concurrent computations. Recently it has been pointed out that the same logic is suited for carefully modeling concepts of action and change in planning problems [19, 16]. In particular linear logic offers a very elegant solution to the so called frame problem [20] The aim of our work is to investigate the use of linear logic as the underlying formal system of linear logic programming languages, and in particular to study the expressiveness of the languages ....

Gerd Große, Steffen Holldobler, Josef Schneeberger, Ute Sigmund, and Michael Thielscher. Equational logic programming, actions, and change. In K. Apt, editor, Joint International Conference and Symposium on Logic Programming, pages 177--191. The MIT Press, 1992.

....from execution traces. Over the last decade, it has been theoretically and experimentally demonstrated that functional logic languages have more expressive power in comparison to functional languages and a better operational behavior in comparison to logic languages [11] One relevant approach [13, 17] to integration is the functional logic programming where the programs are logic programs which are augmented with Horn equational theories. The main semantic properties of logic programs also hold for functional logic programs. Thus, these programs admit least model and fixpoint semantics. The ....

S. Holldobler. Equational Logic Programming. In Proc. Second IEEE Symp. on Logic In Computer Science, pages 335--346. IEEE Computer Society Press, 1987.

....omitting the second equation, we obtain a datatype MSet(ff) for polymorphic multisets. Data structures based on non free constructors, specially sets and multisets, play an important role in several recent proposals for extended logic programming and multiparadigm declarative programming; see e.g. [13, 4, 8, 15]. As a This research has been partially supported by the the Spanish National Project TIC95 0433 C03 01 CPD and the Esprit BRA Working Group EP 22457 CCLII . 2 Note that user defined datatypes are also called algebraic in Haskell. In spite of this terminology, Haskell s data ....

....are intended to solve goals composed of joinability conditions; i.e. goals will have the same form as conditions for defining rules. The expressive power of algebraic constructors in our programs can be used to model action and change problems declaratively, avoiding the so called frame problem [8]. In [8] it has been already shown that planning problems can be modeled by means of equational logic programs, using a binary (ACI) operation ffi, to represent situations as multisets of facts fact 1 ffi : ffi fact n , and a ternary predicate execPlan(initialSit; plan; finalSit) to model ....

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Große G., Holldobler J., Schneeberger J., Sigmund U., Thielscher M.: Equational Logic Programming, Actions, and Change. In Proc. ICLP'92, the MIT Press, pp. 177--191, 1992.

....turned out to be equivalent for a certain class of planning problems [19] In particular, the approach developed in [26] is based on logic programming with an associated equational theory. Since its first formalization, this method has been extended into various directions, e.g. to handle objects [20], specificity [28] concurrent [3] as well as non deterministic actions [5, 45] and ramifications [46] Moreover, its relation to the high level specification languages A [16] and the Ego World Semantics [42] were clarified [45, 44] Although these results illustrate the expressiveness of the ....

.... theory, which requires a non standard unification procedure in conjunction with an extended resolution principle called SLDE resolution [15, 25] In this paper, we follow an alternative direction and investigate a particular program where a unification algorithm which has been proposed in [20] for our special equational theory is integrated by means of additional program clauses while otherwise standard unification is used. On the basis of this logic program, we illustrate two general classes of problems which deserve a successful treatment yet turn out to be unsolvable using ....

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Gerd Große, Steffen Holldobler, Josef Schneeberger, Ute Sigmund, and Michael Thielscher. Equational Logic Programming, Actions, and Change. In K. Apt, editor, Proceedings of the International Joint Conference and Symposium on Logic Programming (IJCSLP), pages 177--191, Washington, 1992. MIT Press.

....possible to represent non deterministic actions like roling a dice, which leads to six successor situations. Based on this representation we may then model sceptical or credulous agents depending on whether they believe a fluent if it holds in each or only one successor situation, respectively [9, 5]. In [4] it was shown how the approach can be extended to parallel actions. In particular, if concurrently executed actions yield conflicting effects then this is interpreted as an implicit uncertainty in the action descriptions. Such actions are treated non deterministically wrt to the ....

G. Große, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher. Equational logic programming, actions, and change. In K. Apt, editor, Proceedings of the International Joint Conference and Symposium on Logic Programming, pages 177--191. MIT Press, 1992.

....are axiomatized by rewrite rules, the difficulties associated with the frame problem disappear [63] In addition, the conservative mappings of Horn logic with equality and of linear logic studied in Sections 4.2 and 4. 3, respectively, directly show how other logics of change recently proposed [48, 38, 39, 65, 66] can be subsumed as special cases. Added benefits include the straightforward support for concurrent change and the logical support for object oriented representation. The paper begins with a summary of the theory of general logics [70] that provides the conceptual basis for our discussion of ....

....saying what changes an action makes, without having to explicitly list all the things it doesn t change as well [ That is the frame problem. Recently, some new logics of action and change have been proposed, among which we can point out the approach of Holldobler and Schneeberger [48, 38, 39], based on Horn logic with equations, and the approach of Masseron, Tollu, and Vauzeilles [65, 66] based on linear logic. The main interest of these formalisms is that they need not explicitly state frame axioms, because they treat facts as resources which are produced and consumed. Having proved ....

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G. Große, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher, Equational logic programming, actions, and change, in: K. Apt (ed.), Proc. Int. Joint Conf. and Symp. on Logic Programming, The MIT Press, 1992, pages 177--191.

....Bibel et al. 33] gave a semantics for this method. Further development of this method was done by Fronhofer [56, 57] A series of works with an emphasis on an equational implementation of the idea is one by Josef Schneeberger, Steffen Holldobler and his students at Techniche Hochschule Darmstadt [86, 87, 78, 79]. They represent the components of state as a multiset term held together by an Associative Commutative with Identity operation (ACI operation) ffi, which can be thought of as multiset union. It is important that ffi is not idempotent. The state of the planner is represented by the predicate ....

....at least for deductive planning, three proposed approaches the linear connection method, the ACI rewriting approach and the LL based approach (see below) are equivalent. They also consider issues of change, action and specificity (selecting the method which matches largest part of the state) [79]. Masseron et al. 127] consider planning in a logic deduction setting. They first describe from philosophical grounds an abstract model of actions (basically relations with specific properties) and then implement this in LL. They consider a blocks world example and the three socks example. In ....

G. Große, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher. Equational logic programming, actions, and change. In Joint Intl. Conf. and Symp. on Logic Programming (JICSLP'92), 1992. Also in Logic and Change, a workshop at GWAI'92.

....to narrowing and reflection. Furthermore, there does not exist a refutation of EP [f(Fg with respect to TRANS if F contains an equation of the form c(s 1 ; s n ) d(t 1 ; t m ) where c and d are different constructors. Such a goal clause is often called a failure. In analogy to [Holldobler, 1987a] we define a function simplify which applies the above mentioned simplification rules to a goal clause as long as possible and tests that it is not a failure. An s derivation is a derivation where each goal clause is simplified. Theorem 12 Let R be a computation rule, EP be a canonical term ....

....extended this result to hold also for higher order E unification. The transformations rules presented herein can be used as a computational method for equational logic programs as proposed by Jaffar et al. 1984; 1986] Goguen Meseguer [1986] by adding a lazy resolution rule as suggested in [Holldobler, 1987a] This rule applied to a selected atom of the form P (s 1 ; s n ) and a program clause of the form P (t 1 ; t n ) D forces the comparison of corresponding arguments, i.e. D [ fP (s 1 ; s n )g Gamma (D [ D [ fs i =t i j 1 i ng; where D and D are sets of atoms and ....

S. Holldobler. Equational logic programming. In Proceedings of the Symposium on Logic Programming, Computer Society, Press of the IEEE, Washington, pages 335--346. IEEE, 1987.

....of simultaneous actions as, for example, presented in [13] Although we have considered only planning problems in this paper, the techniques can also be applied to other domains. Objects and inheritance [1] or database updates [28] are just two examples for such domains, which are discussed in [12]. Logics which treat facts as resources have been invented only recently. Though these logics handle the technical frame problem quite nicely on a theoretical level, it remains to be seen how they deal with the other open problems associated with reasoning about situations, goals, and actions ....

....fetching a sock, we do not know whether the sock is white or black. Obviously, one would like to represent this fact by a disjunction. But the disjuncts should again be treated as resources which are consumed and produced. It seems to be not obvious how this should be done as various examples in [31, 12] suggest. Modelling a world by situations, actions and goals seems to be quite adequate for high level tasks like, for example, explaining to a novice of how to drive a car. At least, this is precisely they way as it is done in driving school. One learns how switching of gears effects the ....

G. Grosse, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher. Equational logic programming, actions, and change. In K. Apt, editor, Proceedings of the International Joint Conference and Symposium on Logic Programming, pages 177--191. MIT Press, 1992.

....in the first clause as it is already engaged in another connection (namely, to satisfy the conditions of the heat FeS action) Hence, although the above matrix is classically tautological, we cannot find a proof given the non classical linear interpretation, which is exactly the desired result. In [14], it has been shown that this calculus, called linear connection method , is in general suitable for conjunctive planning problems, where situations as well as conditions and effects of actions are conjunctions of facts. Moreover, it turned out to be equivalent to a subset of Linear Logic [11] ....

Große, G., Holldobler, S., Schneeberger, J., Sigmund, U., Thielscher, M.: Equational logic programming, actions, and change. In: Apt, K. (ed.): Proceedings of the International Joint Conference and Symposium on Logic Programming (IJCSLP), pp. 177--191, Washington, 1992. MIT Press.

....translated into several formalisms [7, 5, 14, 4, 19] In [1] a sound but, unfortunately, incomplete encoding of AC using extended logic programs following the lines of [7] was presented. In this paper, we extend the work [19] and show how an approach based on equational logic programming (ELP) [10, 9] can be used as a sound and even complete method for encoding AC and A C . ELP is a deductive approach using first order logic for describing actions. Similar to A and AC , the effects of actions are described by adding or removing resources from a single term representing a complete ....

G. Große, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher. Equational Logic Programming, Actions, and Change. In K. Apt, ed., Proc. of IJCSLP, p. 177--191, Washington, 1992. MIT Press.

....and a : C ) E be an action in multiset form such that S 0 = S Gamma oeC) oeE . Then, for each solution oe of the AC1 unification problem Upsilon S =AC1 Upsilon C ffi V we find that oeC S and oeV =AC1 Upsilon S Gamma oeC . Furthermore, Upsilon S 0 = oeV ffi Upsilon E [10]. One should observe that augmenting the equational theory AC1 by the axiom of itempotency, i.e. representing sets of resources instead of multisets, would not guarantee these properties. Based on the term representation of multisets an action with conditions C and effects E is simply represented ....

....restricted. The terms to be unified are always of the form t 1 ffi : ffi t n , where the AC1 symbol ffi does not occur in t i , 1 i n , and for at most one i; 1 i n , t i is a variable. For such restricted AC1 unification problems there is always a finite set of most general unifiers [10]. An analysis of the algorithms presented in [10] shows that such restricted problems can be solved in O(n Theta log n) time, where n is the size of the AC1 unification problem. Proposition 2 All AC1 unification problems encountered in constraint SLDE derivations from (8) are restricted. Proof ....

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G. Große, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher. Equational logic programming, actions, and change. In K. Apt, editor, Proceedings of the International Joint Conference and Symposium on Logic Programming, pages 177--191. MIT Press, 1992.

....is applied as described above and, afterwards, applying p to the resulting situation, E ffi Z) yields g 3 . A major difficulty as regards practical implementations of this approach is caused by the underlying equational theory, which is assumed to be built into the unification procedure. In [10] we argued that the AC1 unification problems that 3 For the sake of an appropriate treatment of equality subgoals, we implicitly add the clause X =AC1 X encoding reflexivity. Note that each SLDE step is intended to be performed with respect to our equational theory, AC1. occur when computing ....

G. Große, S. Holldobler, J. Schneeberger, U. Sigmund and M. Thielscher. Equational Logic Programming, Actions and Change. In K. Apt, editor, Proceedings of IJCSLP, pages 177--191, Washington, 1992. The MIT Press.

....when the action tossing a coin is executed, whereas the term headszheads denotes the set f fjheadsjg g , which represents the fact that there is nothing but exactly one alternative, namely heads . If we regard facts 6 An efficient E Unification procedure for planning problems was presented in [7] as resources and a term of the form X z Y as having either the resources X or the resources Y but not both, then it is natural to require that ffi distributes over z , i.e. that the law of distributivity X ffi (Y z Z) X ffi Y ) z (X ffi Z) D) holds. The reader is invited to observe that z ....

G. Große, S. Holldobler, J. Schneeberger, U. C. Sigmund, and M. Thielscher. Equational Logic Programming, Actions, and Change. In Proc. Joint International Conference and Symposium on Logic Programming JICSLP'92, 1992.

....answer substitution fX 7 loaded ffi aliveg by applying (2) and performing an AC1 unification step. The three recent approaches [3, 22, 16] turned out to be equivalent for planning problems, where situations as well as the conditions and effects of actions are conjunctions of atomic facts [27, 13]. This result does not only provide a standard semantics for fragments of the linear logic and the linear connection method, it also suggests that resources can be treated within classical logic viz. by using equational logic without loosing expressive power. Let us examine the equational ....

....power. Let us examine the equational logic approach of [16] more closely. The axioms AC1 for the function symbol ffi essentially define the data structure multisets. Hence, a situation S is a multiset of facts where consumption means removing elements and production means adding elements [13]. As each description of an action is defined by its name a , the multiset C of its conditions, and the multiset E of its effects, ff = hC; a; Ei is applicable in S if C S and, if executed, transforms S into S 0 = S Gamma C) E . 2 Thus, planning in this approach is closely related to ....

[Article contains additional citation context not shown here]

G. Große, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher. Equational Logic Programming, Actions, and Change. In K. Apt, editor, Proc. of IJCSLP, 177--191. 1992.

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G. Große, S. Holldobler, J. Schneeberger, U.C. Sigmund, and M. Thielscher. Equational Logic Programming, Actions, and Change. In Proc. Joint International Conference and Symposium on Logic Programming JICSLP'92, 1992.

....of movements have been omitted in order to make the basic problems clear. For a complete description, we refer to [20] where the scenario is presented in much more detail. After describing the scenario, two approaches to modeling it are given, one based on the method of deductive planning (cf. [4, 8]) and one based on the method of logical fiberings (cf. 18, 19] The planning model employs conditional planning (because the robots cannot always be sure about the cause of mistakes) and is resource oriented. The description by logical fiberings is a new attempt to formulate planning problems ....

....the two workpieces a and b may have different alternative effects. Therefore, planning and control usually has to deal with a variety of alternative possibilities. In order to be able to describe alternative situations and conditional plans depending on the various alternatives, in [4] [8] and [22] the resource oriented equational logic approach was substantially enhanced by introducing a concept of non determinism. In this extension situations are modeled as multisets of resources as before whereas the sets of possible alternatives correspond to sets of situations, i.e. sets of ....

G. Große, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher. Equational Logic Programming, Actions, and Change. In Proc. Joint International Conference and Symposium on Logic Programming JICSLP'92, 1992.

....logic proofs. Although we have examplified the linear connection method, the equational logic and linear logic approach by planning problems only, the approaches are not restricted to planning problems. They can always be applied whenever situations and actions are to be modelled in logic. In [15] we have shown how objects and databases can be represented and manipulated in an equational logic in much the same way as objects and databases are handled in [1] and [32] respectively. The equational logic approach has been implemented in Prolog. In particular, the AC1 matching and unification ....

....algorithms have been developed and implemented [36] The experience with this implementation has revealed that the features of Prolog nicely interact with the planning process. Formula (5.1) is such an example. The atom new number(n) can easily be implemented by an appropriate Prolog program. [15] contains a database example, where negation by failure is used to conclude that the negation of an atom A in the conditions of an action is true as there is no evidence supporting A . Equation (5.1) also points to a variety of open problems. ffi is an AC1 function symbol which is not idempotent, ....

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G. Große, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher. Equational logic programming, actions, and change. In K. Apt, editor, Proceedings of the International Joint Conference and Symposium on Logic Programming, pages 177--191. MIT Press, 1992.

.... called, say, state calculus where the common successor state argument Result(a; s) is replaced by Result(s) i.e. which no longer depends on the execution of some action Other promising directions are the adaption of resource oriented approaches to reasoning about actions and change [ Gro e et al. 1992 ] dynamic logic [ Harel, 1984 ] or a first order encoding following [ Elkan, 1992 ] Last but not least, though in its current state it covers neither the static momentary distinction nor specificity or nondeterminism, the modal logic approach [ Gro e, 1994 ] should be a candidate worth ....

G. Große, S. Holldobler, J. Schneeberger, U. Sigmund, and M. Thielscher. Equational Logic Programming, Actions, and Change. In K. Apt, ed., Proc. of IJCSLP, p. 177--191, Washington, 1992. MIT Press.

....are quite different at first glance. 2] requires that in a connection proof each literal is connected at most once, 14] is standard first order logic with equality, and [18] is a Gentzen like proof system without weakening and contraction rule. Surprisingly, the three approaches are equivalent [10, 9] for conjunctive planning problems. They are all built on the key idea that facts describing situations are resources which can be consumed and produced. More precisely, a situation is a multiset of facts. The facts are consumed when the conditions of an action are to be satisfied and are produced ....

....is a multiset of facts. The facts are consumed when the conditions of an action are to be satisfied and are produced as the effects of an action. Thus, planning in these approaches is closely related to planning in STRIPS [7, 16] except that multisets are used instead of sets. As argued in [10] multisets represent resources more adequately than sets and, moreover, it is more efficient to compute with multisets instead of sets. Furthermore, planning is performed in a purely deductive system. This allows the integration of arbitrary additional theories and the exploitation of powerful ....

[Article contains additional citation context not shown here]

Gerd Große, Steffen Holldobler, Josef Schneeberger, Ute Sigmund, and Michael Thielscher. Equational Logic Programming, Actions, and Change. In Proc. Joint International Conference and Symposium on Logic Programming JICSLP'92, 1992.

....X , which is employed to solve equality subgoals. 26 AC1 unification is known to be decidable and finitary, i.e. two terms always admit a finite complete set of AC1 unifiers [Stickel, 1981] For efficient unification algorithms see, e.g. B urckert et al. 1988; Lincoln and Christian, 1990; Gro e et al. 1992] 6. S j= F j G iff S j= F and S j= G , or S 6j= F and S 6j= G ; 7. S j= 9X: F iff there exists some o 2 O such that S j= FfX 7 og ; 8. S j= 8X: F iff S j= FfX 7 og for any o 2 O . Here, FfX 7 og denotes the fluent formula resulting from replacing in F all occurrences of X by o . When ....

Gerd Große, Steffen Holldobler, Josef Schneeberger, Ute Sigmund, and Michael Thielscher. Equational logic programming, actions, and change. In K. Apt, editor, Proceedings of the International Joint Conference and Symposium on Logic Programming (IJCSLP), pages 177--191, Washington, 1992. MIT Press.

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