A New Logical Notion of Partial Order Planning

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1 A New Logical Notion of Partial Order Planning Ozan Kahramanoğulları Computer Science Institute, University of Leipzig International Center for Computational Logic, TU Dresden Abstract. We present a new approach to conjunctive planning where an explicit correspondence between partial order plans and multiplicative exponential linear logic proofs is established. As the underlying proof theoretical formalism, we employ the recently developed calculus of structures. This way additional proof theoretical properties become available, particularly interesting from the point of view of computation as proof search. We first express planning problems as structures, then provide a constructive soundness and completeness result of our encoding, and then give a provably correct algorithm for extracting partial order plans from the proofs of these encodings. These partial order plans, extracted from the proofs, admit a noninterleaving, behavioral concurrency semantics. Relying on this fact, we argue that this work is a crucial step for establishing a common language for concurrency and planning that will allow for a meaningful notion of equivalence of plans. 1 Introduction Planning and concurrency are two fields of computer science that evolved independently, aiming at solving tasks that are similar in nature but different in perspective: while planning formalisms focus on finding a plan, if there exists such a plan, that solves a given planning problem; the focus in concurrency theory is on the global behavior of a given concurrent system, resulting in universally quantified queries, e.g., deadlock freeness, verification of a security protocol. In contrast to approaches to planning, in order to be able to handle such queries, languages for concurrency are equipped with a rich arsenal of mathematical methods that allow for an analysis of equivalence of processes. In concurrency theory, parallel and sequential composition are expressed at the same level of representation, since they are equivalently important notions for expressing concurrent processes. However, in planning, although parallel behavior between actions have been studied in partial order planners, e.g., UCPOP [17], Graphplan [2], focusing on increasing the efficiency of the planners, these investigations remained distant from capturing the independence and causality between partially ordered actions, which is crucial from a concurrency theoretic point of view. Another line of research, which aims at capturing the concurrent behavior of actions in the logical AI literature, e.g., in [18], defines concurrency over the parametrized time spans shared by the actions.

2 In this paper, we present a new approach to linear logic planning [15, 14] that aims at providing a common language for planning and concurrency. Such a language will bring these two fields closer and make it possible to import more developed mathematical methods of concurrency theory to planning. This way, by using the methods of concurrency, e.g., bisimulation, a structural analysis of plans, which will result in a notion of plan equivalence, can be obtained. Linear logic, which is widely recognized as a logic of concurrency (see, e.g., [16]) is a natural candidate for establishing such a language, also because of its resource conscious features 1. Furthermore, in [5], the linear logic approach to planning was shown to be equivalent to Bibel s connection method [1] and Hölldobler and Schneeberger s Fluent Calculus [8]. However, in these approaches, any possible partial ordering of the actions is disregarded. In our approach, we establish an explicit correspondence between the proofs of the encodings of the planning problems and the partial order plans that solves these planning problems. These partial order plans, extracted from the proofs, admit a noninterleaving, behavioral concurrency semantics. As the underlying formalism we employ the calculus of structures: the calculus of structures [6] is a proof theoretical formalism, which is a generalization of the sequent calculus where the notion of main connective of sequent calculus disappears. This way, rules become applicable at any depth inside a formula(deep inference), allowing a formula to move into another formula in a way determined by their local structure. This results in proof theoretical properties that are not available in the sequent calculus, and that are interesting from the point of view of computation as proof search: more possibilities in the permutability of the inference rules yield optimized presentations (decomposition) of proofs [20]. In the following, after recapitulating multiset rewriting and linear logic approaches to planning, we first express planning problems as linear logic structures (formulae) where actions and problems become dual notions. Then we check the existence of a plan by searching for a proof of this structure. If succeeded, we extract a constraint set from which we construct a partial order plan of the planning problem. We provide a constructive soundness and completeness result that states the equivalence of the existence of a partial order plan and existence of a proof. We provide comparison of our approach with the existing partial order planners, and conclude with summary and future development. Space restrictions do not permit us to give the proof of the theorems. We refer to [10]. 2 Conjunctive Planning Problems Following [5], a conjunctive planning problem P is given by I, G, A where I : { r 1,..., r m } is a multiset 2 of fluents called initial state. The multiset G : 1 The multiplicative conjunction is not idempotent ( a a a is not provable) where as in classical logic is idempotent. 2 Multisets are denoted by the curly brackets { and }., and denote the multiset operations corresponding to the usual set operations, and, respectively.

3 buy Lem f l d init ch Dollar have Lun f c buy Can bar Fig. 1. (a) A partial order plan h goal d f f f c c, l h cut f l l, f h cut f, f h l f f h cut d h (b) A MLL proof of a planning problem { g 1,..., g n } of fluents is the goal state. A is a finite set of actions of the form a : { c 1,..., c p } { e 1,..., e q }, where { c 1,..., c p } and { e 1,..., e q } are multisets of fluents called conditions and effects, respectively, and a is the name of the action. An action a : { c 1,..., c p } { e 1,..., e q } is applicable to a state S iff { c 1,..., c p } S. The application of such an action a to a state S is defined by the function Φ, where it is applicable, as Φ(a, S) = ( S { c 1,..., c p } ) { e 1,..., e q }. A goal G is satisfied iff there is a plan (structure) p, i.e., a sequence of actions p = a 1 ;... ; a k, which transforms the initial state into the state G such that Φ(a k,..., Φ(a 1, I)...) = G. If there exists such a plan p, then p is a solution for the planning problem P. Then we say p solves P. We denote the empty plan with. If it is more convenient, Φ(a k,..., Φ(a 1, I)...) will be abbreviated with Φ(p, I). The length of a plan is the number of actions in that plan. In [15], Masseron et al. present a multiplicative linear logic (MLL) approach to conjunctive planning problems. In this approach 3, each action of the form { c 1,..., c l } { e 1,..., e k } is represented by a proper axiom of the form c 1,..., c l e 1 e k. To illustrate the above theory on a planning problem, let us look at the following example which is a modification of an example from [5]. Suppose Bert is working on a Sunday at a computer science department, and he feels hungry. He will be happy (h) if he gets a candy-bar (c) and also a lemonade (l) to go with it. There is a wending machine in the department, which offers both the lemonade (l) and the candy-bar (c). The lemonade and the candy-bar cost 50 cents (f) each. Bert has a dollar bill (d) in his pocket, however, because the vending machine accepts only 50 cents coins, he has to get change for his dollar. This scenario can be described as a planning problem with the initial state I : { d }, and the goal state G : { h }. The set A of actions contains the actions { d } { f, f }, { f } { c }, { f } { l }, and { c, l } { h }, respectively, that allow him to change a dollar for two 50 cents coins, buy a candy-bar, buy a lemonade, and have lunch, respectively. We express these actions as the following proper axioms ch-dollar : d f f buy-lem : f l buy-can-bar : f c have-lun : c, l h and the planning problem as d h. Then we get the proof on the right-hand side of Figure 1, which provides a solution to this planning problem: in this approach, 3 For an indepth exposure to this approach, the reader is referred to [15].

4 the plan is computed by means of proof search, where an action is applied by an application of the cut rule. Then the plan is extracted by reading the leaves of the proof tree from left to right. For instance, the below proof leads to the plan ch-dollar ; buy-lem ; buy-can-bar ; have-lun, which is one of the solutions of this planning problem. Clearly, two solutions of this planning problem are the two totally ordered plans which are indicated by the partially ordered plan at the left-hand-side of Figure 1. In fact, due to the explicit representation of resources by multisets, without committing to a totally ordered plan, the partially ordered actions of such a plan can be executed in parallel without causing any resource conflicts. In the following, we will present a different way of expressing planning problems and actions, which will lead to a concurrency semantics, and allow to extract partial order plans from the proofs that capture this intuition. For this purpose, we will employ the calculus of structures presentation of multiplicative exponential linear logic with the gain of availability of properties of proofs and derivations, which are not present in the sequent calculus. 3 The Calculus of Structures and MELL This section re-collects some notions and definitions of the calculus of structures and the system ELS which is the calculus of structures presentation of multiplicative exponential linear logic (MELL), following [20]. Structures are intermediate expressions between formulae and sequents. There are countably many atoms, denoted by a, b, c,... The Structures of the language ELS are denoted by P, Q, R, S... and are generated by R ::= a 1 [ R,..., R ] ( R,..., R )!R?R }{{}}{{} R, >0 >0 where a stands for any atom, 1 and, called one and bottom. A structure [R 1,..., R h ] is called a par structure, (R 1,..., R h ) is called a times structure,! R is called an of-course structure, and? R is called a why-not structure; R is the negation of the structure R. Structures are considered to be equivalent modulo the relation =, which is the smallest congruence relation induced by the equations for associativity and commutativity for par and times structures together with the equations shown in Figure 2. A structure context, denoted as in S{ }, is a structures with a hole that does not appear in the scope of negation. The structure R is a substructure of S{R} and S{ } is its context. Context braces are omitted if no ambiguity is possible. There is a straightforward correspondence between ELS structures and the MELL formulae. For example,! [(?a, b), c,! d] corresponds to!((?a b) c!d ). T In the calculus of structures, an inference rule is a scheme of the kind ρ, R where ρ is the name of the rule, T is its premise and R is its conclusion. A typical S{T } (deep) inference rule has the shape ρ and specifies a step of rewriting, by S{R}

5 Units [, R] = R (1, R) = R Exponentials??R =?R!!R =!R =? 1 =!1 Negation [R, T ] = ( R, T )?R =! R R = R (R, T ) = [ R, T ]!R =? R Fig. 2. Syntactic equivalence = for ELS structures 1 1 S{1} ai S[a, ā] S([R, T ], U) s S[(R, U), T ] S{![R, T ]} p S[!R,?T ] S{ } S[?R, R] w b S{?R} S{?R} Fig. 3. Multiplicative-exponential linear logic system ELS the implication T R inside a generic context S{ }, which is linear implication 4 in our case. An inference rule is called an axiom if its premise is empty. Rules with empty contexts correspond to the case of the sequent calculus. A (formal) system S is a set of inference rules. A derivation in a certain formal system is a finite chain of instances of inference rules in the system. A derivation can consist of just one structure. The topmost structure in a derivation, if present, is called the premise of the derivation, and the bottommost structure is called its conclusion. A derivation whose premise is T, conclusion T Π S is R, and inference rules are in S will be written as S. Similarly, R R will denote a proof Π which is a finite derivation whose topmost inference rule is an axiom. The rules of the system are called one (1 ), atomic interaction (ai ), switch (s), promotion (p ), weakening (w ), and absorption (b ). The system in Figure 3 is called multiplicative Exponential Linear logic in the calculus of Structures, or system ELS. Theorem 1. [20] System ELS is sound and complete for multiplicative exponential linear logic. Theorem 2. [20] (decomposition) For every proof Π in system ELS, there is a proof Π where, seen bottom-up, first system {b }, then {w }, then {p, s}, then {ai }, and then {1 } are applied. For a more detailed discussion on the proof theory of system ELS the reader is referred to [20]. 4 Due to duality between T R and R T, rules come in pairs of dual rules: a down-version and an up-version. For instance, the dual of the ai rule is the cut rule. In this paper, we only consider the down rules, which provide a sound and complete system.

6 4 Encoding the Planning Problem In this section, we will present an encoding of the planning problems in the language of ELS. Instead of using proper axioms for actions, we embed the actions in a structure that represents the planning problem. This way, we will observe cut-free proofs, where the nondeterminism is reduced to the choice of the actions: beside the non-determinism due to the choice of the competing actions, the availability of the cut rule results in an fierce non-determinism. In a bottom-up search, applying a cut rule means guessing a formula to be appropriate to be the cut formula that will result in a proof. By having a cut-free proof system, we reduce this non-determinism in the proof search to the choice of the application of the inference rules. Consider the following sequent encoding of an axiom representing a deterministic action { c 1,..., c p } { e 1,..., e q }. c 1... c p e 1... e q Note that linear implication is defined as s t = s t, where is a par connective. is the times connective. Definition 1 A conjunctive action structure, denoted by A (possibly indexed), is a structure of the form? ( c 1,..., c p, [e 1,..., e q ]). A problem structure, denoted by P, is a structure of the form! [r 1,..., r m, ( t 1,..., t n )]. It is important to observe that an action structure is an encoding of the above deterministic action which is obtained by negating the straight forward translation of the sequent calculus formula into a structure. Because an action can be executed arbitrarily many times, we employ? of linear logic, which retains a controlled contraction and weakening on the action structures. This way we can duplicate an action structure by applying the b rule, or annihilate it by applying the w rule during the proof search. To make the interaction between the planning problems and actions explicit, we prefix a planning problem with!. By applying the p rule in proof search we allow an action structure to get inside and interact with a problem structure. This way, we obtain an explicit logical duality between problem and action structures:? ( c 1,..., c p, [e 1,..., e q ]) =! [c 1,..., c p, (ē 1,..., ē q )] We can now define a planning problem in the language of ELS. Definition 2 Given a conjunctive planning problem P = I, G, A where I = { r 1,..., r m } is the initial state, G = { t 1,..., t n } is the goal state, and A = {act 1,..., act s } is the set of actions such that, for 1 i s, act i = { c i,1,..., c i,pi } { e i,1,..., e i,qi }. The structure [?A 1,...,?A s,! [r 1,..., r m, ( t 1,..., t n )]] is the conjunctive planning problem structure (cpps) that corresponds to P if, for 1 i s, A i = ( c i,1,..., c i,pi, [e i,1,..., e i,qi ]).

7 The encoding of our running example is as follows: [? ( d, [f, f]),? ( f, l),? ( f, c),? ( l, c, h),! [d, h] ] (1) The structures ( d, [f, f]), ( f, l), ( f, c), and ( l, c, h) are the conjunctive action structures for the actions ch-dollar (D), buy-lem (L), buy-can-bar (C), and have-lun (H), respectively. d denotes the dollar at the initial state, and the structure h denotes the goal state. We will now show that proving a cpps in ELS is equivalent to showing that the corresponding conjunctive planning problem has a solution. We first need the following Lemmas, proofs of which can be found in [10]. Lemma 1. [10] (i.)the rule action below is derivable (sound) in ELS. (ii.) Let a : { c 1,..., c p } { e 1,..., e q } be an action, and S = { r 1,..., r m } and S = { t 1,..., t n } be states. For some structure R, T, and E Φ(a, S) = S iff [?( c 1,..., c p, E),! [t 1,..., t n, R], T ] action [?( c1,..., c p, E),! [r 1,..., r m, R], T ]. Lemma 2. [10] The following rule is derivable (sound) in ELS. termination [?A1,...,?A s,! [g 1,..., g m, (ḡ 1,..., ḡ m )]] It is important to observe that the inference rules action and termination provide the operational semantics of a planner: these inference rules can be used as machine instructions in an implementation of this approach. Theorem 3 [10] Let P = [?A 1,...,?A s,! [r 1,..., r m, ( t 1,..., t n )]] be a cpps that corresponds to a planning problem P. There is a proof Π ELS P with k number of applications of the p rule iff there is a plan p with length k that solves the planning problem P, where I { r 1,..., r m } and G { t 1,..., t n }. Sketch of Proof: Proof by induction, for the if direction, on the number of applications of the rule p ; and, for the only if direction, on the length of the plan. The reader might realize that at the if part of the theorem it is possible to state a stronger result than the one stated, that is, the atoms in the problem structure of the cpps being proved are, because of the resource consciousness of linear logic, exactly those that specify the initial and goal state of the planning problem. 5 It is important to observe that the above theorem, by Lemma 1, provides an algorithm which transforms a plan that solves a conjunctive planning problem into an ELS proof. 5 If one wants to give up resource consciousness, she can easily employ? also for expressing the initial and goal states.

8 5 Extracting the Partial Order Plan In this section, we will present an algorithm for extracting partial order plans, which admit a concurrency semantics, from the proofs of the cpps. We first label the atoms in the proof, and then write down constraints for the atoms that get annihilated in a par structure, with respect to their labels, at the application of the ai rule during the proof construction. Putting all these constraints together, we get a partial order. Let us express these ideas formally. Π ELS Definition 4 Let Π be the proof S{T } ρ S{R} where the atoms in an action structure are labeled with the name of that action structure. Furthermore, whenever there is an application of the rule b, the labels of the atoms in the premise, which are copied, are extended with a natural number that does not occur with the same action name elsewhere in the proof. Similarly, in a problem structure, all the positive and negative atoms are labeled with init (I) and goal (G), respectively. Let Label denote the set of all the labels occurring in Π. The function µ on Π is defined as follows. If ρ is the application of the rule ai where R is the structure [a l, ā k ] for an atom a such that l, k Label, then ( ) Π ELS µ(π) = { (l, k) } µ. S{1} If ρ is the application of a rule other than ai and 1, then µ(π) = µ(π ). If ρ is the axiom 1, then µ(π) =. Given a proof Π of P, a constraint set of Π for P (denoted by: C P,Π ) is given with µ(π). We will drop the subscripts when it is obvious from the context which cpps and proof we mean. To illustrate the above definition let us go back to the previous example cpps (1). One of the proofs that one gets for this planning problem with labels instantiated is the following 6 : After plugging this proof into the function µ, we get the constraint set C = {(init, ch-dollar), (ch-dollar, buy-lem), (ch-dollar, buy-can-bar), (buy-lem, have-lun), (buy-can-bar, have-lun), (have-lun, goal)}. Proposition 5 [10] Let C P,Π be a constraint set of a proof Π for a cpps P. (i) There is no label x Label, such that (goal, x) C. (ii) There is no label x Label, such that (x, init) C. 6 In the proof below, expressions such as s n denote n subsequent application of the rule s.

9 1 1 ai! [cbuy-can-bar, c have-lun ] ai! ([lbuy-lem, l have-lun ], [ c H, c C ]) ai! ([hhave-lun, h goal ], [ l H, l L], [ c H, c C ]) ai! ([fch-dollar, f buy-can-bar ], [ h G, h H ], [ l H, l L], [ c H, c C ]) ai! ([fch-dollar, f buy-lem ], [ f C, f D ], [ h G, h H ], [ l H, l L], [ c H, c C ]) ai! ([dinit, d ch-dollar ], [ f L, f D ], [ f C, f D ], [ h G, h H ], [ l H, l L], [ c H, c C ]) s 10! ([ d D, d I ], [ f L, f D ], [ f C, f D ], [ h G, h H ], [([ l H, l L], c H), c C ]) p 4 [?( d D, [f D, f D ]),?( f L, l L),?( f C, c C),?( l H, c H, h H),! [d I, h G]] Fig. 4. The proof of a conjunctive planning problem structure Note that a cpps does not necessarily have a unique constraint set. For instance, consider the following cpps with [?(ā, c),?( b, c),?( c, d),?( c, e),! [a, b, ( d, ē)]] and the two distinct constraint sets C 1 and C 1, respectively: {(in, a 1 ), (init, a 2 ), (a 1, a 3 ), (a 2, a 4 ), (a 3, goal), (a 4, gl)} {(in, a 1 ), (init, a 2 ), (a 1, a 4 ), (a 2, a 3 ), (a 3, goal), (a 4, gl)} However, as a consequence of Theorem 2, it is easy to observe that two different proofs of a cpps have the same constraint set if they decompose to the same proof by permuting the rules, since they have the same applications of the ai rule. Proposition 6 [10] Let P be a cpps defined on the action set A and C P,Π be a constraint set of a proof Π for P. (i) C P,Π is antisymmetric. (ii) C P,Π is irreflexive. A constraint set C is not necessarily a cover relation. For instance, consider the following cpps [? (ā a1, c a1 ),? ( b a2, c a2, d a2 ),! [a int, b int, d gl ]] which results in the constraint set C = {(int, a 1 ), (a 1, a 2 ), (a 2, gl), (int, a 2 )}. Definition 7 Let P be a cpps defined on the action set A and C P,Π be a constraint set of a proof Π for P. Furthermore, let T P,Π denote the set {(x, z) (x, y) C P,Π (y, z) C P,Π }. (i ) The partial order plan of Π for P is Λ P,Π = C P,Π T P,Π. (ii ) The concurrent plan of Π for P is Ω P,Π = C P,Π \ T P,Π. Proposition 8 [10] Let C P,Π be a constraint set. (i) Λ P,Π is a strict partial order. (ii) Ω P,Π is the cover relation of the Λ P,Π. Definition 9 A linearization L of a partial plan Λ is a strict total order defined on Label, such that Λ L. Then a linear plan l induced by L is the sequence of actions that obeys the order defined by a linearization L of Λ so that, from left to right, the actions are sequenced from init to goal, excluding these two labels.

10 Returning back to our running example, the concurrent plan is Ω = C, given in Figure 1, and the partial order plan Λ is the set Λ = C {(I, G), (I, C), (C, G), (I, H), (D, G), (D, H), (I, L), (L, G)}. Then we get the two linear plans l 1 = ch-dollar; buy-lem ; buy-can-bar ; have-lun and l 2 = ch-dollar ; buy-can-bar ; buy-lem ; have-lun that are linearizations of the partial order plan in Figure 1. Following theorem justifies the correctness of our approach. We first need the following Lemma: Lemma 3. [10] Let C be a constraint set of a proof Π for a cpps P, and L be a linearization of Λ P,Π. For an action act Label, if (act, goal) L, then (act, goal) C. Sketch of Proof: Assume that (act, goal) L and (act, goal) / Λ. But this contradicts with Proposition 5, so we have that goal is an upper cover of act also in Λ. Then from Definition 7, we get (act, goal) C. Theorem 10 [10] Let P be a cpps that corresponds to a planning problem P Π ELS such that, and let Λ P,Π be the partial order plan of Π for P. If l is a P linear plan induced by a linearization L of Λ, then l solves P. Sketch of Proof: By induction on the length of l with Lemma 3. 6 Partial Order Plans with a Concurrency Semantics Being driven by efficiency concerns, the partial order planners in the literature, such as, UCPOP [17] or Graphplan [2], provide efficient computations of plans for some domains. However, due to the modeling of the world by means of properties, in contrast to resources, these approaches fail to capture the dependencies between actions in a partial order plan. For instance, consider the following scenario, which is a simplification of the famous dining philosophers problem: there are two hungry philosophers, a and b, sitting at a dinner table. In order for a philosopher to eat, she must have a fork. However, there is only one fork on the table. The problem consists in finding a plan where both philosophers have eaten. The solution of this problem is a plan in which a and b eat in either order. A plan where a and b eat concurrently cannot be a solution for this problem, since a and b cannot have the fork at the same time. Since the fork is a resource, which can not be shared, eating of one is dependent on the other s finishing eating and leaving the fork. Hence, these two actions can be executed in either order but not in parallel. In contrast to partial order planners in the literature, due to the explicit treatment of resources, the approach of the present paper respects the dependency and causality between actions in a planning domain, and results in a noninterleaving, behavioral concurrency semantics, namely, labelled event structure semantics. Labelled event structures (LES) [19] is a model for concurrency. In a

11 LES the causality between actions is expressed as a partial order, and the nondeterminism is obtained by a conflict relation. By using the operational semantics given by the inference rules of system ELS as a labelled transition system, in [10], we provided an algorithm for obtaining a LES from the specification of a planning problem. In this approach, our constraint sets correspond to successful computations in the LES. 7 Discussion We presented a deductive approach to conjunctive planning problems in the multiplicative exponential linear logic, and provided an algorithm to extract partial order plans from the proofs of these structures. This way, we have established an explicit correspondence between proofs of the planning problems and partial order plans, which solve these planning problems. These partial order plans provide a canonical representation of a class of linear plans. Furthermore, because of the explicit treatment of resources, which allows to capture the independence and the causality between the partially ordered actions, in contrast to partial order planners in the literature, our approach respects a noninterleaving, behavioral concurrency semantics. We have implemented proof search for the systems in the calculus of structures, and also a planner which implements the above ideas in the lines of [9, 11]. These implementations, mainly in system Maude [4], are available 7 for download. Systems BV [6] and NEL [7] are extensions of multiplicative linear logic, and multiplicative exponential linear logic, respectively, by a self-dual, noncommutative logical operator. System BV is NP-complete [12], whereas system NEL is Turing-complete [21]. In [13], we employed system NEL to conjunctive planning problems, in a way that captures concurrent plans syntactically, resembling the method in [3], where system BV is used to capture a simple process algebra. Future work includes combining these methods. This way, a common deductive language will be obtained, where parallel and sequential composition of actions are represented at a logical level. The operational semantics of this language will be given by inference rules of system NEL, as in [12], and a noninterleaving, behavioral semantics will be given as in the present paper and [10]. References 1. Wolfgang Bibel. A deductive solution for plan generation. In New Generation Computing, pages A. Blum and Furst M. Fast planning through planning graph analysis. In Artificial Intelligence, volume 90, pages Paola Bruscoli. A purely logical account of sequentiality in proof search. In Peter J. Stuckey, editor, Logic Programming, 18th International Conference, volume 2401 of Lecture Notes in Computer Science, pages Springer-Verlag, ozan/maude cos.html

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