Using Partial Order Plans for Project Management

Transcription

1 Using Partial Order Plans for Project Management Ozan Kahramanogullari Institut für Informatik, University of Leipzig International Center for Computational Logic, TU Dresden Abstract We present a framework in which partial order plans are used for project management tasks where the dependency between each sub task and employee can be observed. For this purpose, extending the resource conscious linear logic planning, we establish an explicit correspondence between multiplicative exponential linear logic proofs and partial order plans. We express planning problems as formulae, and then extract partial order plans from the proofs of these formulae. We provide a constructive soundness and completeness result and show how this approach to planning can be used for project management. 1 Introduction In planning, as a branch of artificial intelligence, exhaustive efforts are made to develop sophisticated planning strategies based on logical inference. In this respect resources are observed to be a key to modeling dynamic change: in contrast to a property, a resource can be produced and consumed in the course of reasoning [18] which establishes a solution to the famous frame problem. Unfortunately these methods have not found broad entrance into real world applications. One generally still unresolved problem of project, skills and knowledge management [3, 16] is to support project management by proposing different subproject compositions to solve a larger project with respect to available work time budgets and skills or knowledge of employees. With respect to resource consciousness, the relation between logic, actions and causality has been studied by various authors: in [1], Bibel imposes a syntactical condition called linearity on proofs, which requires that each literal is engaged in at most one connection. In [10], based on multiset rewriting, Hölldobler and Schneeberger introduce an equational Horn logic where states are represented by an AC1 function symbol. In [17], Masseron et al. applies multiplicative fragment of Girard s linear logic [5] to resource conscious planning by axiomatizing the actions as proper axioms. Linear logic approach to planning is studied further by various authors [12, 15, 4]. In [6], it is shown to be equivalent to the approaches in [1] and [10]. While above mentioned approaches aim at representing a sequence of actions by a proof of the axiomatization of the agent s actions and the world it is acting on, representing a proof by a plan, which captures the parallel behavior of actions, remains a challenge. In this paper, we further elaborate on the linear logic approach to planning and establish an explicit correspondence between partial order plans and proofs. This way, we aim at using multiplicative exponential linear logics proofs to generate partial ordered plans to reach a goal state. Although such a partial ordered plan corresponds to a set of possible plans reaching the goal state, within a concurrency semantics similar to that of Labeled Event Structures [19], this partial order plan can be interpreted as a plan that respects the parallel behavior of actions. Such a semantics allows to use this method on project management problems where limited availability of resources, e.g., human resources in the project cycle, is crucial. Let us consider an introductory example where employees of a company are assigned to distinct sub-projects. The goal of the example project is to establish a corporate content management system (CMS). Table 1 gives an overview over sub-projects, their respective inputs, outputs (also sub-projects) and employee skills required to complete these subprojects. Furthermore, Table 2 collects the information on available human resources, including their profes- LIT 2004 Leipzig, 29. September 1. Oktober 2004

2 Sub-projects Required Inputs Required Skills Outputs Duration (Weeks) T1 - Web-design Design 2 T2 Design Template-creation Template 1 T3 - Server-installation Server 2 T4 Server CMS-deployment CMS 2 T5 - Content-collection Content 2 T6 CMS, Content, Content- Project- 1 Template integration finished Table 1: Separate Tasks (Sub-projects) Employee Skills Available (Weeks) E1 Web-design, Template-creation, 4 Content-collection, Content-integration E2 Server-installation, CMS-deployment, 4 Content-collection E3 Content-collection, Content-integration 3 Table 2: Available Employee sional skills and their availability in weeks. The goal is to reach a state where subproject T6 is completed with output Project finished with respect to the available resources, or state that this will not be possible. In this paper, after introducing the formal framework and its partial plan generation techniques, we will give an encoding of this example as a planning problem within multiplicative exponential linear logic. Then we generate a proof of this planning problem in a purely proof theoretical framework and extract a partial ordered plan. The generated partial ordered plan may be viewed as a maximally parallelized composition of required sub-projects to reach the goal. According to additional constraints (e.g. availability of an employee only in a certain time-frame) this partial plan may also be decomposed in more linearized versions to fit into the constraints. As the underlying formalism we employ the calculus of structures presentation of linear logic: the calculus of structures [8] is a proof theoretical formalism, which can be seen as a generalization of the one-sided sequent calculus where the notion of main connective of sequent calculus disappears. This way rules become applicable deep 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 view of computation as proof search: more possibilities in the permutability of the inference rules yield optimized presentations (decomposition) of proofs [20]. These results point out ways of controlling the non-determinism in proof search by making new normal forms of proofs possible. In the following, after recapitulating multiset rewriting and linear logic approaches to planning, we 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, similar to the method in [11], 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. This way, the implicit correspondence between partial order plans and proofs becomes explicit. We then show how these methods can be applied to project management problems of the above form. We conclude with conjectures and future work. For the proofs of the theorems, we refer to [13].

3 a a Γ s, s t (cut) Γ, t Γ s (1 l ) Γ, 1 s Γ, s, t u Γ s t ( l ) ( r) 1 (1 r) Γ, s t u Γ, s t Figure 1: Multiplicative Linear Logic System MLL 2 Planning Problems Following [6, 17], a planning domain is given by: (1) a set of constants representing atomic properties of the world which we call atoms, denoted by small letters; (2) a set of transition rules (actions) that are multiset 1 rewrite rules and specify actions 2 ; (3) states which are multisets of atoms. A conjunctive planning problem P consists of a multiset I : { a 1,..., a m } of atoms called initial state, a multiset G : { g 1,..., g n } of atoms called goal state and a finite set A of actions of the form act : { c 1,..., c p } { e 1,..., e q }, where { c 1,..., c p } and { e 1,..., e q } are multisets of atoms called conditions and effects, respectively. An action is applicable in state S iff { c 1,..., c p } S. The application of an action leads to a state ( S { c 1,..., c p } ) { e 1,..., e q }. A goal G is satisfied iff there is a plan p, i.e., a sequence of actions [a 1,..., a n], which transforms the initial state into a state S and G S. If there exists such a plan p, then p is called a solution for the planning problem. The length of a plan is the number of actions in that plan. Now, to illustrate the above theory on a planning problem, let us look at the following example which is a modification of an example from [6]. Suppose Bert is thirsty and wants to get some lemonade (l) from a vending machine. The lemonade costs 50 cents (f). Bert has a dollar bill (d) in his pocket. Because the vending machine accepts only 50 cents coins, Bert has to get change for his dollar. The 1 Multisets are denoted by the curly brackets { and }., and denote the multiset operations corresponding to the usual set operations, and, respectively. 2 We consider only propositional actions. problem of getting the lemonade can be described as a planning problem with the initial state I : { d }, the actions { d } { f, f } and { f } { l } that allow him to change a dollar for two 50 cents coins, and to get a lemonade, respectively. The goal state in which Bert got the lemonade is given by G : { l }. Clearly, the solution to the problem is the plan in which Bert changes the dollar and then buys the lemonade: applying this plan to the initial state yields, the state { f, f }, and then { f, l }. As the goal is contained in the last state, the planning problem is solved. 3 Linear Logic and Planning For an indepth exposure to linear logic, we refer to [5]. Following [17], a conjunctive linear theory consists of the axioms and rules in Figure 1, which is the multiplicative fragment of sequent calculus system MLL, together with the proper axioms c 1,..., c l e 1,..., e k for each action. Following [6], a proof of a planning problem Γ t is a solution for a conjunctive planning problem 3 iff Γ is the multiset that represents the initial state, and t is the multiplicative conjunction of the atoms that hold in the goal state. In this approach, because the plan is extracted by reading the leaves of the proof tree from left to right, we must keep track of the premises: in an application of a cut rule, the sequents Γ s and, s t must occur, respectively, on the left and right side of the premise. Furthermore, in an application the r rule one has to decide which premise is written to the left and which one to the right, non-deterministically. These decisions disregard the 3 Only complete information in the planning problems is allowed, i.e., we specify only and exactly what is consumed and produced.

4 Associativity [ R, [ T ]] = [ R, T ] ( R, ( T )) = ( R, T ) Commutativity [ R, T ] = [ T, R] ( R, T ) = ( T, R) Units [, R] = R (1, R) = R Exponentials??R =?R!!R =!R =? 1 =!1 Negation = 1 1 = [R, T ] = (R, T ) (R, T ) = [R, T ] R = R Singleton [R] = (R) = R Figure 2: The equational system underlying ELS. other plans that solve the same planning problem, and in particular any possible partial order between the actions remains hidden. To illustrate these ideas let us return to the example from Section 2. This time Bert is not only thirsty but also hungry. Since he is equipped with the action that allows him to buy a candy-bar (c) for 50 cents from the vending machine, this should not be a problem. Then, once he has a lemonade and a candy bar, he can have lunch which makes him happy (h). We express these actions as the following axioms ch-dollar : d f f buy-can-bar : f c buy-lem : f l have-lun : c, l h and the planning problem is d h. Then we get the following proof d f f and the plan f l d h f c c, l h cut l, f h cut f, f h l f f h cut [ch-dollar, buy-lem, buy-can-bar, have-lun]. Obviously, Bert can as well buy the candy-bar before buying the lemonade, and achieve the same goal. However,the proof above distinguishes this plan from the former. A plan which captures this intuition is the partial order plan in Figure 3 which is not explicit in the above proof. We will now present a different way of expressing planning problems and actions, which will allow us to extract partial order plans from proofs. For this purpose, we will employ the calculus of structures presentation of linear logic with the gain of availability of properties of proofs and derivations, which are not present in the sequent calculus presentation of linear logic. 4 Calculus of Structures and MELL In this section we present multiplicative exponential linear logic (MELL) in its calculus of structures [8] presentation, namely system ELS, following [20]. The reader in a hurry can skip this Section and Section 5 at the first reading, and return later to these sections. With respect to sequent calculus, structures are intermediate expressions between formulae and sequents. There are countably many atoms, denoted by a, b, c,... The Structures of the language ELS Figure 3: An example for a partial order plan d init ch Dollar f f buy Lem buy Can bar c l have Lun h goal

5 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} Figure 4: System ELS are denoted by P,Q,R,... and are generated by R ::= a 1 [ R,..., R ] }{{} ( R,..., R ) }{{} >0 >0!R?R R 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 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. In the calculus of structures, an inference rule is T a scheme of the kind ρ, where ρ is the name of R the rule, T is its premise and R is its conclusion. A S{T } typical (deep) inference rule has the shape ρ S{R} and specifies a step of rewriting, by 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 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. 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 is R, and inference rules are in S will be written as T S R. Similarly, a proof Π which is a finite derivation whose topmost inference rule is an axiom will be denoted by Π R S The system in Figure 4 is called multiplicative Exponential Linear logic in the calculus of Structures, or system ELS. The rules of the system are called one (1 ), atomic interaction (ai ), switch (s), promotion (p ), weakening (w ), and absorption (b ). 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, system {b }, then {w }, then {p, s}, then {ai }, and then {1 } are applied..

6 5 Encoding the Planning Problem Instead of expressing actions as axioms, we embed the actions in a structure that represents the planning problem. A very good reason for this is deriving proofs without cut: beside the non-determinism due to the choice of the competing actions, the availability of the cut rule results in an fierce nondeterminism. 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. For this reason, we will employ multiplicative exponential linear logic. 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. An encoding of the above formula as a structure leads to the definition below which results from negating this formula and then translating the resulting formula into a structure. De nition 3 A conjunctive action structure, denoted by A (possibly indexed), is a structure of the form? ( c 1,..., c p, [e 1,..., e q ]). 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. De nition 4 A problem structure, denoted by P (possibly indexed), is a structure of the form! [r 1,..., r m, ( t 1,..., t n)]. To make the interaction between the planning problems and actions, 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. We obtain an explicit 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. De nition 5 A conjunctive planning problem structure (cpps), denoted by P, is a structure of the form [?A 1,...,?A s, (! P 1,...,! P t)] where A i are conjunctive action structures and P j are problem structures. Given a conjunctive planning problem P with the initial state I, goal state G, and a set A = {act 1,..., act s} of actions where act i = { c i,1,..., c i,pi } { e i,1,..., e i,qi }, for 1 i s. A cpps [?A 1,...,?A s,! [r 1,..., r m, ( t 1,..., t n)]] corresponds to P if A i = ( c i,1,..., c i,pi, [e i,1,..., e i,qi ]), I { r 1,..., r m } and G { t 1,..., t n }. Let us reconsider our running example. This planning problem can be expressed as the following cpps. [? ( 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. Proposition 6 A cpps [?A 1,...,?A s, (! P 1,...,! P t)] has a proof iff, for 1 i t, all the cpps [?A 1,...,?A s,! P i] have proofs. With the next theorem we will show that proving a cpps in ELS is equivalent to showing that the corresponding conjunctive planning problem has a solution.

7 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]] Figure 5: The proof of a cpps Theorem 7 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 P ELS 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 }. 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 The only if part of the proof is constructive in the sense that it defines a procedure which transforms a plan that solves a conjunctive planning problem into the ELS proof. 5.1 Extracting the Partial Order Plan A partial order plan which solves a planning problem can be extracted from the proof of it. We 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. Now, let us express the above ideas formally. 5 If one wants to give up resource consciousness, she can easily employ? also for expressing the initial and goal states. De nition 8 Let Π be the proof Π ELS 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 µ(π) = { (l, k) } µ ( ) Π ELS 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.

8 one gets for this planning problem with labels instantiated is the following 6 in Figure 5. 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 9 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. 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 = {(init, a 1), (init, a 2), (a 1, a 3), (a 2, a 4), (a 3, goal), (a 4, goal)} C 2 = {(init, a 1), (init, a 2), (a 1, a 4), (a 2, a 3), (a 3, goal), (a 4, goal)}. However, as a consequence of Theorem 2, it is easy to observe that two different proofs of a cpps P 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 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 irre exive. 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)}. De nition 11 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. 6 In the proof below, expressions such as s n denote n subsequent application of the rule s. Proposition 12 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. De nition 13 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. Returning back to our running example, the concurrent plan is Ω = C, given in Figure 3, 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 ] which are linearizations of the partial order plan in Figure 3. Theorem 14 Let P be a cpps that corresponds to Π ELS a planning problem P such that, and let P Λ P,Π be the partial order plan of Π for P. If l is a linear plan induced by a linearization L of Λ, then l solves P. 6 Project Management In this section we will show how the approach to partial order planning, presented in the previous sections, can be used for project management. In a resource conscious setting, a project management task can be expressed as a planning problem. Resources are crucial to express the number of weeks required for an employee to accomplish a task. Each task, and assigning each employee to each task becomes an action. We express the different skills of an employee as different actions which treat these skills as resources that are produced as a result of executing this action. This corresponds to assigning a particular employee to a task where he can use these skills. Let us return to our project management example at the Introduction. The actions in Figure 6 are the actions summarizing the information in Table 1 and Table 2: The

9 t 1 : { web-design, web-design } { design } t 2 : { design, template-creation } { template } t 3 : { server-installation, server-installation } { server } t 4 : { server, cms-deployment, cms-deployment } { cms } t 5 : { content-collection, content-collection } { content } t 6 : { cms, content, template, content-integration } { project-finished } - e 2,1 : { emp 2 } { server-installation } e 1,1 : { emp 1 } { web-design } e 2,2 : { emp 2 } { cms-deployment } e 1,2 : { emp 1 } { template-creation } e 2,3 : { emp 2 } { content-collection } e 1,3 : { emp 1 } { content-collection } e 3,1 : { emp 3 } { content-collection } e 1,4 : { emp 1 } { content-integration } e 3,2 : { emp 3 } { content-integration } Figure 6: Actions of a Project Management Problem actions t 1,..., t 6, respectively, correspond to the rows of Table 1 that begin with T1,...,T6, respectively. For instance, action t 1 represents the information that assigning an employee for two weeks to web-design results in a design. t 1 : { web-design, web-design } { design } The actions e 1,1,..., e 3,2 capture the information in Table 2, disregarding how many weeks an employee is available: For instance, the action t 1 : { emp 1 } { web-design } represents the information that the Employee 1 can be assigned to web-design for a week. Then these actions together with the initial state I = { emp 1, emp 1, emp 1, emp 1, and the goal state emp 2, emp 2, emp 2, emp 2, emp 3, emp 3, emp 3 } G = { project-finished } give a planning problem that captures our project management task. It is important to observe that, in the initial state, resource consciousness allows us to express that the Employee 1, 2 and 3 are, respectively, available for 4, 4 and 3 weeks, respectively. By applying the methods from the previous sections, it can be easily seen that cpps for this planning problem has a proof which indicates that this project can be accomplished with these resources in hand, i.e., the employees that are available. It is also easy to see that there is a proof that leads to the concurrent plan Ω, the Hasse-diagram of which is given in Figure 7. Such a concurrent plan serves as a dependency graph of different tasks and employee which shows how many weeks each employee needs to accomplish his part of the project since an employee will be assigned as many weeks as the number of actions that are present in the graph for this employee. For instance, the Employee 1 has to be assigned four weeks to accomplish Task 1 and then Task 2. As shown in Figure 8, it is also possible to extract the dependency graph of each individual employee. 7 Discussion and Future Work We presented a framework in which partial order plans can be used for project management tasks where the dependency between each sub-task and employee can be explicitly observed within a graph. For this purpose we presented a deductive system where the linear logic approach to conjunctive planning problems are extended to partial order plans. By employing the calculus of structures presentation of multiplicative exponential linear logic, we provided an algorithm to extract partial order plans from the proofs of the planning problems. This

10 goal t 6 t 2 t 5 t 4 t 1 t 3 e 3,1 e 3,1 e 3,2 e 1,1 e 1,1 e 1,2 e 1,2 e 2,1 e 2,1 e 2,2 e 2,2 init Figure 7: Concurrent plan Ω of a Project Management Problem way, we established an explicit correspondence between partial order plans and proofs which permits to use our method to project management tasks to observe the explicit dependencies between sub-tasks and employees. In this paper, we implicitly exploited a labeled event structure semantics of proofs which allows to observe the parallel behavior of actions: labeled event structures is a model for concurrency [19] that has been studied for linear logic proofs in the sequent calculus in [7]. Future work includes investigating an explicit labeled event structure semantics of the proofs of the planning problems. System NEL is [8, 9] an extension of multiplicative exponential linear logic by a non-commutative connective. In [14], we employed system NEL to conjunctive planning problems, in a way that captures concurrent actions, resembling the method in [2] that captures a simple process algebra. Future work includes a comparison of these methods. Acknowledgments This work has been supported by the DFG Graduiertenkolleg 446. I would like to thank Sören Auer for providing the introductory example. References [1] Wolfgang Bibel. A deductive solution for plan generation. In New Generation Computing, pages [2] 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, [3] A. Cimitile and G. Visaggio. Managing software projects by structured project planning. International Journal of Software Engineering and Knowledge Engineering, 7(4): , 1997.

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