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Dynamic Logic Programming with Multiple Dimensions João Alexandre Leite José Júlio Alferes Luís Moniz Pereira CENTRIA – Universidade Nova de Lisboa Universidad.

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Apresentação em tema: "Dynamic Logic Programming with Multiple Dimensions João Alexandre Leite José Júlio Alferes Luís Moniz Pereira CENTRIA – Universidade Nova de Lisboa Universidad."— Transcrição da apresentação:

1 Dynamic Logic Programming with Multiple Dimensions João Alexandre Leite José Júlio Alferes Luís Moniz Pereira CENTRIA – Universidade Nova de Lisboa Universidad de la Habana, Cuba, 4 Dec. 2000AGP2000

2 Overview zIntroduction and Motivation z(Overview of Dynamic Logic Programming) zMulti-dimensional Dynamic Logic Programming yFramework ySemantics ySyntactical Transformation zApplication to Multi-agent Systems zConclusions and Future Work Luis Moniz Pereira: actualiza overview Luis Moniz Pereira: actualiza overview

3 Motivation zIn Dynamic Logic Programming (DLP) knowledge is given by a sequence of Programs zEach program represents a different state of our knowledge, where different states may be: ydifferent time points, different hierarchical instances, different viewpoints, etc. zDifferent states may have mutually contradictory or overlapping information. zDLP, using the relations between states, determines the semantics at each one. Luis Moniz Pereira: programs com minúscula -> Luis Moniz Pereira: muttually só com um t -> Luis Moniz Pereira: espaço a mais antes de at -> Luis Moniz Pereira: espaço a mais antes de at ->

4 Motivation (2) zLUPS was presented as a language to build DLPs zIt can been used to: ymodel evolution of knowledge in time yreason about actions yreason about hierarchies, … zBut how to combine several of these aspects in a single system? Luis Moniz Pereira: in -> into -> Luis Moniz Pereira: in -> into ->

5 Motivation Example zThe parliament issues law L1 at time t1. zThe local authorithy issues law L2 at t2 > t1 zParliament laws override local laws, but not vice-versa. zMore recent laws have precendence over older ones L2L1 L2 zHow to combine these two dimension of knowledge precedence? ë DLP with Multiple Dimensions (MDLP) Luis Moniz Pereira: authority e : no fim -> Luis Moniz Pereira: authority e : no fim -> Luis Moniz Pereira: precedence e : no fim -> Luis Moniz Pereira: precedence e : no fim -> Luis Moniz Pereira: dimensions -> Luis Moniz Pereira: dimensions ->

6 DLP with Multiple dimensions zIn MDLP knowledge is given by a set of programs zEach program represents a different state of our knowledge. zStates are connected by a DAG zMDLP, using the relations between states and their precedence in the DAG, determines the semantics at each state. zAllows for combining knowledge which evolve in various dimensions. Luis Moniz Pereira: multiple -> Luis Moniz Pereira: multiple -> Luis Moniz Pereira: faltam pontos em vários fins de linha Luis Moniz Pereira: faltam pontos em vários fins de linha Luis Moniz Pereira: espaço a mais antes de at -> Luis Moniz Pereira: espaço a mais antes de at -> Luis Moniz Pereira: using the DAG precedence relation between states -> Luis Moniz Pereira: using the DAG precedence relation between states -> Luis Moniz Pereira: which evolve -> evolving -> Luis Moniz Pereira: which evolve -> evolving ->

7 2 Dimensional Lattice Luis Moniz Pereira: identificadores subscritos pouco legíveis; estão a cinzento Luis Moniz Pereira: identificadores subscritos pouco legíveis; estão a cinzento Luis Moniz Pereira: Two -> Luis Moniz Pereira: Two ->

8 Acyclic Digraph (DAG) Luis Moniz Pereira: identificadores subscritos pouco legíveis Luis Moniz Pereira: identificadores subscritos pouco legíveis

9 Generalized Logic Programs zTo represent negative information in LP and their updates, we need LPs with not in heads zObject formulae are generalized LP rules: A B 1,…, B k, not C 1,…,not C m not A B 1,…, B k, not C 1,…,not C m zThe semantics is a generalization of SMs Luis Moniz Pereira: negative info ou, melhor, delete info ? Será necessário slide explicando diferença? Mas Teodor já terá explicado. Luis Moniz Pereira: negative info ou, melhor, delete info ? Será necessário slide explicando diferença? Mas Teodor já terá explicado. Luis Moniz Pereira: (syntactic) generalization Luis Moniz Pereira: (syntactic) generalization

10 MDLPs definition Definition: A Multi-dimensional Dynamic Logic Program, P, is a pair ( P D,D) where D=(V,E) is an acyclic digraph and P D ={P V : v V} is a set of generalized logic programs indexed by the vertices v V of D. Luis Moniz Pereira: embelezar formato do slide Luis Moniz Pereira: embelezar formato do slide

11 MDLP - Semantics Definition: Let P =( P D,D) be a Multi-dimensional Dynamic Logic Program, where P D ={P V : v V} and D=(V,E). An interpretation M s is a stable model of the multi-dimensional update at state s V iff: M s =least([ P s – Reject(s, M s )] Defaults ( P s, M s )) P s = j s P i Luis Moniz Pereira: trocar ordem das caixas Luis Moniz Pereira: trocar ordem das caixas Luis Moniz Pereira: tudo com font 24 -> Luis Moniz Pereira: tudo com font 24 ->

12 MDLP - Semantics M s =least([ P s – Reject(s, M s )] Defaults ( P s, M s )) where: Reject(s, M s )= {r P i | r P j, i j s, head(r)=not head(r) M s body(r)} Defaults ( P s, M s )={not A | r P s : head(r)=A M s body(r)} Luis Moniz Pereira: este not é uma função de reescrita -> em que not not L = L Luis Moniz Pereira: este not é uma função de reescrita -> em que not not L = L Luis Moniz Pereira: M s = -> Luis Moniz Pereira: M s = ->

13 Example 1 P s1 P s2 P r1 P r2 P sr {a c} {b}{b} {not a c} {c}{c} {} zSemantics at r1: M = {b, not a, not c} Reject(r1,M) = {} Default( P,M) = {not a, not c} zSemantics at s1: M = {not a, not b, not c} Reject(s1,M) = {} Default( P,M) = M zSemantics at sr: M = {b, not a, c} Reject(sr,M) = {a c} Default( P,M) = {not a} Luis Moniz Pereira: <- troca posição das duas semânticas r1 e s1 ; é mais natural ver s1 primeiro Luis Moniz Pereira: <- troca posição das duas semânticas r1 e s1 ; é mais natural ver s1 primeiro

14 Example 1 (cont) P s1 P s2 P r1 P r2 P sr {a c} {b}{b} {not a c} {c}{c} {} zSemantics at r1: M = {b, not a, not c} Reject(r1,M) = {} Default( P,M) = {not a, not c} zSemantics at s1: M = {a, b, c} Reject(s1,M) = {not a c} Default( P,M) = {} zSemantics at sr: M = {not a, not b, not c} Reject(sr,M) = {} Default( P,M) = M Luis Moniz Pereira: troca posição de semânticas s1 e sr. sr deve ser visto depois de se ver s1 Luis Moniz Pereira: troca posição de semânticas s1 e sr. sr deve ser visto depois de se ver s1

15 Example 2 P t1a1 {p q} {q}{q} {not p q} {} zSemantics at t2a1: M = {p, q} Reject(t2a1,M) = {} Default( P,M) = {} zSemantics at t1a2: M = {not p, not q} Reject(t1a2,M) = {} Default( P,M) = M zSemantics at t2a2: M = {q, not p} Reject(t2a2,M) = {p q} Default( P,M) = {} P t1a2 P t2a2 P t2a1

16 Towards an implementation of MDLP zHow to implement MDLP? zPre-process a MDLP at a state s into a single generalized program, where the stable models at s are the stable models of the single program. zQuery-answering is reduced to that at single programs. Luis Moniz Pereira: espaço a mais antes de at -> Luis Moniz Pereira: espaço a mais antes de at -> Luis Moniz Pereira: single -> Luis Moniz Pereira: single ->

17 MDLP – Syntactical Transformation Definition: Let P =( P D,D) be a Multi-dimensional Dynamic Logic Program, where P D ={P V : v V} and D=(V,E), including a special empty source s0. The dynamic program update over P at the state s S is a logic program s P with: (RP) Rewritten program rules (IR) Inheritance rules (RR) Rejection Rules (CRS) Current State Rules (UR) Update Rules (DR) Default Rules (GR) Graph Rules Luis Moniz Pereira: pôr tamanho único de font -> Luis Moniz Pereira: pôr tamanho único de font -> Luis Moniz Pereira: RP e IR estão em itálico porquê? Ou é para põr todos como vem a seguir? Talvez isso mas sem ser a bold, e nos slides seguintes idem. Luis Moniz Pereira: RP e IR estão em itálico porquê? Ou é para põr todos como vem a seguir? Talvez isso mas sem ser a bold, e nos slides seguintes idem.

18 Syntactical Transformation (RP) Rewritten program rules A Pv B 1, …, B m, C 1, …, C n A´ Pv B 1, …, B m, C 1, …, C n for any rule A B 1, …, B m, not C 1, …, not C n not A B 1, …, B m, not C 1, …, not C n in P v

19 (GR) Graph rules edge(u,v)(for every u < v E ) path(X,Y) edge(X,Y). path(X,Y) edge(X,Z), path(Z,Y). Syntactical Transformation

20 (IR) Inheritance rules A v A u, not reject(A u ), edge(u,v) A´ v A´ u, not reject(A´ u ), edge(u,v) (RR) Rejection rules reject(A u ) A´ P u, path(u,v) reject(A´ u ) A P u, path(u,v) Syntactical Transformation

21 (UP) Update rules A v A Pv (DR) Default rules A s0 (CSR) Current state rules A A s not A A s Syntactical Transformation

22 MDLP - Results zTheorem: The generalized stable models of the program s P coincide with the generalized stable models of the multi-dimensional update at state s according to the semantical characterization. zTheorem: Multi-dimensional Dynamic Logic Programming generalizes Dynamic Logic Programming. Luis Moniz Pereira: state s a bold -> Luis Moniz Pereira: state s a bold ->

23 MDLP applications zCombining agents knowledge yDistributed KBs yProgram composition zEvolution of hierarchical knowledge yLegal reasoning ye-commerce policy integration and evolution yOrganizational decision making zMultiple inheritance zIndividual agents views

24 Future Work zA (LUPS-like) language for building MDLPs yallowing updatable DAGs (new edges and nodes) yallowing for conditions on new edges zSocieties of MDLPs yObservation points (public and private) yInter-MDLP updates and communication yIntegration of MDLPs with common Ps zHypothetical reasoning over MDLPs zRemove the acyclicity condition (??) zApplications and relationships Luis Moniz Pereira: common? blackboard? Luis Moniz Pereira: common? blackboard?


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