Multidimensional Dynamic Knowledge Representation

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Multi-dimensional Dynamic Knowledge Representation

• João Alexandre Leite
• José Júlio Alferes
• Luís Moniz Pereira

CENTRIA New University of Lisbon
LPNMR01
Wien, 18 Sep. 2001
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Motivation

• In Dynamic Logic Programming (DLP) knowledge is
  given by a sequence of Programs
• Each program represents a different state of our
  knowledge, where different states may be
• different time points, different hierarchical
  instances, different viewpoints, etc.
• Different states may have mutually contradictory
  or overlapping information.
• DLP, using the relations between states,
  determines the semantics at each one.

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Motivation (2)

• LUPS was presented as a language to build DLPs
• It can been used to
• model evolution of knowledge in time
• reason about actions
• reason about hierarchies,
• But how to combine several of these aspects in a
  single system?

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Motivation Example

• The parliament issues law L1 at time t1.
• The local authority issues law L2 at t2 gt t1
• Parliament laws override local laws, but not
  vice-versa.

• More recent laws have precedence over older ones

• How to combine these two dimension of knowledge
  precedence?

• DLP with Multiple Dimensions (MDLP)

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Multi-dimensional DLP

• In MDLP knowledge is given by a set of programs
• Each program represents a different state of our
  knowledge.
• States are connected by a DAG
• MDLP, using the relations between states and
  their precedence in the DAG, determines the
  semantics at each state.
• Allows for combining knowledge which evolve in
  various dimensions.

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2 Dimensional Lattice
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Acyclic Digraph (DAG)
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Generalized Logic Programs

• To represent negative information in LP and their
  updates, we need LPs with not in heads
• Object formulae are generalized LP rules
• A B1,, Bk, not C1,,not Cm
• not A B1,, Bk, not C1,,not Cm
• The semantics is a generalization of SMs

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MDLPs definition

• Definition
• A Multi-dimensional Dynamic Logic Program, P, is
  a pair (PD,D) where D(V,E) is an acyclic digraph
  and PDPV v ? V is a set of generalized logic
  programs indexed by the vertices v ? V of D.

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MDLP - Semantics

• Definition
• Let P(PD,D) be a Multi-dimensional Dynamic
  Logic Program, where PDPV v ? V and D(V,E).
  An interpretation Ms is a stable model of P at
  state s?V iff

Msleast(Ps Reject(s, Ms) ? Defaults (Ps, Ms))
Ps ?j?s Pi
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MDLP - Semantics
Mleast(Ps Reject(s, Ms) ? Defaults (Ps, Ms))

• where

Ps ?j?s Pi
Reject(s, Ms) r ? Pi ?r ? Pj , i?j?s,
head(r)not head(r) ? Ms body(r)
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Example 1

• Semantics at r1

Ps1
Ps2

a c
M b, not a, not c Reject(r1,M)
Default(P,M) not a, not c
b
Pr1
Pr2
c
Psr
not a c

• Semantics at s1

• Semantics at sr

M not a, not b, not c Reject(s1,M)
Default(P,M) M
M b, not a, c Reject(sr,M) a
c Default(P,M)
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Example 1 (cont)

• Semantics at r1

Ps1
Ps2

a c
M b, not a, not c Reject(r1,M)
Default(P,M) not a, not c
b
Pr1
Pr2
c
Psr
not a c

• Semantics at s1

M a, b, c Reject(s1,M) not a
c Default(P,M)

• Semantics at sr

M not a, not b, not c Reject(sr,M)
Default(P,M) M
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Example 2

• Semantics at t2a1

p q
Pt1a1
M p, q Reject(t2a1,M) Default(P,M)
q
not p q
Pt1a2
Pt2a1

Pt2a2

• Semantics at t1a2

• Semantics at t2a2

M not p, not q Reject(t1a2,M)
Default(P,M) M
M q, not p Reject(sr,M) not p
q Default(P,M)
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Towards an implementation of MDLP

• How to implement MDLP?
• Pre-process a MDLP at state s into a single
  generalized program, where the stable models at s
  are the stable models of the single program.
• Query-answering is reduced to that at single
  programs.

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MDLP Syntactical Transformation

• Definition
• Let P(PD,D) be a Multi-dimensional Dynamic
  Logic Program, where PDPV 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

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Syntactical Transformation

• (RP) Rewritten program rules
• APv ? B1 , , Bm , C1, , Cn
• APv ? B1 , , Bm , C1, , Cn
• for any rule
• A? B1 , , Bm , not C1, , not Cn
• not A? B1 , , Bm , not C1, , not Cn
• in Pv

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Syntactical Transformation

• (GR) Graph rules
• edge(u,v) (for every u lt v Î E )
• path(X,Y) ? edge(X,Y).
• path(X,Y) ? edge(X,Z), path(Z,Y).

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Syntactical Transformation

• (IR) Inheritance rules
• Av ? Au , not reject(Au), edge(u,v)
• Av ? Au , not reject(Au ), edge(u,v)
• (RR) Rejection rules
• reject(Au) ? APu , path(u,v)
• reject(Au) ? APu , path(u,v)

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Syntactical Transformation

• (UP) Update rules
• Av ? APv Av ? APv
• (DR) Default rules
• As0
• (CSR) Current state rules
• A ? As not A ? As

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MDLP - Results

• Theorem
• The stable models of the program ?s P coincide
  with the stable models of P at state s according
  to the semantical characterization.
• Theorem
• Multi-dimensional Dynamic Logic Programming
  generalizes Dynamic Logic Programming.

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MDLP applications

• Combining agents knowledge
• Distributed (and heterogeneous) KBs
• Program composition
• Evolution of hierarchical knowledge
• Legal reasoning
• e-commerce policy integration and evolution
• Organizational decision making
• Multiple inheritance
• Individual agents views

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Future Work

• A (LUPS-like) language for building MDLPs
• allowing updatable DAGs
• Societies of MDLPs
• Observation points (public and private)
• Inter-MDLP updates and communication
• Hypothetical reasoning over MDLPs
• Remove the acyclicity condition (??)
• Applications and relationships

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Company Hierarchy Example
Situation
type(a,t).
cheap(a).
type(b,t).
reliable(b).
needed(t).
Quality Management Dept. (QMD)
Financial Dept. (FD)
?
buy(X)
t
ype(X,T),needed(T),
?
not buy(X)
not reliable(X).
cheap(X).
Board of Directors (BD)
?
buy(X)
type(X,T), needed(T), not satByOther(T,X).
not buy(X)
type(X,T), needed(T), satByOther(T,X).
?
?
satByOther(T,X)
type(Y,T), buy(Y), X
¹
Y.
President (P)
?
not buy(X)
type(X,T), type(Y,T), X
¹
Y, cheap(Y), not cheap(X).
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Social Representation