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Multi-dimensional Dynamic Knowledge Representation. João Alexandre Leite José Júlio Alferes Luís Moniz Pereira. CENTRIA – New University of Lisbon. LPNMR’01. Wien, 18 Sep. 2001. Motivation. In Dynamic Logic Programming (DLP) knowledge is given by a sequence of Programs

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Multi dimensional dynamic knowledge representation

Multi-dimensional Dynamic Knowledge Representation

João Alexandre Leite

José Júlio Alferes

Luís Moniz Pereira

CENTRIA – New University of Lisbon

LPNMR’01

Wien, 18 Sep. 2001


Motivation
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.

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Motivation 2
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?

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Motivation example

L2

L1

L1

L2

Motivation Example

  • The parliament issues law L1 at time t1.

  • The local authority issues law L2 at t2 > 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)

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Multi dimensional dlp
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.

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


2 dimensional lattice
2 Dimensional Lattice

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Acyclic digraph dag
Acyclic Digraph (DAG)

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Generalized logic programs
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

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Mdlps definition
MDLPs definition

  • Definition:

    A Multi-dimensional Dynamic Logic Program, P, is a pair (PD,D) where D=(V,E) is an acyclic digraph and PD={PV : v  V} is a set of generalized logic programs indexed by the vertices v  V of D.

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Mdlp semantics
MDLP - Semantics

  • Definition:

    Let P=(PD,D) be a Multi-dimensional Dynamic Logic Program, where PD={PV : v  V} and D=(V,E). An interpretation Ms is a stable model of P at state sV iff:

Ms=least([Ps – Reject(s, Ms)]  Defaults (Ps, Ms))

Ps= js Pi

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Mdlp semantics1

Defaults (Ps, Ms)={not A | $r Ps: head(r)=A  Ms |=body(r)}

MDLP - Semantics

M=least([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’)}

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Example 1
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}

{not a ¬ c}

Psr

  • 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) = {}

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Example 1 cont
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}

{not a ¬ c}

Psr

  • 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

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Example 2
Example 2

  • Semantics at t2a1:

{p ¬ q}

Pt1a1

M = {p, q}

Reject(t2a1,M) = {}

Default(P,M) = {}

{not p ¬ q}

Pt1a2

Pt2a1

{q}

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) = {}

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Towards an implementation of mdlp
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.

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Mdlp syntactical transformation

Definition:

Let P=(PD,D) be a Multi-dimensional Dynamic Logic Program, where PD={PV : 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:

MDLP – Syntactical Transformation

  • (RP) Rewritten program rules

  • (IR) Inheritance rules

  • (RR) Rejection Rules

  • (CRS) Current State Rules

  • (UR) Update Rules

  • (DR) Default Rules

  • (GR) Graph Rules

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Syntactical transformation

(RP) Rewritten program rules

APv B1 , … , Bm , C’1, … , C’n

A´Pv  B1 , … , Bm , C’1, … , C’n

for any rule

A B1 , … , Bm , not C1, … , not Cn

not A B1 , … , Bm , not C1, … , not Cn

in Pv

Syntactical Transformation

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Syntactical transformation1
Syntactical Transformation

  • (GR) Graph rules

    edge(u,v) (for every u < v Î E )

    path(X,Y)  edge(X,Y).

    path(X,Y)  edge(X,Z), path(Z,Y).

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Syntactical transformation2
Syntactical Transformation

  • (IR) Inheritance rules

    Av Au , not reject(Au), edge(u,v)

    A´v A´u , not reject(A´u ), edge(u,v)

  • (RR) Rejection rules

    reject(Au)  A´Pu, path(u,v)

    reject(A´u)  APu, path(u,v)

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Syntactical transformation3
Syntactical Transformation

  • (UP) Update rules

    Av APv A’v A’Pv

  • (DR) Default rules

    A’s0

  • (CSR) Current state rules

    A  As not A  A’s

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Mdlp results
MDLP - Results

  • Theorem:

    The stable models of the program s Pcoincide with the stable models of P at state s according to the semantical characterization.

  • Theorem:

    Multi-dimensional Dynamic Logic Programming generalizes Dynamic Logic Programming.

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Mdlp applications
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

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Future work
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

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Company hierarchy example
Company Hierarchy Example

Situation

type(a,t).

cheap(a).

type(b,t).

reliable(b).

needed(t).

Financial Dept. (FD)

Quality Management Dept. (QMD)

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).

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation


Social representation
Social Representation

LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation