Multi-dimensional Dynamic Knowledge Representation

João Alexandre Leite

José Júlio Alferes

Luís Moniz Pereira

CENTRIA – New University of Lisbon

Wien, 18 Sep. 2001LPNMR’01

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 2

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.

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 3

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?

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 4

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

L2 L1

L1 L2

How to combine these two dimension of knowledge precedence?

DLP with Multiple Dimensions (MDLP)

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 5

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 DAGMDLP, 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.

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 6

2 Dimensional Lattice

d1 d2P 11

P 33

P 23P 32

P 31

P 12

P 22

P 21

P 13

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 7

Acyclic Digraph (DAG)

P a

P h

P gP i

P c

P e

P d

P b

P f

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 8

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

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 9

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.

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 10

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 sV iff:

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

Ps= js Pi

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 11

MDLP - Semantics

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

where:

Reject(s, Ms)=

{r Pi | r’ Pj , ijs, head(r)=not head(r’) Ms

body(r’)}

Defaults (Ps, Ms)={not A | r Ps : head(r)=A Ms

body(r)}

Ps= js Pi

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 12

Example 1

Ps1 Ps2

Pr1 Pr2

Psr

{a c}

{b}

{not a c}

{c}

{}Semantics at r1:

M = {b, not a, not c}Reject(r1,M) = {}Default(P,M) = {not a, not b}

Semantics at s1:

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

Semantics at sr:

M = {b, not a, c}Reject(sr,M) = {a c}Default(P,M) = {}

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 13

Example 1 (cont)

Ps1 Ps2

Pr1 Pr2

Psr

{a c}

{b}

{not a c}

{c}

{}Semantics at r1:

M = {b, not a, not c}Reject(r1,M) = {}Default(P,M) = {not a, not b}

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

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 14

Example 2

Pt1a1{p q}

{q}{not p q}

{}

Semantics at t2a1:

M = {p, q}Reject(t2a1,M) = {}Default(P,M) = {}

Semantics at t1a2:

M = {not p, not q}Reject(t1a2,M) = {}Default(P,M) = M

Semantics at t2a2:

M = {q, not p}Reject(sr,M) = {not p q}Default(P,M) = {}

Pt1a2

Pt2a2

Pt2a1

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 15

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.

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 16

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:

(RP) Rewritten program

rules

(IR) Inheritance rules

(RR) Rejection Rules

(CRS) Current State Rules

(UR) Update

Rules

(DR) Default

Rules

(GR) Graph Rules

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 17

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

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 18

(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

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 19

(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)

Syntactical Transformation

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 20

(UP) Update rulesAv APv A’v A’Pv

(DR) Default rules

A’s0

(CSR) Current state rules

A As not A A’s

Syntactical Transformation

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 21

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.

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 22

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 inheritanceIndividual agents’ views

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 23

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 MDLPsRemove the acyclicity condition (??)Applications and relationships

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 24

Company Hierarchy Example

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 Y.

Situation

type(a,t). type(b,t). needed(t).

cheap(a). reliable(b).

Board of Directors (BD)

not buy(X) type(X,T), type(Y,T), XY, cheap(Y), not cheap(X).

President (P)

Quality Management Dept. (QMD)

not buy(X) not reliable(X).buy(X) type(X,T),needed(T),

cheap(X).

Financial Dept. (FD)

19/Sep/2001LPNMR'01 - Multi-dimensional Dynamic Knowledge Representation 25

Social Representation

®

...

...

®'®

¹

°

º ¯

°

° 0

° 1

...

° c

®d

®b

®a

®e

0®d

®b

®a

®e

1®d

®b

®a

®e

c

¯¯ 0 ¯ 1 ... ¯ c

®0 ®1 ®c...