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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
- 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)

- 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

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

- 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

LPNMR\'01 - Multi-dimensional Dynamic Knowledge Representation

Acyclic Digraph (DAG)

LPNMR\'01 - Multi-dimensional Dynamic Knowledge Representation

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

- 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

- 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

LPNMR\'01 - Multi-dimensional Dynamic Knowledge Representation

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

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

where:

Ps= js Pi

Reject(s, Ms)=

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

LPNMR\'01 - Multi-dimensional Dynamic Knowledge Representation

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)

- 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

- 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

- 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

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

(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 TransformationLPNMR\'01 - Multi-dimensional Dynamic Knowledge Representation

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 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 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

- 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

- 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

- 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

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

LPNMR\'01 - Multi-dimensional Dynamic Knowledge Representation

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