Dynamic Logic Programming
with Multiple Dimensions
João Alexandre Leite
José Júlio Alferes
Luís Moniz Pereira
CENTRIA – Universidade Nova de Lisboa
AGP’2000
Universidad de la Habana, Cuba, 4 Dec. 2000
Overview
Introduction and Motivation
(Overview of Dynamic Logic Programming)
Multi-dimensional Dynamic Logic Programming
Framework
Semantics
Syntactical Transformation
Application to Multi-agent Systems
Conclusions and Future Work
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.
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?
Motivation Example
 The parliament issues law L1 at time t1.
 The local authorithy issues law L2 at t2 > t1
 Parliament laws override local laws, but not vice-versa.
L2
L1
 More recent laws have precendence over older ones
L1
L2
How to combine these two dimension of
knowledge precedence?
 DLP with Multiple Dimensions (MDLP)
DLP with Multiple dimensions
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.
P11
P21
P31
d2
d1
2 Dimensional Lattice
P12
P22
P32
P13
P23
P33
Acyclic Digraph (DAG)
Pa
Pb
Pc
Pe
Pd
Pi
Pf
Pg
Ph
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
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.
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 the multi-dimensional update at state sV iff:
Ms=least([Ps – Reject(s, Ms)]  Defaults (Ps, Ms))
Ps=

js
Pi
MDLP - Semantics
Ms=least([Ps – Reject(s, Ms)]  Defaults (Ps, Ms))
where:
Reject(s, Ms)=
{r  Pi | r’  Pj , ijs, head(r)=not head(r’)  Ms |= body(r’)}
Defaults (Ps, Ms)={not A | r  Ps : head(r)=A  Ms |= body(r)}
Example 1
{a  c} Ps1
{b} P
r1
Ps2 {}
Pr2 {c}
{not a  c} Psr
Semantics at sr:
M = {b, not a, c}
Reject(sr,M) = {a  c}
Default(P,M) = {not a}
Semantics at r1:
M = {b, not a, not c}
Reject(r1,M) = {}
Default(P,M) = {not a, not c}
Semantics at s1:
M = {not a, not b, not c}
Reject(s1,M) = {}
Default(P,M) = M
Example 1 (cont)
{a  c} Ps1
{b} P
r1
Ps2 {}
Pr2 {c}
{not a  c} Psr
Semantics at sr:
M = {not a, not b, not c}
Reject(sr,M) = {}
Default(P,M) = M
Semantics at r1:
M = {b, not a, not c}
Reject(r1,M) = {}
Default(P,M) = {not a, not c}
Semantics at s1:
M = {a, b, c}
Reject(s1,M) = {not a  c}
Default(P,M) = {}
Example 2
Semantics at t2a1:
{p  q} P
t1a1
{not p  q} Pt1a2
Pt2a1 {q}
Pt2a2 {}
Semantics at t2a2:
M = {q, not p}
Reject(t2a2,M) = {p  q}
Default(P,M) = {}
M = {p, q}
Reject(t2a1,M) = {}
Default(P,M) = {}
Semantics at t1a2:
M = {not p, not q}
Reject(t1a2,M) = {}
Default(P,M) = M
Towards an implementation
of MDLP
How to implement MDLP?
Pre-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.
Query-answering is reduced to that at
single programs.
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
 (UR) Update Rules
 (RR) Rejection Rules
 (DR) Default Rules
 (CRS) Current State Rules
 (GR) Graph Rules
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
 (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
 (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
 (UP) Update rules
Av  APv
A’v  A’Pv
 (DR) Default rules
A’s0
 (CSR) Current state rules
A  As
not A  A’s
MDLP - Results
Theorem:
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.
Theorem:
Multi-dimensional Dynamic Logic Programming
generalizes Dynamic Logic Programming.
MDLP applications
Combining agents’ knowledge
Distributed KBs
Program composition
Evolution of hierarchical knowledge
Legal reasoning
e-commerce policy integration and evolution
Organizational decision making
Multiple inheritance
Individual agents’ views
Future Work
A (LUPS-like) language for building MDLPs
allowing updatable DAGs (new edges and nodes)
allowing for conditions on new edges
Societies of MDLPs
Observation points (public and private)
Inter-MDLP updates and communication
Integration of MDLPs with common Ps
Hypothetical reasoning over MDLPs
Remove the acyclicity condition (??)
Applications and relationships