1. Complexity of Contextual Reasoning KR, Doctoral Consortium
Bounded Model Property for Multi-Context Systems
Student: Floris Roelofsen
University of Twente, Netherlands
KR, Doctoral Consortium
Whistler, Canada, June 2004
Advisor: Luciano Serafini
ITC-IRST, Trento, Italy
Motivation
Analysis
Formalization
Problem of generality
(McCarthy, 1987)
Principle of locality
(Giunchiglia, 1993)
PLC: Propositional Logic of Context
(McCarthy, Guha, Buvač, et. al.)
MCS: Multi-Context Systems
(Giunchiglia, Serafini, et. al.)
MCS more general than PLC
(Serafini & Bouquet, AIJ 2004)
NP–completeness for PLC (Massacci, AAAI 1996)
Multi-Context Systems
Intuition
Multiple partial, approximate, and perspective representations… (Benerecetti et.al., 2000)
…subject to certain compatibility constraints: if Mr. 1 sees a ball, then Mr. 2 also sees a ball.
Example
Contexts: I = {1,2}
Languages: L1 = L({ l, r }) and L2 = L({ l, c, r })
Bridge rules: : l v r → 2 : l v c v r
Syntax
A set of indices I, each of which denotes a context.
Each context i є I is described by a formal language Li.
Local expressions ~ labeled formulas: i : φ
Relational expressions ~ bridge rules: i : φ ,…, i : φ → i : φ
A multi–context system [ { Li }i єI ,BR] consists of a set of
languages { Li }i єI and a set of bridge rules BR.
Semantics
A chain c = { ci }i є I is a collection of sets of local models.
A chain c satisfies an i – labeled formula φ if all local models in ci (classically) satisfy φ.
A chain complies with a bridge rule if it satisfies its consequence or does not satisfy one of its premises.
A chain c is j – consistent if cj is nonempty.
Bounded Model Property
Contextual Satisfiability Problem
Given a set of formulas Φ (addressing a set of contexts J )
and a set of bridge rules BR. Is there a chain, which:
satisfies Φ
complies with BR
is J – consistent ?
Proof
Take any J – consistent chain c that satisfies Φ and complies with BR.
For every j єJ, there must be at least one local model mj є cj that satisfies
all j – labeled formulas in Φ.
Let BR* be the set of bridge rules whose consequences aren’t satisfied by c.
Every br є BR* must have a premise which is not satisfied by at least one local model mbr in c.
Remove from c all models but: Uj єJ mj U Ubr є BR mbr.
The resulting chain contextually satisfies Φ and contains at most
|J | + |BR| local models.
Bounded Model Property
A set of formulas is contextually satisfiable iff it is satisfied
by a chain which contains at most |J | + |BR| local models.
Implications
Related Work
NP–completeness for MCS.
New complexity bounds:
O (|φ|2) for PLC (previously O (|φ|4))
O ((|J| + |BR|) . N) for MCS (no previous result)
(N is the total size of the formulas in Φ and BR)
Tractable encoding of contextual satisfiability into purely propositional satisfiability.
Contextual Decision Procedures.
(Serafini & Roelofsen, KR 2004)
(Roelofsen, Serafini & Cimatti, ECAI 2004)
Reasoning in Distributed Description Logics.
(Serafini & Tamilin, DL 2004)
Partition–based Reasoning.
(Amir & McIlraith, KR 2000 / AIJ to appear)