sound global caching for abstract modal tableaux

Sound Global Cachingfor Abstract Modal

Tableaux

Rajeev Goré The Australian National University

Linh Anh NguyenUniversity of Warsaw

CS&P’2008

R. Goré & L.A. Nguyen Sound Global Caching for Modal Tableaux 2

Overview Motivation

Examples of tableaux

Abstract modal tableaux

A tableau algorithm with global caching

Soundness of global caching


Motivation Checking satisfiability in description logic ALC:

(whether a concept is satisfiable w.r.t. a TBox) ExpTime-complete

Implemented provers like FaCT or DLP: strongly optimized 2ExpTime (in the worst case)

Goré & Nguyen - DL’07: use sound global caching optimal (ExpTime)

Extend sound global caching for abstract modal tableaux


Example: Tableaux for CPC(Classical Propositional Calculus) Is a formula set X0 satisfiable? NNF: negations occur only before atoms. Tableau rules:

X ;

X ; ; ()

X ;

X ; | X ; ()

X ;

(’)

X ; ;

()


Example: Tableaux for CPC A tableau is a tree ...

p q ; p q

p ; q ; p q

p ; q ; p p ; q ; q

()

()

() ()


Example: Tableaux for CPC A tableau is closed if every branch ends with

p q ; p q

p ; q ; p q

p ; q ; p p ; q ; q

()

()

() ()


Example: Tableaux for CPC A formula set X is inconsistent if

there exists a closed tableau for X.

A formula set X is consistent if all tableaux for X are open.

The calculus is sound and complete:X is satisfiable iff X is consistent


Example: Tableaux for Modal Logic K What is modal logic K?

Formulas: ? Interpretations: ? The satisfaction relation: ?



Formulas: as in the case of CPC,

plus additional constructors: ,



Interpretations

Kripke model

p, rp, q

p, q, r

... ......

possible world



The satisfaction relation

p, rp, q

p, q, q,(p(qr))

... ......


Example: Tableaux for Modal Logic K Is a formula set X0 satisfiable w.r.t.

a set Г of global assumptions?

i.e. Is there a Kripke model M such that X0 is satisfied in some possible world of M,

Г is satisfied in every possible world of M?


Example: Tableaux for Modal Logic K Tableau rules: the rules for CPC plus

X0 is unsatisfiable w.r.t. Г iff

there is a closed tableau with root (X0 ; Г)

X ;

; { : X}; Г()

, , ...

, , ...

transitional


Abstract Modal Tableaux L : logic ID (a finite bit sequence)

representing a name and parameters of a logic

Formulas: finite sequences of symbols

A tableau calculus CL : a finite set of CL-tableau rules: next page a function initCL : initCL(X) is a formula set

computable from X in PTime.

A CL-tableau for X is a tree with root initCL(X), using the rules of CL for expansions.


Abstract Modal Tableaux CL-tableau rules

PTime Denominators: Each Yi is computable from X and L in PTime

Monotonicity: X’ X applying (ρ) to X’ results in Y’i Yi, 1ik

Terminal, Static or Transitional: next page

XY1 | ... | Yk

(ρ)


Abstract Modal Tableaux CL-tableau rules

Cases: ()-rule: only one denominator static rule: X Yi for all 1 i k transitional rule: only one denominator, e.g. ()

XY1 | ... | Yk

(ρ)


Abstract Modal Tableaux Static rules:

Example:

The original and modified rules have the same „effects” in constructing tableaux.

The requirement about static rules gives an easier proof of soundness of global caching.

X ;

X ; | X ;

X ;

X ; ; | X ; ;


Abstract Modal Tableaux A branch in a tableau is closed if it ends with .

A tableau is closed if all of its branches are closed.

A tableau is open if it is not closed.

X is CL-consistent if all CL-tableaux for X are open.

X is CL-inconsistent if any CL-tableau for X is closed.


The Analytic Subformula Property Calculus CL has the analytic subformula

property if for every finite formula set X there is a finite formula set X*

CL such that every formula set carried by a node in a CL-tableau for X is a subset of X*

CL.


A Tableau Algorithmwith Global Caching

Problem: Check whether X is CL-consistent.

Algorithm: Build an and-or graph for X using CL: The root node τ contains initCL(X). Each node is expanded using a CL-tableau rule. Preferences of rules:

1. ()-rule2. unary static rules3. non-unary static rules4. transitional rules

...


A Tableau Algorithmwith Global Caching If a node w is expanded using:

a ()-rule: w receives status incons (inconsistent)

a unary static rule: w is an and-node, 1 successor, status = unkown

a k-ary static rule, k 2: w is an or-node, k successors, status = unknown

transitional rules: apply rules simultaneously in every possible way n possible ways an and-node with n successors status = unknown


A Tableau Algorithmwith Global Caching Global Caching:

Before creating a new node check whether there is an existing node of the same content.

If so, use that node as a proxy.

If no rule is applicable to a node w: w receives status cons (consistent).

When a node receives status cons/incons: propagate the status backward appropriately treating cons = true, incons = false


A Tableau Algorithmwith Global Caching Stop when τ receives status cons or incons Stop when all nodes have been expanded

For every node u with status unknown: Assign u status cons.

Claim: X is CL-consistent iff τ has status cons.


Complexity If CL has the analytic subformula property

then the given algorithm for CL and X runs in exponential time in the size of X*

CL.


Soundness of Global CachingLemma 1: If the root node τ receives status incons

then X is CL-inconsistent.

Sketch: It is an invariant of the given algorithm that for every node v with status incons: either a ()-rule of CL is appl. to v.content, or v is an and-node and there exists an edge

(v,w) such that w v and w.status = incons, or v is an or-node and for every edge (v,w),

w.status = incons.


Saturation Paths In the constructed and-or graph, define a

saturation path of node v to be a sequence v0=v, v1, ..., vk

with k 0 such that, for each 1 i k, we have: vi.status = cons,

the edge (vi-1,vi) was created by a static rule,

vk.content is closed w.r.t. the static rules.

Observe that v0.content ... vk.content.


Soundness of Global CachingLemma 2: If the root node τ receives status cons

then every CL-tableau T for X is open.

Sketch: Maintain a current node cn of T to pin-point an

open branch of T. Initially, set cn to the root of T. Keep a current saturation path v0, v1, ..., vk for

some v0. Initially, v0 = τ (the root of the graph). Maintain the invariant cn.content vk.content by

moving cn along edges of T appropriately and possibly changing the current saturation path.

The branch formed by the instances of cn is an open branch of T.


Soundness of Global Caching Theorem: The root of the graph constructed for

X receives status cons iff X is CL-consistent.

The global caching method is sound.

Corollary: If calculus CL has the analytic subformula property and X*

CL has a polynomial size in the size of X and the length of L, then the given algorithm is an ExpTime decision procedure for checking CL-consistency.

If CL is sound and complete then CL-consistency means L-satisfiability.


Applications We have applied sound global caching for:

regular grammar logics TABLEAUX’05

regular modal logics of agent beliefs CLIMA’07

the description logics ALC and SHI DL’07, TABLEAUX’07


How does global caching co-operate with other optimization techniques? Attend the next talk of Nguyen:

An Efficient Tableau Prover using Global Caching for the Description Logic ALC

CS&P’2008, 1st October

sound global caching for abstract modal tableaux

Documents

modal tableauxexample

modal logic kwhat

modal logic kis

consistentsound global

pagesound global caching

modal logic ktableau

root x0 sound global

set of global assumptions