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Sound Global Caching for Abstract Modal Tableaux. Rajeev Goré The Australian National University  Linh Anh Nguyen University of Warsaw CS&P’2008. Overview. Motivation Examples of tableaux Abstract modal tableaux A tableau algorithm with global caching Soundness of global caching.

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Sound global caching for abstract modal tableaux

Sound Global Cachingfor Abstract Modal Tableaux

Rajeev Goré

The Australian National University

 Linh Anh Nguyen

University of Warsaw

CS&P’2008


Overview
Overview

  • Motivation

  • Examples of tableaux

  • Abstract modal tableaux

  • A tableau algorithm with global caching

  • Soundness of global caching

Sound Global Caching for Modal Tableaux


Motivation
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

Sound Global Caching for Modal Tableaux


Example tableaux for cpc classical propositional calculus

X ;

X ; 

X ; 

(’)

()

X ;  ; 

X ;  | X ; 

X ;  ; 

()

()

Example: Tableaux for CPC(Classical Propositional Calculus)

  • Is a formula set X0 satisfiable?

  • NNF: negations occur only before atoms.

  • Tableau rules:

Sound Global Caching for Modal Tableaux


Example tableaux for cpc

p  q ; p  q

()

p ; q ; p  q

()

p ; q ; p

p ; q ; q

()

()

Example: Tableaux for CPC

  • A tableau is a tree ...

Sound Global Caching for Modal Tableaux


Example tableaux for cpc1

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 

Sound Global Caching for Modal Tableaux


Example tableaux for cpc2
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

Sound Global Caching for Modal Tableaux


Example tableaux for modal logic k
Example: Tableaux for Modal Logic K

  • What is modal logic K?

    • Formulas: ?

    • Interpretations: ?

    • The satisfaction relation: ?

Sound Global Caching for Modal Tableaux


Example tableaux for modal logic k1
Example: Tableaux for Modal Logic K

  • What is modal logic K?

    • Formulas:

      • as in the case of CPC,

      • plus additional constructors:, 

Sound Global Caching for Modal Tableaux


Example tableaux for modal logic k2
Example: Tableaux for Modal Logic K

  • What is modal logic K?

    • Interpretations

      Kripke model

possible world

p, q, r

p, r

p, q

...

...

...

Sound Global Caching for Modal Tableaux


Example tableaux for modal logic k3

p, q, q,

(p(qr))

p, r

p, q

...

...

...

Example: Tableaux for Modal Logic K

  • What is modal logic K?

    • The satisfaction relation

Sound Global Caching for Modal Tableaux


Example tableaux for modal logic k4
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

    • X0is satisfied in some possible world of M,

    • Г is satisfied in every possible world of M?

Sound Global Caching for Modal Tableaux


Example tableaux for modal logic k5

X ; 

, , ...

()

; { : X}; Г

transitional

, , ...

Example: Tableaux for Modal Logic K

  • Tableau rules:the rules for CPC plus

  • X0 is unsatisfiable w.r.t. Гiff

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

Sound Global Caching for Modal Tableaux


Abstract modal tableaux
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.

Sound Global Caching for Modal Tableaux


Abstract modal tableaux1

X

(ρ)

Y1 | ... | Yk

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, 1ik

    • Terminal, Static or Transitional:

      •  next page

Sound Global Caching for Modal Tableaux


Abstract modal tableaux2

X

(ρ)

Y1 | ... | Yk

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. ()

Sound Global Caching for Modal Tableaux


Abstract modal tableaux3

X;

X;

X; | X;

X;; | X;;

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.

Sound Global Caching for Modal Tableaux


Abstract modal tableaux4
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.

Sound Global Caching for Modal Tableaux


The analytic subformula property
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.

Sound Global Caching for Modal Tableaux


A tableau algorithm with global caching
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:

    • ()-rule

    • unary static rules

    • non-unary static rules

    • transitional rules

  • ...

Sound Global Caching for Modal Tableaux


A tableau algorithm with global caching1
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

Sound Global Caching for Modal Tableaux


A tableau algorithm with global caching2
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

Sound Global Caching for Modal Tableaux


A tableau algorithm with global caching3
A Tableau Algorithmwith Global Caching

  • Stop when τreceives 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 τhas status cons.

Sound Global Caching for Modal Tableaux


Complexity
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.

Sound Global Caching for Modal Tableaux


Soundness of global caching
Soundness of Global Caching

Lemma 1: If the root node τreceives status inconsthen 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.

Sound Global Caching for Modal Tableaux


Saturation paths
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.

Sound Global Caching for Modal Tableaux


Soundness of global caching1
Soundness of Global Caching

Lemma 2: If the root node τreceives status consthen 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.

Sound Global Caching for Modal Tableaux


Soundness of global caching2
Soundness of Global Caching

  • Theorem: The root of the graph constructed for X receives status consiff 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.

Sound Global Caching for Modal Tableaux


Applications
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

Sound Global Caching for Modal Tableaux


How does global caching co operate with other optimization techniques
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 Modal Tableaux


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