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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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