Sound Global Caching for Abstract Modal Tableaux Rajeev Goré The Australian National University  Linh Anh Nguyen University of Warsaw CS&P’2008

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux2 Overview  Motivation  Examples of tableaux  Abstract modal tableaux  A tableau algorithm with global caching  Soundness of global caching

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux3 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

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux4 Example: Tableaux for CPC (Classical Propositional Calculus)  Is a formula set X 0 satisfiable?  NNF: negations occur only before atoms.  Tableau rules: X ;    X ;  ;  ()() X ;    X ;  | X ;  ()() X ;   (’) X ;  ;   ()()

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux5 Example: Tableaux for CPC  A tableau is a tree... p  q ; p  q p ; q ; p  q p ; q ; pp ; q ; q  ()() ()() ()()()()

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux6 Example: Tableaux for CPC  A tableau is closed if every branch ends with  p  q ; p  q p ; q ; p  q p ; q ; pp ; q ; q  ()() ()() ()()()()

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux7 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

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux8 Example: Tableaux for Modal Logic K  What is modal logic K? Formulas: ? Interpretations: ? The satisfaction relation: ?

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux9 Example: Tableaux for Modal Logic K  What is modal logic K? Formulas:  as in the case of CPC,  plus additional constructors:  , 

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux10 Example: Tableaux for Modal Logic K  What is modal logic K? Interpretations Kripke model p, r p, q p, q, r... possible world

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux11 Example: Tableaux for Modal Logic K  What is modal logic K? The satisfaction relation p, r p, q  p, q,   q,  (p(qr))...

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux12 Example: Tableaux for Modal Logic K  Is a formula set X 0 satisfiable w.r.t. a set Г of global assumptions?  i.e. Is there a Kripke model M such that X 0 is satisfied in some possible world of M, Г is satisfied in every possible world of M ?

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux13 Example: Tableaux for Modal Logic K  Tableau rules: the rules for CPC plus  X 0 is unsatisfiable w.r.t. Г iff there is a closed tableau with root (X 0 ; Г) X ;  ; { :    X}; Г ()() ,  ,... , ,... transitional

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux14 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 init CL : init CL (X) is a formula set computable from X in PTime.  A CL-tableau for X is a tree with root init CL (X), using the rules of CL for expansions.

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux15 Abstract Modal Tableaux  CL-tableau rules PTime Denominators:  Each Y i is computable from X and L in PTime Monotonicity:  X’ X  applying (ρ) to X’ results in Y’ i  Y i, 1ik Terminal, Static or Transitional:   next page X Y 1 |... | Y k (ρ)(ρ)

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux16 Abstract Modal Tableaux  CL-tableau rules Cases: ()-rule: only one denominator  static rule: X  Y i for all 1  i  k transitional rule: only one denominator, e.g. () X Y 1 |... | Y k (ρ)(ρ)

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux17 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 ;    X ;    ;  | X ;    ; 

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux18 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.

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux19 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.

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux20 A Tableau Algorithm with Global Caching Problem: Check whether X is CL-consistent. Algorithm: Build an and-or graph for X using CL:  The root node τ contains init CL (X).  Each node is expanded using a CL-tableau rule.  Preferences of rules: 1. ()-rule 2. unary static rules 3. non-unary static rules 4. transitional rules ...

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux21 A Tableau Algorithm with 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

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux22 A Tableau Algorithm with 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

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux23 A Tableau Algorithm with 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.

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux24 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.

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux25 Soundness of Global Caching Lemma 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.

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux26 Saturation Paths  In the constructed and-or graph, define a saturation path of node v to be a sequence v 0 =v, v 1,..., v k with k  0 such that, for each 1  i  k, we have: v i.status = cons, the edge (v i-1, v i ) was created by a static rule, v k.content is closed w.r.t. the static rules.  Observe that v 0.content ...  v k.content.

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux27 Soundness of Global Caching Lemma 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 v 0, v 1,..., v k for some v 0. Initially, v 0 = τ (the root of the graph).  Maintain the invariant cn.content  v k.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.

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux28 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.

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux29 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

R. Goré & L.A. NguyenSound Global Caching for Modal Tableaux30 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