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A Cut-free Tableau Calculus for the Logic of Common Knowledge

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In this paper we focus on the development of a cut-free nitary tableau calculus with histories for n-agent modal logics with common knowledge (LCK). Thus, we get a proof system where proof-search becomes feasible and we lay the basis for developing a
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  A Cut-free Tableau Calculus for the Logic of Common Knowledge Pietro Abate and Rajeev Gor´eAustralian National University, Canberra, Australia, [Pietro.Abate, Rajeev.Gore ]@anu.edu.au December 1, 2006 Abstract In this paper we focus on the development of a cut-free finitary tableaucalculus with histories for  n -agent modal logics with common knowledge(LCK). Thus, we get a proof system where proof-search becomes feasibleand we lay the basis for developing a uniform framework for the treatmentof the family of logics of common knowledge. Unlike two-pass decisionmethods like those for temporal logics, our calculus gives a single-passdecision procedure which is space-optimal. 1 Introduction Reasoning about knowledge, in particular about the knowledge of agents whoreason about the world and each other’s knowledge, plays an important rolein several areas of computer science, philosophy, game theory and many otherfields. The notion of Common Knowledge (CK), where everybody knows, every-one knows that everyone knows, etc has also been proved fundamental in fieldswhich deal with the analysis of interacting groups of agents: see [FHMV95] foran extensive overview.In this paper we focus on the development of a cut-free finitary tableaucalculus with histories for  n -agent modal logics with common knowledge (LCK).Thus, we get a proof system where proof-search becomes feasible and we lay thebasis for developing a uniform framework for the treatment of the family of logics of common knowledge. Unlike two-pass proof calculi based on temporallogics, or the the existing solutions in [AJ02], [vD02], we give a single-pass,finitary and cut-free decision procedure for LCK 2 Syntax and Semantics We consider countably many atomic propositions  AP  , a finite non-empty set of agents  A  with  AP   ∩  A  =  ∅ , the connectives  ¬ , ∧ , [ a ] for every  a  ∈  A  and the1  constants  ⊥  and   . The formulae of   LCK   are defined using the BNF grammar:  p  ::=  p 0  |  p 1  |  p 2  | ··· ϕ  ::=  p  |  | ⊥ | ¬ ϕ  |  ϕ 1  ∨ ϕ 2  |  ϕ 1  ∧ ϕ 2 |  [ a ] ϕ  |  a  ϕ  |  [ C  ] ϕ  |  C   ϕ We define [ E  ] and   E    and [ E  ] k with  k  ≥  1 as:[ E  ] ϕ  :=  a ∈ A [ a ] ϕ   E   ϕ  :=  a ∈ A  a  ϕ  [ E  ] 0 ϕ  :=  ϕ  [ E  ] k ϕ  := [ E  ][ E  ] k − 1 ϕ Thus the common knowledge operator is [ C  ] and has a dual   C   . We usethe term “iterated eventuality” for formulae of the form   C   ϕ  and we use theterm “eventualities” for formulae of the form   i  ϕ . Definition 2.1.  An LCK model is a triple   W,R,V    where  W   is a non-emptyset of worlds,  R  is a function that assigns to each agent  a  ∈  A  an accessibilityrelation  R a  ⊆  W  x W   and  V   a valuation that assigns to each atom  p  ∈  P   a set V  (  p )  ⊆ P  ( W  ) where  P  ( W  ) is the subset of all subset of   W  . Definition 2.2.  Let  R A  := (  a ∈ A R a ) + . Given an LCK model   W,R,V   , anR-path  π  is a sequence  w 0 ,w 1 ,...  of worlds from  W   such that  w i R A w i +1  forall  i  ≥  0. We sometimes write  π  as  w 0 R A w 1 R A ...  write  π i  for  w i , and writeΠ( w ) as the set of all  R -paths starting at  w . Definition 2.3.  Let  M   =   W,R,V    be an LCK model and  w  ∈  W   then thevalue of   M,w    ϕ  of a formula  ϕ  in a world  w  is inductively defined as follows:1.  M,w    p  iff   w  ∈  V  (  p )2.  M,w    ϕ ∨ ψ  iff   M,w    ϕ  or  M,w    ψ 3.  M,w    ϕ ∧ ψ  iff   M,w    ϕ  and  M,w    ψ 4.  M,w     a  ϕ  iff   ∃ v  ∈  W,wR a v  and  M,v    ϕ 5.  M,w    [ a ] ϕ  iff   ∀ v  ∈  W,wR a v  implies  M,v    ϕ 6.  M,w     C   ϕ  iff there exists an  R -path  wR A w 1 R A ··· R A w n  ∈  Π( w ) with n  ≥  1 such that  π n    ϕ  and  ∀ i , 1  ≤  i < n  ⇒  π i    ¬ ϕ .7.  M,w    [ C  ] ϕ  iff   ∀  R -paths  π  ∈  Π( w ), and  ∀ i  ≥  1,  π i    [ C  ] ϕ  and  π i    ϕ . Definition 2.4.  A formula  ϕ  is LCK-satisfiable iff there exists an LCK model M   =   W,R,V    and a world  w  ∈  W   such that  M,w    ϕ . A formula  ϕ  is validiff   ¬ ϕ  is not LCK-satisfiable.2  3 Tableau Rules for  LCK  n In this section we give a tableau calculus to determine LCK-satisfiability usingtraditional tableau extended with histories to pass extra information from par-ents to children and variables to pass information from children to parents: see[Gor99] for an overview, and also [HSZ96, Sch98] for similar calculi.As is usual, the rules are categorised into static rules and transitional rules[Gor99]. We assume a strategy for rule applications that applies all the staticrules until they are no longer applicable, giving a (saturated) state, and thenapplies the transitional rules to obtain (the cores of the) new pre-states.Each rule is composed of a numerator and a list of denominators where thenumerator and each denominator is of the form Γ ::  H   ::  V   where:Γ is a set of formulae as usual, H   consists of two histories called  Fev  and  Br  where Fev  is a set of (eventuality) formulae fulfilled between two consecutiveapplications of the transitional rules, Br  is a list of pairs, of the form ( Core,Fev ) where the first component of the pair is a set representing the (pre-state) core that led to a state,and the second component is a set of the iterated eventualities thatwere fulfilled by this branch in creating that state out of the  Core , V   consists of a variable  uev  which is a list of pairs (  C   ϕ,n ) where  n  is aninteger.  uev  tracks iterated eventualities unfulfilled in this branch, butwhich may be fulfilled by sibling branches as we pop the recursion stack.To focus only on locally relevant parts, we use “ ... ” for the not important parts.If “ ... ”appears at the same position in the numerator and denominator(s) of arule, then we mean that the corresponding parts are the same.A tableau for a set Γ is a tree of nodes with root Γ ::  H   ::  V   where thechildren of a node are obtained by instantiating a rule: we say that a rule isassociated with the parent node and conversely, a node is associated with thenumerator of a rule. A node is terminal if it is associated with an applicationof a terminal rule (which has no denominator). We say that the tableau is expanded   if each leaf node is obtained by an application of a terminal rule.In the following we present a tableau calculus for the logic of common knowl-edge when LCK  2  with two agents  A  =  { 1 , 2 }  to keep things simple. We use  A  Γ to stand for a set   1  Γ 1  ∪  2  Γ 2  and [ A ]∆ to stand for an arbitrary set[1]∆ 1  ∪ [2]∆ 2 . To minimise the number of rules, we assume that all formulaeare in negation normal form. A literal is an atom or a negated atom. We use Λto stand for a set of literals. Static Linear Rules. ( ∧ )  ϕ ∧ ψ  ; Γ ::  ··· ϕ  ;  ψ  ; Γ ::  ···  ([ C  ]) [ C  ] ϕ  ; Γ ::  ··· [ E  ] ϕ  ; [ E  ][ C  ] ϕ  ; Γ ::  ···  ([ E  ]) [ E  ] ϕ  ; Γ ::  ··· [1] ϕ  ; [2] ϕ  ; Γ ::  ··· 3  Static Universal Branching Rules. ( ∨ )  ϕ ∨ ψ  ; Γ ::  ···  ::  uevϕ  ; Γ ::  ···  ::  uev 1  |  ψ  ; Γ ::  ···  ::  uev 2 (  C   )   C   ϕ  ; Γ ::  Fev,Br  ::  uev  E   ϕ  ; Γ ::  { C   ϕ }∪ Fev,Br  ::  uev 1  |  E   C   ϕ  ; Γ ::  Fev,Br  ::  uev 2 (  E   )   E   ϕ  ; Γ ::  Fev,Br  ::  uev 1  1  ϕ  ; Γ ::  Fev,Br  ::  uev  |  2  ϕ  ; Γ ::  Fev,Br  ::  uev 2 where in the (  C   ), (  E   ) and ( ∨ ) rules : f  ( x,y ) =  {  j  |  ( x,j )  ∈  y } UC   =  { (  C   ψ,n )  | f  (  C   ψ,uev 1 )   =  ∅ &  f  (  C   ψ,uev 2 )   =  ∅ &  n  =  min  ( f  (  C   ψ,uev 1 )  ∪  f  (  C   ψ,uev 2 ))  } uev  :=  uev 1  if   uev 2  =  { ( false ,  ) } uev 2  if   uev 1  =  { ( false ,  ) } UC   otherwise Transitional Existential Branching Rule.  For each  a  ∈  A  =  { 1 , 2 } , a rule:(  a  )   a  ϕ 1  ; ... ;  a  ϕ n  ;   a  Γ ; [ a ]∆ ; Σ ::  Fev,Br  ::  uevϕ 1  ; ∆::  ∅ ,Br 1  ::  uev 1 ‡ (1) || ··· || ϕ n  ; ∆::  ∅ ,Br n  ::  uev n ‡ ( n ) || a  Γ ; [ a ]∆ ; Σ::  Fev,Br  ::  uev n +1 † with  n  ≥  1 and for 1  ≤  i  ≤  n : ‡ ( i ) is the condition  ∀  j .  0  ≤  j  ≤  len ( Br )  ⇒ { ϕ i }∪ ∆   =  Br [  j ] .core †  is the (blocking) condition  ∀ ψ  ∈  Γ  .  ∃  j  ≥  0 s.t.  { ψ }∪ ∆ =  Br [  j ] .core and where Br i  = ( { ϕ i }∪ ∆ ,Fev ) .Br  for 1  ≤  i  ≤  nl  =  len ( Br ) − 1 UEV   =  n +1 j =1  uev j uev  :=   UEV   if ( false ,  )  / ∈  UEV  &  ∀ (  ,n )  ∈  UEV . n  ≤  l { ( false ,l ) }  otherwise4  Static Terminal Rules. ( id )   A  Γ ; [ A ]∆ ; Λ ::  Fev,Br  ::  uevStop  {¬  p ;  p } ⊆  Λwhere  uev  :=  { ( false ,len ( Br )) } ( block )   A  Γ ; [ A ]∆ ; Λ ::  Fev,Br  ::  uevStop  {¬  p ;  p } ⊆  Λ and  † where  †  is the blocking condition  ∀ ψ  ∈  Γ  .  ∃  j  ≥  0  .  { ψ }∪ ∆ =  Br [  j ] .core , and uev  :=  {  (  C   ϕ,i )  | ∃  i  such that  { C   ϕ }∪ ∆ =  Br [ i ] .core &  ∀  j . i  ≤  j  ≤  len ( Br )  ⇒  C   ϕ  ∈  Br [  j ] .core  &   C   ϕ / ∈  Br [  j ] .fev  } Proposition 3.1 (Termination).  Every LCK-tableau for a node   ϕ  ::  ∅ , [] :: uev  is finite. Theorem 3.2 (Soundness).  If   T    is an expanded tableau for   ϕ  ::  ∅ , [] ::  uev ,with   uev  =  ∅  then   ϕ  is LCK-satisfiable. Theorem 3.3 (Completeness).  If a formula   ϕ  is LCK-satisfiable then there exists an expanded tableau for   ϕ  ::  ∅ , [] ::  uev  where   uev  =  ∅ . 4 Conclusions We presented a cut-free tableau calculus based on histories to reason about LCK.Our calculus is easy to extend to the logic of common knowledge based upon T  ,  S  4 and  S  5. An implementation of   LCK   with two agents using the tableauxworkbench ( TWB ) [AG03] is available at  http:\\twb.rsise.anu.edu.au . References [AG03] P Abate and R Gor´e. System description: The tableaux workbench. In TABLEAUX  , LNCS, Springer, 2003.[AJ02] L Alberucci and G J¨aeger. About cut elimination for logics of commonknowledge. TR, University of Berne, Switzerland, 2002.[FHMV95] R. Fagin, J. Halpern, Y. Moses, and M. Vardi.  Reasoning about Knowledge  .The MIT Press, 1995.[Gor99] R Gor´e. Chapter 6: Tableau methods for modal and temporal logics. In Handbook of Tableau Methods  , pages 297–396. Kluwer, 1999.[HSZ96] A Heuerding, M Seyfried, and H Zimmermann. Efficient loop-checkfor backward proof search in some non-classical propositional logics. InTABLEAUX96, LNCS pages 210–225, 1996.[Sch98] S Schwendimann. A new one-pass tableau calculus for PLTL. In TABLEAUX  , LNCS 1397:277–291, Springer, 1998.[vD02] Hans van Ditmarsch and Roy Dyckhoff. Sequent calculi for logics withcommon knowledge. manuscript, June 2002. 5
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