A Tableau Calculus for Regular Grammar Logics with Converse€¦ · A Tableau Calculus for REGc An...

A Tableau Calculus forRegular Grammar Logics with Converse

Linh Anh Nguyen and Andrzej Sza las

Institute of InformaticsUniversity of Warsaw

CADE-22, Montreal, August 7, 2009

Regular Grammar Logics with Converse (REGc )A Tableau Calculus for REGc

An ExpTime Tableau Decision Procedure for REGc

Motivations

The class REG c of regular grammar logics with converse containsa considerable number of common and useful modal logics.

Examples of REG c logics

all 15 basic monomodal logics obtained from K by adding anarbitrary combination of axioms D, T , B, 4, 5,

regular modal logics of agent beliefs(Gore & Nguyen, CLIMA’07),

description logic SHI,

REG c logics used as description logics.

Linh Anh Nguyen and Andrzej Sza las Tableaux for Regular Grammar Logics with Converse

Motivations

The general satisfiability problem of regular grammar logics

without converse:

Demri, 2001:ExpTime-completeness (by a transformation into PDL)

Gore & Nguyen, 2005:an ExpTime tableau decision procedure

with converse

Demri & de Nivelle, 2005:ExpTime-completeness (by a transformation into GF2)

this work: an ExpTime tableau decision procedure

Outline

1 Regular Grammar Logics with Converse (REG c)Grammar LogicsSymmetric Regular Semi-Thue SystemsRegular Grammar Logics with Converse

2 A Tableau Calculus for REG c

The Main IdeasTableau Rules of the CalculusSoundness and Completeness

3 An ExpTime Tableau Decision Procedure for REG c

The Basic AlgorithmOptimizations

Grammar LogicsSymmetric Regular Semi-Thue SystemsRegular Grammar Logics with Converse

Grammar Logics

Grammar logic

a normal multimodal logic characterized by axioms of the form

[σ1] . . . [σh]ϕ→ [%1] . . . [%k ]ϕ

Such an axiom corresponds to the grammar rule

σ1 . . . σh → %1 . . . %k

Context-free grammar logic

a normal multimodal logic characterized by axioms of the form

[σ]ϕ→ [%1] . . . [%k ]ϕ

Symmetric Regular Semi-Thue Systems

Alphabet with converse (for indices of modal operators)

Σ = Σ+ ∪ Σ−, where

Σ+ is a finite set of symbols,

Σ− = {σ | σ ∈ Σ+},Σ− ∩ Σ+ = ∅,if % = σ ∈ Σ− then %

def= σ.

Symmetric Regular Semi-Thue Systems

Context-free semi-Thue system

a set S of production rules over alphabet Σ of the form

σ → %1 . . . %k

Symmetric regular semi-Thue system

a context-free semi-Thue system S such that:

symmetry: (σ → %1 . . . %k) ∈ S implies (σ → %k . . . %1) ∈ S ,

regularity: for every σ ∈ Σ, the set of words derivable from σis recognized by a finite automaton Aσ.

Regular Grammar Logics with Converse

Let S be a symmetric regular semi-Thue system over Σ.

The regular grammar logic with converse L corresponding to S

An L-model is a Kripke model M = 〈W ,R, h〉 such that

(σ → %1 . . . %k) ∈ S implies R%1 ◦ · · · ◦ R%k⊆ Rσ

Satisfiability

X is L-satisfiable w.r.t. Γ if there exists an L-model that

validates Γ (at every possible world),

satisfies X (at some possible world).

Example

The only inclusion axiom: [σ]ϕ→ [%][%]ϕ

Is X = {¬p, 〈%〉〈%〉[σ]p} satisfiable w.r.t. Γ = ∅ ?

u : ¬p, 〈%〉〈%〉[σ]p

Example

u : ¬p, 〈%〉〈%〉[σ]p

Example

u : ¬p, 〈%〉〈%〉[σ]p

R% // v : 〈%〉[σ]p

, [%]p

Example

u : ¬p, 〈%〉〈%〉[σ]p

R% // v : 〈%〉[σ]p

, [%]p

Example

u : ¬p, 〈%〉〈%〉[σ]p

R% // v : 〈%〉[σ]p

, [%]p

R% // w : [σ]p

, [%][%]p

Example

u : ¬p, 〈%〉〈%〉[σ]p

R% // v : 〈%〉[σ]p

, [%]p

R% // w : [σ]p

, [%][%]p

Example

u : ¬p, 〈%〉〈%〉[σ]p

R% // v : 〈%〉[σ]p

, [%]p

R% // w : [σ]p, [%][%]p

Example

u : ¬p, 〈%〉〈%〉[σ]p

R% // v : 〈%〉[σ]p

, [%]p

R% // w : [σ]p, [%][%]p

wwTY_ej

Example

u : ¬p, 〈%〉〈%〉[σ]p

R% // v : 〈%〉[σ]p, [%]pR% // w : [σ]p, [%][%]p

wwTY_ej

Example

u : ¬p, 〈%〉〈%〉[σ]p

R% // v : 〈%〉[σ]p, [%]pR% //

vvUZ_di

w : [σ]p, [%][%]p

wwTY_ej

Example

u : ¬p, 〈%〉〈%〉[σ]p, pR% // v : 〈%〉[σ]p, [%]p

vvUZ_di

w : [σ]p, [%][%]p

wwTY_ej

Example

u : ¬p, 〈%〉〈%〉[σ]p, pR% // v : 〈%〉[σ]p, [%]p

vvUZ_di

w : [σ]p, [%][%]p

wwTY_ej

Idea 1: Automaton-Modal Operators

Problem:

an arbitrarily long word %1 . . . %n may be derivable from σ(i.e. accepted by Aσ)

having [σ]ϕ, we may need to derive [%1] . . . [%n]ϕ,which may be arbitrarily long

Solution: Use automaton-modal operators to control behaviors ofuniversal modalities (Gore & Nguyen [TABLEAUX’05]).

Let Aσ = 〈Σ,Qσ, Iσ, δσ,Fσ〉 and Iσ = {q1, . . . , qk}.Replace [σ]ϕ by [Aσ, q1]ϕ, . . . , [Aσ, qk ]ϕ.

If [Aσ, q]ϕ ∈ w , wR%−→ w ′, and q

%−→ q′ is a transition of Aσ

then add [Aσ, q′]ϕ to w ′.

If [Aσ, q]ϕ ∈ w and q ∈ Fσ then add ϕ to w .

Problem:

Idea 2: Global Caching

To increase efficiency and to achieve optimal complexity.

For each possible contents, at most one node with thatcontents appears in the search space, and it is expanded atmost once.

The idea appeared first in Pratt’s work on PDL.

Global caching has been formalized and proved sound by Goreand Nguyen for traditional tableaux in a number of modal anddescription logics [TABLEAUX’05&07, DL’07, CLIMA’07,CS&P’08].

In this work, tableaux are formulated directly as “and-or”graphs with global caching.

Idea 3: Analytic Cuts for Dealing with Converse

In a traditional (non-prefixed) tableau

we can guess the future (using cuts),but cannot modify the past.

Consider the situation when

wR%−→ w ′ and [Aσ, q

′]ϕ ∈ w ′ and

q′ %−→ q is a transition of Aσ.

We have w ′R%−→ w , which causes [Aσ, q]ϕ ∈ w .

The cut rule for w is: either [Aσ, q]ϕ or [%]¬[Aσ, q′]ϕ

under certain conditions.

We do not need to automatize the operator [%],so we replace [%] by its “blocked” version 2%.

Writing 2%¬[Aσ, q′]ϕ in NNF, we have 2%〈Aσ, q′〉ϕ.

′]ϕ ∈ w ′ and

Idea 4: Fulfilling Eventualities

Operators 〈Aσ, q〉 are like operators 〈α〉 of PDL,which may contain the ∗ constructor.

Adopt Pratt’s technique of fulfilling eventualities(which was formulated for PDL).

Use notions of “marking” and “trace”, which are similar to theones introduced by Niwinski and Walukiewicz for µ-calculus.

”Or”-Rules and “And”-Rules

“Or”-rules

Z1 | . . . | Zk(k ≥ 1)

If Y is L-satisfiable w.r.t. Γ then so is some Zi .

Expanding a node with contents Y using the above “or”-rulemakes the node become an “or”-node with k successors withcontents Z1, . . . , Zk , respectively.

”Or”-Rules and “And”-Rules

“And”-rules

Z1 & . . . & Zkor

&{Zi such that . . .}

If Y is L-satisfiable w.r.t. Γ then all Zi are also L-satisfiablew.r.t. Γ, possibly at different possible worlds.

Expanding a node with contents Y using the above“and”-rule makes the node become an “and”-node with ksuccessors with contents Z1, . . . , Zk , respectively.

The Rules of the Calculus (1)

Assume that formulas are in NNF.

(⊥0)Y ,⊥⊥

(⊥)Y , p,¬p

(∧)Y , ϕ ∧ ψ

Y , ϕ ∧ ψ,ϕ, ψ(∨)

Y , ϕ ∨ ψY , ϕ ∨ ψ,ϕ | Y , ϕ ∨ ψ,ψ

The only “and”-rule / transitional rule is:

(trans)Y

&{ ({ϕ} ∪ {ψ s.t. 2σψ ∈ Y } ∪ Γ) s.t. 〈σ〉ϕ ∈ Y }

An instance of this rule w.r.t. Γ = {s} :

〈σ〉p, 〈σ〉q,2σr

p, r , s & q, r , s

(aut)Y , [σ]ϕ

Y , [σ]ϕ, [Aσ, q1]ϕ, . . . , [Aσ, qk ]ϕif Iσ = {q1, . . . , qk}

if δσ(q) = {(%1, q1), . . . , (%k , qk)} and q /∈ Fσ :

([A])Y , [Aσ, q]ϕ

Y , [Aσ, q]ϕ,2%1 [Aσ, q1]ϕ, . . . ,2%k[Aσ, qk ]ϕ

(〈A〉) Y , 〈Aσ, q〉ϕY , 〈%1〉〈Aσ, q1〉ϕ | . . . | Y , 〈%k〉〈Aσ, qk〉ϕ

if δσ(q) = {(%1, q1), . . . , (%k , qk)} and q ∈ Fσ :

([A]f )Y , [Aσ, q]ϕ

Y , [Aσ, q]ϕ,2%1 [Aσ, q1]ϕ, . . . ,2%k[Aσ, qk ]ϕ,ϕ

(〈A〉f )Y , 〈Aσ, q〉ϕ

Y , 〈%1〉〈Aσ, q1〉ϕ | . . . | Y , 〈%k〉〈Aσ, qk〉ϕ | Y , ϕ

(cut)Y

Y , [Aσ, q]ϕ | Y ,2%〈Aσ, q′〉ϕif

Y contains some 〈%〉ψ,[Aσ, q

′]ϕ ∈ clL(Y ∪ Γ),(q′, %, q) ∈ δσ

where clL(Y ∪ Γ) is the closure Y ∪ Γ,which is a finite set of formulas dependent only on Y ∪ Γ.

The Rules of the Calculus

Assumptions:

The rules (∧), (∨), (aut), ([A]), ([A]f ), (cut) are applicableonly when the premise is a proper subset of each of thepossible conclusions.

The rules (⊥0) and (⊥) have the highest priority.

The static rules have a higher priority than the transitionalrule (trans).

We denote the calculus for a REG c logic L by CL.

Tableaux

Purpose: checking whether X is L-satisfiable w.r.t. Γ.

CL-tableau for (X , Γ)

an “and-or” graph such that

The contents of the initial node (the root) is X ∪ Γ.

The nodes are expanded using the rules of CL

accordingly to the preference of the rules,using global caching.

Soundness and Completeness

Marking of an “and-or” graph G

a subgraph G ′ of G such that:

The root of G is the root of G ′.

If v is a node of G ′ and is an “or”-node of G then at leastone edge (v ,w) of G is an edge of G ′.

If v is a node of G ′ and is an “and”-node of G then everyedge (v ,w) of G is an edge of G ′.

If (v ,w) is an edge of G ′ then v and w are nodes of G ′.

Consistent marking of an “and-or” graph G

a marking G ′ of G such that:

local consistency: G ′ does not contain any node withcontents {⊥};global consistency: for every node v of G ′, every formula ofthe form 〈Aσ, q〉ϕ of the contents of v has a 3-realizationin G ′ (i.e., is fulfilled).

Theorem (Soundness and Completeness)

Let L be a REG c logic and let G be an “and-or” graph for (X , Γ)w.r.t. CL. Then X is L-satisfiable w.r.t. the set Γ of globalassumptions iff G has a consistent marking.

The Basic Decision Procedure for REG c

The basic algorithm

Construct an “and-or” graph G for (X , Γ) w.r.t. CL.

Try to construct a consistent marking G ′ of G by startingfrom G and repeatedly eliminating nodes that violate the localconsistency property or the global consistency property.

To find nodes that violate the global consistency property:

construct the graph of traces of G ′;analyze “productiveness” in this graph of traces.

If such a G ′ exists then answer “yes”, else answer “no”.

The basic algorithm

Optimizations

Many optimization techniques for “and-or” graphs with globalcaching can be applied. See:

Gore & Nguyen, manuscript: “Optimised ExpTime Tableauxfor ALC Using Sound Global Caching, Propagation andCutoffs”

Nguyen, FI(93): “An Efficient Tableau Prover using GlobalCaching for the Description Logic ALC”

For example:

propagating inconsistency on-the-fly

cutoffs

Further Work

Nguyen & Sza las, ICCCI’2009:ExpTime Tableaux for Checking Satisfiability of a KnowledgeBase in the Description Logic ALC

Nguyen & Sza las, manuscript (Arxiv):Optimal Tableau Decision Procedures for PDL

Nguyen & Sza las, KSE’2009:An Optimal Tableau Decision Procedure for Converse-PDL

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