Dealing with Symmetries in Modal Tableaux

Dealing with Symmetries in Modal Tableaux
Carlos Areces and Ezequiel Orbe
Universidad Nacional de C´ordoba, Argentina
CONICET, Argentina
Frontiers of Combining Systems 2013, Nancy, France

Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions
Introduction
A symmetry is a permutation of the variables (literals) of a
problem that preserves its structure and its set of solutions.
For instance:
ϕ= (¬p∨r)∧(q∨r)∧(¬p∨q)
has symmetry:
ρ= (¬p q)(¬q p)
We may improve the performance of theorem proving if:
symmetry detection is cheap
this information pays off in terms of performance
We present a tableau optimization called ”symmetry blocking”.

Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions
Syntax
Modal Conjunctive Normal Form:
Clausal representation of modal formulas:
(¬p∨r)∧(q∨r)∧(r∨(¬p∨q)))
→ {{¬p, r},{q, r},{r, {¬p, q}}}
Disregard order and multiplicity: formulas as set of sets.
Symmetry:
Permutations of literals, ρ:PLIT 7→ PLIT
ρis a symmetry of ϕif ρ(ϕ) = ϕ.

Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions
Semantics
Models:
Kripke model: M=hW, R, V i
Wis the domain
R⊆W×W
V:W7→ P(PROP)
Pointed Models: M=hw, W, R, V i,w∈W
Satisfaction Relation:
M |=ϕiff M |=Cfor all clauses C∈ϕ
M |=Ciff M |=lfor some literal l∈C
M |=piff p∈V(w)for p∈PROP
M |=Ciff hw0, W, R, V i |=Cfor all w0s.t. wRw0

Introduction Definitions Symmetry Detection Symmetry Blocking Conclusions
Permutation Sequences
In modal logics that have the tree model property, a notion of
layer is induced:
∧
¬�
q
r
∨
∨
∨
¬p
�
q
¬r
Layer 1
Layer 2
Layer 3
Model
M
Formula
ϕ
p:= true
p:= true
r :=true
p
modal depth = 0
modal depth = 1
modal depth = 2
depth = 0
depth = 1
depth = 2
We can consider a different permutation at each layer.
Permutation Sequence: ¯ρ=hρ1, . . . , ρni
Enables to find more symmetries.