Complexity Results on Dependence Logic

Complexity
Results on
Dependence
Logic
Juha Kontinen
Complexity Results on Dependence Logic
Juha Kontinen
University of Helsinki
3.3.2014
Outline of the talk
Complexity
Results on
Dependence
Logic
Juha Kontinen
1
Short introduction to dependence logic
2
Some variants of the dependence atoms
3
The Strict and the Lax semantics
4
Recent results on expressive power, satisfiability and model
checking
5
Conclusion
Dependence logic
Complexity
Results on
Dependence
Logic
Juha Kontinen
Definition
The syntax of dependence logic (D) extends the syntax of FO,
defined in terms of ∨, ∧, ¬, ∃ and ∀, by new atomic
(dependence) formulas of the form
=(t1 , . . . , tn ),
(1)
where t1 , . . . , tn are terms.
In (1), n is called the arity (or width) of the dependence atom.
Semantics of D
Complexity
Results on
Dependence
Logic
Juha Kontinen
The semantics of D is defined in terms of teams (sets of
assignments):
Definition
Let A be a set and {x1 , . . . , xk } variables. A team X of A with
domain {x1 , . . . , xk } is a set of assignments s,
s : {x1 , . . . , xk } → A.
Semantics of D
Complexity
Results on
Dependence
Logic
Juha Kontinen
The following operations are used to interpret quantifiers in D.
Below, s(a/xn ) is the assignment that agrees otherwise with s,
but maps xn to a.
Definition
Suppose A is a set, X is a team of A, and F : X → A.
supplementation: X (F /x) = {s(F (s)/x) : s ∈ X }.
Duplication: X (A/x) = {s(a/x) : s ∈ X and a ∈ A}.
Supplementation
Complexity
Results on
Dependence
Logic
Juha Kontinen
Let A = {0, 1} and X be
s0
s1
x0 x1
1 0
0 1
Let F : X → A be such that F (s0 ) = 1 and F (s1 ) = 0, then
X (F /x2 ) is
x0 x1 x2
s2 1 0 1
s3 0 1 0
Duplication
Complexity
Results on
Dependence
Logic
Let A = {0, 1} and X be
Juha Kontinen
s0
s1
x0 x1
1 0
0 1
Then X (A/x2 ) is
s2
s3
s4
s5
x0 x1 x2
1 0 1
1 0 0
0 1 1
0 1 0
Satisfaction for NNF-formulas
Complexity
Results on
Dependence
Logic
Juha Kontinen
Definition
Below φ(t1 , . . . , tn ) is atomic or negated atomic FO-formula:
A |=X φ(t1 , . . . , tn ) ⇔ for all s ∈ X : A |=s φ(t1 , . . . , tn )
A |=X =(t1 , . . . , tn ) ⇔ for all s, s 0 ∈ X : if tiA hsi = tiA hs 0 i
for 1 ≤ i ≤ n − 1, then tnA hsi = tnA hs 0 i.
A |=X ψ ∧ φ ⇔ A |=X ψ and A |=X φ.
A |=X ψ ∨ φ ⇔ X = Y ∪ Z such that A |=Y ψ and
A |=Z φ
A |=X ∃xψ ⇔ A |=X (F /x) ψ for some F : X → A.
A |=X ∀xψ ⇔ A |=X (A/x) ψ.
Finally, a sentence ϕ of D is true in A if A |={∅} ϕ.
Example
Complexity
Results on
Dependence
Logic
Juha Kontinen
Not all familiar propositional equivalences of connectives hold
for D, e.g., idempotence of disjunction, and the distributivity
laws of disjunction and conjunction fail.
Example
Let A = {0, 1, 2} and X be
s0
s1
s2
x0 x1 x2
1 2 2
2 1 2
2 0 2
(2)
Now A 6|=X x0 = x2 and A 6|=X ¬x0 = x2 . Also A 6|=X =(x2 , x0 ),
but A |=X (=(x2 , x0 )∨ =(x2 , x0 )).
Basic properties of D
Complexity
Results on
Dependence
Logic
Juha Kontinen
Proposition
Let φ be a formula of D without dependence atoms. Then for
all A and X :
A |=X φ ⇔ for all s ∈ X : A |=s φ
Proposition (Downward closure)
Let Y ⊆ X teams. Then A |=X φ implies A |=Y φ.
Proposition (Locality)
Let Fr(φ) ⊆ V . Then, for all models A and teams X , A |=X φ
if and only if A |=X V φ.
Variants of dependence logic
Independence Logic
Complexity
Results on
Dependence
Logic
Juha Kontinen
Definition (Grädel and Väänänen, 2013)
Conditional independence atom
y ⊥x z
interpreted as: A |=X y ⊥x z iff for all s, s 0 ∈ X s.t.
s(x) = s 0 (x) there exists s 00 ∈ X s.t. s 00 (xy ) = s(xy ), and
s 00 (xz) = s 0 (xz).
The version y ⊥ z, where x = ∅, is called Pure.
FO(⊥c ) (FO(⊥)) is FO with (pure) independence atoms.
Variants of dependence logic
Inclusion and exclusion logics
Complexity
Results on
Dependence
Logic
Definition (Galliani 2012)
Juha Kontinen
1
Inclusion atom
x ⊆y
interpreted as: A |=X x ⊆ y iff for all s ∈ X there exists
s 0 ∈ X s.t. s(x) = s 0 (y ).
2
Exclusion atom
x|y
interpreted as: A |=X x|y iff s(x) 6= s 0 (y ) for all s, s 0 ∈ X
The variants of D with the atoms x ⊆ y and x|y , are denoted
by FO(⊆), FO(|), and FO(⊆, |).
Strict and Lax semantics
Galliani 2012
Complexity
Results on
Dependence
Logic
Juha Kontinen
The Strict semantics is obtained by changing the clause for ∨
to:
A |=X ψ ∨ θ iff there are Y and Z such that Y ∪ Z = X ,
Y ∩ Z = ∅, and A |=Y ψ and A |=Z θ
The Lax semantics is obtained by changing the clause for ∃ to:
A |=X ∃xψ iff there exists F : X → P(M)\{∅} such that
A |=X (F /x) ψ.
Downward closure renders the strict and the lax semantics
equivalent for all D-formulas.
Comparing the strict and the lax semantics I
Complexity
Results on
Dependence
Logic
Juha Kontinen
The Locality property holds for all of the the aforementioned
logics under the lax semantics:
Proposition (Locality for the lax semantics)
Let φ be a formula of FO(=(. . .), ⊥c , ⊆) whose free variables
Fr(φ) are contained in V . Then, for all models A and teams
X , A |=X φ if and only if A |=X V φ.
Comparing the strict and the lax semantics II
Complexity
Results on
Dependence
Logic
Under the strict semantics,
Juha Kontinen
X |= x ⊆ y ∨ z ⊆ y
but
X {x, y , z} 6|= x ⊆ y ∨ z ⊆ y ,
where X is
s0
s1
s2
s3
x
0
1
1
2
y
1
0
0
1
z
2
1
1
0
v
3
3
4
4
Comparing the strict and the lax semantics III
Complexity
Results on
Dependence
Logic
Juha Kontinen
Note that X {x, y , z} is the team
s0
s1
s3
x
0
1
2
y
1
0
1
z
2
1
0
The claim follows because the full team X {x, y , z}, and none
of its singletons, satisfy x ⊆ y or z ⊆ y .
Relations among logics
Complexity
Results on
Dependence
Logic
Juha Kontinen
For both versions of the semantics, compositional translations
of formulas show:
D = FO(|)
FO(⊥c ) = FO(⊥) = FO(⊆, |)
Theorem (Galliani and Hella 2013)
Under the lax semantics, FO(⊆) = GFP+ .
Hence, over (ordered) finite structures,
FO(⊆) = LFP = PTIME. On the other hand, for the strict
semantics:
Theorem (Hannula, K., Galliani 2013)
Under the strict semantics, FO(⊆) = ESO.
Complexity of syntactic fragments
Complexity
Results on
Dependence
Logic
Juha Kontinen
Theorem (Jarmo Kontinen 2013)
Define
Let ϕ be =(x, y )∨ =(z, u)
Let ψ be =(x, y )∨ =(z, u)∨ =(z, u)
Deciding whether a finite team X satisfies ϕ is NL-complete
and, for ψ, NP-complete.
By the above the universal D-sentence of vocabulary {R}:
∀xyzy (¬R(x, y , z, u) ∨ ψ),
defines an NP-complete problem. On the other hand, purely
existential sentences of D are equivalent to FO-sentences.
Relevant fragments of ESO I
Complexity
Results on
Dependence
Logic
Juha Kontinen
Let ESOf (k-ary) denote the class of ESO-sentences
∃f1 . . . ∃fn ψ
in which the function symbols fi are at most k-ary and ψ
is first-order.
Let ESOf (k∀) denote the class of skolem normal form
ESO-sentences (i.e., ψ is quantifier free)
∃f1 . . . ∃fn ∀x1 . . . ∀xm ψ,
where m ≤ k.
Relevant fragments of ESO II
Complexity
Results on
Dependence
Logic
Juha Kontinen
Theorem (Ajtai 1983)
Let R be a k + 1-ary relation symbol. Then the property ”|R|
even” cannot be defined in the logic ESOf (k-ary) but is
definable in ESOf (k + 1-ary).
Theorem (Grandjean and Olive 2004)
ESOf (k∀) = NTIMERAM (nk )
Note that NTIMERAM (nk ) < NTIMERAM (nk+1 ).
Fragments in Logics with Teams Semantics
Complexity
Results on
Dependence
Logic
Juha Kontinen
Definition
Let C ⊆ {⊆, =(. . .), ⊥c , ⊥}.
FO(C)(k∀) is the class of FO(C) formulae in which at
most k universal quantifiers may appear,
FO(C)(k-inc) is the class of FO(C) formulae in which
inclusion atoms of the form ~x1 ⊆ ~x2 where ~x1 and ~x2 are
sequences of length at most k, may appear,
FO(C)(k-dep) is the class of FO(C) formulae in which
dependence atoms of the form =(~x1 , x2 ) where ~x1 x2 is a
sequence of length at most k + 1, may appear,
FO(C)(k-ind) is the class of FO(C) formulae in which
conditional independence atoms of the form ~x2 ⊥~x1 ~x3
where ~x1 ~x2 ~x3 is a sequence listing at most k + 1 distinct
variables, may appear.
The expressive power of the fragments I
Under the Lax semantics
Complexity
Results on
Dependence
Logic
By restricting the arity we get:
Juha Kontinen
Theorem (Galliani, Hannula, and K. 2013; Durand and K.
2012)
FO(⊥c )(k-ind) = ESOf (k-ary) = D(k-dep)
On the other hand, for inclusion logic the following holds:
Theorem (Hannula 2014)
Over graphs,
FO(⊆)(k-ind) < FO(⊆)(k + 1-ind)
The expressive power of the fragments II
Under the Lax semantics
Complexity
Results on
Dependence
Logic
Juha Kontinen
By restricting the number of universal quantifiers:
Theorem (Durand and K. 2012)
ESOf (k∀) ≤ D(2k∀) ≤ ESOf (2k∀).
In the presence of inclusion or independence atoms, universal
quantifiers can be simulated by existential quantifiers:
Theorem (Hannula 2014; Galliani, Hannula and K. 2013)
1
If ⊆∈ C then the hierarchy collapses at level 1:
FO(C) = FO(C)(1∀);
2
If ⊥ ∈ C then it collapses at level 2: FO(C) = FO(C)(2∀).
Hierarchy theorems for the lax semantics
Complexity
Results on
Dependence
Logic
Juha Kontinen
D
FO(⊥c )
Arity of atoms
Strict
strict
FO(⊆)
strict
Number of ∀
Infinite
collapse at
k=2
collapse at
k=1
The expressive power of the fragments
Under the strict semantics
Complexity
Results on
Dependence
Logic
Juha Kontinen
The situation is more complicated because of the locality
property failing.
Theorem (Hannula and K. 2014)
1
FO(⊆)(k∀) = ESOf (k∀) = NTIMERAM (nk )
2
FO(⊥c )(k∀) ≤ ESOf (k + 1∀)
3
ESOf (k∀) ≤ FO(⊥c )(2k∀)
This implies the following hierarchy theorems:
FO(⊥c )
FO(⊆)
Arity of atoms
?
?
Number of ∀
infinite
strict
The 2-variable fragment of D
Complexity
Results on
Dependence
Logic
Juha Kontinen
Denote by D2 the sentences of D in which only variables x and
y appear.
Theorem (K., Kuusisto, Lohmann and Virtema 2011)
1
2
3
The Satisfiability (and Finite Satisfiability) problem of D2
is NEXPTIME-complete.
The logic D2 is quite expressive being able to express, e.g.,
”A infinite”, ”|P| = |Q|”, and some NP-complete
problems.
In contrast, the satisfiability (and finite satisfiability)
problem of IF 2 is undecidable.
Remark
The complexity of the validity problem for D2 is open.
Complexity of Model Checking
Complexity
Results on
Dependence
Logic
Juha Kontinen
E. Grädel (2013) formulated a general model checking game for
logics with team semantics.
Recall that the model checking problem, with input (φ, A, X ),
is to decide whether A |=X φ.
Theorem (Grädel 2013)
The model-checking problem for D is NEXPTIME-complete.
Futhermore, containment in NEXPTIME was shown to hold for
any variant FO(C) of D s.t. the atoms in C are
PTIME-computable.
Other complexity results
Complexity
Results on
Dependence
Logic
Juha Kontinen
A certain Horn fragment of D captures PTIME over
successor structures (Ebbing, K. Müller and Vollmer 2012).
Intuitionistic implication → makes D equivalent to full
second-oder logic (Yang 2013). A similar result holds when
dependence logic is extended by the classical negation.

Download Report