Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Finite Model Reasoning in Expressive
Fragments of First-Order Logic
Lidia Tendera
Institute of Mathematics and Informatics
Opole University, Poland
M4M Kanpur
8.–9. January 2017
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
O UTLINE
Introduction/Motivation
Standard translation of ML
Base Fragments
Definitions
Properties and Complexity
Extensions of Base Fragments
More operators
Special classes of structures
Deciding (Fin)Sat
More or less natural reductions
Finitary unravellings
Linear/Integer Programming
Conclusion
Deciding (Fin)Sat
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
C LASSICAL D ECISION P ROBLEM
L – any logic
FO– first-order logic
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Sat(L):
given a formula ϕ ∈ L, does ϕ admit a model ?
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FinSat(L):
given a formula ϕ ∈ L, does ϕ admit a finite model ?
Theorem (Church, Turing, Trahtenbrot)
Sat(FO) and FinSat(FO) are undecidable and recursively
inseparable.
Possible response:
devise incomplete algorithms
identify decidable fragments
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
W HY F IN S AT ?
Databases, systems etc. are often considered to be finite.
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L has the finite model property (FMP) iff every satisfiable
ϕ ∈ L has a finite model.
Observation
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If L has FMP then Sat(L) and FinSat(L) coincide.
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If L is a fragment of FO and L has FMP then Sat(L) is
decidable.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
D ECIDABLE F RAGMENTS OF FO
Note: one cannot study all possible fragments!
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defined by restrictions on signatures
e.g. [Löwenheim-Skolem 1915] monadic theories (FMP)
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prenex classes defined by quantifier prefix
∃∗ ∀∗ , ∃∗ ∀∃∗ , ∃∗ ∀∀∃∗ (equality free)
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defined by other syntactic restrictions and suitably
motivated
Also: we want to identify reasons for a logic to be
(un)decidable, (in)tractable etc.
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Can we decide whether a formula is satisfiable without
actually seeing a model?
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Some formulas have only infinite models.
Can we decide whether they are satisfiable?
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
M OTIVATION : M ODAL L OGIC
[VARDI 1996]: W HY IS MODAL LOGIC SO ROBUSTLY DECIDABLE ?
I
Propositional modal logic:
Boolean logic + operators ♦ (possibly) and (necessary)
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Good model-theoretical and algorithmic properties,
robustly decidable
Variants and extensions of modal logics have applications
in various areas of computer science:
I
I
I
I
I
verification of hardware and software
artificial intelligence
distributed systems
knowledge representation
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
S TANDARD T RANSLATION OF M ODAL L OGIC (1)
I
Modal logic can be translated into FO:
P∧♦(Q∨¬P)
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!
Px ∧ ∃y(Rxy ∧ (Qy ∨ ∀z(Ryz → ¬Pz)))
FO3 undecidable [Kahr, Moore, Wang, 1959]
[Gabbay, 1981] TWO variables suffice!
Observation
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ML can be embedded in the two-variable fragment FO2
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
S TANDARD T RANSLATION OF M ODAL L OGIC (1)
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Modal logic can be translated into FO:
P∧♦(Q∨¬P)
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!
Px ∧ ∃y(Rxy ∧ (Qy ∨ ∀x(Ryx → ¬Px)))
FO3 undecidable [Kahr, Moore, Wang, 1959]
[Gabbay, 1981] TWO variables suffice!
Observation
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ML can be embedded in the two-variable fragment FO2
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
S TANDARD T RANSLATION OF M ODAL L OGIC (2)
P ∧ ♦(Q ∨ ¬P)
!
Px ∧ ∃y(Rxy ∧ (Qy ∨ ∀z(Ryz → ¬Pz)))
The translation suggests other restrictions of FO:
I
fluted fragment FL:
variables appear in some fixed order and no
quantifier-rescoping occurs; order of quantification of
variables matches order of appearance in predicates.
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guarded fragment GF:
quantifiers are relativized by atomic formulas
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unary negation fragment UN F:
negation is applied only to subformulas with a single free
variable.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
P ROBLEMS R EDUCING TO (F IN )S AT
E XAMPLE : Q UERY A NSWERING
A knowledge base hD, Oi:
database D (a set of facts, i.e. ground atoms),
ontology O (i.e. a logical formula).
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Query Answering:
given a knowledge base hD, Oi and a query Q:
does hD, Oi entail Q, i.e.
D ∧ O |= Q?
Observation
D ∧ O |= Q
iff
D ∧ O ∧ ¬Q is unsatisfiable
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
B ASE F RAGMENTS
FRAGMENTS EMBEDDING MODAL LOGIC
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two-variable fragment FO2
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fluted fragment FL
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guarded fragment GF
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unary negation fragment UN F
Theorem
All four base fragments enjoy the finite model property.
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FMP often gives a bound on the size of minimal models.
Hence:
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FMP often gives an upper bound for the computational
complexity of Sat(L)=FinSat(L).
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
FMP AND C OMPLEXITY OF FO2
Theorem (Mortimer, 75)
FO2 has doubly exponential model property:
every satisfiable ϕ ∈ FO2 has a model of size at most doubly
exponential in |ϕ|.
Theorem (Grädel, Kolaitis, Vardi, 97)
FO2 has exponential model property:
every satisfiable ϕ ∈ FO2 has a model of size at most exponential in
|ϕ|.
Corollary
Sat(FO2 ) is NE XP T IME-complete.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
F LUTED F RAGMENT FL
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First identified by W.V.Quine in 1968:
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I
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homogeneous m-adic formulas (generalization of monadic
fragment)
later generalized to fluted fragment
Examples of fluted formulas:
No student admires every professor
∀x1 (student(x1 ) → ¬∀x2 (prof(x2 ) → admires(x1 , x2 )))
No lecturer introduces any professor to every student
∀x1 (lecturer(x1 ) →
¬∃x2 (prof(x2 )∧
∀x3 (student(x3 ) → intro(x1 , x2 , x3 )))).
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Order of quantification of variables matches order of
appearance in predicates.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
F LUTED F RAGMENT FL
I
First identified by W.V.Quine in 1968:
I
I
I
homogeneous m-adic formulas (generalization of monadic
fragment)
later generalized to fluted fragment
Examples of fluted formulas:
No student admires every professor
∀x1 (student(x1 ) → ¬∀x2 (prof(x2 ) → admires(x1 , x2 )))
No lecturer introduces any professor to every student
∀x1 (lecturer(x1 ) →
¬∃x2 (prof(x2 )∧
∀x3 (student(x3 ) → intro(x1 , x2 , x3 )))).
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Order of quantification of variables matches order of
appearance in predicates.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
F LUTED F RAGMENT FL
I
First identified by W.V.Quine in 1968:
I
I
I
homogeneous m-adic formulas (generalization of monadic
fragment)
later generalized to fluted fragment
Examples of fluted formulas:
No student admires every professor
∀x1 (student(x1 ) → ¬∀x2 (prof(x2 ) → admires(x1 , x2 )))
No lecturer introduces any professor to every student
∀x1 (lecturer(x1 ) →
¬∃x2 (prof(x2 )∧
∀x3 (student(x3 ) → intro(x1 , x2 , x3 )))).
I
Order of quantification of variables matches order of
appearance in predicates.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
F LUTED F RAGMENT FL
I
First identified by W.V.Quine in 1968:
I
I
I
homogeneous m-adic formulas (generalization of monadic
fragment)
later generalized to fluted fragment
Examples of fluted formulas:
No student admires every professor
∀x1 (student(x1 ) → ¬∀x2 (prof(x2 ) → admires(x1 , x2 )))
No lecturer introduces any professor to every student
∀x1 (lecturer(x1 ) →
¬∃x2 (prof(x2 )∧
∀x3 (student(x3 ) → intro(x1 , x2 , x3 )))).
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Order of quantification of variables matches order of
appearance in predicates.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
FL FORMAL DEFINITION
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Let x1 , x2 , . . . be a fixed sequence of variables.
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The fluted fragment with k free variables, FL[k] , is defined
by simultaneous induction for all k:
- any atom p(x` , . . . , xk ) is in FL[k] ;
- FL[k] is closed under Boolean operations;
- FL[k] contains ∃xk+1 ϕ and ∀xk+1 ϕ for any ϕ ∈ FL[k+1] .
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The fluted fragment, FL[k] is the union:
[
FL =
FL[k] .
k≥0
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For all m > 0, we define FLm , to be the set of fluted
formulas containing at most the variables x1 , . . . , xm , free
or bound.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
FMP AND C OMPLEXITY OF FL
P URDY 1996, P RATT-H ARTMANN , S ZWAST, T. 2016
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Any satisfiable formula of FLm has a model of m-tuply
exponential size, that is, of size bounded by a function
2p(kϕk) 

.
..
2
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I
I
m 2’s

= t(m, p(kϕk))
where p is a polynomial.
Hence, Sat(FLm ) is in m-NE XP T IME.
On the other hand, satisfiable formulas of FL2m force
models of m-tuply exponential size.
Essentially the same proof shows that Sat(FL2m ) is
m-NE XP T IME-hard.
Therefore, for m ≥ 1, the complexity of Sat(FLm ) lies
between
bm/2c-NE XP T IME-hard
and
m-NE XP T IME.
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
G UARDED F RAGMENT GF
A NDRÉKA , VAN B ENTHEM AND N ÉMETI , 1996
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Restricting the use of quantifiers:
∀x(G(x, y) → ϕ(x, y))
∃x(G(x, y) ∧ ϕ(x, y))
G(x, y) – atomic formula, guard of the quantifier.
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
FMP AND C OMPLEXITY OF GF
Theorem (Grädel, 99)
GF has doubly exponential model property1 . Moreover,
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Sat(GF) is 2-E XP T IME-complete
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Sat(GF k ) is E XP T IME-complete, for any fixed k.
The above complexity bounds do not follow from the bound on
size of finite models.
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1
GF has the tree model property.
Nice application of combinatorial results by Hrushowski, Herwig and
Lascar about extensions of certain graphs.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
U NARY N EGATION F RAGMENT UN F
TEN
C ATE , S EGOUFIN 2011
Idea: start from existential FO (unions of conjunctive queries)
allow only ¬ϕ(x), where x is a single variable.
Formally:
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any atom of the form R(x̄) or x = y is in UN F;
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UN F is closed under ∨, ∧ and ∃;
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if ϕ(x) is a formula of UN F with no free variables besides
(possibly) x, then ¬ϕ(x) belongs to UN F.
Examples.
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∀x∀y∀z(Pxyz → Rxyz) ∈ GF (but not in UN F)
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∀x∃y∃z(Rxy ∧ Ryz ∧ Rzx) ∈ UN F (but not in GF or FO2 )
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∃x∃y¬Rxy ∈ FL2 (but not in UN F or GF)
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
E XTENSIONS
Trace the limits of decidability: study properties of extensions
of base fragments obtained by adding e.g.
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counting operators / functions
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built-in relations (e.g. orderings, equivalences)
and going beyond FO:
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transitive (or equivalence) closure
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fixed-points
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...
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
M ORE OPERATORS AND L OSS OF FMP
I NFINITY A XIOMS
∃x∀y¬Ryx ∧ ∀x∃yRxy ∧ ∀x∃≤1 yRyx
◦
/◦
/◦
/◦
/◦
/◦
(1)
...
∀x¬Rxx ∧ ∀x∃yRxy ∧ ∀x∀y(TC(R)xy ↔ Rxy)
(2)
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
O VERVIEW : E XTENSIONS OF B ASE F RAGMENTS
MOTIVATED BY EXTENSIONS OF MODAL / TEMPORAL / DESCRIPTION LOGICS
Logic
GF
GF
2
FO2
IUN F I
IFLI
Transitive closure
undecidable
G 99
decidable∗
M 09
undecidable
GOR 97
mon-Fixed Points
2-E XP T IME
Sat: GW 99
FinSat: BB 12
E XP T IME
Sat: GW 99
FinSat: BB 12
undecidable
GOR 97
Counting
undecidable
G 99
E XP T IME
P-H 05
NE XP T IME
Sat: PST 97, P-H 05
FinSat: P-H 05
?
2-E XP T IME
StC 13
?
?
?
?
Bárány, Bojańczyk, Grädel, Michaliszyn, Otto, Pratt-Hartmann, Rosen,
Pacholski, Segoufin, Szwast, ten Cate, T., Walukiewicz
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
R ESTRICTED C LASSES OF S TRUCTURES
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None of the base fragments can express transitivity
e.g. transitivity allows one to write infinity axioms:
∀x¬(x < x) ∧ ∀x∃y (x < y) with transitive <
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Transitivity is a useful property
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Solution: consider satisfiability in classes of structures
with predefined interpretation of some binary symbols (as
transitive relations, orders, equivalences, etc.)
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Corresponds to (multi-)modal logics K4, S4, S5
Theorem
FO2 and GF 2 undecidable with several transitive, order or
equivalence relations (Grädel, Otto, Rosen, Ganzinger, Meyer,
Veanes, Kazakov, Kieroński, T....)
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
FO2 AND GF 2 OVER SPECIAL CLASSES OF STRUCTURES
Logic
Special symbols
Transitivity
2-E XP T IME
undecidable
undecidable
K05
GMV99
FMP, NE XP T IME
KO05
K05, Kaz06
Sat: KO05
2-E XP T IME
FinSat: KP-HT15
undecidable
KO05
Sat: ST13
in 2-NE XP T IME∗)
FinSat: ?
undecidable
K05, Kaz06
undecidable
GOR99
Linear order
NE XP T IME
Ott01
Sat: ?
E XP S PACE∗)
FinSat: SchZ10
undecidable
Ott01, K11
Equivalence
FMP, NE XP T IME
KO05
2-NE XP T IME
KMP-HT12
undecidable
KO05
Equivalence
Closure
FMP, NE XP T IME
KMP-HT12
2-NE XP T IME
KMP-HT12
undecidable
KO05
GF 2
FMP
E XP T IME
Number of special symbols in the signature
1
2
3 or more
Equivalence
Transitivity
FO2
GKV97
FMP
NE XP T IME
. . . , Ganzinger, Meyer, Veanes, Kazakov, Schwentick, Zeume
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
D ECIDING F INITE S ATISFIABILITY
1. Finite Model Property and Tree Model Property.
2. More or less Natural Reductions.
3. Locally Acyclic Covers.
4. via Linear/Integer Programming.
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
FMP AND TMP
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FMP often gives natural upper complexity bounds.
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FMP does not help when L contains infinity axioms.
L has tree (tree-like) model property iff every satisfiable ϕ ∈ L
has a tree (tree-like) model.
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Advantages:
I
I
Useful for logics without FMP.
Often gives better complexity bounds.
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Positive examples:
FO2 , GF, UN F.
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Disadvantage:
Tree-like models are usually infinite, so TMP is not suitable
to decide FinSat.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
R EDUCTIONS TO OTHER FRAGMENTS
Theorem
[Segoufin, ten Cate 2015] There is an exponential reduction from
UNF with fixed-points to µ-calculus preserving finiteness of models.
Hence:
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UN F with fixed-points has TMP and is 2-E XP T IME-complete.
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UN F has FMP.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
R EDUCING FinSat(L) TO Sat(L)
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DL-Lite [Rosati 2008]
I
Horn-SHIQ [Garcia, Lutz, Schneider 2013]
Idea. Complete a given TBox T to Tfin by adding new axioms
(reversing cycles in T ) and show that
T is finitely satisfiable iff Tfin is satisfiable.
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Advantage:
allows to run existing reasoners for Sat.
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
F INITARY U NRAVELLINGS
Roughly: TMP in the finite
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[Otto 2004]: ”Finitary unravelling” – construction that
make a structure locally acyclic (avoiding short cycles).
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[Bárány, Gottlob, Otto 2009]:
Every finite structure is GF-bisimilar to a finite structure
whose hypergraph is locally acyclic (suitably defined).
Applied to obtain:
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small model property for GF.
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decidability of FinSat for GF with fixed points.
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correctness of the reduction from UN F with fixed points
to µ-calculus.
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
R EDUCTION TO L INEAR /I NTEGER P ROGRAMMING
Idea: Depending on the logic:
identify (finitely many types of) building blocks of a
potential model and connecting conditions for them,
describe them in a succinct way by a set of (in)equalities.
Advantages:
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Useful for solving simultaneously Sat and FinSat.
We look for solutions over N (FinSat) or over N ∪ {∞} (Sat),
e.g.
x+1=x
has a solution x = ∞.
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Does not depend on TMP.
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Gives better (optimal) complexity bounds.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
LP/IP: FROM M ODELS TO B IPARTITE G RAPHS
Example: FO2 with two equivalence relations E1 , E2
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Lemma (small intersections property): in a model of ϕ we
can replace every equivalence class of E1 ∩ E2 by a class
bounded exponentially in |ϕ|
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Think about E1 -classes and E2 -classes as about nodes of a
bipartite graph
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Intersections are represented by edges of the graph
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Colors of edges represent isomorphism types of
intersections
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A question, whether ϕ is satisfiable becomes a question
about the existence of a graph satisfying some constraints.
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These constraints can be expressed in terms of linear
inequalities.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
FO2 +{E1 , E2 }: MODEL CONSTRUCTION ( IDEA )
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
FO2 +{E1 , E2 }: MODEL CONSTRUCTION ( IDEA )
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
FO2 +{E1 , E2 }: C OMPLEXITY
Construct a system of linear inequalities
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variables correspond to types of classes: triply
exponentially many
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inequalities correspond to intersections: doubly
exponentially many
As integer programming is in NP this gives
3-NE XP T IME-upper bound.
I
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A Caratheodory-type result of Eisenbrand and Shmonin
(2006) says that any system of linear inequalities, solvable
over integers, has a solution in which the number of
non-zero variables is polynomial in the number of
inequalities.
This allows to show 2-NE XP T IME-upper bound:
I
Just guess relevant variables and construct a system of
doubly exponential size.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
FO2 +{E1 , E2 }: C OMPLEXITY
Construct a system of linear inequalities
I
variables correspond to types of classes: triply
exponentially many
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inequalities correspond to intersections: doubly
exponentially many
As integer programming is in NP this gives
3-NE XP T IME-upper bound.
I
I
A Caratheodory-type result of Eisenbrand and Shmonin
(2006) says that any system of linear inequalities, solvable
over integers, has a solution in which the number of
non-zero variables is polynomial in the number of
inequalities.
This allows to show 2-NE XP T IME-upper bound:
I
Just guess relevant variables and construct a system of
doubly exponential size.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
LP/IP A PPROACH
Successfully used to get optimal complexity bounds for:
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FO2 and GF 2 with counting quantifiers [Pratt-Hartmann
2005]
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FO2 with two equivalence relations [Kieroński,
Michaliszyn, Pratt-Hartmann, T. 2013]
I
FO2 with counting quantifiers and one equivalence
relation [Pratt-Hartmann 2013]
I
...
Disadvantages:
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not ideal for implementation.
I
not yet clear how to extend to logics with predicates of
higher arity.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
LP/IP A PPROACH
Successfully used to get optimal complexity bounds for:
I
FO2 and GF 2 with counting quantifiers [Pratt-Hartmann
2005]
I
FO2 with two equivalence relations [Kieroński,
Michaliszyn, Pratt-Hartmann, T. 2013]
I
FO2 with counting quantifiers and one equivalence
relation [Pratt-Hartmann 2013]
I
...
Disadvantages:
I
not ideal for implementation.
I
not yet clear how to extend to logics with predicates of
higher arity.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
F UTURE R ESEARCH (1)
I
answer remaining open question
I
optimise known algorithms towards practical
implementation
combine fragments
I
I
I
[Bárány, ten Cate, Segoufin 2011]
Guarded Negation Fragment
a common generalisation of GF and UN F.
[Kuusisto, Hella 2014]
Uniform One-Dimensional FO
generalization of FO2 to contexts with relations of arbitrary
arity
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identify useful (and tractable) subfragments
I
...
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
F UTURE R ESEARCH (2)
Query Answering:
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D ∧ O |= Query
iff
D ∧ O ∧ ¬Query is unsatisfiable
Classes of queries:
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positive existential
I
conjunctive queries
I
unions of conjunctive queries
Depending on LO and considered class of queries, ¬Query is
not necessarily in LO , so existing procedures deciding
satisfiability cannot be applied directly.
Conclusion
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
QA: C OMPLEXITY M EASURES
D ∧ O |= Query
I
Data complexity: only the size of the database matters.
The ontology O and Query are considered fixed.
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Schema complexity: only the size of the ontology matters.
D and Query are considered fixed.
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Combined complexity: no parameter is considered fixed.
In practise one often assumes that the size of the data largely
dominates the size of the ontology (and of the query) and
considers data complexity as the relevant complexity measure.
Identify fragments with low data complexity.
Introduction/Motivation
Base Fragments
Extensions of Base Fragments
Deciding (Fin)Sat
Conclusion
T HANK Y OU !
D ZIȨKUJȨ !
Questions?