Finite Model Reasoning in Expressive Fragments of First-Order Logic - - PowerPoint PPT Presentation

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

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

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

OUTLINE

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

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

CLASSICAL DECISION PROBLEM

L – any logic FO– first-order logic

◮ Sat(L):

given a formula ϕ ∈ L, does ϕ admit a model ?

◮ 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

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

WHY FINSAT?

Databases, systems etc. are often considered to be finite.

◮ L has the finite model property (FMP) iff every satisfiable

ϕ ∈ L has a finite model.

Observation

◮ If L has FMP then Sat(L) and FinSat(L) coincide. ◮ 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 Conclusion

DECIDABLE FRAGMENTS OF FO

Note: one cannot study all possible fragments!

◮ defined by restrictions on signatures

e.g. [Löwenheim-Skolem 1915] monadic theories (FMP)

◮ prenex classes defined by quantifier prefix

∃∗∀∗, ∃∗∀∃∗, ∃∗∀∀∃∗ (equality free)

◮ defined by other syntactic restrictions and suitably

motivated Also: we want to identify reasons for a logic to be (un)decidable, (in)tractable etc.

◮ Can we decide whether a formula is satisfiable without

actually seeing a model?

◮ Some formulas have only infinite models.

Can we decide whether they are satisfiable?

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

MOTIVATION: MODAL LOGIC

[VARDI 1996]: WHY IS MODAL LOGIC SO ROBUSTLY DECIDABLE?

◮ Propositional modal logic:

Boolean logic + operators ♦ (possibly) and (necessary)

◮ Good model-theoretical and algorithmic properties,

robustly decidable

◮ Variants and extensions of modal logics have applications

in various areas of computer science:

◮ verification of hardware and software ◮ artificial intelligence ◮ distributed systems ◮ knowledge representation

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

STANDARD TRANSLATION OF MODAL LOGIC (1)

◮ Modal logic can be translated into FO:

P∧♦(Q∨¬P)

• Px ∧ ∃y(Rxy ∧ (Qy ∨ ∀z(Ryz → ¬Pz)))

◮ FO3 undecidable [Kahr, Moore, Wang, 1959]

[Gabbay, 1981] TWO variables suffice!

Observation

◮ ML can be embedded in the two-variable fragment FO2

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

STANDARD TRANSLATION OF MODAL LOGIC (1)

◮ Modal logic can be translated into FO:

P∧♦(Q∨¬P)

• Px ∧ ∃y(Rxy ∧ (Qy ∨ ∀x(Ryx → ¬Px)))

◮ FO3 undecidable [Kahr, Moore, Wang, 1959]

[Gabbay, 1981] TWO variables suffice!

Observation

◮ ML can be embedded in the two-variable fragment FO2

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

STANDARD TRANSLATION OF MODAL LOGIC (2)

P ∧ ♦(Q ∨ ¬P)

• Px ∧ ∃y(Rxy ∧ (Qy ∨ ∀z(Ryz → ¬Pz)))

The translation suggests other restrictions of FO:

◮ 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.

◮ guarded fragment GF:

quantifiers are relativized by atomic formulas

◮ unary negation fragment UNF:

negation is applied only to subformulas with a single free variable.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

PROBLEMS REDUCING TO (FIN)SAT

EXAMPLE: QUERY ANSWERING

A knowledge base D, O: database D (a set of facts, i.e. ground atoms),

• ntology O (i.e. a logical formula).

◮ Query Answering:

given a knowledge base D, O and a query Q: does D, O entail Q, i.e. D ∧ O | = Q?

Observation

D ∧ O | = Q iff D ∧ O ∧ ¬Q is unsatisfiable

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

BASE FRAGMENTS

FRAGMENTS EMBEDDING MODAL LOGIC

◮ two-variable fragment FO2 ◮ fluted fragment FL ◮ guarded fragment GF ◮ unary negation fragment UNF

Theorem

All four base fragments enjoy the finite model property.

◮ FMP often gives a bound on the size of minimal models.

Hence:

◮ FMP often gives an upper bound for the computational

complexity of Sat(L)=FinSat(L).

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FMP AND COMPLEXITY 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 NEXPTIME-complete.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FLUTED FRAGMENT FL

◮ First identified by W.V.Quine in 1968:

◮ 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)))).

◮ Order of quantification of variables matches order of

appearance in predicates.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FLUTED FRAGMENT FL

◮ First identified by W.V.Quine in 1968:

◮ 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)))).

◮ Order of quantification of variables matches order of

appearance in predicates.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FLUTED FRAGMENT FL

◮ First identified by W.V.Quine in 1968:

◮ 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)))).

◮ Order of quantification of variables matches order of

appearance in predicates.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FLUTED FRAGMENT FL

◮ First identified by W.V.Quine in 1968:

◮ 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)))).

◮ 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

◮ Let x1, x2, . . . be a fixed sequence of variables. ◮ The fluted fragment with k free variables, FL[k], is defined

by simultaneous induction for all k:

• any atom p(xℓ, . . . , xk) is in FL[k];
• FL[k] is closed under Boolean operations;
• FL[k] contains ∃xk+1ϕ and ∀xk+1ϕ for any ϕ ∈ FL[k+1].

◮ The fluted fragment, FL[k] is the union:

FL =

• k≥0

FL[k].

◮ For all m > 0, we define FLm, to be the set of fluted

formulas containing at most the variables x1, . . . , xm, free

• r bound.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FMP AND COMPLEXITY OF FL

PURDY 1996, PRATT-HARTMANN, SZWAST, T. 2016

◮ Any satisfiable formula of FLm has a model of m-tuply

exponential size, that is, of size bounded by a function 2

...2p(ϕ)  m 2’s

= t(m, p(ϕ)) where p is a polynomial. Hence, Sat(FLm) is in m-NEXPTIME.

◮ 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-NEXPTIME-hard.

◮ Therefore, for m ≥ 1, the complexity of Sat(FLm) lies

between ⌊m/2⌋-NEXPTIME-hard and m-NEXPTIME.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

GUARDED FRAGMENT GF

ANDRÉKA, VAN BENTHEM AND NÉMETI, 1996

◮ 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.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FMP AND COMPLEXITY OF GF

Theorem (Grädel, 99)

GF has doubly exponential model property1. Moreover,

◮ Sat(GF) is 2-EXPTIME-complete ◮ Sat(GFk) is EXPTIME-complete, for any fixed k.

The above complexity bounds do not follow from the bound on size of finite models.

◮ GF has the tree model property.

1Nice 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

UNARY NEGATION FRAGMENT UNF

TEN CATE, SEGOUFIN 2011

Idea: start from existential FO (unions of conjunctive queries) allow only ¬ϕ(x), where x is a single variable. Formally:

◮ any atom of the form R(¯

x) or x = y is in UNF;

◮ UNF is closed under ∨, ∧ and ∃; ◮ if ϕ(x) is a formula of UNF with no free variables besides

(possibly) x, then ¬ϕ(x) belongs to UNF. Examples.

◮ ∀x∀y∀z(Pxyz → Rxyz) ∈ GF (but not in UNF) ◮ ∀x∃y∃z(Rxy ∧ Ryz ∧ Rzx) ∈ UNF (but not in GF or FO2) ◮ ∃x∃y¬Rxy ∈ FL2 (but not in UNF or GF)

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

EXTENSIONS

Trace the limits of decidability: study properties of extensions

• f base fragments obtained by adding e.g.

◮ counting operators / functions ◮ built-in relations (e.g. orderings, equivalences)

and going beyond FO:

◮ transitive (or equivalence) closure ◮ fixed-points ◮ . . .

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

MORE OPERATORS AND LOSS OF FMP

INFINITY AXIOMS

∃x∀y¬Ryx ∧ ∀x∃yRxy ∧ ∀x∃≤1yRyx (1)

. . . ∀x¬Rxx ∧ ∀x∃yRxy ∧ ∀x∀y(TC(R)xy ↔ Rxy) (2)

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

OVERVIEW: EXTENSIONS OF BASE FRAGMENTS

MOTIVATED BY EXTENSIONS OF MODAL/TEMPORAL/DESCRIPTION LOGICS

Logic

Transitive closure mon-Fixed Points Counting undecidable 2-EXPTIME undecidable

GF

G 99 Sat: GW 99 G 99 FinSat: BB 12 decidable∗ EXPTIME EXPTIME

GF2

M 09 Sat: GW 99 P-H 05 FinSat: BB 12 undecidable undecidable NEXPTIME

FO2

GOR 97 GOR 97 Sat: PST 97, P-H 05 FinSat: P-H 05

IUNFI

? 2-EXPTIME ? StC 13

IFLI

? ? ?

Bárány, Boja´ nczyk, 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

RESTRICTED CLASSES OF STRUCTURES

◮ 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 <

◮ Transitivity is a useful property ◮ Solution: consider satisfiability in classes of structures

with predefined interpretation of some binary symbols (as transitive relations, orders, equivalences, etc.)

◮ Corresponds to (multi-)modal logics K4, S4, S5

Theorem

FO2 and GF2 undecidable with several transitive, order or equivalence relations (Grädel, Otto, Rosen, Ganzinger, Meyer, Veanes, Kazakov, Kiero´ nski, T....)

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FO2 AND GF2 OVER SPECIAL CLASSES OF STRUCTURES

Logic

Special symbols Number of special symbols in the signature 1 2 3 or more Transitivity 2-EXPTIME undecidable undecidable

GF2

K05 K05, Kaz06 GMV99 FMP Sat: KO05 EXPTIME Equivalence FMP, NEXPTIME 2-EXPTIME undecidable KO05 FinSat: KP-HT15 KO05 Sat: ST13 Transitivity in 2-NEXPTIME∗) undecidable undecidable FinSat: ? K05, Kaz06 GOR99

FO2

Sat: ? Linear order NEXPTIME EXPSPACE∗) undecidable GKV97 Ott01 FinSat: SchZ10 Ott01, K11 FMP NEXPTIME Equivalence FMP, NEXPTIME 2-NEXPTIME undecidable KO05 KMP-HT12 KO05 Equivalence FMP, NEXPTIME 2-NEXPTIME undecidable Closure KMP-HT12 KMP-HT12 KO05 ..., Ganzinger, Meyer, Veanes, Kazakov, Schwentick, Zeume

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

DECIDING FINITE SATISFIABILITY

• 1. Finite Model Property and Tree Model Property.
• 2. More or less Natural Reductions.
• 3. Locally Acyclic Covers.
• 4. via Linear/Integer Programming.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FMP AND TMP

◮ FMP often gives natural upper complexity bounds. ◮ 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.

◮ Advantages:

◮ Useful for logics without FMP. ◮ Often gives better complexity bounds.

◮ Positive examples:

FO2, GF, UNF.

◮ 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

REDUCTIONS 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:

◮ UNF with fixed-points has TMP and is 2-EXPTIME-complete. ◮ UNF has FMP.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

REDUCING FinSat(L) TO Sat(L)

◮ DL-Lite [Rosati 2008] ◮ 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.

◮ Advantage:

allows to run existing reasoners for Sat.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FINITARY UNRAVELLINGS

Roughly: TMP in the finite

◮ [Otto 2004]: ”Finitary unravelling” – construction that

make a structure locally acyclic (avoiding short cycles).

◮ [Bárány, Gottlob, Otto 2009]:

Every finite structure is GF-bisimilar to a finite structure whose hypergraph is locally acyclic (suitably defined). Applied to obtain:

◮ small model property for GF. ◮ decidability of FinSat for GF with fixed points. ◮ correctness of the reduction from UNF with fixed points

to µ-calculus.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

REDUCTION TO LINEAR/INTEGER PROGRAMMING

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:

◮ 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 = ∞.

◮ Does not depend on TMP. ◮ Gives better (optimal) complexity bounds.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

LP/IP: FROM MODELS TO BIPARTITE GRAPHS

Example: FO2 with two equivalence relations E1, E2

◮ 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|

◮ Think about E1-classes and E2-classes as about nodes of a

bipartite graph

◮ Intersections are represented by edges of the graph ◮ Colors of edges represent isomorphism types of

intersections

◮ A question, whether ϕ is satisfiable becomes a question

about the existence of a graph satisfying some constraints.

◮ These constraints can be expressed in terms of linear

inequalities.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FO2+{E1, E2}: MODEL CONSTRUCTION (IDEA)

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FO2+{E1, E2}: MODEL CONSTRUCTION (IDEA)

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FO2+{E1, E2}: COMPLEXITY

Construct a system of linear inequalities

◮ variables correspond to types of classes: triply

exponentially many

◮ inequalities correspond to intersections: doubly

exponentially many As integer programming is in NP this gives 3-NEXPTIME-upper bound.

◮ A Caratheodory-type result of Eisenbrand and Shmonin

(2006) says that any system of linear inequalities, solvable

• ver integers, has a solution in which the number of

non-zero variables is polynomial in the number of inequalities.

◮ This allows to show 2-NEXPTIME-upper bound:

◮ 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}: COMPLEXITY

Construct a system of linear inequalities

◮ variables correspond to types of classes: triply

exponentially many

◮ inequalities correspond to intersections: doubly

exponentially many As integer programming is in NP this gives 3-NEXPTIME-upper bound.

◮ A Caratheodory-type result of Eisenbrand and Shmonin

(2006) says that any system of linear inequalities, solvable

• ver integers, has a solution in which the number of

non-zero variables is polynomial in the number of inequalities.

◮ This allows to show 2-NEXPTIME-upper bound:

◮ 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 APPROACH

Successfully used to get optimal complexity bounds for:

◮ FO2 and GF2 with counting quantifiers [Pratt-Hartmann

2005]

◮ FO2 with two equivalence relations [Kiero´

nski, Michaliszyn, Pratt-Hartmann, T. 2013]

◮ FO2 with counting quantifiers and one equivalence

relation [Pratt-Hartmann 2013]

◮ . . .

Disadvantages:

◮ not ideal for implementation. ◮ 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 APPROACH

Successfully used to get optimal complexity bounds for:

◮ FO2 and GF2 with counting quantifiers [Pratt-Hartmann

2005]

◮ FO2 with two equivalence relations [Kiero´

nski, Michaliszyn, Pratt-Hartmann, T. 2013]

◮ FO2 with counting quantifiers and one equivalence

relation [Pratt-Hartmann 2013]

◮ . . .

Disadvantages:

◮ not ideal for implementation. ◮ 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

FUTURE RESEARCH (1)

◮ answer remaining open question ◮ optimise known algorithms towards practical

implementation

◮ combine fragments

◮ [Bárány, ten Cate, Segoufin 2011]

Guarded Negation Fragment a common generalisation of GF and UNF.

◮ [Kuusisto, Hella 2014]

Uniform One-Dimensional FO generalization of FO2 to contexts with relations of arbitrary arity

◮ identify useful (and tractable) subfragments ◮ . . .

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

FUTURE RESEARCH (2)

Query Answering:

◮ D ∧ O |

= Query iff D ∧ O ∧ ¬Query is unsatisfiable Classes of queries:

◮ positive existential ◮ conjunctive queries ◮ 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.

Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion

QA: COMPLEXITY MEASURES

D ∧ O | = Query

◮ Data complexity: only the size of the database matters.

The ontology O and Query are considered fixed.

◮ Schema complexity: only the size of the ontology matters.

D and Query are considered fixed.

◮ 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

THANK YOU!

DZIE

¸ KUJE ¸ !

Questions?