1. Complexity of Reasoning
Hamid Mahini

2. Outline Brief description of complexity classes. OR-branching.
AND-branching.
Combination of sources.
Using axioms.
Undecidability.

3. Complexity Classes Deterministic turing machine.
2-Sat.
Non-deterministic turing machine.
3-Sat
Worse case complexity.
C-Complete meaning.
C-Hard meaning.
C is an abbrivation for P, NP or other problem space

4. Complexity Classes P Class. NP Class Co-NP Class. PSPACE Class.
EXP-TIME Class.
P<NP<CO-NP<PSPACE<EXP-TIME

5. Complexity Classes Example of complete problem.
Circuit evaluation. (P-Complete)
3-CNF. (NP-Complete)
3-DNF. (CO-NP complete)
QBF. (PSPACE)
How can we determine complexity class of a problem?

6. Complexity Classes Relation between complexity classes. P  NP.
P  co-NP.
NP  PSPACE.
Co-NP  PSPACE.
PSPACE  EXP_TIME.

7. Complexity Classes Relation between  and NP class.
Relation between  and co-NP class.
Combination of  , and PSPACE class.
Intuitively connection between DL and complexity classes.
OR – AND branching.

8. AL Language. We discuss about AL and its extension. AL :
Atomic concept.
C ∏ D. C.
R.C.
R.T ≡ R.

9. AL extension Union. ( U ) Full existential quantification. ( ε )
Number restrictions. ( N )
Negation. ( C )
Inverse role. ( I )
C ≡ N , ε.

10. FL and FL- Definition of FL : FL- is FL with out role restriction.
Atomic concept.
C ∏ D.
R.C.
R.T ≡ R.
R|c. (role restriction)
FL- is FL with out role restriction.

11. Subsumption in FL Subsumption in FL in co-NP hard.
We reduce 3DNF to it.
We create a FL :
is : if

12. Subsumption in FL Some useful equivalence :
F is tautology if and only if
Subsumption in FL- can be solve in polynomial time.
FL-εN is equivalent to ALεN.
A and its negation can be simulated by Role R with number restriction.

13. Satisfiabiligy in FL-εN
Satisfiability in FL-εN is NP-hard.
We Solve set splitting problem.
Given a collection C of subset of a set S, decide if there exists partitioning of S into S1 and S2 such that no subset of C is entirely in S1 or S2.
given S and subset of it.
S has partitioning if and only if is satisfiable.
are pairwise disjiont.

14. Subsumption in FL-εN Non-Subsumption is like Satisfiability.
is satisfiable if and only if is not subsume by
We prove non-subsumption in ALεN is coNP-hard.
We know subsumption in ALεN is PSPACE-complete.
Similar prove is apply to prove :
Subsumption in ALNI is coNP-hard.

15. And Simulation Replacing with is not change satisfiability of concept.
By this simulation we have :
Satisfiability and non-subsumption in ALN(∏) is NP-hard.
We prove it for FL-εN and by applying And simulation we have the result.

16. ALε and AL(∏) Satisfiability is ALε is coNP-complete.
We solve exact cover.
Unsatisfiability in ALε in NP-hard.
Use And simulation :
Unsatisfiability in AL(∏) is NP-hard.
Safisfiability and subsumption of concepts are NP-hard in AL(∏).

17. FL- and its extension Subsumtion in FL-ε NP-hard.
FL- with role conjuction and role inverse :
Tableaux role for inverse.
Subsumption in FL-(∏,¯) is NP-hard.
FL- with role conjection and role chain.
Subsumption in FL-(∏,○) is NP-hard.

18. FL- and its extension FL- with role chain and role inverse.
Simulatin R.C via role chain and role inverse (o Simulation):
Replace R.C with (R○Q)∏(R○Q○Q¯).C will not change satisfiability.
This does not true for subsumption.

19. FL- and its extension Prove :
A open tableau for D is also and open tableau for D.
And open tableau for D can transform to and open tableau for D by:
For each R(x,y) which add to satisfy R.C in D , add Q(y,u).

20. FL- and its extension
In C is an ALε concept its o simulation an AL(○,¯) concept.
Subsumption in FL-(○,¯) is NP-hard.

21. Reasoning in different cases
The Following will complex our problem and our complexity class will be change with respect to these :
Set of axiom.
Role-value map.
Reasoning w.r.t to ABOX.

22. Subsumption in PTIME.
FL-.
AL.
ALN.
AL(○).
AL(‾).
FL-(∏)

23. NP Subsumption and unsatisfiability in ALε are NP-complete.
Subsumption and unsatisfiability in AL(∏) and ALε(∏) and FL-ε are NP-complete.
Subsumption and unsatisfiability in FL-(∏,‾) , FL-(∏,○) and FL-(○,‾) are NP-hard.

24. Co-NP Subsumption and satisfiability in ALU in coNP-complete.
ALN(‾) subsumption in coNP-complete while satisfiability is decidable in PTIME.
Satisfiability is ALε is coNP-complete.

25. PSPACE. Satisfiability and subsumption in ALC is PSPACE-hard.
Satisfiability and subsumption in ALεN is PSPACE-hard.
Satisfiability and subsumption in FL , ALN(∏) and ALU(∏) are PSPACE-hard.

26. EXPTIME and NEXPTIME.
Satisfiability and subsumption in AL w.r.t a set of axioms is in EXPTIME-hard.
ALC( ) satisfiability in NEXPTIME-hard.
Satisfiability and subsumption in ALCNR w.r.t a set of axioms is NEXPTIME-hard.

27. Undecidability
Subsumption in FL-(○,) which is a subset of the language of the knowledge representation suystem KL-ONE is undecidable.
ALCN(○,∏) satisfiability w.r.t to a set of axiom if undecidable.

28. Overview We explain meaning of complexity classes.
We define different DL language.
We show the result about complexity of reasoning of each language.