Complexity of Reasoning Hamid Mahini Outline l Brief

Complexity of Reasoning Hamid Mahini

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

Complexity Classes l Deterministic turing machine. ¡ 2 -Sat. l Non-deterministic turing machine. ¡ 3 -Sat l Worse case complexity. l C-Complete l C-Hard meaning.

Complexity Classes l. P Class. l NP Class l Co-NP Class. l PSPACE Class. l EXP-TIME Class.

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

Complexity Classes l Relation between complexity classes. ¡P NP. ¡P co-NP. ¡ NP PSPACE. ¡ Co-NP PSPACE. ¡ PSPACE EXP_TIME.

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

AL Language. l We l AL discuss about AL and its extension. : ¡ Atomic concept. ¡ C ∏ D. ¡ C. ¡ R. T ≡ R.

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

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

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

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

Satisfiabiligy in FL-εN l Satisfiability in FL-εN is NP-hard. ¡ We Solve set splitting problem. l Given a collection C of subset of a set S, decide if there exists partitioning of S into S 1 and S 2 such that no subset of C is entirely in S 1 or S 2. given S and subset of it. ¡ S has partitioning if and only if is satisfiable. ¡ l l l are pairwise disjiont.

Subsumption in FL-εN l Non-Subsumption is like Satisfiability. ¡ is satisfiable if and only if subsume by. is not l We prove non-subsumption in ALεN is co. NPhard. l We know subsumption in ALεN is PSPACEcomplete. l Similar prove is apply to prove : ¡ Subsumption in ALNI is co. NP-hard.

And Simulation l Replacing with is not change satisfiability of concept. l 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.

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

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

FL- and its extension l FL- with role chain and role inverse. l 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.

FL- and its extension l 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: l For each R(x, y) which add to satisfy R. C in D , add Q(y, u).

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

Reasoning in different cases l 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.

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

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

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

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

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

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

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