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Complexity basics EXPTIME-membership Complexity basics EXPTIME-membership Goal for today & tomorrow Automated reasoning plays an important role for DLs. Description Logics: a Nice Family of Logics It allows the development of intelligent applications. — Complexity, Part 1 — Uli Sattler1 1 School The expressivity of DLs is strongly tailored towards this goal. Requirements for automated reasoning: Thomas Schneider 2 Decidability of the relevant decision problems of Computer Science, University of Manchester, UK 2 Department Low complexity if possible of Computer Science, University of Bremen, Germany Algorithms that perform well in practice ESSLLI, 17 August 2016 Yesterday & today: 1 & 3 Now: 2 Uli Sattler, Thomas Schneider Complexity basics DL: Complexity (1) EXPTIME-membership 1 And now . . . Uli Sattler, Thomas Schneider Complexity basics DL: Complexity (1) EXPTIME-membership 2 Cognitive versus Computational Complexity Consider decision problems for reasoning, e.g. O |=? C v D ] (more on Friday) Cognitive complexity How hard is it, for a human, to decide or understand ] ? 1 interesting, little understood topic Complexity basics relevant to provide tool support for ontology engineers 2 Computational complexity EXPTIME-membership (today) How much time/space is needed to decide ] ? interesting, well understood topic loads of results thanks to relationships DL – FOL – ML relevant to understand trade-off: expressivity of a DL ↔ complexity of reasoning whether a given algorithm is optimal/can be improved Uli Sattler, Thomas Schneider DL: Complexity (1) 3 Uli Sattler, Thomas Schneider DL: Complexity (1) 4 Complexity basics EXPTIME-membership Complexity basics Decidability EXPTIME-membership Computational complexity A (decision) problem . . . is a subset P ⊆ M m∈M Examples: P = set of all prime numbers, M = N m∈P? Output A „yes“ ⇒ m ∈ P „no“ ⇒ m ∈ / P Exponential time: Number of computation steps is 6 2pol(|m|) ... (Programming language and machine model are largely irrelevant) DL: Complexity (1) EXPTIME-membership 5 Some standard complexity classes Restriction Example problem L NL P logarithmic space nondeterministic log. space polynomial time graph connectivity graph accessibility prime numbers NP PSPACE nondeterm. polynomial time polynomial space (propositional) SAT QBF-SAT EXPTIME NEXPTIME EXPSPACE .. . exponential time nondeterm. exponential time exponential space .. . CTL-SAT undecidable first-order SAT DL: Complexity (1) Uli Sattler, Thomas Schneider Complexity basics DL: Complexity (1) EXPTIME-membership 6 Reductions Name Uli Sattler, Thomas Schneider A „yes“ ⇒ m ∈ P „no“ ⇒ m ∈ / P Polynomial space: Number of memory cells used is 6 pol(|m|) Decidability: P is decidable if there is an algorithm A that implements the black box. Uli Sattler, Thomas Schneider Complexity basics Output Polynomial time: Number of computation steps is 6 pol(|m|), for some polynomial function pol think of it as a black box: Input m∈P? Complexity: measures time/space needed by A in the worst case, depending on the length of the input |m| P = set of triples (O, C , D) with O |= C v D, M = set of all triples (O, C , D) from ALC m∈M Input m∈M Input m∈P? Output A „yes“ ⇒ m ∈ P „no“ ⇒ m ∈ / P A (polynomial) reduction of P ⊆ M to P 0 ⊆ M 0 is a (poly-time computable) function π : M → M 0 with m∈P m∈M π iff π(m) ∈ P 0 m∈P? yes π(m) π(m) ∈ P 0 ? no yes no If P reducible to P 0 then P is “at most as hard” as P 0 . 7 If all problems from a complexity class C are reducible to P, then P is hard for C. Uli Sattler, Thomas Schneider DL: Complexity (1) 8 Complexity basics EXPTIME-membership Complexity basics Determining the complexity EXPTIME-membership Worst-case complexity Worst case: algorithm runs, for all m ∈ M, in at most C resources, e.g., like this on all problems of size 7: Usually one shows that a problem P ⊆ M is . . . in a complexity class C, by designing/finding an algorithm A that solves P, showing that A is sound, complete, and terminating showing that A runs, for every m ∈ M, in at most C ressources . . . A can be, e.g., a reduction to a problem known to be in C hard for C, by finding a suitable problem P 0 ⊆ M 0 that is known to be hard for C and a reduction of P 0 to P complete for C, by showing that P is in C and hard for C Uli Sattler, Thomas Schneider Complexity basics DL: Complexity (1) EXPTIME-membership 9 Worst-case complexity DL: Complexity (1) 10 EXPTIME-membership Worst-case complexity Worst case: algorithm runs, for all m ∈ M, in at most C resources, e.g.,or like this on all problems of size 7: Uli Sattler, Thomas Schneider Uli Sattler, Thomas Schneider Complexity basics DL: Complexity (1) Worst case: algorithm runs, for all m ∈ M, in at most C resources, e.g.,or like this on all problems of size 7: 10 Uli Sattler, Thomas Schneider DL: Complexity (1) 10 Complexity basics EXPTIME-membership Worst-case complexity Complexity basics EXPTIME-membership Known complexity results from Days 2–3 Worst case: algorithm runs, for all m ∈ M, in at most C resources, e.g.,or like this on all problems of size 7: From the tableau technique, we know that all considered reasoning problems are decidable for ALCQI because the tableau algorithm is sound, complete, terminating consistency of ALC ontologies is in EXPSPACE and so are satisfiability and subsumption w.r.t. ontologies ü We can do better: we’ll show they are EXPTIME-complete satisfiability and subsumption of ALC concepts are in PSPACE ü We cannot do better: we’ll show that they are PSPACE-hard Uli Sattler, Thomas Schneider Complexity basics DL: Complexity (1) 10 EXPTIME-membership And now . . . Uli Sattler, Thomas Schneider Complexity basics DL: Complexity (1) 11 EXPTIME-membership EXPTIME-membership We start with an EXPTIME upper bound for concept satisfiability in ALC relative to TBoxes. 1 2 Theorem The following problem is in EXPTIME. Complexity basics Input: Question: EXPTIME-membership an ALC concept C0 and an ALC TBox T is there a model I |= T with C I 6= ∅ ? We’ll use a technique known from modal logic: type elimination [Pratt 1978] The basis is a syntactic notion of a type. Uli Sattler, Thomas Schneider DL: Complexity (1) 12 Uli Sattler, Thomas Schneider DL: Complexity (1) 13 Complexity basics EXPTIME-membership Syntactic types Complexity basics EXPTIME-membership General idea We assume that the input concept C0 is in NNF the input TBox is T = {> v CT } with CT in NNF General idea of type elimination for input C0 , T : Generate all types for C0 and T Let sub(C0 , T ) be the set of subconcepts of C0 and CT . Repeatedly eliminate types that cannot occur in any model of C0 and T . A type for C0 and T is a subset t ⊆ sub(C0 , T ) such that 1. A ∈ t iff ¬A ∈ / t 2. C u D ∈ t iff C ∈ t and D ∈ t 3. C t D ∈ t iff C ∈ t or D ∈ t 4. CT ∈ t for all ¬A ∈ sub(C0 , T ) for all C u D ∈ sub(C0 , T ) Uli Sattler, Thomas Schneider Check whether some type containing C0 has survived. If yes, return “satisfiable”; otherwise “unsatisfiable”. for all C t D ∈ sub(C0 , T ) Intuition: Types describe domain elements completely, up to sub(C0 , T ). DL: Complexity (1) (exponentially many). • 14 Uli Sattler, Thomas Schneider DL: Complexity (1) 15