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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)
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Uli Sattler, Thomas Schneider
DL: Complexity (1)
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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)
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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:
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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)
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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)
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Uli Sattler, Thomas Schneider
DL: Complexity (1)
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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).
•
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Uli Sattler, Thomas Schneider
DL: Complexity (1)
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