Logical Aspects of Artificial Intelligence Tableaux Calculi and - - PowerPoint PPT Presentation

Logical Aspects of Artificial Intelligence Tableaux Calculi and Complexity

St´ ephane Demri demri@lsv.fr December 16th, 2019

Plan of the lecture

◮ Recapitulation of the previous lecture. ◮ Decision procedures using model-theoretical properties. ◮ Tableaux proof system for ALC. ◮ Complexity of decision problems for ALC and variants. ◮ Exercises session.

2

Recapitulation of the previous lecture

3

ALC in a nutshell

C ::= ⊤ | ⊥ | A | ¬C | C ⊓ C | C ⊔ C | ∃r.C | ∀r.C

◮ Interpretation I = (∆I, ·I). ◮ TBox T = {C ⊑ D, . . .}. ◮ ABox A = {a : C, (b, b′) : r, . . .}. ◮ Knowledge base K = (T , A).

(a.k.a. ontology)

◮ Decision problems include concept satisfiability, knowledge

base consistency, and other problems for classification.

4

⊤I

def

= ∆I ⊥I

def

= ∅ (¬C)I

def

= ∆I \ CI (C1 ⊔ C2)I

def

= CI

1 ∪ CI 1

(C1 ⊓ C2)I

def

= CI

1 ∩ CI 1

(∃r.C)I

def

= {a ∈ ∆I | r I(a) ∩ CI = ∅} (∀r.C)I

def

= {a ∈ ∆I | r I(a) ⊆ CI}

5

A few properties about ALC

◮ Concept satisfiability problem is PSPACE-complete. ◮ Knowledge base consistency problem is

EXPTIME-complete.

◮ ALC has many well-known fragments and extensions,

some of them to deal with

◮ inverse roles, ◮ number restrictions, ◮ properties on the role interpretations, ◮ inclusions between the composition of roles, ◮ etc.. 6

DLs and ontologies

◮ W3C’s OWL 2 is based on the queen description logic

SROIQ.

◮ OWL reasoners: implement decision procedures for

consistency and ontology classification.

◮ Open-source ontology editor Prot´

eg´ e.

◮ Interaction with DL reasoners (FaCT++, Pellet, Racer) via

the OWL API.

◮ Show results about ontology classification. 7

Tree interpretations

◮ I is a tree interpretation for C with respect to T iff the

conditions below hold:

◮ TI = (∆I,

r r I) is a tree,

◮ the root of TI belongs to CI, ◮ I |

= T .

◮ ALC has the tree interpretation property and the finite

interpretation property.

◮ Path in I: finite sequence (a1, . . . , an) ∈ (∆I)+ such that

for all i ∈ [1, n − 1], we have (ai, ai+1) ∈

r r I. ◮ a-path: path such that a1 = a.

8

Unravelling an interpretation with a single role

◮ Unravelling of I at a ∈ ∆I: U = (∆U, ·U) with

◮ ∆U is the set of a-paths in I. ◮ For all A, AU def

= {(a1, . . . , an) ∈ ∆U | an ∈ AI},

◮ For all role names r, we have

r U def = {((a1, . . . , an), (a1, . . . , an, an+1)) | (an, an+1) ∈ r I}

01 011 0111 . . . . . . 0112 . . . 012 0121 . . . . . . 02 021 . . . . . . 1 2

◮ C is satisfiable with respect to a TBox T implies C has a

tree interpretation with respect to T .

9

Small interpretation property

10

Subconcepts

◮ Set of subconcepts sub(C) and size size(C):

Concept C sub(C) size(C) A {A} 1 ⊤/⊥ {⊤}/{⊥} 1 ¬C1 sub(C1) ∪ {¬C1} 1 + size(C1) C1 ⊓ C2 sub(C1) ∪ sub(C2) ∪ {C1 ⊓ C2} 1 + size(C1) + size(C2) C1 ⊔ C2 sub(C1) ∪ sub(C2) ∪ {C1 ⊔ C2} 1 + size(C1) + size(C2) ∃r.C1 sub(C1) ∪ {∃r.C1} 1 + size(C1) ∀r.C1 sub(C1) ∪ {∀r.C1} 1 + size(C1) sub(T ) =

• C⊑D∈T

sub(C)∪sub(D) sub(A) =

• a:C∈A

sub(C)

◮ size(T ), size(A): sum of the sizes of its elements. ◮ card(sub(T ) ∪ sub(A)) ≤ size(T ) + size(A).

11

Type

◮ A set of concepts X is closed under subconcepts iff

sub(X) = X.

◮ sub(C), sub(T ), sub(A) are closed under subconcepts. ◮ X-type of a in I:

typeX(a)

def

= {C ∈ X | a ∈ CI}

◮ If X is finite, then

card({typeX(a) | a ∈ ∆I}) ≤ 2card(X)

◮ Small interpretation property is established by showing that

no need to keep too many individuals with the same X-type with X = sub(C) ∪ sub(T ) ∪ sub(A).

12

Filtration

◮ Set X closed under subconcepts, interpretation I. ◮ a ≈X b

def

⇔ typeX(a) = typeX(b), and equivalence class [a]X.

◮ X-filtration J = (∆J , ·J ):

◮ ∆J def

= {[a]X | a ∈ ∆I}.

◮ AJ def

= {[a]X | there is a′ ∈ [a]X such that a′ ∈ AI}. (A ∈ X)

◮ r J def

= {([a]X, [b]X) | there is a′ ∈ [a]X, b′ ∈ [b]X such that (a′, b′) ∈ r I}.

◮ card(∆J ) ≤ 2card(X). ◮ For all C ∈ X and a ∈ ∆I, we have a ∈ CI iff [a]X ∈ CJ .

13

Induction step for ∃r.C ∈ X

◮ First, suppose that a ∈ ∃r.CI.

◮ By definition of ·I, there is b such that (a, b) ∈ r I and

b ∈ CI.

◮ By definition of J , ([a]X, [b]X) ∈ r J . ◮ By X closed under subconcepts and (IH), [b]X ∈ CJ . ◮ By definition of ·J , [a]X ∈ (∃r.C)J .

◮ Suppose that [a]X ∈ (∃r.C)J .

◮ By definition of ·J , there is [b]X such that ([a]X, [b]X) ∈ r J

and [b]X ∈ CJ .

◮ By definition of r J , there is (a′, b′) ∈ r I such that a′ ∈ [a]X

and b′ ∈ [b]X.

◮ By X closed under subconcepts and (IH), b ∈ CI and as

b′ ∈ [b]X, b′ ∈ CI too.

◮ By definition of ·I, a′ ∈ (∃r.C)I. ◮ As a′ ∈ [a]X, a ∈ (∃r.C)I. 14

Small interpretation property

◮ C, K = (T , A), X closed under subconcepts with

(sub(C) ∪ sub(T ) ∪ sub(A)) ⊆ X.

◮ If C is satisfiable w.r.t. K, then there is an interpretation J

such that J | = K, CJ = ∅ and card(∆J ) ≤ 2card(X).

◮ J is an X-filtration based on some interpretation I such

that for all a : C ∈ A, we have aJ

def

= [aI]X.

◮ The equivalence “a ∈ CI iff [a]X ∈ CJ ” leads to the

satisfaction of K in J .

15

Generalities about decision procedures

◮ A decision problem P is a subset of Σ∗.

(Σ is a finite alphabet)

◮ Alternatively, given w ∈ Σ∗, is w in P? ◮ An algorithm for P is sound if whenever it answers

“w ∈ P”, then w ∈ P.

◮ An algorithm for P is complete if whenever w ∈ P, it

answers “w ∈ P”.

◮ An algorithm for P is terminating if it stops after finitely

many steps for all w ∈ Σ∗.

◮ Decision procedure: sound, complete and terminating.

16

A brute force decision procedure

◮ Input: C, K = (T , A). ◮ Guess an interpretation I such that card(∆I) ≤ 2card(X)

with X = sub(C) ∪ sub(T ) ∪ sub(A).

◮ Compute the set CI using a labelling algorithm based on

the definition of ·I for complex concepts.

◮ Check the satisfaction of I |

= K using again a labelling algorithm.

◮ Checking CI = ∅ and I |

= K can be done in NEXPTIME.

17

A PSPACE algorithm for ALC satisfiability

18

Closure

◮ ∼ C

def

= D if C = ¬D, otherwise ∼ C

def

= ¬C.

◮ Closure cl(C) of a concept C: least set closed under

subconcepts containing sub(C) and closed under ∼.

◮ card(cl(C)) ≤ 2 × card(sub(C)). ◮ A set X is closed

def

⇔ X =

C∈X cl(C). ◮ A set Y is patently inconsistent

def

⇔ Y contains ⊥, ¬⊤, or a pair of concepts of the form either C and ¬C or C and ∼ C.

19

Maximally consistent sets

◮ Given a closed set of concepts X, the set Y ⊆ X is

maximally consistent

def

⇔

◮ Y is not patently inconsistent. ◮ For all C ∈ X, either C ∈ Y or ∼ C ∈ Y. ◮ For all ¬¬C ∈ X, C ∈ Y iff ¬¬C ∈ Y. ◮ For all C1 ⊓ C2 ∈ X, C1 ⊓ C2 ∈ Y iff {C1, C2} ⊆ Y. ◮ For all C1 ⊔ C2 ∈ X, C1 ⊔ C2 ∈ Y iff {C1, C2} ∩ Y = ∅.

◮ Consequently, for all ¬(C1 ⊓ C2) ∈ Y, we have ∼ C1 ∈ Y or

∼ C2 ∈ Y.

◮ {C | a ∈ CI, C ∈ X} is maximally consistent.

20

Extended closure

◮ ecl(n, C): subconcepts occurring in ∃/∀-depth at least n. ◮ The ecl(n, C)’s are the least sets of concepts satisfying the

conditions below:

◮ ecl(n, C) is closed. ◮ ecl(0, C) def

= cl(C).

◮ If ∃r.D or ∀r.D occurs in some concept of ecl(n, C), then

D ∈ ecl(n + 1, C).

◮ Y ⊆ cl(C) is n-maximally consistent

def

⇔ Y is maximally consistent w.r.t. ecl(n, C).

◮ ecl(size(C), C) = ∅.

21

Example

◮ C = ∃r.⊤ ⊔ (∀r.∃s.A).

ecl(0, C) = {C, ¬C, ∃r.⊤, ¬∃r.⊤, ∀r.∃s.A, ¬∀r.∃s.A, ⊤, ¬⊤, ∃s.A, ¬∃s.A, A, ¬A}

◮ ecl(1, C) = {⊤, ¬⊤, ∃s.A, ¬∃s.A, A, ¬A}. ◮ ecl(2, C) = {A, ¬A}. ◮ ecl(3, C) = ∅.

22

Nondeterministic algorithm for concept satisfiability

1: procedure SATALC(Y, d) 2:

if Y is not d-maximally consistent then abort

3:

end if

4:

if Y contains only ∃/∀-free concepts then return true

5:

end if

6:

for ∃r.D ∈ Y do

7:

Guess Z ⊆ ecl(d + 1, C) such that D ∈ Z and {∼ D′ : ¬∃r.D′ ∈ Y} ∪ {D′ : ∀r.D′ ∈ Y} ⊆ Z

8:

if not SATALC(Z, d + 1) then abort

9:

end if

10:

end for

11:

for ¬∀r.D ∈ Y do

12:

Guess Z ⊆ ecl(d + 1, C) such that ∼ D ∈ Z and {∼ D′ : ¬∃r.D′ ∈ Y} ∪ {D′ : ∀r.D′ ∈ Y} ⊆ Z

13:

if not SATALC(Z, d + 1) then abort

14:

end if

15:

end for

16: end procedure

23

Computational properties and correctness

◮ As ecl(size(C), C) = ∅, the recursive depth of SATALC is

bounded by size(C).

◮ Each call requires linear space in size(C). ◮ If Y ⊆ cl(C), then SATALC(Y, d) runs in nondeterministic

polynomial space. (PSPACE = NPSPACE)

◮ C is satisfiable iff there is a 0-maximally consistent set Y

such that C ∈ Y and SATALC(Y, 0) has an accepting computation.

◮ Concept satisfiability problem for ALC is in PSPACE.

24

Miscellaneous remarks

◮ The correctness proof establishes a tree interpretation

property as an accepting computation leads to a tree interpretation.

◮ The algorithm can be easily extended to admit GCIs but

the recursive depth becomes exponential.

◮ The algorithm does not assume any proof system but it

has brutal nondeterministic steps.

◮ The forthcoming tableaux-style proof systems are able to

better control the nondeterministic steps and admit strategies leading to optimal complexity upper bounds.

25

Tableaux for ALC

26

Automated reasoning for non-classical logics

◮ Direct methods:

◮ Analytical calculi: tableaux, sequents, hypersequents, etc. ◮ Resolution. ◮ Automata-based decision procedures.

◮ Translation into

◮ other modal logics (PDL, modal µ-calculus, . . . ) ◮ decidable fragments of first-order logic (FO2, GF, . . . ) ◮ second-order monadic logics (S2S,. . . ) 27

Tableaux-based proof systems

◮ Analytical method: the rules of the calculi perform a

syntactic decomposition of the concepts (formulae, etc.).

◮ Method developped initially by R. Smullyan for first-order

logic (circa 1968).

◮ Close relationships with sequent-style proof systems. ◮ Modular approach as new conditions or new ingredients

may correspond to the addition of new rules.

◮ Labels are sometimes used in such proof systems.

◮ Labels are interpreted as entities of the domain under

construction.

◮ Expressions external to the original logical language. ◮ Labels can be also used as control data structures. 28

Methodology

◮ To design a sound and complete proof system for

knowledge base consistency, we start by designing a calculus when the TBox T is empty.

◮ The extension with a TBox is treated in a second part. ◮ This will lead to a decision procedure (terminating) for

knowledge base consistency in which a few optimisations leads to EXPTIME (not presented today).

◮ The proof system works with ABoxes and rewrite it with the

intention to build an interpretation from the ABoxes.

◮ The presentation follows the usual way to present

tableaux-style proof systems for description logics but

• ther presentations exist for modal and temporal logics.

29

Negation normal form

◮ Having ∼ in the algorithm for concept satisfiability was fine

but an alternative way to proceed it to use concepts in negation normal form.

◮ C is in negation normal form (NNF)

def

⇔ the negation ¬

• ccurs only in front of concept names.

◮ Every concept has an equivalent concept in NNF:

¬(C ⊔ D) ≡ ¬C ⊓ ¬D ¬(C ⊓ D) ≡ ¬C ⊔ ¬D ¬∃r.C ≡ ∀r.¬C ¬∀r.C ≡ ∃r.¬C ¬¬C ≡ C ¬⊤ ≡⊥ ¬ ⊥≡ ⊤

◮ Transforming a concept into an equivalent concept in NNF

takes polynomial time (only).

◮ NNFs are not a must but this simplifies forthcoming

developments.

30

Principles of the tableaux-style proof systems

◮ The calculus is made of rewriting rules that transform an

ABox A into another ABox A′ nondeterministically. (ex: ⊓-rule)

◮ The order of rule applications is irrelevant, except when

• ptimal strategies are designed.

◮ To guarantee termination, provisos are added to the

application of the rewriting rules. (ex. blocking technique)

◮ Modular approach as new ingredients in the logic leads to

new rules, and the provisos are refined (when possible).

◮ ABoxes with no contradiction and for which no rule

application adds value correspond to interpretations.

31

Example: the ⊓-rule

⊓-rule: If a : C ⊓ D ∈ A and {a : C, a : D} ⊆ A then A − → A ∪ {a : C, a : D}

◮ Applying the rule can be viewed as repairing locally the

non maximal consistency.

◮ Satisfaction of {a : C, a : D} ⊆ A avoids void rule

applications.

◮ The other rules are designed on the same pattern.

32

Expansion rules for ALC ABox consistency

⊓-rule: If a : C ⊓ D ∈ A and {a : C, a : D} ⊆ A then A − → A ∪ {a : C, a : D} ⊔-rule: If a : C ⊔ D ∈ A and {a : C, a : D} ∩ A = ∅ then A − → A ∪ {a : E} for some E ∈ {C, D} ∃-rule: If a : ∃r.C ∈ A and there is no b such that {(a, b) : r, b : C} ⊆ A then A − → A ∪ {(a, c) : r, c : C} where c is fresh ∀-rule: If {(a, b) : r, a : ∀r.C} ⊆ A and b : C ∈ A, then A − → A ∪ {b : C}

33

Complete and clash-free ABox

◮ An ABox A contains a clash if {a : A, a : ¬A} ⊆ A or

a :⊥∈ A.

Notion of clash to be extended if the concepts are not in

NNF .

◮ An ABox A is clash-free if it does not contain a clash. ◮ An ABox A is complete if it contains a clash or if no rule is

applicable. Objective: to show that A is consistent iff A ∗ − → A′ for some complete and clash-free ABox A′.

◮ The only nondeterministic rule is the ⊔-rule.

34

Example

A = {(a, b) : s, (a, c) : r}∪ {a : A1 ⊓ ∃s.A5, a : ∀s.¬A5 ⊔ ¬A2, b : A2, c : A3 ⊓ ∃s.A4} A ∗ − → A ∪ {a : A1, a : ∃s.A5, anew : A5, (a, anew) : s b : ¬A5 ⊔ ¬A2, anew : ¬A5 ⊔ ¬A2, b : ¬A5, anew : ¬A2, c : A3, c : ∃s.A4, cnew : A4, (c, cnew) : s}

a anew c b cnew s s r s

35

Termination

◮ The ∃-weight of C is the number of its subconcepts of the

form ∃r.D. w∃(C)

def

= card({∃r.D | ∃r.D ∈ sub(C)})

The definition assumes that C is in NNF

.

◮ w∃(A)

def

=

a:C∈A w∃(C). ◮ The ∀∃-depth of C, written d∀∃(C), is the maximal number

• f imbrications of ∃r. and ∀s. in C.

◮ d∀∃(∃r.⊤ ⊔ ∀r.∃s.A) = 2 ◮ d∀∃(A) = max{d∀∃(C) | a : C ∈ A}.

36

Labelling the individual names

◮ Let A be an ABox with W = w∃(A), D = d∀∃(A) and N is

the number of distinct individual names in A.

◮ Let A0 be the variant of A where a : C is replaced by

a0 : C.

⊓-rule: If ai : C ⊓ D ∈ A and {ai : C, ai : D} ⊆ A then A − → A ∪ {ai : C, ai : D} ⊔-rule: If ai : C ⊔ D ∈ A and {ai : C, ai : D} ∩ A = ∅ then A − → A ∪ {ai : E} for some E ∈ {C, D} ∃-rule: If ai : ∃r.C ∈ A and there is no bj such that {(ai, bj) : r, bj : C} ⊆ A then A − → A ∪ {(ai, ci+1) : r, ci+1 : C} where c is fresh ∀-rule: If {(ai, bj) : r, ai : ∀r.C} ⊆ A and bj : C ∈ A, then A − → A ∪ {bj : C}

37

Quantities about A0

∗

− → A′

◮ If ai : C ∈ A′, then i + d∀∃(C) ≤ D.

Trees from individual names labelled by zero have depth at most D.

◮ ai : C ∈ A′ implies

card({(ai, bj) | (ai, bj) : r ∈ A′}) ≤ N + W. The maximum branching degree of nodes in the trees is at most N + W.

◮ ai : C ∈ A′ implies C ∈ sub(A). ◮ The length of the derivation A0 ∗

− → A′ is at most N × (D + 1) × (N + W)D × card(sub(A)) (why?)

38

The auxiliary function exp

◮ Expansion function exp(A, R, X) taking as arguments

◮ an ABox A, ◮ an expansion rule R, ◮ a subset X of A (with one or two elements) allowing the

application of R

◮ . . . and returning the set of ABoxes obtained from A by

applying the rule R with main assertions in X.

◮ exp({a : E, a : C ⊔ D}, ⊔-rule, a : C ⊔ D) is equal to

{{a : E, a : C ⊔ D, a : C}, {a : E, a : C ⊔ D, a : D}}

39

Main algorithm

◮ We shall show that A is consistent iff A ∗

− → A′ for some complete and clash-free ABox A′.

◮ Existence of A′ amounts to explore a finite tree of bounded

depth and bounded degree.

1: procedure EXPAND(A) 2:

if A has a clash then return ∅

3:

end if

4:

if A is clash-free and complete then return A

5:

end if

6:

for applicable R, X on A and A′ ∈ exp(A, R, X) do

7:

if expand(A′) = ∅ then return expand(A′)

8:

end if

9:

end for

10:

return ∅

11: end procedure

40

1: procedure EXPAND(A) 2:

if A has a clash then return ∅

3:

end if

4:

if A is clash-free and complete then return A

5:

end if

6:

for applicable R, X on A and A′ ∈ exp(A, R, X) do

7:

if expand(A′) = ∅ then return expand(A′)

8:

end if

9:

end for

10:

return ∅

11: end procedure

A1 A2 A3 A4 A5 A6 A7

41

Root individuals and tree individuals

◮ Tree individuals are generated by application of the

∃-rule.

◮ If (a, b) : r is added by application of the ∃-rule, b is an

r-successor of a.

◮ Root individuals have no predecessors or ancestors.

42

Soundness

◮ Let A be a finite ABox with at least one concept assertion,

complete, clash-free and all the concepts in NNF . Then, A is consistent.

◮ For each individual name a occurring in A, we write

conA(a) to denote the set {C | a : C ∈ A}.

◮ Let us define I

def

= (∆I, ·I) as follows.

◮ ∆I def

= {a | a : C ∈ A}.

◮ aI def

= a for all individual names a in A.

◮ AI def

= {a | A ∈ conA(a)} for all concept names A ∈ sub(A).

◮ r I def

= {(a, b) | (a, b) : r ∈ A}.

◮ Let us show that for all a : C ∈ A, we have aI ∈ CI.

43

Proof by structural induction

◮ The base case with concept assertions a : A is immediate

by definition of AI.

◮ The base case with concept assertions a : ¬A is

immediate by definition of AI as A is clash-free.

◮ Case a : C ⊔ D in the induction step.

◮ As A is complete, a : C ∈ A or a : D ∈ A. ◮ W.l.o.g., suppose a : C ∈ A. By (IH), aI ∈ CI. ◮ By definition of ·I, we conclude aI ∈ (C ⊔ D)I.

◮ Case a : ∃r.C in the induction step.

◮ As A is complete, {(a, b) : r, b : C} ⊆ A for some b. ◮ By definition of r I, (a, b) ∈ r I. ◮ By (IH), bI ∈ CI. ◮ By definition of ·I, we conclude aI ∈ (∃r.C)I. 44

Concluding the soundness

◮ The cases in the induction step for ⊓-concept assertions

and ∀-concept assertions are similar.

◮ If expand(A) = ∅, then A is consistent. ◮ Indeed, expand(A) = ∅ if there is some A′ with A ⊆ A′

such that A′ is complete and clash-free.

◮ Consistency of A′ leads to the consistency of A.

45

Completeness

◮ If A is consistent, then A ∗

− → A′ for some complete and clash-free ABox A′.

◮ Let I

def

= (∆I, ·I) be such that I | = A.

◮ If A is complete, we are done. Otherwise, at least one rule

is application to A preserving consistency.

◮ Otherwise, if A is not complete, we show that there is A′

such that A − → A′ and A′ is consistent.

◮ As the length of a derivation from A is bounded by an

exponential in the size of A, there is A′ such that A ∗ − → A′ and A′ is complete, clash-free (and consistent).

◮ It remains to prove that non-completeness implies the

existence of one expansion preserving consistency.

46

Single steps in the completeness proof

◮ If the ⊔-rule is applicable on a : C ⊔ D, then there is

E ∈ {C, D} such that I | = A ∪ {a : E}.

◮ A −

→ A ∪ {a : E} and I | = A ∪ {E}.

◮ If the ∃-rule is applicable on a : ∃r.C, then we use the fact

that aI ∈ (∃r.C)I.

◮ There is a ∈ ∆I such that a ∈ CI and (aI, a) ∈ r I. ◮ Let I′ be equal to I except that I′(c) = a for some fresh c. ◮ Then, A −

→ A ∪ {c : C, (a, c) : r} and I′ | = A ∪ {c : C, (a, c) : r}

47

Decision procedure of ABox consistency

◮ A is consistent iff A ∗

− → A′ for some complete and clash-free ABox A′.

◮ Derivations A ∗

− → A′ have length bounded by an exponential in size(A).

◮ Existence of A′ amounts to explore a tree of bounded

depth and bounded degree.

48

Adding a TBox – First properties

◮ I |

= C ⊑ D iff I | = ⊤ ⊑ ¬C ⊔ D.

◮ I |

= C ≡ D iff I | = ⊤ ⊑ (¬C ⊔ D) ⊓ (¬D ⊔ C).

◮ In the sequel, GCIs are of the form ⊤ ⊑ E with E in NNF

. ⊑-rule: If a : C ∈ A, ⊤ ⊑ D ∈ T and a : D ∈ A, then A − → A ∪ {a : D}

◮ The termination argument for ABox consistency does not

work anymore. (Why?)

49

Blocking

◮ Given A ∗

− → A′, a is an ancestor of b in A′ iff {(a1, a2) : r1, . . . , (ak, ak+1) : rk} ⊆ A′ with a1 = a, ak+1 = b and b is a tree individual.

The notion of ancestor assumes that one can distinguish

the root individuals (individual names from A) from the tree individuals (those introduced by applying the ∃-rule).

◮ Termination can be regained thanks to the blocking

technique.

◮ An individual name b in A′ is blocked by a if

◮ a is an ancestor of b, ◮ conA′(b) ⊆ conA′(a). 50

Expansion rules with blocking

⊓-rule: If a : C ⊓ D ∈ A, a is not blocked and {a : C, a : D} ⊆ A then A − → A ∪ {a : C, a : D}. ⊔-rule: If a : C ⊔ D ∈ A, a is not blocked and {a : C, a : D} ∩ A = ∅ then A − → A ∪ {a : E} for some E ∈ {C, D}. ∃-rule: If a : ∃r.C ∈ A, a is not blocked and there is no b such that {(a, b) : r, b : C} ⊆ A then A − → A ∪ {(a, c) : r, c : C} where c is fresh ∀-rule: If {(a, b) : r, a : ∀r.C} ⊆ A, a is not blocked and b : C ∈ A, then A − → A ∪ {b : C}. ⊑-rule: If a : C ∈ A, ⊤ ⊑ D ∈ T , a is not blocked and a : D ∈ A, then A − → A ∪ {a : D}.

51

Termination

◮ K = (T , A) with concepts in NNF

, a : C ∈ A, and CGIs of the form ⊤ ⊑ D.

◮ N: number of root individuals in A, M = card(sub(K)),

W = w∃(K).

◮ A ∗

− → A′ and a : C ∈ A′ imply card({(a, b) | (a, b) : r ∈ A′}) ≤ N + W.

◮ A ∗

− → A′ and a : C ∈ A′ imply C ∈ sub(K).

◮ {(a1, a2) : r1, . . . , (ak, ak+1) : rk} ⊆ A′ and a2 is a tree

individual imply k ≤ 2M.

◮ The length of the derivation A ∗

− → A′ is at most N × (2M + 1) × (N + W)2M × M

52

Soundness

◮ K = (T , A) with concepts in NNF

, a : C ∈ A, and CGIs of the form ⊤ ⊑ D.

◮ A ∗

− → A′ with A′ complete and clash-free.

◮ We construct A′′ as the ABox made of the following

assertions {a : C | a : C ∈ A′, a is not blocked} ∪ {(a, b) : r | (a, b) : r ∈ A′, b is not blocked} ∪ {(a, b′) : r | (a, b) : r ∈ A′, a is not blocked and b is blocked by b′}

53

Properties of A′′

◮ A ⊆ A′′ as root individual cannot be blocked and A ⊆ A′. ◮ None of the individual names occurring in A′′ is blocked. ◮ For all a in A′, we have conA′′(a) = conA′(a). ◮ A′′ is complete and clash-free.

54

More about the soundness proof

◮ A ∗

− → A′ with A′ complete and clash-free and A′′ computed as above.

◮ Let us define I

def

= (∆I, ·I) as follows.

◮ ∆I def

= {a | a : C ∈ A′′}.

◮ aI def

= a for all individual names a in A′′.

◮ AI def

= {a | A ∈ conA(a)} for all concept names A ∈ sub(A′′).

◮ r I def

= {(a, b) | (a, b) : r ∈ A′′}.

◮ One can show that for all a : C ∈ A′′, we have aI ∈ CI.

55

Completeness (bis)

◮ If K = (T , A) is consistent, then A ∗

− → A′ for some complete and clash-free ABox A′.

◮ Let I

def

= (∆I, ·I) be such that I | = A.

◮ If A is complete, we are done. Otherwise, at least one rule

is application to A preserving consistency.

◮ Otherwise (A is not complete), we show there is A′ such

that A − → A′ and A′ is consistent.

◮ As the length of a derivation from A is bounded by a

double-exponential in the size of A, there is A′ such that A ∗ − → A′ and A′ is complete, clash-free (and consistent).

◮ One can prove that non-completeness implies the

existence of one expansion preserving consistency.

56

Complexity issues

◮ ALC concept satisfiability in PSPACE, knowledge base

consistency in EXPTIME.

◮ The algorithm for ABox consistency runs in exponential

space:

◮ Because of the nondeterministic ⊔-rule, exponentially many

ABoxes may be generated.

◮ Complete ABoxes may be exponentially large.

◮ PSPACE bound for ABox consistency can be regained by

exploring the tree-like interpretations in a depth-first manner having only one path at a time.

57

Tableaux for ALCI (ALC + inverse)

{(a, b) : r, b : ∀r −.D, a : ∀r.C} | =ALCI {b : C, a : D}

◮ b is an r-neightbour of a if (a, b) : r or (b, a) : r −. ◮ b is an r −-neightbour of a if (a, b) : r − or (b, a) : r. ◮ Below, R is either some r or some r −.

∃-rule: If a : ∃R.C ∈ A and there is no b such that b : C ∈ A and b is an R-neightbour of a then A − → A ∪ {(a, c) : R, c : C} where c is fresh ∀-rule: If a : ∀R.C ∈ A and b is an R-neightbour of a, then A − → A ∪ {b : C}

58

Equality blocking

◮ We need to strenghten the blocking (inclusion of sets of

concepts is not anymore sufficient).

◮ An individual name b in A′ is blocked by a if

◮ a is an ancestor of b, ◮ conA′(b) = conA′(a) (equality blocking).

◮ Termination, soundness and completeness can be

established in a similar fashion, though adaptations are needed.

59

Recapitulation: Tableaux for ALC knowledge base consistency

◮ Tableaux-based algorithm to decide ALC knowledge base

consistency.

◮ All other standard decision problems can be handled too. ◮ Termination is guaranteed thanks to the blocking

technique.

◮ In the worst-case, exponential space is needed but

• ptimisations exist to meet the optimal complexity upper

bounds.

60

Complexity of problems for ALC and variants

61

Recapitulation of upper bounds

◮ The concept satisfiability problem for ALC is in PSPACE. ◮ The knowledge base consistency problem for ALC is in

EXPTIME.

◮ Same upper bounds for ALCI. ◮ In the sequel, we focus on complexity lower bounds.

62

Why hardness results using tiling problems?

◮ Ideally, master reductions from Turing machines. ◮ Tiling problems form a family of problems complete for

numerous complexity classes.

◮ Tiling an arena with tile types naturally corresponds to

computations in Turing machines.

◮ Tiling problems have little structure.

63

Tiling system

◮ Tiling system: (T, H, V, t0) where

◮ T is a finite set of tiles and t0 ∈ T, ◮ H, V ⊆ T × T are two relations referred to as the

horizontal, resp. vertical matching relation.

◮ A set of tiles

t1 =

2 2 1

t2 =

1 1 2 2

t3 =

2 1

t4 =

2 1 2 ◮ . . . with its matching relations

◮ H = {(t1, t3), (t1, t4), (t2, t1), (t3, t2), (t4, t1)}, ◮ V = {(t1, t2), (t1, t4), (t2, t3), (t4, t1), (t4, t2)}. 64

A tiling for the ([0, 3] × [0, 2])-arena

2 1 2 2 2 1 2 1 2 2 2 1 1 1 2 2 2 2 1 2 1 2 2 2 1 2 1 1 1 2 2 2 2 1 2 1 2

65

A PSPACE-complete tiling game problem

◮ Let (T, H, V, t0) be a tiling system. ◮ (n × n)-tiling game on (T, H, V, t0) is played on a finite

number of rounds between Player 1 and Player 2 to construct a tiling τ : [0, n − 1] × [0, n − 1] → T.

◮ At the jth, Player 1 chooses τ(0, j), and then Player 2

chooses τ(1, j), . . . , τ(n − 1, j).

◮ Player 1 loses immediately in

◮ round 0, if τ(0, 0) = t0; ◮ round j > 0, if

• τ(0, j − 1), τ(0, j)
• ∈ V.

◮ Player 2 loses immediately in

◮ round j ≥ 0, if there is an i < n − 1 such that

• τ(i, j), τ(i + 1, j)
• ∈ H;

◮ round j > 0, if there is an i with 0 < i < n and

• τ(i, j − 1), τ(i, j)
• ∈ V.

◮ A player wins if the opponent loses.

66

Complexity of the (n × n)-tiling game problem

◮ Tiling game problems are closely related to alternating

Turing machines where Player 1/ Player 2, corresponds to universal/existential states.

◮ Tiling problems are closely related to nondeterministic

Turing machines but of no help to characterise complexity classes with deterministic Turing machines.

◮ The (n × n)-tiling game problem is APTIME-complete. ◮ . . . and therefore the (n × n)-tiling game problem is

PSPACE-complete as PSPACE = APTIME.

67

An EXPTIME-complete tiling game problem

◮ (n × ∞)-tiling game on (T, H, V, t0) is played on an infinite

number of rounds between Player 1 and Player 2 to construct a tiling τ : [0, n − 1] × N → T.

◮ At the jth, Player 1 chooses τ(0, j), and then Player 2

chooses τ(1, j), . . . , τ(n − 1, j).

◮ Player 1 loses immediately in

◮ round 0, if τ(0, 0) = t0; ◮ round j > 0, if

• τ(0, j − 1), τ(0, j)
• ∈ V.

◮ Player 2 loses immediately in

◮ round j ≥ 0, if there is an i < n − 1 such that

• τ(i, j), τ(i + 1, j)
• ∈ H;

◮ round j > 0, if there is an i with 0 < i < n and

• τ(i, j − 1), τ(i, j)
• ∈ V.

◮ A player wins if the opponent loses.

68

Complexity of the (n × ∞)-tiling game problem

◮ Given (T, H, V, t0) and n ≥ 1 in unary, has Player 2 has a

winning strategy on the ([0, n − 1] × N)-arena ?

◮ Tiling game problems are closely related to alternating

Turing machines where Player 1/ Player 2, corresponds to universal/existential states.

◮ The (n × ∞)-tiling game problem is APSPACE-complete. ◮ . . . and therefore the (n × ∞)-tiling game problem is

EXPTIME-complete as EXPTIME = APSPACE.

69

An undecidable tiling problem

◮ The (∞ × ∞)-tiling problem.

Input: A tiling system (T, H, V, t0). Question: Is there a tiling τ : N × N → T such that for all i, j ∈ N,

(hori) if τ(i, j) = t and τ(i + 1, j) = t′, then (t, t′) ∈ H, (verti) if τ(i, j) = t and τ(i, j + 1) = t′, then (t, t′) ∈ V,

◮ The (∞ × ∞)-tiling problem is undecidable.

70

Concept satisfiability for ALC is PSPACE-hard

◮ Reduction from the (n × n)-tiling game problem. ◮ Tiling system T = (T, H, V, t0) and n ∈ N. ◮ We construct an ALC concept Cn T such that Cn T is

satisfiable iff Player 2 has a winning strategy for the tiling game on T on the ([0, n − 1]2)-arena.

71

Strategies are finite trees

t0 =

2 1 2

t1 =

1 1 2 2

t2 =

2 1

t3 =

2 2 1

Player 2 has a winning strategy with initial tile t0 on the ([0, 1] × [0, 2])-arena.

2 1 2 2 2 1 2 2 1 2 1 2 1 1 2 2 2 2 1 2 1 2 2 2 1 2 1 2 2 2 1 1 1 2 2 2 2 1 2 2 1 2 1 2 2 1 1 1 2 2 2 1 2 2 2 1 1 1 2 2 2 2 1 2 2 1 2 1 2 72

The reduction

◮ Every individual belongs to a unique tile type.

uni

def

=

• t∈T

(t⊓

• t′=t

¬t′) (tile types understood as concept names)

◮ Each individual “at distance” at most n2 has a unique tile

type (arbitrary role name r). Cn

uni

def

= t0 ⊓

• t=t0

¬t ⊓ ∀r.(uni ⊓ ∀r.(uni ⊓ . . . ∀r.(uni ⊓ ∀r.

• n2−1 occurrences of ∀r

uni) · · · ))

◮ Local horizontal/vertical matching relation:

hm

def

=

• (t,t′)∈H

¬t⊔∀r.¬t′ vm

def

=

• (t,t′)∈V

¬t⊔(∀r)n.¬t′ (∀r)i+1D

def

= (∀r)i.∀r.D

73

Satisfying the matching relations everywhere !

◮ The horizontal matching relation is respected everywhere:

hm⊓ ∀r(hm ⊓ . . . ∀r

• n−2 occurrences of ∀r

(hm ⊓ ∀r∀r(hm ⊓ ∀r(hm ⊓ . . . ∀r(hm ⊓ ∀r

• n2−1 occurrences of ∀r

hm) . . .))))

◮ The vertical matching relation is respected everywhere:

vm ⊓ ∀r · (vm ⊓ ∀r · (vm ⊓ . . . ∀r · (vm ⊓ ∀r

• n2−n−1 occurrences of ∀r

vm) . . .))

74

Avoiding a single individual interpretation

◮ Every first individual of a row has a chain of n − 1

r-successors and the last one has an r-successor for all possible matching choices of Player 1. chain

def

=

• t∈T

¬t ⊔ (∃r)n−1 · (

• (t,t′)∈V

∃r · t′) Cn

struct

def

= chain⊓(∀r)n(chain ⊓ . . . (∀r)n(chain ⊓ (∀r)n

• n−2 occurrences of(∀r)n

chain) . . .)

75

The properties

◮ Cn T defined as the conjunction of the above concepts is

constructed in logarithmic space in n and card(T).

◮ A winning strategy for Player 2 on the ([0, n − 1]2)-arena

induces an interpretation I such that (Cn

T)I is non-empty. ◮ Similarly, every interpretation I such that (Cn T)I is

non-empty, leads to a winning strategy for Player 2.

◮ Consequently, the concept satisfiability problem for ALC is

PSPACE-hard.

76

Conclusion

◮ Lecture 1 (last week): Introduction to description logics ◮ Lecture 2: Tableaux proof systems and complexity .

◮ Model-theoretical properties. ◮ Complete calculi for ALC and variants. ◮ Complexity results, a bit of undecidability.

◮ Lecture 3: Introduction to temporal logics for multi-agents

systems.

77