[PPT] - More on Computational Issues COMP34512 Rafael Gonalves Bijan PowerPoint Presentation

SLIDE 1

More on Computational Issues COMP34512

Rafael Gonçalves Bijan Parsia bparsia@cs.man.ac.uk

Tuesday, 11 March 14

SLIDE 2

Tuning our Logic

Both Prop & Pred Logic had issues

– Even for our simple use case!

We would like the formalism to be

– Representationally adequate

(Propositional fails)

– Computationally reasonable

(Both have issues)

– Usable (cognitively adequate)

Predicate logic has too much choice
Propositional logic has “wrong expressivity”

– So we are forced into hacky workarounds

Proposal: RDFS--

– Tuned for expressing terminologies

2

Expressivity (Representational Adequacy) Usability (Weak Cognitive Adequacy vs. Cognitive Complexity) Computability (vs. Computational and Implementational Complexity)

More Expressivity hurt Usability Less Expressivity hurt Usability (in different ways)

Tuesday, 11 March 14

SLIDE 3

RDFS--: Syntax

Super simple syntax!

– Tiny bit of propositional logic – Tiny bit of predicate logic

Variables & Quantifiers hidden! No pointless choice there!
We divide the theory into 2 parts:

– TBox: “Terminological box”

Universally quantified (bi)conditionals between monadic predicates

– ABox: “Assertional Box”

Ground monadic predicates

– TBox ≈ Schema; ABox ≈ Data

3

Individual (f) ::= [a-z]+ Class (A) ::= [A-Z][a-zA-Z]* (in camel case) TBoxAxiom (A1, A2) :: = A1 ⊑ A2 | A1 ≡ A2 ABoxAxiom (A1,f) :: = f:A1 | A1(f) A1 SubClassOf: A2 | A1 EquivalentTo: A2 | f Types: A1

Tuesday, 11 March 14

SLIDE 4

RDFS--: Semantics

Two possible approaches

– Direct Model Theory (an interpretation function) – By Translation (a translation function)

4

Syntax Translation to FOL Interpretation (assuming Δ) fluffy fluffy fluffyI ∈ Δ Cat Cat/1 i.e., Cat(x) CatI ⊆ Δ Cat(fluffy) fluffy:Cat Cat(fluffy) fluffyI ∈ CatI Cat ⊑ Mammal ∀x(Cat(x) → Mammal(x)) CatI ⊆ MammalI Animal ≡ Animalia ∀x(Animal(x) ↔ Animalia(x)) AnimalI = AnimaliaI

Tuesday, 11 March 14

SLIDE 5

RDFS--: Semantics

Model theory is simpler for the simpler language!

5

RFDS-- RFDS-- Interpretation FOL FOL Interpretation Cat(fluffy) fluffy:Cat True, since fluffyI ∈ CatI Cat(fluffy) True, since fluffyI ∈ CatI Cat ⊑ Mammal True, since CatI ⊆ MammalI ∀x(Cat(x) → Mammal(x)) True, since every possible σ is such that if σ(C, x) is True then σ(M, x) is also True I1: Δ = {1,2,3...} CatI = {1,4,5} MammalI = {1,2,3,4,5} fluffyI = 5

Exactly the same Equivalent

Tuesday, 11 March 14

SLIDE 6

RDFS--: Expressivity

Can captures some stuff better than propositional!

6

English Prop Logic Predicate Logic A Term Logic (RDFS--) Mammals are Animals Mf → Af Ms → As ∀x(M(x) → A(x)) M ⊑ A Cats are Mammals Cf → Mf Cs → Ms ∀y(C(y) → M(y)) C ⊑ M Dogs are Mammals D → M ∀y∀x(D(x) → M(x)) D ⊑ M Fluffy is a Cat Cf C(f) C(f) Sparkles is a Cat Cs C(s) C(s)

No need to hack names and multiply axioms

Tuesday, 11 March 14

SLIDE 7

English Prop Logic Predicate Logic RDFS-- Mammals are Animals M → A ∀x(M(x) → A(x)) M ⊑ A Cats are Mammals C → M ∀y(C(y) → M(y)) C ⊑ M Dogs are Mammals D → M ∀y∀x(D(x) → M(x)) D ⊑ M Cats are not Dogs C → ¬D ∀y(C(y) → ¬D(y)) C ⊑ NotD Cats eat some dogs C → ED ∀y(C(y) → ∃x(eats(y,x) ∧ D(x)) C ⊑ ED English Prop Logic Predicate Logic RDFS-- Mammals are Animals M → A ∀x(M(x) → A(x)) M ⊑ A Cats are Mammals C → M ∀y(C(y) → M(y)) C ⊑ M Dogs are Mammals D → M ∀y∀x(D(x) → M(x)) D ⊑ M Cats are not Dogs C → ¬D ∀y(C(y) → ¬D(y)) C ⊑ NotD Cats eat some dogs C → ED ∀y(C(y) → ∃x(eats(y,x) ∧ D(x)) C ⊑ ED

RDFS--: Expressivity

Does as bad or worse in other cases

7

Tuesday, 11 March 14

SLIDE 8

RDFS--: Computational Complexity

Consistency?

– WCC: 0

They are all always consistent
How can we know this?
Entailment?

– There are only three sorts of possible entailment – Subsumption (C SubClassOf: D)

Find a (directed) path between C and D
Only tricky bit is cycle detection

– Not that tricky!

Polynomial (what’s a tighter bound?)

– Instantiation (f Types: C)

Same! Just hang f off its asserted type node

– Equivalence (C EquivalentTo: D)

Second kind of edge and/or find a cycle

8

C D C1 C2 C3 C4

Tuesday, 11 March 14

SLIDE 9

RDFS--: Cognitively?

Easy to learn and use and analyse
Severe expressivity limitations a PITA

9

English Prop Logic Predicate Logic RDFS-- Mammals are Animals M → A ∀x(M(x) → A(x)) M ⊑ A Cats are Mammals C → M ∀y(C(y) → M(y)) C ⊑ M Dogs are Mammals D → M ∀y∀x(D(x) → M(x)) D ⊑ M Cats are not Dogs C → ¬D ∀y(C(y) → ¬D(y)) C ⊑ NotD Cats eat some dogs C → ED ∀y(C(y) → ∃x(eats(y,x) → D(x)) C ⊑ ED

Tuesday, 11 March 14

SLIDE 10

ALC: Syntax

Somewhat simple syntax!

– All of propositional logic – Tiny bit of predicate logic – Same basic structure as RDFS--

Similar semantics, by translation or direct model theory
Still decidable

– But, depending on the exact flavor and problem v. HARD

PSPACE to EXPTIME

10

Individual (f) ::= [a-z]+ Class (A) ::= [A-Z][a-zA-Z]* (in camel case) Property (P) ::= [a-z][a-zA-Z]* (in camel case) ClassExpression (C, D) :: = A | ¬C | C ⊔ D | C ⊓ D | ∃P.C | ∀P.C TBoxAxiom (C, D) :: = A1 ⊑ A2 | A1 ≡ A2 ABoxAxiom (C,f,g) :: = f:C | C(f) | <f,g>:P | P(f,g)

Tuesday, 11 March 14

SLIDE 11

ALC: Expressivity

Does great! (on our example)

11

English Prop Logic Predicate Logic ALC Mammals are Animals M → A ∀x(M(x) → A(x)) M ⊑ A Cats are Mammals C → M ∀y(C(y) → M(y)) C ⊑ M Dogs are Mammals D → M ∀y∀x(D(x) → M(x)) D ⊑ M Cats are not Dogs C → ¬D ∀y(C(y) → ¬D(y)) C ⊑ ¬D Cats eat some dogs C → ED ∀y(C(y) → ∃x(eats(y,x) ∧ D(x)) C ⊑ ∃eats.D Fluffy is a Cat. F → C C(f) C(f)

Tuesday, 11 March 14

SLIDE 12

Computational Interlude

Do we have a hard constraint?

– Polynominal? I.e., “tractable”? – Does this rule out ALC immediately?

Not necessarily!

– Constants can matter – Worst case != Every case

Or even my case!
Key point

– For a given input (or set of inputs)

we must terminate in reasonable time

– for our context’s value of “reasonable”

– Will this happen with ALC?

12

Tuesday, 11 March 14

SLIDE 13

Exercise

On the linear y-axis chart:

– Draw a line representing

A linear function (L)
A quadratic function (Q)
An exponential function (E)
On the log y-axis chart

– Do the same!

Each of these represents the WWC of an algo

– for a single problem! (they are named, L, Q, and E :)) – what do we know about the WWC of this problem?

Using different dots (xs vs. os vs +s, or different colors)

– plot a set of inputs for each algo s.t. – we would prefer using E to Q and Q to L

13

Tuesday, 11 March 14

SLIDE 14

Linear Y-Axis

14

2.7x 0.8x2+20 1.3x+50 exponential (E) quadratic (Q) linear (L)

Each is preferable w.r.t. a certain input size range

Tuesday, 11 March 14

SLIDE 15

Logarithmic Y-Axis

15

2.7x 0.8x2+20 1.3x+50 exponential (E) now linear quadratic (Q) now looks almost linear linear (L) still linear

Tuesday, 11 March 14

SLIDE 16

Robustness

Usability obviously is affected by computation time

– Roughly, the more time to compute, the less usable

But often predictability is more important

– Even with “hard” problems,

if the hardness is predictable, the user might be ok
predictability gives some user control
We say that the problem is predictable if

– the time to solve the problem is “proportionate” to the size – very dependent on user knowledge!

16

Tuesday, 11 March 14

SLIDE 17

Made Up Illustration

17

0" 50000" 100000" 150000" 200000" 250000" 300000" 0" 100" 200" 300" 400" 500" 600" WCC" Actual"Case"

Tuesday, 11 March 14