Sebastian Bitzer ([email protected])[email protected] Seminar Knowledge Representation University of Osnabrueck 10.07.2003 Applications of the Channel Theory.

Sebastian Bitzer ([email protected])Seminar Knowledge RepresentationUniversity of Osnabrueck10.07.2003

Applications of the Channel Theory

Commonsense Reasoning

Representations

mailto:[email protected]

10.07.2003 Applications of Channel Theory 2

Overview

• Repetition

• Commonsense Reasoning / Nonmonotonicity

• (Imperfect) Representations

Repetition

10.07.2003 Repetition 4

Basics

• tokens (particulars, instances): things in the world (in time) - a, b, c

• types (in state spaces: states): α, β, γ, σ

• classification: A, set of tokens (A) is classified in set of types

• if a is of type α we write: a╞A α (with respect to A)


Constraints and Infomorphisms

• Γ, Δ are sets of types, e.g. Γ={α, β}, Δ = {γ}Γ, Δ is a constraint, e.g. if a is of types α, β

then a has to be of type γ

• AC is an infomorphism between classifications

A C

ΣA ΣC

╞A ╞C

fˆ

fˇ

fˇ(c) ╞A α

c ╞C fˆ(α)


Information Channels and Local Logics

• C is an information channel, consists of core C and infomorphisms from parts to C:

• L is a local logic, consists of A, a set of constraints of A and a set of normal tokens

• normal tokens: satisfy all constraints in L

C

A B

f g f: ACg: BC


State Spaces

• S is a state space, consists of a set of tokens (S), a set of states (ΩS) and a function mapping between them: S = S, ΩS, state

• Evt(S) is the according classification

• Log(S) is the local logic on Evt(S)

Commonsense Reasoning

10.07.2003 Commonsense Reasoning 9

Overview

• Problem of Nonmonotonicity

• State Spaces, enhanced

• Background Conditions

• Relativising to a Background Condition


The Problem of Nonmonotonicity

Baseball:

(α1) pitcher throws ball to batter

(β) ball will arrive at batter

α1 ├ β

monotonicity:

α1, α2 ├ β

but:

(α2) ball hits bird

α1, α2 ├ ¬β


Real valued State Spaces

• S = S, Ω Rn, state• each state is a vector σ with dimension n

• σ has input and output coordinates (σ = σi×σo) = observables

• outputs can be computed from inputs


Judith’s heating system

• a state σ = (σ1, σ2, σ3, σ4, σ5, σ6, σ7)

σ1: thermostat setting 55 ≤ σ1 ≤ 80

σ2: room temperature 20 ≤ σ2 ≤ 110 σ3: power on (σ3 = 1) or off (σ3 = 0)

σ4: exhaust vents blocked (=0) or clear (=1)

σ5: op. conditions cooling (=-1), off (=0) or heating (=1)

σ6: running yes (=1) or no (=0)

σ7: output air temperature 20 ≤ σ7 ≤ 110

σi

σo


Background Conditions

• B is a function from domain P to real numbers

• P in inputs of S– is called set of parameters of B

• a state σ satisfies B if the corresponding inputs of σ have same value as given by B (σi=B(i) iP)

• B1 ≤ B2 P1 P2


Silence

• α is silent on B, if it does not tell anything about the parameters of B:– σ =B σ‘ if σt = σt‘ tB

σ,σ‘: if σ =B σ‘ and σα then σ‘α

if we are reasoning about an observable t, then t must be either an explicit input or output of the system (and not a parameter)



(α1) thermostat: 65 ≤ σ1 ≤ 70

(α2) room temperature: σ2 = 58

(α3) power: σ3 = 0

(β) hot air is coming out of the vents

α1, α2, β are silent about σ3, σ4, σ5 (which are supposed to be the parameters)

but: α3 is not silent about σ3

α1, α2 ├ β

α1, α2 , α3 ├ ¬β


Weakening

• Γ is a set of types

• the weakening of B by Γ (B↑Γ) is the greatest lower bound of all B0 ≤ B such that every type αΓ is silent on B0

B↑α is restriction of B to the set of inputs iP such that α is silent on i





(α3) power: σ3 = 0

(β) hot air is coming out of the vents

B = {σ3= σ4= σ5 = 1}

B↑α3 = {σ4= σ5 = 1}


Relativising to a Background Condition

• SB is relativisation of S to B– subspace of S

– only states that satisfy B

• Log(SB) is the local logic on Evt(S) supported by B– consistent states are those satisfying B

– entailment only over states satisfying B• Γ├B Δ σ sat. B (if σ p p Γ then σ q for some qΔ)

– normal tokens are those satisfying B





(α3) power: σ3 = 0(β) hot air is coming out of the vents

• α1, α2 ├ β holds in Log(SB)• because α3 not silent about σ3: switch to B↑α3

α1, α2 ├ β does not hold in Log(SB↑α3) (is no constraint there)

α1, α2 , α3 ├ ¬β is constraint in Log(SB↑α3)


Strict Entailment

• Γ B Δ: Γ strictly entails Δ relative to B– Γ├Log(SB) Δ

– all types in Γ Δ are silent on B

• then (conclusions):– Γ B Δ is a better model of human reasoning than Γ├B Δ

– Γ B Δ is monotonic in Γ and Δ (only with weakening)

– if you have a type α not silent on B it is natural to weaken B: B↑α

– Γ B Δ does not entail: Γ, α B↑α Δ or Γ B↑α Δ, α

Representations

10.07.2003 Representations 22

Overview

• The Problem of Imperfect Representations

• Representation Systems

• Explaining Imperfect Representations


The bridge


The Map


The bridge


The bridge


The Map

correct?


Representation Systems

• R = C,L is a representation system– C = {f: AC, g:

BC} is a binary channel

– L is the local logic on the core C

L

A B

f g

Representations Targets


Representations• a is a representation of b,

if a,b are connected by some cC– a is accurate representation

of b, if c is normal token (cNL)

• content of a: a╞A Γ f[Γ]├L g(Δ)

• a represents b as being of type β, if β in content of a

L

A B

f g

Representations Targetsa╞A α b╞B β

c

if a is accurate representation of b and a represents b as being of type β, then b╞B β


Explaining Imperfect Representations

• tokens in target classification (B) are really regions at times

• if b0 changes to b1 this gives rise to a new connection c1 between a and b1

a represents both: b0 and b1

but c1 supports not all of the constraints of R

b0

b1

10.07.2003 Applications of Channel Theory 31

References

• Jon Barwise and Jerry Seligman, Information Flow, The Logic of Distributed Systems, Cambridge University Press, 1997

Sebastian Bitzer ([email protected])[email protected] Seminar Knowledge Representation University of Osnabrueck 10.07.2003 Applications of the Channel Theory.

Documents

b c slide

c c f slide

commonsense reasoning

repetition slide

evts slide

oo slide

parameter slide

parameters of b