sebastian bitzer ([email protected])[email protected] seminar knowledge representation university of...
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Sebastian Bitzer ([email protected])Seminar Knowledge RepresentationUniversity of Osnabrueck10.07.2003
Applications of the Channel Theory
Commonsense Reasoning
Representations
10.07.2003 Applications of Channel Theory 2
Overview
• Repetition
• Commonsense Reasoning / Nonmonotonicity
• (Imperfect) Representations
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)
10.07.2003 Repetition 5
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)
10.07.2003 Repetition 6
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
10.07.2003 Repetition 7
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)
10.07.2003 Commonsense Reasoning 9
Overview
• Problem of Nonmonotonicity
• State Spaces, enhanced
• Background Conditions
• Relativising to a Background Condition
10.07.2003 Commonsense Reasoning 10
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
10.07.2003 Commonsense Reasoning 11
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
10.07.2003 Commonsense Reasoning 12
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
10.07.2003 Commonsense Reasoning 13
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
10.07.2003 Commonsense Reasoning 14
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)
10.07.2003 Commonsense Reasoning 15
Judith’s heating system
(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
10.07.2003 Commonsense Reasoning 16
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
10.07.2003 Commonsense Reasoning 17
Judith’s heating system
(α1) thermostat: 65 ≤ σ1 ≤ 70
(α2) room temperature: σ2 = 58
(α3) power: σ3 = 0
(β) hot air is coming out of the vents
B = {σ3= σ4= σ5 = 1}
B↑α3 = {σ4= σ5 = 1}
10.07.2003 Commonsense Reasoning 18
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
10.07.2003 Commonsense Reasoning 19
Judith’s heating system
(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 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)
10.07.2003 Commonsense Reasoning 20
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
10.07.2003 Representations 22
Overview
• The Problem of Imperfect Representations
• Representation Systems
• Explaining Imperfect Representations
10.07.2003 Representations 28
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
10.07.2003 Representations 29
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
10.07.2003 Representations 30
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