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Sebastian Bitzer Seminar Knowledge Representation University of Osnabrueck 10.07.2003 Applications of the Channel Theory.

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Presentation on theme: "Sebastian Bitzer Seminar Knowledge Representation University of Osnabrueck 10.07.2003 Applications of the Channel Theory."— Presentation transcript:

1 Sebastian Bitzer Seminar Knowledge Representation University of Osnabrueck Applications of the Channel Theory Commonsense Reasoning Representations

2 Applications of Channel Theory2 Overview Repetition Commonsense Reasoning / Nonmonotonicity (Imperfect) Representations

3 Repetition

4 Repetition4 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)

5 Repetition5 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 A  C is an infomorphism between classifications A C ΣAΣA ΣCΣC ╞A╞A ╞C╞C fˆfˆ fˇfˇ fˇ(c) ╞ A α  c ╞ C fˆ( α)

6 Repetition6 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 AB fg f: A  C g: B  C

7 Repetition7 State Spaces S is a state space, consists of a set of tokens (S), a set of states (Ω S ) and a function mapping between them: S =  S, Ω S, state  Evt(S) is the according classification Log(S) is the local logic on Evt(S)

8 Commonsense Reasoning

9 Commonsense Reasoning9 Overview Problem of Nonmonotonicity State Spaces, enhanced Background Conditions Relativising to a Background Condition

10 Commonsense Reasoning10 The Problem of Nonmonotonicity Baseball: (α 1 ) pitcher throws ball to batter (β) ball will arrive at batter  α 1 ├ β monotonicity: α 1, α 2 ├ β but: ( α 2 ) ball hits bird  α 1, α 2 ├ ¬β

11 Commonsense Reasoning11 Real valued State Spaces S =  S, Ω  R n, state  each state is a vector σ with dimension n σ has input and output coordinates (σ = σ i ×σ o ) = observables outputs can be computed from inputs

12 Commonsense Reasoning12 Judith’s heating system a state σ = (σ 1, σ 2, σ 3, σ 4, σ 5, σ 6, σ 7 ) σ 1 : thermostat setting 55 ≤ σ 1 ≤ 80 σ 2 : room temperature 20 ≤ σ 2 ≤ 110 σ 3 : power on (σ 3 = 1) or off (σ 3 = 0) σ 4 : exhaust vents blocked (=0) or clear (=1) σ 5 : op. conditions cooling (=-1), off (=0) or heating (=1) σ 6 : running yes (=1) or no (=0) σ 7 : output air temperature 20 ≤ σ 7 ≤ 110 σiσi σoσo

13 Commonsense Reasoning13 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 (σ i =B(i)  i  P) B 1 ≤ B 2  P 1  P 2

14 Commonsense Reasoning14 Silence α is silent on B, if it does not tell anything about the parameters of B: –σ = B σ‘ if σ t = σ t ‘  t  B –  σ,σ‘: if σ = B σ‘ and σ  α 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)

15 Commonsense Reasoning15 Judith’s heating system (α 1 ) thermostat: 65 ≤ σ 1 ≤ 70 (α 2 ) room temperature: σ 2 = 58 (α 3 ) power: σ 3 = 0 (β) hot air is coming out of the vents  α 1, α 2, β are silent about σ 3, σ 4, σ 5 (which are supposed to be the parameters) but: α 3 is not silent about σ 3 α 1, α 2 ├ β α 1, α 2, α 3 ├ ¬β

16 Commonsense Reasoning16 Weakening Γ is a set of types the weakening of B by Γ (B↑Γ) is the greatest lower bound of all B 0 ≤ B such that every type α  Γ is silent on B 0  B↑α is restriction of B to the set of inputs i  P such that α is silent on i

17 Commonsense Reasoning17 Judith’s heating system (α 1 ) thermostat: 65 ≤ σ 1 ≤ 70 (α 2 ) room temperature: σ 2 = 58 (α 3 ) power: σ 3 = 0 (β) hot air is coming out of the vents B = {σ 3 = σ 4 = σ 5 = 1}  B↑α 3 = {σ 4 = σ 5 = 1}

18 Commonsense Reasoning18 Relativising to a Background Condition S B is relativisation of S to B –subspace of S –only states that satisfy B Log(S B ) 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 q  Δ) –normal tokens are those satisfying B

19 Commonsense Reasoning19 Judith’s heating system (α 1 ) thermostat: 65 ≤ σ 1 ≤ 70 (α 2 ) room temperature: σ 2 = 58 (α 3 ) power: σ 3 = 0 (β) hot air is coming out of the vents α1, α2 ├ β holds in Log(S B ) because α 3 not silent about σ 3 : switch to B↑α 3  α1, α2 ├ β does not hold in Log(S B↑α3 ) (is no constraint there)  α1, α2, α3 ├ ¬β is constraint in Log(S B↑α3 )

20 Commonsense Reasoning20 Strict Entailment Γ  B Δ: Γ strictly entails Δ relative to B –Γ├ Log(S B ) 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

21 Representations

22 Representations22 Overview The Problem of Imperfect Representations Representation Systems Explaining Imperfect Representations

23 Representations23 The bridge

24 Representations24 The Map

25 Representations25 The bridge

26 Representations26 The bridge

27 Representations27 The Map correct?

28 Representations28 Representation Systems R =  C,L  is a representation system – C = {f: A  C, g: B  C} is a binary channel – L is the local logic on the core C L AB fg RepresentationsTargets

29 Representations29 Representations a is a representation of b, if a,b are connected by some c  C –a is accurate representation of b, if c is normal token (c  N L ) 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 AB fg Representations Targets a╞ A αb╞ B β c if a is accurate representation of b and a represents b as being of type β, then b╞ B β

30 Representations30 Explaining Imperfect Representations tokens in target classification (B) are really regions at times if b 0 changes to b 1 this gives rise to a new connection c 1 between a and b 1  a represents both: b 0 and b 1  but c 1 supports not all of the constraints of R b0b0 b1b1

31 Applications of Channel Theory31 References Jon Barwise and Jerry Seligman, Information Flow, The Logic of Distributed Systems, Cambridge University Press, 1997


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