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Sebastian Bitzer (sbitzer@uos.de)sbitzer@uos.de Seminar Knowledge Representation University of Osnabrueck 10.07.2003 Applications of the Channel Theory Commonsense Reasoning Representations

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10.07.2003Applications of Channel Theory2 Overview Repetition Commonsense Reasoning / Nonmonotonicity (Imperfect) Representations

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Repetition

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10.07.2003Repetition4 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)

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10.07.2003Repetition5 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)

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10.07.2003Repetition6 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

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10.07.2003Repetition7 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)

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Commonsense Reasoning

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10.07.2003Commonsense Reasoning9 Overview Problem of Nonmonotonicity State Spaces, enhanced Background Conditions Relativising to a Background Condition

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10.07.2003Commonsense 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

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10.07.2003Commonsense 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

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10.07.2003Commonsense 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

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10.07.2003Commonsense 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

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10.07.2003Commonsense 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)

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10.07.2003Commonsense 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

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10.07.2003Commonsense 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

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10.07.2003Commonsense 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}

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10.07.2003Commonsense 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

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10.07.2003Commonsense 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 )

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10.07.2003Commonsense 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

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Representations

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10.07.2003Representations22 Overview The Problem of Imperfect Representations Representation Systems Explaining Imperfect Representations

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10.07.2003Representations23 The bridge

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10.07.2003Representations24 The Map

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10.07.2003Representations25 The bridge

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10.07.2003Representations26 The bridge

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10.07.2003Representations27 The Map correct?

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10.07.2003Representations28 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

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10.07.2003Representations29 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

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10.07.2003Representations30 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

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10.07.2003Applications of Channel Theory31 References Jon Barwise and Jerry Seligman, Information Flow, The Logic of Distributed Systems, Cambridge University Press, 1997

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