Applications of the Channel Theory

1. Applications of the Channel Theory Commonsense Reasoning Representations

2. Overview • Repetition • Commonsense Reasoning / Nonmonotonicity • (Imperfect) Representations Applications of Channel Theory

3. 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╞Aα (with respect to A) 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 • AC is an infomorphism between classifications ΣA ΣC fˇ(c)╞A α  c╞C fˆ(α) fˆ ╞A ╞C A C fˇ Repetition

6. C f: AC g: BC f g A B 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 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) Repetition

8. Commonsense Reasoning

9. Overview • Problem of Nonmonotonicity • State Spaces, enhanced • Background Conditions • Relativising to a Background Condition 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 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 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 ≤ 110σ3: 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 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) iP) • B1≤ B2 P1  P2 Commonsense Reasoning

14. Silence • α is silent on B, if it does not tell anything about the parameters of B: • σ =Bσ‘ if σt = σt‘ tB • σ,σ‘: if σ =Bσ‘ andσαthenσ‘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) 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 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 αΓ is silent on B0  B↑α is restriction of B to the set of inputs iP such that α is silent on i 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} 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├BΔ σ sat. B (if σ p p Γ thenσ q for some qΔ) • normal tokens are those satisfying B 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) Commonsense Reasoning

20. Strict Entailment • Γ BΔ: Γ strictly entails Δ relative to B • Γ├Log(SB)Δ • all types in ΓΔ are silent on B • then (conclusions): • Γ BΔ is a better model of human reasoning than Γ├BΔ • Γ BΔ 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 BΔdoes not entail: Γ, αB↑αΔ or Γ B↑αΔ, α Commonsense Reasoning

21. Representations

22. Overview • The Problem of Imperfect Representations • Representation Systems • Explaining Imperfect Representations Representations

23. The bridge Representations

24. The Map Representations

25. The bridge Representations

26. The bridge Representations

27. The Map correct? Representations

28. L f g A B Representations Targets 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 Representations

29. c L f g A B Representations Targets a╞A α b╞B β 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NL) • 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 if a is accurate representation of b and a represents b as being of type β, then b╞B β 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 Representations

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