Epistemic Oughts in Stit Semantics. John Horty. University of Maryland

Transcription

1 Epistemic Oughts in Stit Semantics John Horty University of Maryland Version of: April 15,

2 Outline 1. Stit semantics [α stit: A] = α (an agent) sees to it that A 2. Oughts in stit semantics 3. Some problems with knowledge 4. Labeled stit semantics 5. Epistemic oughts 6. Some variations 2

3 Stit semantics 1. Concepts: Tree < m h H m = {h : m h} Question: Is FA true at m 1? Answer: Who knows? Conclusion: Must relativize truth to moments and histories or m/h indices. 3

4 2. Branching time model: Tree, <, v with v mapping sentence letters into sets of m/h pairs. 3. Evaluation rules: booleans, P, F, m/h = A iff m/h v(a), for A atomic m/h = A iff m/h = A, etc. m/h = PA iff there is an m m < m and m /h = A m/h = FA iff there is an m m < m and m /h = A h such that h such that m/h = A iff m/h = A for each h H m. 4. Moment determinate: m = A iff m/h = A for each h H m. 5. Propositions A = {m/h : m/h = A} A m = {h H m : m/h A } A = {m : h(m/h A )} 4

5 6. Stit concepts: Agent Choice 7. Examples: Choice m 1 α = {K 1, K 2, K 3 } Choice m 1 α (h 4 ) = K 3 Note: these are action tokens 5

6 8. Stit model: Tree, <, Agent, Choice, v 9. Stit evaluation rule: m/h = [α stit: A] iff Choice m α (h) A m Example: [α stit: A] true at m/h 1, not at m 1 /h 3 6

7 Oughts in stit semantics 1. Deontic stit model: Tree, <, Agent, Choice, V alue, v, where V alue maps histories into numbers, representing values 2. Standard deontic operator: m/h = A iff m/h = A for each best h H m. Note: moment determinate m/h = A iff m = A 3. Meinong/Chisholm analysis: S ought to bring it about that p defined as It ought to be that S brings it about that p (Chisholm) [α cstit: A] = It ought to be that α sees to it that A = α ought to see to it that A 7

8 4. Examples: A without [α stit: A] [α stit: A] 8

9 5. The gambling problem: m = [α stit: G] 9

10 6. More gambling: m = [α stit: G] 10

11 7. Ordering the action tokens: Where K, K Choice m α K K iff For all h K, h K : [Value(h) Value(h )] K < K iff K K and (K K) 8. Optimal action tokens: K-Optimal m α = {K Choice m α : K Choice m α (K < K )} 9. A new deontic operator: m/h = [α cstit: A] iff For all K K-Optimal m α : K A m Also moment determinate: m/h = [α cstit: A] iff m = [α cstit: A] 11

12 10. The gambling problem, resolved: m = [α stit: G], but m = [α stit: G] K-Optimal m α = {K 1, K 2 } 12

13 11. More gambling, resolved: m = [α stit: G], but m = [α stit: G] K-Optimal m α = {K 2 } 13

14 Knowledge and oughts 1. m 1 = [α stit: BH], but is that right? Maybe it is, but agent just doesn t know it?? 14

15 2. Indistinguishability: equivalence relation between moments: m α m 3. Epistemic deontic stit models: Tree, <, Agent, Choice, V alue, { α } α Agent, v 4. Knowledge operator: m/h = K α A iff For all m /h such that m α m : m /h = A Moment determinate: m/h = K α A iff m = K α A 15

16 5. m 1 = [α stit: BH], but m 1 = K α [α stit: BH] K-Optimal m 1 α = {K 1 } K-Optimal m 2 α = {K 5 } 16

17 6. Problem #1: m 1 = K α [α stit: G], but that s wrong K-Optimal m 1 α = {K 1 } K-Optimal m 2 α = {K 5 } 17

18 7. Problem #2: m 1 = K α [α stit: G], but that s wrong too K-Optimal m 1 α = {K 1, K 2 } K-Optimal m 2 α = {K 4 } 18

19 8. Problem #3: m 1 = K α [α stit: W], but what action to take? K-Optimal m 1 α = {K 1 } K-Optimal m 2 α = {K 4 } Oughts should be action guiding Ought implies can 19

20 Labeled stit logic 1. Type = {τ 1, τ 2,...} a set of action types Intuitions: Basic robot actions Agent performs a token by executing a type Types are repeatable Types (not tokens) lie within agent control 2. Partial execution function [ ] mapping τ to [τ] m α Choice m α token resulting when τ is executed by α at m. 3. Label function Label mapping K Choice m α to This function is one-one Label(K) Type 20

21 4. Execution/label constraints: If K Choice m α, then [Label(K)] m α = K If τ Type then Label([τ] m α ) = τ (Note: requires [τ] m α defined) 5. Action types available to α at moment m: Type m α = {Label(K) : K Choice m α } The action type executed by α at m/h is Type m α (h) = Label(Choice m α (h)) 6. Labeled stit model: Tree, <, Agent, Choice, { α } α Agent, Type,[ ], Label, v subject to constraint If m α m, then Type m α = Typem α 21

22 7. Kstit operator: m/h = [α kstit: A] iff For all m such that m α m : Example: [Type m α (h)] m α A m M m 1 /h 1 = [α kstit: A] m 1 /h 2 = [α stit: A], but m 1 /h 2 = [α kstit: A] 22

23 8. Some notes on the kstit logic: S5 operator Properly between K α + stit and stit: K α [α stit: A] [α kstit: A] [α kstit: A] [α stit: A] and converses fail Collapses to stit if m α m implies m = m Do you know what you re doing? Ex ante and ex interim knowledge K α A is ex ante [α kstit: A] is ex interim Relations: K α A [α kstit: A] K α A [α kstit: A] 23

24 Epistemic oughts 1. Information set: a set I of moments subject to If m α m, then Type m α = Type m α In particular, I m α = {m : m α m } is an information set, representing information available to α at m Example: I m 1 α = {m 1, m 2 } 24

25 2. Goal: rank action types, relative to I 3. One idea: take [τ] I α = {[τ] m α : m I} and the define τ better than τ based on I iff [τ] I α < [τ ] I α 4. Example: τ 2 better than τ 1 based on I = {m 1, m 2 }, since [τ 1 ] I α < [τ 2 ] I α 25

26 5. Problem: we do not have [τ 1 ] I α < [τ 2 ] I α but it seems by sure-thing reasoning that τ 2 is better than τ 1 26

27 6. Instead: where τ, τ Type m α τ m,i α τ iff For all m I : [τ] m α [τ ] m α τ m,i α τ iff τ m,i α τ and (τ m,i α τ) 27

28 7. Optimal action types: T-Optimal m,i α = {τ Type m α : τ Type m α (τ m,i α τ )}. 8. Note: in this and previous definiton, I is unspecified, but Iα m is particularly interesting 9. Labeled deontic stit model: Add V alue to labeled stit models, and then Epistemic ought: m/h = [α kstit: A] iff For each τ T-Optimal m,im α α : For each m I m α : [τ] m α A m Note: here, I is bound to I m α Note: moment determinate m/h = [α kstit: A] iff m = [α kstit: A] 28

29 11. Problem #1: m 1 = K α [α stit: G], but m1 = [α kstit: G] K-Optimal m 1 α = {K 1} K-Optimal m 2 α = {K 5} T-Optimal m 1,I m 1 α α = {τ 1, τ 2, τ 3 } 29

30 12. Problem #2: m 1 = K α [α stit: G], but m1 = [α kstit: G] K-Optimal m 1 α = {K 1, K 2 } K-Optimal m 2 α = {K 4} T-Optimal m 1,I m 1 α α = {τ 2 } 30

31 13. Problem #3: m 1 = K α [α stit: W], but what action to take?? Here, do not have m 1 = [α kstit: W] K-Optimal m 1 α = {K 1 } K-Optimal m 2 α = {K 4 } T-Optimal m 1,I m 1 α α = {τ 1, τ 2 } 31

32 13. Observations on the epistemic ought: Normal operator supporting [α kstit: A] [α kstit: A] No relations between two oughts; neither [α stit: A] [α kstit: A] [α kstit: A] [α stit: A] But everything collapses if I m α = {m}: [α stit: A] [α kstit: A], since K-Optimal m α = {[τ] m α : τ T-Optimal m,im α α } T-Optimal m,im α α = {Label(K) : K K-Optimal m α } Finally, knowledge of epistemic oughts: [α kstit: A] Kα [α kstit: A] 32

33 Conditional epistemic oughts 1. If A is moment determinate ie, A A then A = {m/h : m/h = A} meets the constraint: If m/h A, then m/h A for each h H m so that A can be represented as the set A = {m : h(m/h A )} 2. Local restriction: can conditionalize only on moment determinate propositions 3. Conditional epistemic ought: m/h = ([α kstit: A]/B) iff For each τ T-Optimal m,im α B α : For each m I m α B : [τ] m α A m 33

34 4. Reasoning by cases fails: don t have ([α kstit: G]/H) ([α kstit: G]/T) H T [α kstit: G] T-Optimal m 1,I m α H α = {τ 1 } T-Optimal m 1,I m α T α = {τ 2 } T-Optimal m 1,I m α α = {τ 1, τ 2, τ 3 } 34

35 5. Note similarity to conditional oughts in ADL 6. Note similarity to miners 7. A special case: ([α kstit: A τ α ]/B) ([α kstit: A τ α ]/C) B C [α kstit: A τ α ] is valid 35

36 8. Modus ponens fails: don t have ([α kstit: BH]/H) H [α kstit: BH] T-Optimal m 1,I m α H α = {τ 1 } T-Optimal m 1,I m α α = {τ 1, τ 2, τ 3 } But works if we have K α H 36

37 Perspectival oughts 1. Recall that m,i α, m,i α T-Optimal m,i α relativized to I, and then I set to I m α in evaluation rule for epistemic ought: m/h = [α kstit: A] iff For each τ T-Optimal m,i α : [τ] m α A m for each m I m α 2. Another route: define context c = m/h/i and then evaluate oughts at contexts: m/h/i = [α kstit: A] iff For each τ T-Optimal m,i α : [τ] m α A m for each m I 37

38 3. Perspectivalism: where c = m/h/i c = m /h /I are two contexts, define A true at c from the perspective of c or, A true at c as assessed from c iff m/h/i = A 4. In particular, [α kstit: A] true at c as assessed from c iff m/h/i = [α kstit: A] ie, iff [τ] m α A m for each τ T-Optimal m,i α m I and 38

39 5. Three information sets I m 1 α = {m 1, m 2, m 3 } I = {m 1, m 2 } I = {m 1 } leading to contexts c m 1 α = m 1 /h 1 /I m 1 α c = m 1 /h 1 /I c = m 1 /h 1 /I 39

40 6. What is optimal? T-Optimal m 1,I m 1 α α = {τ 1, τ 2, τ 3 } T-Optimal m 1,I α = {τ 1, τ 2 } T-Optimal m 1,I α = {τ 1 } 40

41 7. So have m 1 /h 1 /I m 1 α = [α kstit: A B C] m 1 /h 1 /I m 1 α = [α kstit: A B] m 1 /h 1 /I = [α kstit: A B] m 1 /h 1 /I = [α kstit: A] m 1 /h 1 /I = [α kstit: A] 41

42 8. And finally Only [α kstit: A B C] is true at c m 1 α from the perspective of c m 1 α [α kstit: A B] is true at c m 1 α from the perspective of c, but [α kstit: A] is not [α kstit: A] is true at c m 1 from the perspective of c α 42

43 Today s summary 1. Reviewed previous work 2. Illustrated problems mixing oughts/knowledge 3. Reviewed labeled stit logics 4. Extended deontic ideas to new framework 5. Some variations 6. Much work to be done: Multiple agents Relax assumptions Connections: eg, Broersen, Tamminga 43