Logic and Artificial Intelligence Lecture 20

Transcription

1 Logic and Artificial Intelligence Lecture 20 Eric Pacuit Currently Visiting the Center for Formal Epistemology, CMU Center for Logic and Philosophy of Science Tilburg University ai.stanford.edu/ epacuit November 14, 2011 Logic and Artificial Intelligence 1/23

2 Actions 1. Actions as transitions between states, or situations: Logic and Artificial Intelligence 2/23

3 Actions 1. Actions as transitions between states, or situations: a s t Logic and Artificial Intelligence 2/23

4 Actions 1. Actions as transitions between states, or situations: a s t 2. Actions restrict the set of possible future histories. Logic and Artificial Intelligence 2/23

5 J. van Benthem, H. van Ditmarsch, J. van Eijck and J. Jaspers. Chapter 6: Propositional Dynamic Logic. Logic in Action Online Course Project, Logic and Artificial Intelligence 3/23

6 pty, this means that the action a has aborted in state s. If the set has a s 0, this means that the action a is deterministic on state s, and if the set re elements, this means that action a is non-deterministic on state s. The is: s 1 s 2 s s 3 s n e extend this picture to the whole set S, what emerges is a binary relation rrow from s to s 0 (or equivalently, a pair (s, s 0 ) in the relation) just in case ion a in state s may have s 0 as result. Thus, we can view binary relations rpretations of basic action symbols a. Logic and Artificial Intelligence 2 4/23

7 Propositional Dynamic Logic Language: The language of propositional dynamic logic is generated by the following grammar: p ϕ ϕ ψ [α]ϕ where p At and α is generated by the following grammar: a α β α; β α ϕ? where a Act and ϕ is a formula. Logic and Artificial Intelligence 5/23

8 Propositional Dynamic Logic Language: The language of propositional dynamic logic is generated by the following grammar: p ϕ ϕ ψ [α]ϕ where p At and α is generated by the following grammar: a α β α; β α ϕ? where a Act and ϕ is a formula. Semantics: M = W, {R a a P}, V where for each a P, R a W W and V : At (W ) Logic and Artificial Intelligence 5/23

9 Propositional Dynamic Logic Language: The language of propositional dynamic logic is generated by the following grammar: p ϕ ϕ ψ [α]ϕ where p At and α is generated by the following grammar: a α β α; β α ϕ? where a Act and ϕ is a formula. Semantics: M = W, {R a a P}, V where for each a P, R a W W and V : At (W ) [α]ϕ means after doing α, ϕ will be true α ϕ means after doing α, ϕ may be true Logic and Artificial Intelligence 5/23

10 M, w = [α]ϕ iff for each v, if wr α v then M, v = ϕ M, w = α ϕ iff there is a v such that wr α v and M, v = ϕ Logic and Artificial Intelligence 6/23

11 Union R α β := R α R β 6-12 CHAPTER 6. LOGIC AND ACTION s 1 s 2 s 3 s n s s 0 1 s 0 2 s 0 3 s 0 m Logic and Artificial Intelligence 7/23

12 Sequence Rcture: α;β := R α R β s 11 s 12 s 1 s 13 s s 2 s 3 s 1m s n Logic and Artificial Intelligence 8/23

13 Test R ϕ? = {(w, w) M, w = ϕ} N RELATIONS s 1 s 2 s s 3 s n t 1 t 2 t t 3 t m Logic and Artificial Intelligence 9/23

14 Iteration R α := n 0 R n α Logic and Artificial Intelligence 10/23

15 Propositional Dynamic Logic 1. Axioms of propositional logic 2. [α](ϕ ψ) ([α]ϕ [α]ψ) 3. [α β]ϕ [α]ϕ [β]ϕ 4. [α; β]ϕ [α][β]ϕ 5. [ψ?]ϕ (ψ ϕ) 6. ϕ [α][α ]ϕ [α ]ϕ 7. ϕ [α ](ϕ [α]ϕ) [α ]ϕ 8. Modus Ponens and Necessitation (for each program α) Logic and Artificial Intelligence 11/23

16 Propositional Dynamic Logic 1. Axioms of propositional logic 2. [α](ϕ ψ) ([α]ϕ [α]ψ) 3. [α β]ϕ [α]ϕ [β]ϕ 4. [α; β]ϕ [α][β]ϕ 5. [ψ?]ϕ (ψ ϕ) 6. ϕ [α][α ]ϕ [α ]ϕ (Fixed-Point Axiom) 7. ϕ [α ](ϕ [α]ϕ) [α ]ϕ (Induction Axiom) 8. Modus Ponens and Necessitation (for each program α) Logic and Artificial Intelligence 12/23

17 Actions and Ability An early approach to interpret PDL as logic of actions was put forward by Krister Segerberg. Segerberg adds an agency program to the PDL language δa where A is a formula. K. Segerberg. Bringing it about. JPL, Logic and Artificial Intelligence 13/23

18 Actions and Agency The intended meaning of the program δa is that the agent brings it about that A : formally, δa is the set of all paths p such that Logic and Artificial Intelligence 14/23

19 Actions and Agency The intended meaning of the program δa is that the agent brings it about that A : formally, δa is the set of all paths p such that 1. p is the computation according to some program α, and 2. α only terminates at states in which it is true that A Logic and Artificial Intelligence 14/23

20 Actions and Agency The intended meaning of the program δa is that the agent brings it about that A : formally, δa is the set of all paths p such that 1. p is the computation according to some program α, and 2. α only terminates at states in which it is true that A Interestingly, Segerberg also briefly considers a third condition: 3. p is optimal (in some sense: shortest, maximally efficient, most convenient, etc.) in the set of computations satisfying conditions (1) and (2). Logic and Artificial Intelligence 14/23

21 Actions and Agency The intended meaning of the program δa is that the agent brings it about that A : formally, δa is the set of all paths p such that 1. p is the computation according to some program α, and 2. α only terminates at states in which it is true that A Interestingly, Segerberg also briefly considers a third condition: 3. p is optimal (in some sense: shortest, maximally efficient, most convenient, etc.) in the set of computations satisfying conditions (1) and (2). The axioms: 1. [δa]a 2. [δa]b ([δb]c [δa]c) Logic and Artificial Intelligence 14/23

22 Actions and Agency in Branching Time Alternative accounts of agency do not include explicit description of the actions: t 0 t 1 t 2 t 3 Logic and Artificial Intelligence 15/23

23 STIT Each node represents a choice point for the agent. Logic and Artificial Intelligence 16/23

24 STIT Each node represents a choice point for the agent. A history is a maximal branch in the above tree. Logic and Artificial Intelligence 16/23

25 STIT Each node represents a choice point for the agent. A history is a maximal branch in the above tree. Formulas are interpreted at history moment pairs. Logic and Artificial Intelligence 16/23

26 STIT Each node represents a choice point for the agent. A history is a maximal branch in the above tree. Formulas are interpreted at history moment pairs. At each moment there is a choice available to the agent (partition of the histories through that moment) Logic and Artificial Intelligence 16/23

27 STIT Each node represents a choice point for the agent. A history is a maximal branch in the above tree. Formulas are interpreted at history moment pairs. At each moment there is a choice available to the agent (partition of the histories through that moment) The key modality is [i stit]ϕ which is intended to mean that the agent i can see to it that ϕ is true. [i stit]ϕ is true at a history moment pair provided the agent can choose a (set of) branch(es) such that every future history-moment pair satisfies ϕ Logic and Artificial Intelligence 16/23

28 STIT We use the modality to mean historic possibility. [i stit]ϕ: the agent has the ability to bring about ϕ. Logic and Artificial Intelligence 17/23

29 STIT Model A STIT models is M = T, <, Choice, V where Logic and Artificial Intelligence 18/23

30 STIT Model A STIT models is M = T, <, Choice, V where T, < : T a set of moments, < a tree-like ordering on T (irreflexive, transitive, linear-past) Logic and Artificial Intelligence 18/23

31 STIT Model A STIT models is M = T, <, Choice, V where T, < : T a set of moments, < a tree-like ordering on T (irreflexive, transitive, linear-past) Let Hist be the set of all histories, and H t = {h Hist t h} the histories through t. Logic and Artificial Intelligence 18/23

32 STIT Model A STIT models is M = T, <, Choice, V where T, < : T a set of moments, < a tree-like ordering on T (irreflexive, transitive, linear-past) Let Hist be the set of all histories, and H t = {h Hist t h} the histories through t. Choice : A T ( (H)) is a function mapping each agent to a partition of H t Choicei t K for each K Choicei t For all t and mappings s t : A (H t ) such that s t (i) Choicei t, we have i A s t(i) Logic and Artificial Intelligence 18/23

33 STIT Model A STIT models is M = T, <, Choice, V where T, < : T a set of moments, < a tree-like ordering on T (irreflexive, transitive, linear-past) Let Hist be the set of all histories, and H t = {h Hist t h} the histories through t. Choice : A T ( (H)) is a function mapping each agent to a partition of H t Choicei t K for each K Choicei t For all t and mappings s t : A (H t ) such that s t (i) Choicei t, we have i A s t(i) V : At (T Hist) is a valuation function assigning to each atomic proposition Logic and Artificial Intelligence 18/23

34 STIT Model A STIT models is M = T, <, Choice, V where T, < : T a set of moments, < a tree-like ordering on T (irreflexive, transitive, linear-past) Let Hist be the set of all histories, and H t = {h Hist t h} the histories through t. Choice : A T ( (H)) is a function mapping each agent to a partition of H t Choicei t K for each K Choicei t For all t and mappings s t : A (H t ) such that s t (i) Choicei t, we have i A s t(i) V : At (T Hist) is a valuation function assigning to each atomic proposition Logic and Artificial Intelligence 18/23

35 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Logic and Artificial Intelligence 19/23

36 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Logic and Artificial Intelligence 19/23

37 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Independence of agents Logic and Artificial Intelligence 19/23

38 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Independence of agents Logic and Artificial Intelligence 19/23

39 Many Agents The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Independence of agents Logic and Artificial Intelligence 19/23

40 STIT Language ϕ = p ϕ ϕ ψ [i stit]ϕ [i dstit : ϕ] ϕ Logic and Artificial Intelligence 20/23

41 STIT Language ϕ = p ϕ ϕ ψ [i stit]ϕ [i dstit : ϕ] ϕ M, t/h = p iff t/h V (p) Logic and Artificial Intelligence 20/23

42 STIT Language ϕ = p ϕ ϕ ψ [i stit]ϕ [i dstit : ϕ] ϕ M, t/h = p iff t/h V (p) M, t/h = ϕ iff M, t/h = ϕ Logic and Artificial Intelligence 20/23

43 STIT Language ϕ = p ϕ ϕ ψ [i stit]ϕ [i dstit : ϕ] ϕ M, t/h = p iff t/h V (p) M, t/h = ϕ iff M, t/h = ϕ M, t/h = ϕ ψ iff M, t/h = ϕ and M, t/h = ψ Logic and Artificial Intelligence 20/23

44 STIT Language ϕ = p ϕ ϕ ψ [i stit]ϕ [i dstit : ϕ] ϕ M, t/h = p iff t/h V (p) M, t/h = ϕ iff M, t/h = ϕ M, t/h = ϕ ψ iff M, t/h = ϕ and M, t/h = ψ M, t/h = ϕ iff M, t/h = ϕ for all h H t Logic and Artificial Intelligence 20/23

45 STIT Language ϕ = p ϕ ϕ ψ [i stit]ϕ [i dstit : ϕ] ϕ M, t/h = p iff t/h V (p) M, t/h = ϕ iff M, t/h = ϕ M, t/h = ϕ ψ iff M, t/h = ϕ and M, t/h = ψ M, t/h = ϕ iff M, t/h = ϕ for all h H t M, t/h = [i stit]ϕ iff M, t/h = ϕ for all h Choicei t(h) Logic and Artificial Intelligence 20/23

46 STIT Language ϕ = p ϕ ϕ ψ [i stit]ϕ [i dstit : ϕ] ϕ M, t/h = p iff t/h V (p) M, t/h = ϕ iff M, t/h = ϕ M, t/h = ϕ ψ iff M, t/h = ϕ and M, t/h = ψ M, t/h = ϕ iff M, t/h = ϕ for all h H t M, t/h = [i stit]ϕ iff M, t/h = ϕ for all h Choice t i (h) M, t/h = [i dstit]ϕ iff M, t/h = ϕ for all h Choice t i (h) and there is a h H t such that M, t/h = ϕ Logic and Artificial Intelligence 20/23

47 STIT: Example The following are false: A [stit]a and [stit](a B) [stit]a [stit]b. h 1 h 2 h 3 A B A B A B t K 1 K 2 J. Horty. Agency and Deontic Logic Logic and Artificial Intelligence 21/23

48 STIT: Axiomatics S5 for : (ϕ ψ) ( ϕ ψ), ϕ ϕ, ϕ ϕ, ϕ ϕ Logic and Artificial Intelligence 22/23

49 STIT: Axiomatics S5 for : (ϕ ψ) ( ϕ ψ), ϕ ϕ, ϕ ϕ, ϕ ϕ S5 for [i stit]: [i stit](ϕ ψ) ([i stit]ϕ [i stit]ψ), [i stit]ϕ ϕ, [i stit]ϕ [i stit][i stit]ϕ, [i stit]ϕ [i stit] [i stit]ϕ Logic and Artificial Intelligence 22/23

50 STIT: Axiomatics S5 for : (ϕ ψ) ( ϕ ψ), ϕ ϕ, ϕ ϕ, ϕ ϕ S5 for [i stit]: [i stit](ϕ ψ) ([i stit]ϕ [i stit]ψ), [i stit]ϕ ϕ, [i stit]ϕ [i stit][i stit]ϕ, [i stit]ϕ [i stit] [i stit]ϕ ϕ [i stit]ϕ Logic and Artificial Intelligence 22/23

51 STIT: Axiomatics S5 for : (ϕ ψ) ( ϕ ψ), ϕ ϕ, ϕ ϕ, ϕ ϕ S5 for [i stit]: [i stit](ϕ ψ) ([i stit]ϕ [i stit]ψ), [i stit]ϕ ϕ, [i stit]ϕ [i stit][i stit]ϕ, [i stit]ϕ [i stit] [i stit]ϕ ϕ [i stit]ϕ ( i A [i stit]ϕ i) ( i A [i stit]ϕ i) Logic and Artificial Intelligence 22/23

52 STIT: Axiomatics S5 for : (ϕ ψ) ( ϕ ψ), ϕ ϕ, ϕ ϕ, ϕ ϕ S5 for [i stit]: [i stit](ϕ ψ) ([i stit]ϕ [i stit]ψ), [i stit]ϕ ϕ, [i stit]ϕ [i stit][i stit]ϕ, [i stit]ϕ [i stit] [i stit]ϕ ϕ [i stit]ϕ ( i A [i stit]ϕ i) ( i A [i stit]ϕ i) Modus Ponens and Necessitation for M. Xu. Axioms for deliberative STIT. Journal of Philosophical Logic, Volume 27, pp , P. Balbiani, A. Herzig and N. Troquard. Alternative axiomatics and complexity of deliberative STIT theories. Journal of Philosophical Logic, 37:4, pp , Logic and Artificial Intelligence 22/23

53 Recap: Logics of Action and Ability F ϕ: ϕ is true at some moment in the future Logic and Artificial Intelligence 23/23

54 Recap: Logics of Action and Ability F ϕ: ϕ is true at some moment in the future F ϕ: there is a history where ϕ is true some moment in the future Logic and Artificial Intelligence 23/23

55 Recap: Logics of Action and Ability F ϕ: ϕ is true at some moment in the future F ϕ: there is a history where ϕ is true some moment in the future [α]ϕ: after doing action α, ϕ is true Logic and Artificial Intelligence 23/23

56 Recap: Logics of Action and Ability F ϕ: ϕ is true at some moment in the future F ϕ: there is a history where ϕ is true some moment in the future [α]ϕ: after doing action α, ϕ is true [δϕ]ψ: after bringing about ϕ, ψ is true Logic and Artificial Intelligence 23/23

57 Recap: Logics of Action and Ability F ϕ: ϕ is true at some moment in the future F ϕ: there is a history where ϕ is true some moment in the future [α]ϕ: after doing action α, ϕ is true [δϕ]ψ: after bringing about ϕ, ψ is true [i stit]ϕ: the agent can see to it that ϕ is true Logic and Artificial Intelligence 23/23

58 Recap: Logics of Action and Ability F ϕ: ϕ is true at some moment in the future F ϕ: there is a history where ϕ is true some moment in the future [α]ϕ: after doing action α, ϕ is true [δϕ]ψ: after bringing about ϕ, ψ is true [i stit]ϕ: the agent can see to it that ϕ is true [i stit]ϕ: the agent has the ability to bring about ϕ Logic and Artificial Intelligence 23/23