1 / 26

PTR: A Probabilistic Transaction Logic

PTR: A Probabilistic Transaction Logic. A logic for reasoning about action under uncertainty. A mathematically sound foundation for building software that requires such reasoning. Julian Fogel. Applications. Uncertainty in workflow Unreliable circuits AI planning Game theory.

kalea
Télécharger la présentation

PTR: A Probabilistic Transaction Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. PTR:A Probabilistic Transaction Logic A logic for reasoning about action under uncertainty. A mathematically sound foundation for building software that requires such reasoning Julian Fogel

  2. Applications • Uncertainty in workflow • Unreliable circuits • AI planning • Game theory

  3. Uncertainty in Workflow

  4. Unreliable Circuits

  5. AI Planning

  6. Game Theory

  7. Logic = Syntax + Semantics + Reasoning • Syntax: a formal language – just meaningless symbols • Semantics: giving meaning to symbols – truth in a structure ⊧ • Reasoning: what else is true given what is known – logical implication⇒, entailment ⊦

  8. Semantics • Possible worlds, initial state • Actions and transactions, paths • Path distribution

  9. Possible Worlds Example One propositional variable H, which is true when the coin is heads. H H Two possible worlds: H and H. H: the coin is not heads in this state (it’s tails) H: the coin is heads in this state Coin is hidden, and even chance of it being heads or tails: PW assigns probability 0.5 to each state. PW(H) = 0.5and PW(H) = 0.5. 0.5 H 0.5 H

  10. Possible Worlds Definitions • Propositional symbols:Each symbol can be either True or False • State: a particular assignment of True or False to the propositional symbols • PW: A probability distribution over states [Fagin and Halpern 90]

  11. Path Example One atomic action symbol F: flip a fair coin 0.5 H H 0.5 H H 0.5 0.5 H H H H PA(F,H) PA(F,H) One transaction symbol F2: flip a fair coin twice 0.25 H H H 0.25 H H H 0.25 0.25 H H H H H H 0.25 0.25 H H H H H H 0.25 0.25 H H H H H H PT(F2,H) PT(F2,H)

  12. Path Definitions • Atomic action symbols:trigger transitions between two states • Transaction symbols: allows intermediate states [Bonner and Kifer 94] • Path: sequence of states (W1,…,Wn) • PA and PT: probability distributions over paths

  13. Path Distribution Example 0.125 H H H Given Pw, PA, and PT as in the previous examples, the path distribution P shown here makes the transaction formula F2 true. 0.125 H H H 0.125 H H H 0.125 H H H 0.125 H H H 0.125 H H H 0.125 H H H 0.125 H H H

  14. Path Distribution: The Heart of PTR • Given the initial probabilistic knowledge about the world encoded in Pw, PA, and PT, aPTR formulaistrueorfalse(succeeds or fails)on a path distribution P. • If a formula succeeds on a path distribution, then it executes along one of the paths that have non-zero probability.

  15. Formulas • Probabilistic state formula: Pr(Q)  c where Q is an ordinary propositional formula • Transaction Formula: • Atomic action or transaction symbol • Serial conjunction f   • Disjunction f   • Negation f • Pre/postcondition []-f-[] where  and  are probabilistic state formulas

  16. Pre/postcondition Example SomeSuccessfulTransactions F2 [Pr(H)=0.25]-F2-[Pr(H)=0.5] [Pr(H)0.7]-F2 F2-[Pr(HH)1.0] 0.0625 H H H 0.0625 H H H 0.0625 H H H 0.0625 H H H Some Failed Transactions [Pr(H)=0.25]-F2-[Pr(H)=0.55] [Pr(H)0.8]-F2 [Pr((HH))>0]-F2 F2-[Pr((HH))<0] 0.1875 H H H 0.1875 H H H 0.1875 H H H 0.1875 H H H

  17. Pre/postcondition []-f-[] • Precondition: constrains the distribution of the initial state of a transaction, describes what must be known before f can execute • Postcondition: constrains the distribution of the final state of a transaction, describes something known to be true after the transaction f executes

  18. Serial Conjunction Example 0.125 H H H H 0.125 H H H H Assume that PW(H) = 1. The path distribution P tothe left makes transaction formula (F2  F) true. 0.125 H H H H 0.125 H H H H 0.125 H H H H 0.125 H H H H 0.125 H H H H 0.125 H H H H

  19. Serial Conjunction f   • First execute f followed by • Both conjuncts need to succeed • Probabilities along paths are combined like a cross-product, then normalized

  20. Disjunction f   • Executeone off or  nondeterministically • Succeeds if either disjunct succeeds • Useful in defining other connectives such as conditional and biconditional • Not parallel execution

  21. Negation • f • Succeeds on any path distribution on which f fails • Mainly useful in defining other connectives, or in conjunction with them

  22. A Small Example A B Actions: OA, OB, MC Propositions: a, b, mc Transaction: MIX K={MIXOA(OBMC)} Query: MIX-[Pr(mc)0.8] C

  23. PA(MC)

  24. PA(OA), PA(OB), PW PW(MAB) = 1

  25. PT(MIX) 0.989 0.989 0.85 , , , 0.989 0.989 0.15 , , , 0.989 0.011 1 , , , 0.011 0.989 1 , , , 0.011 0.011 1 , , , We can verify that for any path distribution P, S and P make MIXOA(OBMC) true, and that if P is set to PT(MIX) then S and P make MIX-[Pr(mc)0.8] true.

  26. Proposed Directions • Adding observations to the logic • Proof theory • Allowing concurrent transactions • PTR logic programming • Investigating applications • Comparison with other probabilistic logics

More Related