Friday, April 17, PTR: A Probabilistic Transaction Logic Julian Fogel A logic for reasoning about action under uncertainty. A mathematically sound foundation for building software that requires such reasoning
Friday, April 17, Applications Uncertainty in workflow Unreliable circuits AI planning Game theory
Friday, April 17, Uncertainty in Workflow
Friday, April 17, Unreliable Circuits
Friday, April 17, AI Planning
Friday, April 17, Game Theory
Friday, April 17, 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 ⊦
Friday, April 17, Semantics Possible worlds, initial state Actions and transactions, paths Path distribution
Friday, April 17, Possible Worlds Example One propositional variable H, which is true when the coin is heads. 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 HH Coin is hidden, and even chance of it being heads or tails: P W assigns probability 0.5 to each state. P W (H) = 0.5 and P W (H) = 0.5. H H 0.5
Friday, April 17, 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 P W : A probability distribution over states [Fagin and Halpern 90]
Friday, April 17, Path Example One atomic action symbol F: flip a fair coin H H H HHH HH 0.5 P A ( F,H) One transaction symbol F2: flip a fair coin twice H HH H 0.25 P T ( F2,H) H H H H 0.25 H HH H H HH H P T ( F2,H) H H H H 0.25 H HH H
Friday, April 17, Path Definitions Atomic action symbols: trigger transitions between two states Transaction symbols: allows intermediate states [Bonner and Kifer 94] Path: sequence of states (W 1,…,W n ) P A and P T : probability distributions over paths
Friday, April 17, Path Distribution Example H HH H H H H H H HH H H HH H H H H H H HH H Given P w, P A, and P T as in the previous examples, the path distribution P shown here makes the transaction formula F2 true.
Friday, April 17, Path Distribution: The Heart of PTR Given the initial probabilistic knowledge about the world encoded in P w, P A, and P T, a PTR formula is true or false (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.
Friday, April 17, Formulas Probabilistic state formula: Pr (Q) c where Q is an ordinary propositional formula Transaction Formula: –Atomic action or transaction symbol –Serial conjunction –Disjunction –Negation –Pre/postcondition [ ]- -[ ] where and are probabilistic state formulas
Friday, April 17, Pre/postcondition Example H HH H H H H H H HH H H HH H H H H H H HH H Some Successful Transactions F2 [ Pr (H)=0.25]- F2 -[ Pr (H)=0.5] [ Pr ( H) 0.7]- F2 F2 -[ Pr ( H H) 1.0] 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]
Friday, April 17, Pre/postcondition Precondition: constrains the distribution of the initial state of a transaction, describes what must be known before can execute Postcondition: constrains the distribution of the final state of a transaction, describes something known to be true after the transaction executes [ ]- -[ ]
Friday, April 17, Serial Conjunction Example H HH H H H H H H HH H H HH H H H H H H HH H H H H H H H H H Assume that P W ( H ) = 1. The path distribution P to the left makes transaction formula (F2 F) true
Friday, April 17, Serial Conjunction First execute followed by Both conjuncts need to succeed Probabilities along paths are combined like a cross-product, then normalized
Friday, April 17, Disjunction Execute one of or nondeterministically Succeeds if either disjunct succeeds Useful in defining other connectives such as conditional and biconditional Not parallel execution
Friday, April 17, Negation Succeeds on any path distribution on which fails Mainly useful in defining other connectives, or in conjunction with them
Friday, April 17, A Small Example A B C Actions: OA, OB, MC Propositions: a, b, mc Transaction: MIX K={MIX OA (OB MC)} Query: MIX-[Pr( mc ) 0.8 ]
Friday, April 17, P A ( MC ) to from mab mab mab mab mab
Friday, April 17, P A ( OA ), P A ( OB ), P W to from ma ma ma P W (MAB) = 1
Friday, April 17, P T ( MIX ) path P T ( MIX ) description mab mab success mab mab mixing fails mab mab valve A fails mab mab valve B fails mab mab both valves fail 0.989,, 0.85, 0.989,, 0.15, 0.989, 0.011, 1,, 0.989, 1, 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 P T (MIX) then S and P make MIX-[Pr( mc ) 0.8 ] true.
Friday, April 17, Proposed Directions Adding observations to the logic Proof theory Allowing concurrent transactions PTR logic programming Investigating applications Comparison with other probabilistic logics