PDT Logic A Probabilistic Doxastic Temporal Logic for Reasoning - PowerPoint PPT Presentation

PDT Logic A Probabilistic Doxastic Temporal Logic for Reasoning about Beliefs in Multi-agent Systems Dissertation Presentation Dipl.-Ing. Karsten Martiny Chairman: Prof. Dr. Stefan Fischer Reviewers: Prof. Dr. Ralf M¨ oller Prof. Dr. R¨ udiger Reischuk October 4, 2016

PDT Logic A representation formalism to reason about probabilistic beliefs over time in multi-agent systems Agents’ beliefs are quantified with imprecise probabilities (i.e., probability intervals) Time is modeled in discrete steps for a finite set of time points Agents’ subjective beliefs change upon observing facts PDT Logic 2 / 12

Main Contribution Combine and extend results from different fields of formal logic Temporal Logic [SPSS11] Epistemic Logic [FHVM95] Probabilistic Dynamic Epistemic logic [Koo03] Create a semantically rich representation formalism for beliefs [MM15a] Develop specialized decision procedures [MM16a] [FHMV95] R. Fagin, J. Halpern, Y. Moses, M. Vardi: Reasoning About Knowledge MIT Press, 1995 [Koo03] B. Kooi: Probabilistic Dynamic Epistemic Logic Journal of Logic, Language and Information, Volume 12, pages 381-408, September 2003 [SPSS11] P. Shakarian, A. Parker, G. Simari, V. S. Subrahmanian: Annotated Probabilistic Temporal Logic , ACM Transactions on Computational Logic, Volume 13, pages 1-33, April 2012 [MM15a] K. Martiny, R. M¨ oller: A Probabilistic Doxastic Temporal Logic for Reasoning about Beliefs in Multi-agent Systems 7th International Conference on Agents and Artificial Intelligence (ICAART), Lisbon, Portugal, 2015 [MM16a] K. Martiny, R. M¨ oller: PDT Logic: A Probabilistic Doxastic Temporal Logic for Reasoning about Beliefs in Multi- agent Systems , Journal of Artificial Intelligence Research (JAIR), Volume 57, pages 39-112, September 2016 PDT Logic 3 / 12

Representation Describing possible worlds A propositional language describes ontic facts Observation Atoms Obs G ( F ) specify that a group of agents G observes some ontic fact F Time Temporal evolution ⇔ sequence of possible worlds ( thread Th ) Probabilistic temporal relations expressed as temporal rules using frequency functions Probabilistic Beliefs Each thread Th has a prior probability ( “interpretation” ) I ( Th ) Probabilistic beliefs depend on observations of the respective agent ⇒ different threads yield different belief evolutions PDT Logic 4 / 12

Interpretation Updates Example: two agents 1, 2, six threads Th 1 ,..., Th 6 : I ( Th 1 )=0 . 1 F F F F F I ( Th 2 )=0 . 2 F G F G F I ( Th 3 )=0 . 1 F F F G G I ( Th 4 )=0 . 3 G F F G G I ( Th 5 )=0 . 1 G F F G F I ( Th 6 )=0 . 2 G F G G F t PDT Logic 5 / 12

Interpretation Updates Example: two agents 1, 2, six threads Th 1 ,..., Th 6 : I ( Th 1 )=0 . 1 F F F F F I ( Th 2 )=0 . 2 F G F G F I ( Th 3 )=0 . 1 F F F G G I ( Th 4 )=0 . 3 G F F G G I ( Th 5 )=0 . 1 G F F G F I ( Th 6 )=0 . 2 G F G G F t Obs 1 ( F ) PDT Logic 5 / 12

Interpretation Updates Example: two agents 1, 2, six threads Th 1 ,..., Th 6 : I ( Th 1 )=0 . 1 I 1 , 1 ( Th 1 )=0 . 25 F F F F F I ( Th 2 )=0 . 2 I 1 , 1 ( Th 2 )=0 . 50 F G F G F I 1 , 1 ( Th 3 )=0 . 25 I ( Th 3 )=0 . 1 F F F G G I 1 , 1 ( Th 4 )=0 . 00 I ( Th 4 )=0 . 3 G F F G G I ( Th 5 )=0 . 1 I 1 , 1 ( Th 5 )=0 . 00 G F F G F I ( Th 6 )=0 . 2 I 1 , 1 ( Th 6 )=0 . 00 G F G G F t Obs 1 ( F ) PDT Logic 5 / 12

Interpretation Updates Example: two agents 1, 2, six threads Th 1 ,..., Th 6 : I ( Th 1 )=0 . 1 I 1 , 1 ( Th 1 )=0 . 25 F F F F F I ( Th 2 )=0 . 2 I 1 , 1 ( Th 2 )=0 . 50 F G F G F I 1 , 1 ( Th 3 )=0 . 25 I ( Th 3 )=0 . 1 F F F G G I 1 , 1 ( Th 4 )=0 . 00 I ( Th 4 )=0 . 3 G F F G G I ( Th 5 )=0 . 1 I 1 , 1 ( Th 5 )=0 . 00 G F F G F I ( Th 6 )=0 . 2 I 1 , 1 ( Th 6 )=0 . 00 G F G G F t Obs 1 ( F ) Obs 2 ( F ) PDT Logic 5 / 12

Interpretation Updates Example: two agents 1, 2, six threads Th 1 ,..., Th 6 : I ( Th 1 )=0 . 1 I 1 , 2 ( Th 1 )=0 . 25 I 2 , 2 ( Th 1 )=0 . 125 F F F F F I ( Th 2 )=0 . 2 I 1 , 2 ( Th 2 )=0 . 50 I 2 , 2 ( Th 2 )=0 . 000 F G F G F I 1 , 2 ( Th 3 )=0 . 25 I 2 , 2 ( Th 3 )=0 . 125 I ( Th 3 )=0 . 1 F F F G G I 1 , 2 ( Th 4 )=0 . 00 I 2 , 2 ( Th 4 )=0 . 375 I ( Th 4 )=0 . 3 G F F G G I ( Th 5 )=0 . 1 I 1 , 2 ( Th 5 )=0 . 00 I 2 , 2 ( Th 5 )=0 . 125 G F F G F I ( Th 6 )=0 . 2 I 1 , 2 ( Th 6 )=0 . 00 I 2 , 2 ( Th 6 )=0 . 250 G F G G F t Obs 1 ( F ) Obs 2 ( F ) PDT Logic 5 / 12

Interpretation Updates Example: two agents 1, 2, six threads Th 1 ,..., Th 6 : I ( Th 1 )=0 . 1 I 1 , 2 ( Th 1 )=0 . 25 I 2 , 2 ( Th 1 )=0 . 125 F F F F F I ( Th 2 )=0 . 2 I 1 , 2 ( Th 2 )=0 . 50 I 2 , 2 ( Th 2 )=0 . 000 F G F G F I 1 , 2 ( Th 3 )=0 . 25 I 2 , 2 ( Th 3 )=0 . 125 I ( Th 3 )=0 . 1 F F F G G I 1 , 2 ( Th 4 )=0 . 00 I 2 , 2 ( Th 4 )=0 . 375 I ( Th 4 )=0 . 3 G F F G G I ( Th 5 )=0 . 1 I 1 , 2 ( Th 5 )=0 . 00 I 2 , 2 ( Th 5 )=0 . 125 G F F G F I ( Th 6 )=0 . 2 I 1 , 2 ( Th 6 )=0 . 00 I 2 , 2 ( Th 6 )=0 . 250 G F G G F t Obs 1 ( F ) Obs { 1 , 2 } ( F ) Obs { 1 , 2 } ( F ) Obs 2 ( F ) PDT Logic 5 / 12

Interpretation Updates Example: two agents 1, 2, six threads Th 1 ,..., Th 6 : I ( Th 1 )=0 . 1 I 1 , 5 ( Th 1 ) ≈ 0 . 33 I 2 , 5 ( Th 1 )=0 . 25 F F F F F I ( Th 2 )=0 . 2 I 1 , 5 ( Th 2 ) ≈ 0 . 67 I 2 , 5 ( Th 2 )=0 . 00 F G F G F I 1 , 5 ( Th 3 )=0 . 00 I 2 , 5 ( Th 3 )=0 . 00 I ( Th 3 )=0 . 1 F F F G G I 1 , 5 ( Th 4 )=0 . 00 I 2 , 5 ( Th 4 )=0 . 00 I ( Th 4 )=0 . 3 G F F G G I ( Th 5 )=0 . 1 I 1 , 5 ( Th 5 )=0 . 00 I 2 , 5 ( Th 5 )=0 . 25 G F F G F I ( Th 6 )=0 . 2 I 1 , 5 ( Th 6 )=0 . 00 I 2 , 5 ( Th 6 )=0 . 50 G F G G F t Obs 1 ( F ) Obs { 1 , 2 } ( F ) Obs { 1 , 2 } ( F ) Obs 2 ( F ) PDT Logic 5 / 12

Belief Operators - Definitions Agents can have beliefs of three different types, all quantified with a probability interval [ ℓ, u ], seen from thread Th ′ : Belief in facts B ℓ, u i , t ′ ( F t ): � I Th ′ ℓ ≤ i , t ′ ( Th ) ≤ u Th : Th ( t ) | = F Belief in temporal rules B ℓ, u i , t ′ ( r fr ∆ t ( F , G )): � I Th ′ ℓ ≤ i , t ′ ( Th ) · fr( Th , F , G , ∆ t ) ≤ u Th i , t ′ ( B ℓ j , u j Nested beliefs B ℓ, u j , t ( · )): � I Th ′ ℓ ≤ i , t ′ ( Th ) ≤ u ℓ j , uj Th , I Th j , t | = B ( · ) j , t PDT Logic 6 / 12

Belief Operators - Example I 1 , 2 ( Th 1 )=0 . 25 I ( Th 1 )=0 . 1 F F F F F I 1 , 2 ( Th 2 )=0 . 50 I ( Th 2 )=0 . 2 F G F G F I ( Th 3 )=0 . 1 I 1 , 2 ( Th 3 )=0 . 25 F F F G G I ( Th 4 )=0 . 3 I 1 , 2 ( Th 4 )=0 . 00 G F F G G I ( Th 5 )=0 . 1 I 1 , 2 ( Th 5 )=0 . 00 G F F G F I ( Th 6 )=0 . 2 I 1 , 2 ( Th 6 )=0 . 00 G F G G F t B . 1 ,. 3 1 , 2 ( G 5 ) PDT Logic 7 / 12

Belief Operators - Example I 1 , 2 ( Th 1 )=0 . 25 I ( Th 1 )=0 . 1 F F F F F I 1 , 2 ( Th 2 )=0 . 50 I ( Th 2 )=0 . 2 F G F G F I ( Th 3 )=0 . 1 I 1 , 2 ( Th 3 )=0 . 25 F F F G G I ( Th 4 )=0 . 3 I 1 , 2 ( Th 4 )=0 . 00 G F F G G I ( Th 5 )=0 . 1 I 1 , 2 ( Th 5 )=0 . 00 G F F G F I ( Th 6 )=0 . 2 I 1 , 2 ( Th 6 )=0 . 00 G F G G F t B . 1 ,. 3 1 , 2 ( G 5 ) agent PDT Logic 7 / 12

Belief Operators - Example I 1 , 2 ( Th 1 )=0 . 25 I ( Th 1 )=0 . 1 F F F F F I 1 , 2 ( Th 2 )=0 . 50 I ( Th 2 )=0 . 2 F G F G F I ( Th 3 )=0 . 1 I 1 , 2 ( Th 3 )=0 . 25 F F F G G I ( Th 4 )=0 . 3 I 1 , 2 ( Th 4 )=0 . 00 G F F G G I ( Th 5 )=0 . 1 I 1 , 2 ( Th 5 )=0 . 00 G F F G F I ( Th 6 )=0 . 2 I 1 , 2 ( Th 6 )=0 . 00 G F G G F t B . 1 ,. 3 1 , 2 ( G 5 ) agent time of the belief PDT Logic 7 / 12

Belief Operators - Example I 1 , 2 ( Th 1 )=0 . 25 I ( Th 1 )=0 . 1 F F F F F I 1 , 2 ( Th 2 )=0 . 50 I ( Th 2 )=0 . 2 F G F G F I ( Th 3 )=0 . 1 I 1 , 2 ( Th 3 )=0 . 25 F F F G G I ( Th 4 )=0 . 3 I 1 , 2 ( Th 4 )=0 . 00 G F F G G I ( Th 5 )=0 . 1 I 1 , 2 ( Th 5 )=0 . 00 G F F G F I ( Th 6 )=0 . 2 I 1 , 2 ( Th 6 )=0 . 00 G F G G F t B . 1 ,. 3 1 , 2 ( G 5 ) belief object agent time of the belief PDT Logic 7 / 12

Belief Operators - Example I 1 , 2 ( Th 1 )=0 . 25 I ( Th 1 )=0 . 1 F F F F F I 1 , 2 ( Th 2 )=0 . 50 I ( Th 2 )=0 . 2 F G F G F I ( Th 3 )=0 . 1 I 1 , 2 ( Th 3 )=0 . 25 F F F G G I ( Th 4 )=0 . 3 I 1 , 2 ( Th 4 )=0 . 00 G F F G G I ( Th 5 )=0 . 1 I 1 , 2 ( Th 5 )=0 . 00 G F F G F I ( Th 6 )=0 . 2 I 1 , 2 ( Th 6 )=0 . 00 G F G G F t probability quantification B . 1 ,. 3 1 , 2 ( G 5 ) belief object agent time of the belief PDT Logic 7 / 12

Belief Operators - Example I 1 , 2 ( Th 1 )=0 . 25 I ( Th 1 )=0 . 1 F F F F F I 1 , 2 ( Th 2 )=0 . 50 I ( Th 2 )=0 . 2 F G F G F I ( Th 3 )=0 . 1 I 1 , 2 ( Th 3 )=0 . 25 F F F G G � ∈ [0 . 1 , 0 . 3] I ( Th 4 )=0 . 3 I 1 , 2 ( Th 4 )=0 . 00 G F F G G I ( Th 5 )=0 . 1 I 1 , 2 ( Th 5 )=0 . 00 G F F G F I ( Th 6 )=0 . 2 I 1 , 2 ( Th 6 )=0 . 00 G F G G F t probability quantification B . 1 ,. 3 � 1 , 2 ( G 5 ) belief object agent time of the belief PDT Logic 7 / 12

Belief Operators - Example I 1 , 5 ( Th 1 ) ≈ 0 . 33 I ( Th 1 )=0 . 1 F F F F F I 1 , 5 ( Th 2 ) ≈ 0 . 67 I ( Th 2 )=0 . 2 F G F G F I ( Th 3 )=0 . 1 I 1 , 5 ( Th 3 )=0 . 00 F F F G G I ( Th 4 )=0 . 3 I 1 , 5 ( Th 4 )=0 . 00 G F F G G I ( Th 5 )=0 . 1 I 1 , 5 ( Th 5 )=0 . 00 G F F G F I ( Th 6 )=0 . 2 I 1 , 5 ( Th 6 )=0 . 00 G F G G F t probability quantification B . 1 ,. 3 � 1 , 2 ( G 5 ) belief object agent time of the belief B . 1 ,. 3 1 , 5 ( G 5 ) PDT Logic 7 / 12