A First-Order Logic of Probability and Only Knowing in Unbounded Domains

Only knowing captures the intuitive notion that the beliefs of an agent are precisely those that follow from its knowledge base. It has previously been shown to be useful in characterizing knowledge-based reasoners, especially in a quantified setting. While this allows us to reason about incomplete knowledge in the sense of not knowing whether a formula is true or not, there are many applications where one would like to reason about the degree of belief in a formula. In this work, we propose a new general first-order account of probability and only knowing that admits knowledge bases with incomplete and probabilistic specifications. Beliefs and non-beliefs are then shown to emerge as a direct logical consequence of the sentences of the knowledge base at a corresponding level of specificity.

[1]  Gerhard Lakemeyer,et al.  Situations, Si! Situation Terms, No! , 2004, KR.

[2]  Johan van Benthem,et al.  Dynamic Update with Probabilities , 2009, Stud Logica.

[3]  Robert C. Moore.  Semantical Considerations on Nonmonotonic Logic , 1985, IJCAI.

[4]  Joseph Y. Halpern.  An Analysis of First-Order Logics of Probability , 1989, IJCAI.

[5]  D. Monderer,et al.  Approximating common knowledge with common beliefs , 1989 .

[6]  Hector J. Levesque,et al.  Reasoning about Noisy Sensors in the Situation Calculus , 1995, IJCAI.

[7]  Martín Abadi,et al.  Decidability and expressiveness for first-order logics of probability , 1989, 30th Annual Symposium on Foundations of Computer Science.

[8]  Gerhard Lakemeyer,et al.  Levesque's Axiomatization of only Knowing is Incomplete , 1995, Artif. Intell..

[9]  Gerhard Lakemeyer,et al.  Multi-agent Only-Knowing , 1996, CLIMA.

[10]  Ronald Fagin,et al.  Reasoning about knowledge , 1995 .

[11]  Aviad Heifetz,et al.  Probability Logic for Type Spaces , 2001, Games Econ. Behav..

[12]  Hector J. Levesque,et al.  Reasoning about Noisy Sensors and Effectors in the Situation Calculus , 1995, Artif. Intell..

[13]  Joseph Y. Halpern,et al.  Knowledge, probability, and adversaries , 1993, JACM.

[14]  Joseph Y. Halpern,et al.  Towards a Theory of Knowledge and Ignorance: Preliminary Report , 1989, NMR.

[15]  Sebastian Thrun,et al.  Probabilistic robotics , 2002, CACM.

[16]  David Poole,et al.  First-order probabilistic inference , 2003, IJCAI.

[17]  Vaishak Belle,et al.  Multi-Agent Only-Knowing Revisited , 2010, AlgoSyn.

[18]  Martin Meier,et al.  Probability Logic for Type Spaces , 2001 .

[19]  Pedro M. Domingos,et al.  A Tractable First-Order Probabilistic Logic , 2012, AAAI.

[20]  Joseph Y. Halpern,et al.  A logical characterization of iterated admissibility , 2009, TARK '09.

[21]  Miodrag Raskovic,et al.  Some first-order probability logics , 2000, Theor. Comput. Sci..

[22]  David Kaplan.  Quantifying in , 2004, Synthese.

[23]  Ian Pratt-Hartmann,et al.  Total Knowledge , 2000, AAAI/IAAI.

[24]  Fahiem Bacchus,et al.  Representing and reasoning with probabilistic knowledge , 1988 .

[25]  Hector J. Levesque,et al.  All I Know: A Study in Autoepistemic Logic , 1990, Artif. Intell..

[26]  Robert J. Aumann,et al.  Interactive epistemology II: Probability , 1999, Int. J. Game Theory.

[27]  Joseph Y. Halpern,et al.  Updating Sets of Probabilities , 1998, UAI.

[28]  Dexter Kozen,et al.  Semantics of probabilistic programs , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[29]  Gerhard Lakemeyer,et al.  ESP: A Logic of Only-Knowing, Noisy Sensing and Acting , 2007, AAAI.

[30]  Gerhard Lakemeyer,et al.  Only-Knowing: Taking It Beyond Autoepistemic Reasoning , 2005, AAAI.

[31]  Glenn Shafer,et al.  Perspectives on the theory and practice of belief functions , 1990, Int. J. Approx. Reason..

[32]  Arild Waaler,et al.  Semantics for multi-agent only knowing: extended abstract , 2005, TARK.

[33]  Joseph Y. Halpern.  Reasoning about uncertainty , 2003 .

[34]  Robert C. Moore.  A Formal Theory of Knowledge and Action , 1984 .

[35]  Ronald Fagin,et al.  Reasoning about knowledge and probability , 1988, JACM.

[36]  Fahiem Bacchus,et al.  Representing and reasoning with probabilistic knowledge - a logical approach to probabilities , 1991 .

[37]  Gerhard Lakemeyer,et al.  The logic of knowledge bases , 2000 .

[38]  Riccardo Rosati,et al.  On the decidability and complexity of reasoning about only knowing , 2000, Artif. Intell..

[39]  H. Gaifman.  Concerning measures in first order calculi , 1964 .

[40]  Nils J. Nilsson,et al.  Probabilistic Logic * , 2022 .