Epistemic Probability Logic Simplified

We propose a logic for reasoning about (multi-agent) epistemic probability models, and for epistemic probabilistic model checking. Epistemic probability models are multi-agent Kripke models that assign to each agent an equivalence relation on worlds, together with function from worlds to positive rationals (a lottery). The difference with the usual approach is that probability is linked to knowledge rather than belief, and that knowledge is equated with certainty. Contributions of this paper are a semantics with a single lottery, an adaptation of the Hennessy-Milner Theorem, a completeness result for epistemic probability logic, and information about its model checking complexity. In particular, we state the PSPACEcompleteness of the model checking in the dynamic version with action models of this framework. One of the purposes of the logic is model checking for epistemic probability logic. A prototype model checker for the logic exists. This program can be used to keep track of information flow about aleatory acts among multiple agents. 1. PROBABILITY AS A FUNCTION OF DEGREE OF INFORMATION A classical view of probability theory is that probability measures degree of information. Here is a characteristic quote from [16]: Dans les choses qui ne sont que vraisemblables, la différence des données que chaque homme a sur elles, est une des causes principales de la diversité des opinions que l’on voit régner sur les mêmes objects. (Laplace) This paper presents a logic of probability and knowledge where the two are related as follows: Agent a knows φ iff the probability a assigns to φ equals 1. Let Paφ be the probability that agent a assigns to φ. Certainty implies Truth (Paφ = 1)→ φ. Positive Introspection into Certainty (Paφ = 1)→ Pa(Paφ = 1) = 1. Appears in: Proceedings of the 13th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2014), Lomuscio, Scerri, Bazzan, Huhns (eds.), May, 5–9, 2014, Paris, France. Copyright c © 2014, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved. Negative Introspection into Certainty (Paφ < 1)→ Pa(Paφ < 1) = 1. Our proposal has obvious relations to earlier proposals on combining knowledge and probability [8, 14, 13, 5, 4, 11], and many more. A key difference is that these proposals do not equate knowledge with certainty. Although, in real applications, knowledge and certainty are strongly related. We deal with such examples in section 2 when we present lotteries. This notably simplify the framework of epistemic probability logic we present in section 3. In particular, we will present models with a single lottery and in section 4 we prove that semantics with a single lottery and with several lotteries are the same. We then prove the Hennessy-Milner Theorem for epistemic probability logic in section 5. In section 6, we give an axiomatization for our epistemic probabilistic logic based on [8] and we prove that S5 axioms can be retrieved. In section 7, we deal with the model checking procedure that runs in polynomial time. In section 8, we see how to add action models of Dynamic epistemic logic and we provide a PSPACE-completeness proof for the model checking problem with a dynamic operator in the language. 2. LOTTERIES A W -lottery l is a function from a (finite) set of worlds W to the set of positive (non-zero) rationals, i.e., l : W → Q. If we have a lottery l : W → Q and a blockB ⊆W in a partition of W , then this determines a probability distribution P on B, by means of (we assume that B 6= ∅): P (w) = l(w) ∑ {l(w′) | w′ ∈ B} . Example 1 Say there are two urns,U and V . U contains one black marble and two white marbles, V contains one black marble and one white marble.