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

Logic, Probability and Action: A Situation Calculus Perspective

The unification of logic and probability is a long-standing concern in AI, and more generally, in the philosophy of science. In essence, logic provides an easy way to specify properties that must hold in every possible world, and probability allows us to further quantify the weight and ratio of the worlds that must satisfy a property. To that end, numerous developments have been undertaken, culminating in proposals such as probabilistic relational models. While this progress has been notable, a general-purpose first-order knowledge representation language to reason about probabilities and dynamics, including in continuous settings, is still to emerge. In this paper, we survey recent results pertaining to the integration of logic, probability and actions in the situation calculus, which is arguably one of the oldest and most well-known formalisms. We then explore reduction theorems and programming interfaces for the language. These results are motivated in the context of cognitive robotics (as envisioned by Reiter and his colleagues) for the sake of concreteness. Overall, the advantage of proving results for such a general language is that it becomes possible to adapt them to any special-purpose fragment, including but not limited to popular probabilistic relational models

Learning and Discovery

We formulate a dynamic framework for an individual decision-maker within which discovery of previously unconsidered propositions is possible. Using a standard game-theoretic representation of the state space as a tree structure generated by the actions of agents (including acts of nature), we show how unawareness of propositions can be represented by a coarsening of the state space. Furthermore we develop a semantics rich enough to describe the individual's awareness that currently undiscovered propositions may be discovered in the future. Introducing probability concepts, we derive a representation of ambiguity in terms of multiple priors, reflecting implicit beliefs about undiscovered proposition, and derive conditions for the special case in which standard Bayesian learning may be applied to a subset of unambiguous propositions. Finally, we consider exploration strategies appropriate to the context of discovery, comparing and contrasting them with learning strategies appropriate to the context of justification, and sketch applications to scientific research and entrepreneurship.

Excursions in first-order logic and probability: infinitely many random variables, continuous distributions, recursive programs and beyond

The unification of the first-order logic and probability has been seen as a long-standing concern in philosophy, AI and mathematics. In this talk, I will briefly review our recent results on revisiting that unification. Although there are plenty of approaches in communities such as statistical relational learning, automated planning, and neuro-symbolic AI that leverage and develop languages with logical and probabilistic aspects, they almost always restrict the representation as well as the semantic framework in various ways that does not fully explain how to combine first-order logic and probability theory in a general way. In many cases, this restriction is justified because it may be necessary to focus on practicality and efficiency. However, the search for a restriction-free mathematical theory remains ongoing. In this article, we discuss our recent results regarding the development of languages that support arbitrary quantification, possibly infinitely many ran- dom variables, both discrete and continuous distributions, as well as programming languages built on top of such features to include recursion and branching control