🤖 AI Summary
Formal verification of bias between probabilistic outputs (e.g., sampling results) of AI systems and their target distributions remains challenging. Method: This paper introduces TPTND—the first probabilistic type-theoretic natural deduction system in which “trustworthiness” is defined as a decidable logical predicate. Its core innovation is the Trust operator and associated inference rules, enabling syntactic, derivable trust verification; trust judgments are modeled as statistical hypothesis tests on the distance between empirical frequencies and target probabilities, rigorously grounded via integrated probabilistic, computational-term, and logical semantics. Contribution/Results: We establish a complete metatheory, proving structural preservation of trustworthiness under both evaluation and logical inference. The framework enables rigorous, machine-checkable trust certification for outputs of complex probabilistic programs.
📝 Abstract
In this paper we present the probabilistic typed natural deduction calculus TPTND, designed to reason about and derive trustworthiness properties of probabilistic computational processes, like those underlying current AI applications. Derivability in TPTND is interpreted as the process of extracting $n$ samples of possibly complex outputs with a certain frequency from a given categorical distribution. We formalize trust for such outputs as a form of hypothesis testing on the distance between such frequency and the intended probability. The main advantage of the calculus is to render such notion of trustworthiness checkable. We present a computational semantics for the terms over which we reason and then the semantics of TPTND, where logical operators as well as a Trust operator are defined through introduction and elimination rules. We illustrate structural and metatheoretical properties, with particular focus on the ability to establish under which term evaluations and logical rules applications the notion of trustworthiness can be preserved.