Related papers: Deduction Theorem: The Problematic Nature of Commo…
In deduction modulo, a theory is not represented by a set of axioms but by a congruence on propositions modulo which the inference rules of standard deductive systems---such as for instance natural deduction---are applied. Therefore, the…
In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of…
Inference is the process of using facts we know to learn about facts we do not know. A theory of inference gives assumptions necessary to get from the former to the latter, along with a definition for and summary of the resulting…
As large language models become increasingly capable, it is critical that their outputs can be easily checked by less capable systems. Prover-verifier games can be used to improve checkability of model outputs, but display a degradation in…
Motivated by emerging decentralized applications, the \emph{game of coding} framework has been recently introduced to address scenarios where the adversary's control over coded symbols surpasses the fundamental limits of traditional coding…
We study the strength of set-theoretic axioms needed to prove Rabin's theorem on the decidability of the MSO theory of the infinite binary tree. We first show that the complementation theorem for tree automata, which forms the technical…
We study when a sound arithmetic theory $\mathcal S{\supseteq}S^1_2$ with polynomial-time decidable axioms efficiently proves the bounded consistency statements $Con_{\mathcal S{+}\phi}(n)$ for a true sentence $\phi$. Equivalently, we ask…
A distortion function, which captures the payoff gap between a player's actual payoff and her true payoff, is introduced and used to analyze games. In our proposed framework, we argue that players' actual payoff functions should be used to…
In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a…
Investigations of causality in operational systems aim at providing human-understandable explanations of why a system behaves as it does. There is, in particular, a demand to explain what went wrong on a given counterexample execution that…
Investigations of causality in operational systems aim at providing human-understandable explanations of why a system behaves as it does. There is, in particular, a demand to explain what went wrong on a given counterexample execution that…
Games offer a compelling paradigm for developing general reasoning capabilities in language models, as they naturally demand strategic planning, probabilistic inference, and adaptive decision-making. However, existing self-play approaches…
In this paper, we give a counter-example, in the general case, Kronecker theorem will derive contradiction. Kronecker theorem be correct after removing some conditions.
The basic problem in the cooperation theory is to justify the cooperation. Here we propose a new approach, where players are driven by their altruism to cooperate or not. The probability of cooperation depends also on the co-player's…
Deceptive games are games where the reward structure or other aspects of the game are designed to lead the agent away from a globally optimal policy. While many games are already deceptive to some extent, we designed a series of games in…
Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between $\Pi^1_2$ principles over $\omega$-models of…
Across machine learning (ML) sub-disciplines researchers make mathematical assumptions to facilitate proof-writing. While such assumptions are necessary for providing mathematical guarantees for how algorithms behave, they also necessarily…
We develop a general game-theoretic framework for reasoning about strategic agents performing possibly costly computation. In this framework, many traditional game-theoretic results (such as the existence of a Nash equilibrium) no longer…
We extend Kuhn's Theorem to games of the extensive form with unawareness. We prove that if a game of the extensive form with unawareness has perfect recall, then for each mixed strategy there is an equivalent behavior strategy. We show that…
The growing use of machine learning models in consequential settings has highlighted an important and seemingly irreconcilable tension between transparency and vulnerability to gaming. While this has sparked sizable debate in legal…