Related papers: A formalization of Borel determinacy in Lean
As automated reasoning systems advance rapidly, there is a growing need for research-level formal mathematical problems to accurately evaluate their capabilities. To address this, we present Formal Conjectures, an evolving benchmark of…
We recently introduced p-automata, automata that read discrete-time Markov chains. We used turn-based stochastic parity games to define acceptance of Markov chains by a subclass of p-automata. Definition of acceptance required a cumbersome…
A precise description of the convexity of Gaussian measures is provided by sharp Brunn-Minkowski type inequalities due to Ehrhard and Borell. We show that these are manifestations of a game-theoretic mechanism: a minimax variational…
Let D = { d_n } be a countable collection of Delta^1_3 degrees. Assuming that all co-analytic games on integers are determined (or equivalently that all reals have ``sharps''), we prove that either D has a Delta^1_3-minimal upper bound, or…
We tell the story of how schemes were formalised in three different ways in the Lean theorem prover.
In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone's representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof…
We study a game first introduced by Martin (actually we use a slight variation of this game) which plays a role for measure analogous to the Banach-Mazur game for category. We first present proofs for the basic connections between this game…
Conway's Game of Life (GOL) is a cellular automaton that has captured the interest of hobbyists and mathematicians alike for more than 50 years. The Game of Life is Turing complete, and people have been building increasingly sophisticated…
We give a direct and elementary proof of the theorem on formal functions by studying the behaviour of the Godement resolution of a sheaf of modules under completion.
We study tree games developed recently by Matteo Mio as a game interpretation of the probabilistic $\mu$-calculus. With expressive power comes complexity. Mio showed that tree games are able to encode Blackwell games and, consequently, are…
In this article we prove results on logaritmic convexity of fixed points of stochastic kernel operators. These results are expected to play a key role in the economic application to strategic market games.
Game Logic is an excellent setting to study proofs-about-programs via the interpretation of those proofs as programs, because constructive proofs for games correspond to effective winning strategies to follow in response to the opponent's…
As mathematical induction is applied to prove statements on natural numbers, {\it continuous induction} (or, {\it real induction}) is a tool to prove some statements in real analysis.(Although, this comparison is somehow an overstatement.)…
We propose a new determinacy hypothesis for transfinite games, use the hypothesis to extend the perfect set theorem, prove relationships between various determinacy hypotheses, expose inconsistent versions of determinacy, and provide a…
We study the underlying mathematical properties of various partial order models of concurrency based on transition systems, Petri nets, and event structures, and show that the concurrent behaviour of these systems can be captured in a…
This paper deals with different concepts for characterizing the size of mathematical objects. A game theoretic investigation and generalization of two size concepts, which can both be formulated in topological terms, is provided: the so…
Game semantics aim at describing the interactive behaviour of proofs by interpreting formulas as games on which proofs induce strategies. In this article, we introduce a game semantics for a fragment of first order propositional logic. One…
We prove two determinacy and decidability results about two-players stochastic reachability games with partial observation on both sides and finitely many states, signals and actions.
This paper presents a new mathematical formalism that describes the quantization of games. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published in Physics Review Letters in 1999.…
We define a general framework of partition games for formulating two-player pebble games over finite structures. We show that one particular such game, which we call the invertible-map game, yields a family of polynomial-time approximations…