Assuming that no family of polynomial-size Boolean circuits can factorize a constant fraction of all products of two $n$-bit primes, we show that the bounded arithmetic theory $\text{PV}_1$, even when augmented by the sharply bounded choice scheme $BB(Σ^b_0)$, cannot prove that every number has some prime divisor. By the completeness theorem, it follows that under this assumption there is a model $M$ of $\text{PV}_1$ that contains a nonstandard number $m$ which has no prime factorization.
Topology may be interpreted as the study of verifiability, where opens correspond to semi-decidable properties. In this paper we make a distinction between verifiable properties themselves and processes which carry out the verification procedure. The former are simply opens, while we call the latter \emph{machines}. Given a frame presentation $\mathcal{O} X = \langle G \mid R\rangle$ we construct a space of machines $Σ^{Σ^G}$ whose points are given by formal combinations of basic machines corresponding to generators in $G$. This comes equipped with an `evaluation' map making it a weak exponential with base $Σ$ and exponent $X$. When it exists, the true exponential $Σ^X$ occurs as a retract of machine space. We argue this helps explain why some spaces are exponentiable and others not. We then use machine space to study compactness by giving a purely topological version of Escardó's algorithm for universal quantification over compact spaces in finite time. Finally, we relate our study of machine space to domain theory and domain embeddings.
We introduce a fragment of second-order unification, referred to as \emph{Second-Order Ground Unification (SOGU)}, with the following properties: (i) only one second-order variable is allowed, and (ii) first-order variables do not occur. We study an equational variant of SOGU where the signature contains \textit{associative} binary function symbols (ASOGU) and show that Hilbert's 10$^{th}$ problem is reducible to ASOGU unifiability, thus proving undecidability. Our reduction provides a new lower bound for the undecidability of second-order unification, as previous results required first-order variable occurrences, multiple second-order variables, and/or equational theories involving \textit{length-reducing} rewrite systems. Furthermore, our reduction holds even in the case when associativity of the binary function symbol is restricted to \emph{power associative}, i.e. f(f(x,x),x)= f(x,f(x,x)), as our construction requires a single constant.
We present a data-driven approach to the quantitative verification of probabilistic programs and stochastic dynamical models. Our approach leverages neural networks to compute tight and sound bounds for the probability that a stochastic process hits a target condition within finite time. This problem subsumes a variety of quantitative verification questions, from the reachability and safety analysis of discrete-time stochastic dynamical models, to the study of assertion-violation and termination analysis of probabilistic programs. We rely on neural networks to represent supermartingale certificates that yield such probability bounds, which we compute using a counterexample-guided inductive synthesis loop: we train the neural certificate while tightening the probability bound over samples of the state space using stochastic optimisation, and then we formally check the certificate's validity over every possible state using satisfiability modulo theories; if we receive a counterexample, we add it to our set of samples and repeat the loop until validity is confirmed. We demonstrate on a diverse set of benchmarks that, thanks to the expressive power of neural networks, our method yields smaller or comparable probability bounds than existing symbolic methods in all cases, and that our approach succeeds on models that are entirely beyond the reach of such alternative techniques.
