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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.

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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.

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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.

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The synthesis of reactive systems aims for the automated construction of strategies for systems that interact with their environment. Whereas the synthesis approach has the potential to change the development of reactive systems significantly due to the avoidance of manual implementation, it still suffers from a lack of efficient synthesis algorithms for many application scenarios. The translation of the system specification into an automaton that allows for strategy construction (if a winning strategy exists) is nonelementary in the length of the specification in S1S and doubly exponential for LTL, raising the need of highly specialized algorithms. In this article, we present an approach on how to reduce this state space explosion in the construction of this automaton by exploiting a monotonicity property of specifications. For this, we introduce window counting constraints that allow for step-wise refinement or abstraction of specifications. In an iterative synthesis procedure, those window counting constraints are used to construct automata representing over- or under-approximations (depending on the counting constraint) of constraint-compliant behavior. Analysis results on winning regions of previous iterations are used to reduce the size of the next automaton, leading to an overall reduction of the state space explosion extent. We present the implementation results of the iterated synthesis for a zero-sum game setting as proof of concept. Furthermore, we discuss the […]

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In this work, we study the computability of topological graphs, which are obtained by gluing arcs and rays together at their endpoints. We prove that every semicomputable graph in a computable metric space can be approximated, with arbitrary precision, by its computable subgraph with computable endpoints.

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