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Exactly solving first-order constraints (i.e., first-order formulas over a certain predefined structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers it is promising to compute approximate…
We investigate quantifier alternation hierarchies in first-order logic on finite words. Levels in these hierarchies are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a…
Image colorization aims to bring colors back to grayscale images. Automatic image colorization methods, which requires no additional guidance, struggle to generate high-quality images due to color ambiguity, and provides limited user…
This paper seeks to apply categorical logic to the design of artificial intelligent agents that reason symbolically about objects more richly structured than sets. Using Johnstone's sequent calculus of terms- and formulae-in-context, we…
In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] with pair related quantifiers and constructs, in view of possible applications in the field…
The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road…
A coloring of edges of a finite directed graph turns the graph into finite-state automaton. The synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton…
Complex systems are difficult to study not only because they are nonlinear, multiscale, and often nonstationary, but because their scientifically relevant organization is often invisible at the level of individual components, pairwise…
In this paper, we present improved algorithms for the $(\Delta+1)$ (vertex) coloring problem in the Congested-Clique model of distributed computing. In this model, the input is a graph on $n$ nodes, initially each node knows only its…
Automated theorem proving in first-order logic is an active research area which is successfully supported by machine learning. While there have been various proposals for encoding logical formulas into numerical vectors -- from simple…
Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for…
We combine Recurrent Neural Networks with Tensor Product Representations to learn combinatorial representations of sequential data. This improves symbolic interpretation and systematic generalisation. Our architecture is trained end-to-end…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
When the color distribution of input images changes at inference, the performance of conventional neural network architectures drops considerably. A few researchers have begun to incorporate prior knowledge of color geometry in neural…
We analyze a model of dynamically broken topcolor in the limit in which the number of colors is large. We show that the second order nature of the phase transition, necessary for the success of topcolor models, passes the nontrivial check…
We count the number of countable homogeneous colored linear orderings in $k$ colors. Relatedly, we count the number of countable $C_{n,m}$-homogeneous linear orderings. $C_{n,m}$-homogeneity is a strong homogeneity notion that approximates…
We regard explanations as a blending of the input sample and the model's output and offer a few definitions that capture various desired properties of the function that generates these explanations. We study the links between these…
We investigate the expressive power of first-order quantifications in the context of monadic second-order logic over pictures. We show that k+1 set quantifier alternations allow to define a picture language that cannot be defined using k…
First-order relational languages have been used in MDP planning and reinforcement learning (RL) for two main purposes: specifying MDPs in compact form, and representing and learning policies that are general and not tied to specific…
Language models are useful adjuncts to optical models for producing accurate optical character recognition (OCR) results. One factor which limits the power of language models in this context is the existence of many specialized domains with…