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Chase algorithms are indispensable in the domain of knowledge base querying, which enable the extraction of implicit knowledge from a given database via applications of rules from a given ontology. Such algorithms have proved beneficial in…
Typically, real-world stochastic processes are not easy to analyze. In this work we study the representation of any stochastic process as a memoryless innovation process triggering a dynamic system. We show that such a representation is…
Combining abstract, symbolic reasoning with continuous neural reasoning is a grand challenge of representation learning. As a step in this direction, we propose a new architecture, called neural equivalence networks, for the problem of…
The $Q$-learning algorithm is a simple and widely-used stochastic approximation scheme for reinforcement learning, but the basic protocol can exhibit instability in conjunction with function approximation. Such instability can be observed…
We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed…
We show that, if an integer sequence is given by a linear recurrence of constant rational coefficients, then it can be represented as the difference of two arithmetic terms with exponentiation, which do not contain any irrational constant.…
The multitime multiple recurrences are common in analysis of algorithms, computational biology, information theory, queueing theory, filters theory, statistical physics etc. The theoretical part about them is little or not known. That is…
Quantum models of computation are widely believed to be more powerful than classical ones. Efforts center on proving that, for a given problem, quantum algorithms are more resource efficient than any classical one. All this, however,…
We study log-concavity properties of real sequences $(a_n)_{n \ge 0}$ satisfying a $d$-th order linear recurrence whose coefficients are linear functions of $n$; the so-called P-recursive (or holonomic) sequences. Writing the recurrence in…
We define a triangular array closely related to Stern's diatomic array and show that for a fixed integer $r\geq 1$, the sum $u_r(n)$ of the $r$th powers of the entries in row $n$ satisfy a linear recurrence with constant coefficients. The…
Inductive proofs can be represented as proof schemata, i.e. as parameterized sequences of proofs defined in a primitive recursive way. Applications of proof schemata can be found in the area of automated proof analysis where the schemata…
The ability to abstract, count, and use System~2 reasoning are well-known manifestations of intelligence and understanding. In this paper, we argue, using the example of the ``Look and Say" puzzle, that although deep neural networks can…
The advent of Scientific Machine Learning has heralded a transformative era in scientific discovery, driving progress across diverse domains. Central to this progress is uncovering scientific laws from experimental data through symbolic…
We consider dynamical systems arising from substitutions over a finite alphabet. We prove that such a system is linearly repetitive if and only if it is minimal. Based on this characterization we extend various results from primitive…
Identifying patterns of relations among the units of a complex system from measurements of their activities in time is a fundamental problem with many practical applications. Here, we introduce a method that detects dependencies of any…
As a cornerstone of automated reasoning, equational reasoning finds equivalences between symbolic expressions and fuels advances across scientific disciplines. Yet, its potential remains limited by the exponential growth of equivalent…
Large Language Models (LLMs) excel in complex reasoning tasks but struggle with consistent rule application, exception handling, and explainability, particularly in domains like legal analysis that require both natural language…
Lothar Collatz had proposed in 1937 a conjecture in number theory called Collatz conjecture. Till today there is no evidence of proving or disproving the conjecture. In this paper, we propose an algorithmic approach for verification of the…
Coalgebras for analytic functors uniformly model graph-like systems where the successors of a state may admit certain symmetries. Examples of successor structure include ordered tuples, cyclic lists and multisets. Motivated by goals in…
Growing interest in modelling complex systems from brains to societies to cities using networks has led to increased efforts to describe generative processes that explain those networks. Recent successes in machine learning have prompted…