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Automata learning is a popular technique used to automatically construct an automaton model from queries. Much research went into devising ad hoc adaptations of algorithms for different types of automata. The CALF project seeks to unify…
We introduce a large family of combinatorial objects, called standard puzzles, defined by very simple rules. We focus on the standard puzzles for which the enumeration problems can be solved by explicit formulas or by classical numbers,…
Code generation, defined as automatically writing a piece of code to solve a given problem for which an evaluation function exists, is a classic hard AI problem. Its general form, writing code using a general language used by human…
In this paper we use a natural iteration technique to prove existence of solutions to nonlinear Dirichlet problems. Among the examples included is the prescribed mean curvature equation. The nature of the technique allows applications to…
Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions…
Szemeredi's regularity lemma can be viewed as a rough structure theorem for arbitrary dense graphs, decomposing such graphs into a structured piece (a partition into cells with edge densities), a small error (corresponding to irregular…
In analogy with the regularity lemma of Szemer\'edi, regularity lemmas for polynomials shown by Green and Tao (Contrib. Discrete Math. 2009) and by Kaufman and Lovett (FOCS 2008) modify a given collection of polynomials \calF =…
In automata theory, while determinisation provides a standard route to solving many common problems in automata theory, some weak forms of nondeterminism can be dealt with in some problems without costly determinisation. For example, the…
Finite automata (FA) are a fundamental computational abstraction that is widely used in practice for various tasks in computer science, linguistics, biology, electrical engineering, and artificial intelligence. Given an input word, an FA…
Objects or structures that are regular take uniform dimensions. Based on the concepts of regular models, our previous research work has developed a system of a regular ontology that models learning structures in a multiagent system for…
In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a…
A differential algebra of nonlinear generalized functions is presented as a tool for a wide range of nonsmooth nonlinear problems. The power of the differential algebra is used to do mathematical calculations or proofs; then the final…
The HOM problem, which asks whether the image of a regular tree language under a given tree homomorphism is again regular, is known to be decidable [Godoy & Gim\'enez: The HOM problem is decidable. JACM 60(4), 2013]. However, the problem…
The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's…
In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value…
In this paper one shows if the number of natural solutions of a general linear equation is limited or not. Also, it is presented a method of solving the Diophantine equation $ax-by=c$ in the set of natural numbers, and an example of solving…
Analogous to regular string and tree languages, regular languages of directed acyclic graphs (DAGs) are defined in the literature. Although called regular, those DAG-languages are more powerful and, consequently, standard problems have a…
Data assisted reconstruction algorithms, incorporating trained neural networks, are a novel paradigm for solving inverse problems. One approach is to first apply a classical reconstruction method and then apply a neural network to improve…
There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions…
Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…