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Recently, it was shown that any theory of strings containing the string-replace function (even the most restricted version where pattern/replacement strings are both constant strings) becomes undecidable if we do not impose some kind of…
We show that certain determinantal functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. These are shown to be true in complete generality; that is, no properties of…
We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of…
We address the following decision problem. Given a numeration system $U$ and a $U$-recognizable set $X\subseteq\mathbb{N}$, i.e. the set of its greedy $U$-representations is recognized by a finite automaton, decide whether or not $X$ is…
Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is of type 1 if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative…
We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding…
We are lifting classical problems from single instances to regular sets of instances. The task of finding a positive instance of the combinatorial problem $P$ in a potentially infinite given regular set is equivalent to the so called…
The classical decision problem, as it is understood today, is the quest for a delineation between the decidable and the undecidable parts of first-order logic based on elegant syntactic criteria. In this paper, we treat the concept of…
MapReduce is a popular programming model for data parallel computation. In MapReduce, the reducer produces an output from a list of inputs. Due to the scheduling policy of the platform, the inputs may arrive at the reducers in different…
In recent years there has been considerable interest in theories over string equations, length function, and string-number conversion predicate within the formal verification, software engineering, and security communities. SMT solvers for…
First-order linear real arithmetic enriched with uninterpreted predicate symbols yields an interesting modeling language. However, satisfiability of such formulas is undecidable, even if we restrict the uninterpreted predicate symbols to…
Let $\mathbb{K}$ be a finite commutative ring, and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Let $A$ and $B$ be two $n \times n$-matrices over $\mathbb{L}$ that have the same characteristic polynomial. The main result of this…
A weighted automaton is functional if any two accepting runs on the same finite word have the same value. In this paper, we investigate functional weighted automata for four different measures: the sum, the mean, the discounted sum of…
Characteristic formulae give a complete logical description of the behaviour of processes modulo some chosen notion of behavioural semantics. They allow one to reduce equivalence or preorder checking to model checking, and are exactly the…
Quotient is a basic operation of formal languages, which plays a key role in the construction of minimal deterministic finite automata (DFA) and the universal automata. In this paper, we extend this operation to formal power series and…
In this work, the commutator of any two reasonable functions of several pairs of canonical conjugate operators is obtained as a sum of terms of partial derivatives of those functions (equations 9, 10 or 11). When applied to quantum…
The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be…
Recently, symbolic structures were proposed as finite representations of potentially infinite first-order structures, where Linear Integer Arithmetic terms and formulas define the domain and interpretations of a structure. We generalize…
Previously, mathematicians Steven Krantz and Jeffery McNeal studied a type of positive numbers permutation called $\lambda$-permutation. This type of permutation, when applied to the index of terms of a series, is defined to be both…
We call an operator algebra A {\em reversible} if A with reversed multiplication is also an abstract operator algebra (in the modern operator space sense). This class of operator algebras is intimately related to the {\em symmetric operator…