We consider shifted equality sets of the form $E_G(a,g_1,g_2)=\{w\mid g_1\left(w\right)=a{g}_2\left(w\right)\}$, where $g_1$ and $g_2$ are nonerasing morphisms and $a$ is a letter. We are interested in the family consisting of the languages $h\left(E_G\left(J\right)\right)$, where $h$ is a coding and $E_G\left(J\right)$ is a shifted equality set. We prove several closure properties for this family. Moreover, we show that every recursively enumerable language $L\subseteq A^{*}$ is a projection of a shifted equality set, that is, $L={\pi }_A\left(E_G(a,g_1,g_2)\right)$ for some (nonerasing) morphisms $g_1$ and $g_2$ and a letter $a$, where ${\pi }_A$ deletes the letters not in $A$. Then we deduce...

We consider shifted equality sets of the form EG(a,g1,g2) = {ω | g1(ω) = ag2(ω)}, where g1 and g2 are nonerasing morphisms and a is a letter. We are interested in the family consisting of the languages h(EG(J)), where h is a coding and (EG(J)) is a shifted equality set. We prove several closure properties for this family. Moreover, we show that every recursively enumerable language L ⊆ A* is a projection of a shifted equality set, that is, L = πA(EG(a,g1,g2)) for some (nonerasing) morphisms g1...

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities....

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities. As...

It is well-known that some of the most basic properties of words, like the commutativity ($xy=yx$) and the conjugacy ($xz=zy$), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation $x^m{y}^n=z^p$ has only periodic solutions in a free monoid, that is, if $x^m{y}^n=z^p$ holds with integers $m,n,p\ge 2$, then there exists a word $w$ such that $x,y,z$ are powers of $w$. This result, which received a lot of attention, was first proved by Lyndon and...

It is well-known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation xMyN = zP has only periodic solutions in a free monoid, that is, if xMyN = zP holds with integers m,n,p ≥ 2, then there exists a word w such that x, y, z are powers of w. This result, which received a lot...

This paper considers a distributed aggregative game problem for a group of players with misinformation, where each player has a different perception of the game. Player’s deception behavior is inevitable in this situation for reducing its own cost. We utilize hypergame to model the above problems and adopt $ϵ$-Nash equilibrium for hypergame to investigate whether players believe in their own cognition. Additionally, we propose a distributed deceptive algorithm for a player implementing deception and...

We generalize Jiroušek’s (right) composition operator in such a way that it can be applied to distribution functions with values in a “semifield“, and introduce (parenthesized) compositional expressions, which in some sense generalize Jiroušek’s “generating sequences” of compositional models. We say that two compositional expressions are equivalent if their evaluations always produce the same results whenever they are defined. Our first result is that a set system $\mathscr{H}$ is star-like with centre $X$ if...

Taking the view that infinite plays are draws, we study Conway non-terminating games and non-losing strategies. These admit a sharp coalgebraic presentation, where non-terminating games are seen as a final coalgebra and game contructors, such as disjunctive sum, as final morphisms. We have shown, in a previous paper, that Conway’s theory of terminating games can be rephrased naturally in terms of game (pre)congruences. Namely, various...

Taking the view that infinite plays are draws, we study Conway non-terminating games and non-losing strategies. These admit a sharp coalgebraic presentation, where non-terminating games are seen as a final coalgebra and game contructors, such as disjunctive sum, as final morphisms. We have shown, in a previous paper, that Conway’s theory of terminating games can be rephrased naturally in terms of game (pre)congruences. Namely, various...