Máquina de Turing Analógica para Ensino de Linguagens Formais e Autômatos

The article describes the development of a practical device for teachingin the area of Computer Theory. In the study, an adaptationof the Turing Machine is presented, using hardware and softwareintegration to interpret Formal Languages. Simulating an Automaton,sensors and motors are used to move the device head to the leftand right and to read and write the input tape. The developmentof the mechanism is described in two parts, the first includes thehardware that consists of the construction and adaptation of theTuring Machine, the second the implementation of the software andcommunication part between both. The developed device, allowsthe interpretation of a binary alphabet (0, 1), where an input word isaccepted, and as an output result, such device rejected or acceptedthe word.

On the number of distinct k-decks: Enumeration and bounds

<p style='text-indent:20px;'>The <i><inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-deck</i> of a sequence is defined as the multiset of all its subsequences of length <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id="M4">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> denote the number of distinct <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula>-decks for binary sequences of length <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>. For binary alphabet, we determine the exact value of <inline-formula><tex-math id="M7">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> for small values of <inline-formula><tex-math id="M8">\begin{document}$ k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ n $\end{document}</tex-math></inline-formula>, and provide asymptotic estimates of <inline-formula><tex-math id="M10">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M11">\begin{document}$ k $\end{document}</tex-math></inline-formula> is fixed.</p><p style='text-indent:20px;'>Specifically, for fixed <inline-formula><tex-math id="M12">\begin{document}$ k $\end{document}</tex-math></inline-formula>, we introduce a trellis-based method to compute <inline-formula><tex-math id="M13">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> in time polynomial in <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math></inline-formula>. We then compute <inline-formula><tex-math id="M15">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M16">\begin{document}$ k \in \{3,4,5,6\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ k \leqslant n \leqslant 30 $\end{document}</tex-math></inline-formula>. We also improve the asymptotic upper bound on <inline-formula><tex-math id="M18">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula>, and provide a lower bound thereupon. In particular, for binary alphabet, we show that <inline-formula><tex-math id="M19">\begin{document}$ D_k(n) = O\bigl(n^{(k-1)2^{k-1}+1}\bigr) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}$ D_k(n) = \Omega(n^k) $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M21">\begin{document}$ k = 3 $\end{document}</tex-math></inline-formula>, we moreover show that <inline-formula><tex-math id="M22">\begin{document}$ D_3(n) = \Omega(n^6) $\end{document}</tex-math></inline-formula> while the upper bound on <inline-formula><tex-math id="M23">\begin{document}$ D_3(n) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M24">\begin{document}$ O(n^9) $\end{document}</tex-math></inline-formula>.</p>

New algebraic and geometric constructs arising from Fibonacci numbers

Fibonacci numbers are the basis of a new geometric construction that leads to the definition of a family $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ N } of octagons that come very close to the regular octagon. Such octagons, in some previous articles, have been given the name of Carboncettus octagons for historical reasons. Going further, in this paper we want to introduce and investigate some algebraic constructs that arise from the family $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ N } and therefore from Fibonacci numbers: From each Carboncettus octagon $$C_n$$ C n , it is possible to obtain an infinite (right) word $$W_n$$ W n on the binary alphabet $$\{0,1\}$$ { 0 , 1 } , which we will call the nth Carboncettus word. The main theorem shows that all the Carboncettus words thus defined are Sturmian words except in the case $$n=5$$ n = 5 . The fifth Carboncettus word $$W_5$$ W 5 is in fact the only word of the family to be purely periodic: It has period 17 and periodic factor 000 100 100 010 010 01. Finally, we also define a further word $$W_{\infty }$$ W ∞ named the Carboncettus limit word and, as second main result, we prove that the limit of the sequence of Carboncettus words is $$W_{\infty }$$ W ∞ itself.

Adapting Logic to Physics: The Quantum-Like Eigenlogic Program

Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet { 0 , 1 } using projection operators to the binary alphabet { + 1 , − 1 } employing reversible involution operators. Also, many-valued logical operators are synthesized, for whatever alphabet, using operator methods based on Lagrange interpolation and on the Cayley–Hamilton theorem. Considering a superposition of logical input states one gets a fuzzy logic representation where the fuzzy membership function is the quantum probability given by the Born rule. Historical parallels from Boole, Post, Poincaré and Combinatory Logic are presented in relation to probability theory, non-commutative quaternion algebra and Turing machines. An extension to first order logic is proposed inspired by Grover’s algorithm. Eigenlogic is essentially a logic of operators and its truth-table logical semantics is provided by the eigenvalue structure which is shown to be related to the universality of logical quantum gates, a fundamental role being played by non-commutativity and entanglement.

Parikh q-Matrices and q-Ambiguous Words

The Parikh matrix mapping plays an important role in the study of words through numerical properties. The Parikh [Formula: see text]-matrix mapping, introduced by Egecioglu and Ibarra (2004) as an extension of the Parikh matrix mapping, maps words to matrices with polynomial entries in [Formula: see text] A word [Formula: see text] over an ordered alphabet [Formula: see text] is said to be [Formula: see text]-ambiguous if there exists another word [Formula: see text] over [Formula: see text] such that both the words have same Parikh [Formula: see text]-matrix. Here we derive several properties of [Formula: see text]-ambiguous words, in particular, for a binary alphabet.

Surface and Curve Languages

In the area of combinatorial studies of string languages under formal grammars, the concept of Generalized Parikh Vector (GPV) gives the positions of symbols in linear strings. It has been proved that GPVs of the strings of the same length lie on a hyper plane. The studies on GPV of strings over a binary alphabet gave rise to the concept of line languages. The concept has been extended to surface languages in this paper.

Reset Complexity of Ideal Languages Over a Binary Alphabet

We prove PSPACE-completeness of checking whether a given ideal language serves as the language of reset words for some automaton with at most four states over a binary alphabet. We compare the reset complexity and the state complexity for languages related to slowly synchronizing automata.

Existence of words over a binary alphabet free from squares with mismatches

The paper is concerned with the problem of existence of periodic structures in words from formal languages. Squares (that is, fragments of the form xx, where x is an arbitrary word) and Δ-squares (that is, fragments of the form xy, where a word x differs from a word y by at most Δ letters) are considered as periodic structures. We show that in a binary alphabet there exist arbitrarily long words free from Δ-squares with length at most 4Δ+4. In particular, a method of construction of such words for any Δ is given.

Introduction

In a short article written in Latin in 1875 [C1], Elwin Bruno Christoffel introduced a class of words on a binary alphabet, now called Christoffel words. They were followed in the twentieth century by the theory of Sturmian sequences, introduced by Morse and Hedlund in 1940 [...