Search results for " Formal"
showing 10 items of 273 documents
Abelian combinatorics on words: A survey
2022
We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these results.
Splicing Systems from Past to Future: Old and New Challenges
2014
A splicing system is a formal model of a recombinant behaviour of sets of double stranded DNA molecules when acted on by restriction enzymes and ligase. In this survey we will concentrate on a specific behaviour of a type of splicing systems, introduced by P\u{a}un and subsequently developed by many researchers in both linear and circular case of splicing definition. In particular, we will present recent results on this topic and how they stimulate new challenging investigations.
Computational Limitations of Affine Automata
2019
We present two new results on the computational limitations of affine automata. First, we show that the computation of bounded-error rational-values affine automata is simulated in logarithmic space. Second, we give an impossibility result for algebraic-valued affine automata. As a result, we identify some unary languages (in logarithmic space) that are not recognized by algebraic-valued affine automata with cutpoints.
Monoids and Maximal Codes
2011
In recent years codes that are not Uniquely Decipherable (UD) are been studied partitioning them in classes that localize the ambiguities of the code. A natural question is how we can extend the notion of maximality to codes that are not UD. In this paper we give an answer to this question. To do this we introduce a partial order in the set of submonoids of a monoid showing the existence, in this poset, of maximal elements that we call full monoids. Then a set of generators of a full monoid is, by definition, a maximal code. We show how this definition extends, in a natural way, the existing definition concerning UD codes and we find a characteristic property of a monoid generated by a maxi…
Quantum, stochastic, and pseudo stochastic languages with few states
2014
Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all th…
Abelian Repetitions in Sturmian Words
2012
We investigate abelian repetitions in Sturmian words. We exploit a bijection between factors of Sturmian words and subintervals of the unitary segment that allows us to study the periods of abelian repetitions by using classical results of elementary Number Theory. We prove that in any Sturmian word the superior limit of the ratio between the maximal exponent of an abelian repetition of period $m$ and $m$ is a number $\geq\sqrt{5}$, and the equality holds for the Fibonacci infinite word. We further prove that the longest prefix of the Fibonacci infinite word that is an abelian repetition of period $F_j$, $j>1$, has length $F_j(F_{j+1}+F_{j-1} +1)-2$ if $j$ is even or $F_j(F_{j+1}+F_{j-1}…
Abelian Powers and Repetitions in Sturmian Words
2016
Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period $m$ starting at a given position in any Sturmian word of rotation angle $\alpha$. vAs an analogue of the critical exponent, we introduce the abelian critical exponent $A(s_\alpha)$ of a Sturmian word $s_\alpha$ of angle $\alpha$ as the quantity $A(s_\a…
Enumeration and Structure of Trapezoidal Words
2013
Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated …
The Descriptive Complexity Approach to LOGCFL
1998
Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's ``hardest context-free language'' is LOGCFL-complete under quantifier-free BIT-free proj…
New Results on Vector and Homing Vector Automata
2019
We present several new results and connections between various extensions of finite automata through the study of vector automata and homing vector automata. We show that homing vector automata outperform extended finite automata when both are defined over $ 2 \times 2 $ integer matrices. We study the string separation problem for vector automata and demonstrate that generalized finite automata with rational entries can separate any pair of strings using only two states. Investigating stateless homing vector automata, we prove that a language is recognized by stateless blind deterministic real-time version of finite automata with multiplication iff it is commutative and its Parikh image is …