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Combinatorics
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
It can generally be assumed that a password uses some combination of upper and lowercase letters, digits, and some set of special characters. This leads to an alphabet size of around 90 (depending on the special characters allowed). It can also be assumed that passwords are generally between 4 and 20 characters long. Does this mean that every password has strength equal to log2(904+905+⋯9020)≈1324?
Introduction
Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002
John N. Mordeson, Davender S. Malik
Let X be a finite set of symbols called an alphabet. Let L be a lattice with minimum element 0 . An L-fuzzy language over X is defined to be a function from X* into L. At times, we omit the symbol L if its presence is clear. If μ is an L-fuzzy language over X and x∈X*, then μ(x) represents the membership grade for x to be in the L-fuzzy language.
The Early Shaping of Cognitive Science by Artificial Intelligence
Published in Alessio Plebe, Pietro Perconti, The Future of the Artificial Mind, 2021
Alessio Plebe, Pietro Perconti
There is no doubt that the combinatorics on words is rich with difficult problems, what Thue did not foresee was the tremendous impact of his pleasant speculations on the domains of linguistics and computer science. Words, even if still called ‘words’, are not only meaningless, but can also be made up of symbols other than the letters of the alphabet. The symbols should be defined by a finite set, still called ‘alphabet’. The basic operation on words is the concatenation, just putting words one after the other. Being the most common operation of words, it has no specific symbol, quite like the product in algebra: if w and v are two words, their concatenation is simply written as wv. Given an alphabet of basic symbols A, by repeatedly applying concatenation to those symbols, and to the results of concatenations, infinite different words can be generated. In modern abstract algebra jargon the set of all these words is denoted by A*, and called the Kleene star of A, after the American mathematician Stephen Kleene (1952). The set A*, together with the concatenation operation and the empty word ε is the type of object called monoid in abstract algebra, that has the following properties: associativity: ∀w, v, u ∈ A* (wv)u = w(vu);closure: ∀w, v ∈ A* wv ∈ A*;identity: ∀w ∈ A*wε = w.
Most-intersection of countable sets
Published in Journal of Applied Non-Classical Logics, 2021
An alphabet is a finite set of symbols. A string is a finite sequence of symbols over an alphabet. A language is a set of strings. The length of a string w, denoted by , is the number of symbols in w. The unique string of length 0 is called the empty string and is denoted by ϵ. The concatenation of two strings u and v is simply denoted by uv. Note that for any string u, we have that . For a set of symbols Σ and a given , denotes the set of all strings of length i over Σ.
Bounded topological speedups
Published in Dynamical Systems, 2018
Lori Alvin, Drew D. Ash, Nicholas S. Ormes
Let denote a finite set which we will refer to as an alphabet, the elements of which we will refer to as symbols. Let denote the set of finite concatenations of symbols in which we will refer to as words. For a word w = w1w2…wn, we let |w| = n denote the length of w.
Synthesis of regular expression problems and solutions
Published in International Journal of Computers and Applications, 2020
A symbol is an item. A symbol maybe terminal or nonterminal symbol. Terminal symbols (denoted by lowercase letters, e.g. a) are not rewritable while nonterminal symbols (denoted by uppercase letters, e.g. A) are rewritable. An alphabet (denoted by Σ) is a finite set of symbols. A string is the concatenation of zero or more symbols. A language is a set of strings.