2,361 research outputs found
The Magic Number Problem for Subregular Language Families
We investigate the magic number problem, that is, the question whether there
exists a minimal n-state nondeterministic finite automaton (NFA) whose
equivalent minimal deterministic finite automaton (DFA) has alpha states, for
all n and alpha satisfying n less or equal to alpha less or equal to exp(2,n).
A number alpha not satisfying this condition is called a magic number (for n).
It was shown in [11] that no magic numbers exist for general regular languages,
while in [5] trivial and non-trivial magic numbers for unary regular languages
were identified. We obtain similar results for automata accepting subregular
languages like, for example, combinational languages, star-free, prefix-,
suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free
languages, showing that there are only trivial magic numbers, when they exist.
For finite languages we obtain some partial results showing that certain
numbers are non-magic.Comment: In Proceedings DCFS 2010, arXiv:1008.127
Remarks on separating words
The separating words problem asks for the size of the smallest DFA needed to
distinguish between two words of length <= n (by accepting one and rejecting
the other). In this paper we survey what is known and unknown about the
problem, consider some variations, and prove several new results
Quantum Tetrahedra
We discuss in details the role of Wigner 6j symbol as the basic building
block unifying such different fields as state sum models for quantum geometry,
topological quantum field theory, statistical lattice models and quantum
computing. The apparent twofold nature of the 6j symbol displayed in quantum
field theory and quantum computing -a quantum tetrahedron and a computational
gate- is shown to merge together in a unified quantum-computational SU(2)-state
sum framework
Quotient Complexity of Regular Languages
The past research on the state complexity of operations on regular languages
is examined, and a new approach based on an old method (derivatives of regular
expressions) is presented. Since state complexity is a property of a language,
it is appropriate to define it in formal-language terms as the number of
distinct quotients of the language, and to call it "quotient complexity". The
problem of finding the quotient complexity of a language f(K,L) is considered,
where K and L are regular languages and f is a regular operation, for example,
union or concatenation. Since quotients can be represented by derivatives, one
can find a formula for the typical quotient of f(K,L) in terms of the quotients
of K and L. To obtain an upper bound on the number of quotients of f(K,L) all
one has to do is count how many such quotients are possible, and this makes
automaton constructions unnecessary. The advantages of this point of view are
illustrated by many examples. Moreover, new general observations are presented
to help in the estimation of the upper bounds on quotient complexity of regular
operations
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