225 research outputs found
Factorizations of languages and commutativity conditions
Representations of languages as a product (catenation) of languages are investigated, where the factor languages are "prime", that is, cannot be decomposed further in a nontrivial manner. In general, such prime decompositions do not necessarily exist. If they exist, they are not necessarily unique - the number of factors can vary even exponentially. The paper investigates prime decompositions, as well as the commuting of the factors, especially for the case of finite languages. In particular, a technique about commuting is developed in Section 4, where the factorization of languages L1 and L2 is discussed under the assumption L1L2 = L2L1
LIPIcs
A regular language L of finite words is composite if there are regular languages Lâ,Lâ,âŚ,L_t such that L = â_{i = 1}^t L_i and the index (number of states in a minimal DFA) of every language L_i is strictly smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced and studied in [O. Kupferman and J. Mosheiff, 2015], where the complexity of deciding the primality of the language of a given DFA was left open, with a doubly-exponential gap between the upper and lower bounds. We study primality for unary regular languages, namely regular languages with a singleton alphabet. A unary language corresponds to a subset of â, making the study of unary prime languages closer to that of primality in number theory. We show that the setting of languages is richer. In particular, while every composite number is the product of two smaller numbers, the number t of languages necessary to decompose a composite unary language induces a strict hierarchy. In addition, a primality witness for a unary language L, namely a word that is not in L but is in all products of languages that contain L and have an index smaller than Lâs, may be of exponential length. Still, we are able to characterize compositionality by structural properties of a DFA for L, leading to a LogSpace algorithm for primality checking of unary DFAs
Using knot Floer invariants to detect prime knots
We present knot primality tests that are built from knot Floer homology. The
most basic of these is a simply stated and elementary consequence of Heegaard
Floer theory: if the two-variable knot Floer polynomial of a knot K is
irreducible, then K is prime. Improvements in this test yield a primality
condition that has been over 90 percent effective in identifying prime knots of
up to 30 crossings. As another illustration of the strength of these tools,
there are 1,315 non-hyperbolic prime knots with crossing number 20 or less; the
tests we develop prove the primality of over 96 percent of them.
The filtered chain homotopy class of the knot Floer hat complex of a knot K
has a unique minimal-dimension representative that is the direct sum of a
one-dimensional complex and two-dimensional complexes, each of which can be
assigned a parity. Let delta(K), b_e(K), and b_o(K) denote the dimension of
this minimal representative and the number of even and odd two-dimensional
summands, respectively. For a composite knot K, we observe that there is a
non-trivial factorization delta(K) = mn satisfying (m-1)(n-1) \le 4 min(b_e(K),
b_o(K)). This yields another knot primality test. One corollary is a simple
proof of Krcatovich's result that L-space knots are prime.Comment: 6 pages. The primality tests have been improved by using results of
Baldwin-Sivek showing that specific knots, including 5_2, are determined by
their knot Floer homolog
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