scholarly journals Abelian maximal pattern complexity of words

2013 ◽  
Vol 35 (1) ◽  
pp. 142-151 ◽  
Author(s):  
TETURO KAMAE ◽  
STEVEN WIDMER ◽  
LUCA Q. ZAMBONI
Keyword(s):  
Lower Bound ◽  
Pattern Complexity ◽  
Infinite Words ◽  
Abelian Equivalence

AbstractIn this paper, we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection. We show that in the case of binary words, the bound is actually achieved and gives a characterization of recurrent aperiodic words.

scholarly journals On a generalization of Abelian equivalence and complexity of infinite words

2013 ◽  
Vol 120 (8) ◽  
pp. 2189-2206 ◽  
Author(s):  
Juhani Karhumaki ◽  
Aleksi Saarela ◽  
Luca Q. Zamboni
Keyword(s):  
Infinite Words ◽  
Abelian Equivalence

Some results involving HNBUE distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1038-1044
Author(s):  
A. P. Basu ◽  
S. N. U. A. Kirmani
Keyword(s):  
Lower Bound ◽  
Exponential Distribution ◽  
Renewal Process ◽  
Renewal Function ◽  
Underlying Distribution

A characterization of the exponential distribution in the class of all distributions which are HNBUE or HNWUE is proved. An upper (a lower) bound is obtained on the renewal function of a renewal process when the underlying distribution is HNBUE (HNWUE).

scholarly journals Heuristic for estimation of multiqubit genuine multipartite entanglement

2015 ◽  
Vol 13 (03) ◽  
pp. 1550023 ◽  
Author(s):  
Paulo E. M. F. Mendonça ◽  
Marcelo A. Marchiolli ◽  
Gerard J. Milburn
Keyword(s):  
Lower Bound ◽  
Density Matrix ◽  
Lower Bounds ◽  
Multipartite Entanglement ◽  
Zero Temperature ◽  
Mixed States ◽  
X State ◽  
Computational Basis ◽  
Genuine Multipartite Entanglement

For every N-qubit density matrix written in the computational basis, an associated "X-density matrix" can be obtained by vanishing all entries out of the main- and anti-diagonals. It is very simple to compute the genuine multipartite (GM) concurrence of this associated N-qubit X-state, which, moreover, lower bounds the GM-concurrence of the original (non-X) state. In this paper, we rely on these facts to introduce and benchmark a heuristic for estimating the GM-concurrence of an arbitrary multiqubit mixed state. By explicitly considering two classes of mixed states, we illustrate that our estimates are usually very close to the standard lower bound on the GM-concurrence, being significantly easier to compute. In addition, while evaluating the performance of our proposed heuristic, we provide the first characterization of GM-entanglement in the steady states of the driven Dicke model at zero temperature.

scholarly journals BÜCHI COMPLEMENTATION MADE TIGHTER

2006 ◽  
Vol 17 (04) ◽  
pp. 851-867 ◽  
Author(s):  
EHUD FRIEDGUT ◽  
ORNA KUPFERMAN ◽  
MOSHE Y. VARDI
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Blow Up ◽  
Careful Analysis ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Theoretical Point ◽  
Infinite Words ◽  
Büchi Automata ◽  
Buchi Automata

The complementation problem for nondeterministic word automata has numerous applications in formal verification. In particular, the language-containment problem, to which many verification problems is reduced, involves complementation. For automata on finite words, which correspond to safety properties, complementation involves determinization. The 2n blow-up that is caused by the subset construction is justified by a tight lower bound. For Büchi automata on infinite words, which are required for the modeling of liveness properties, optimal complementation constructions are quite complicated, as the subset construction is not sufficient. From a theoretical point of view, the problem is considered solved since 1988, when Safra came up with a determinization construction for Büchi automata, leading to a 2O(n log n) complementation construction, and Michel came up with a matching lower bound. A careful analysis, however, of the exact blow-up in Safra's and Michel's bounds reveals an exponential gap in the constants hiding in the O( ) notations: while the upper bound on the number of states in Safra's complementary automaton is n2n, Michel's lower bound involves only an n! blow up, which is roughly (n/e)n. The exponential gap exists also in more recent complementation constructions. In particular, the upper bound on the number of states in the complementation construction of Kupferman and Vardi, which avoids determinization, is (6n)n. This is in contrast with the case of automata on finite words, where the upper and lower bounds coincides. In this work we describe an improved complementation construction for nondeterministic Büchi automata and analyze its complexity. We show that the new construction results in an automaton with at most (0.96n)n states. While this leaves the problem about the exact blow up open, the gap is now exponentially smaller. From a practical point of view, our solution enjoys the simplicity of the construction of Kupferman and Vardi, and results in much smaller automata.

scholarly journals MINIMAL NON-INTEGER ALPHABETS ALLOWING PARALLEL ADDITION

2018 ◽  
Vol 58 (5) ◽  
pp. 285 ◽  
Author(s):  
Jan Legerský
Keyword(s):  
Lower Bound ◽  
Necessary Conditions ◽  
Parallel Addition ◽  
Consecutive Integers ◽  
Numeration Systems

Parallel addition, i.e., addition with limited carry propagation has been so far studied for complex bases and integer alphabets. We focus on alphabets consisting of integer combinations of powers of the base. We give necessary conditions on the alphabet allowing parallel addition. Under certain assumptions, we prove the same lower bound on the size of the generalized alphabet that is known for alphabets consisting of consecutive integers. We also extend the characterization of bases allowing parallel addition to numeration systems with non-integer alphabets.

scholarly journals Semiarcs with Long Secants

10.37236/3771 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Bence Csajbók
Keyword(s):  
Lower Bound ◽  
Projective Plane ◽  
Complete Characterization ◽  
Symmetric Difference ◽  
Blocking Set ◽  
Point Set

In a projective plane $\Pi_q$ of order $q$, a non-empty point set $\mathcal{S}_t$ is a $t$-semiarc if the number of tangent lines to $\mathcal{S}_t$ at each of its points is $t$. If $\mathcal{S}_t$ is a $t$-semiarc in $\Pi_q$, $t<q$, then each line intersects $\mathcal{S}_t$ in at most $q+1-t$ points. Dover proved that semiovals (semiarcs with $t=1$) containing $q$ collinear points exist in $\Pi_q$ only if $q\leq 3$. We show that if $t>1$, then $t$-semiarcs with $q+1-t$ collinear points exist only if $t\geq \sqrt{q-1}$. In $\mathrm{PG}(2,q)$ we prove the lower bound $t\geq(q-1)/2$, with equality only if $\mathcal{S}_t$ is a blocking set of Rédei type of size $3(q+1)/2$.We call the symmetric difference of two lines, with $t$ further points removed from each line, a $V_t$-configuration. We give conditions ensuring a $t$-semiarc to contain a $V_t$-configuration and give the complete characterization of such $t$-semiarcs in $\mathrm{PG}(2,q)$.

scholarly journals Efficient Search for Optimal Diffusion Layers of Generalized Feistel Networks

2019 ◽  
pp. 218-240
Author(s):  
Patrick Derbez ◽  
Pierre-Alain Fouque ◽  
Baptiste Lambin ◽  
Victor Mollimard
Keyword(s):  
Lower Bound ◽  
Open Problem ◽  
Building Block ◽  
Efficient Algorithm ◽  
Main Idea ◽  
Diffusion Layers ◽  
Round Function ◽  
Feistel Networks ◽  
Impossibility Proof

The Feistel construction is one of the most studied ways of building block ciphers. Several generalizations were then proposed in the literature, leading to the Generalized Feistel Network, where the round function first applies a classical Feistel operation in parallel on an even number of blocks, and then a permutation is applied to this set of blocks. In 2010 at FSE, Suzaki and Minematsu studied the diffusion of such construction, raising the question of how many rounds are required so that each block of the ciphertext depends on all blocks of the plaintext. They thus gave some optimal permutations, with respect to this diffusion criteria, for a Generalized Feistel Network consisting of 2 to 16 blocks, as well as giving a good candidate for 32 blocks. Later at FSE’19, Cauchois et al. went further and were able to propose optimal even-odd permutations for up to 26 blocks.In this paper, we complete the literature by building optimal even-odd permutations for 28, 30, 32, 36 blocks which to the best of our knowledge were unknown until now. The main idea behind our constructions and impossibility proof is a new characterization of the total diffusion of a permutation after a given number of rounds. In fact, we propose an efficient algorithm based on this new characterization which constructs all optimal even-odd permutations for the 28, 30, 32, 36 blocks cases and proves a better lower bound for the 34, 38, 40 and 42 blocks cases. In particular, we improve the 32 blocks case by exhibiting optimal even-odd permutations with diffusion round of 9. The existence of such a permutation was an open problem for almost 10 years and the best known permutation in the literature had a diffusion round of 10. Moreover, our characterization can be implemented very efficiently and allows us to easily re-find all optimal even-odd permutations for up to 26 blocks with a basic exhaustive search

NONDETERMINISTIC STATE COMPLEXITY OF PROPORTIONAL REMOVALS

2014 ◽  
Vol 25 (07) ◽  
pp. 823-835 ◽  
Author(s):  
DANIEL GOČ ◽  
ALEXANDROS PALIOUDAKIS ◽  
KAI SALOMAA
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
State Complexity ◽  
Nondeterministic State

The language [Formula: see text] consists of first halfs of strings in L. Many other variants of a proportional removal operation have been considered in the literature and a characterization of removal operations that preserve regularity is known. We consider the nondeterministic state complexity of the operation [Formula: see text] and, more generally, of polynomial removals as defined by Domaratzki (J. Automata, Languages and Combinatorics 7(4), 2002). We give an O(n2) upper bound for the nondeterministic state complexity of polynomial removals and a matching lower bound in cases where the polynomial is a sum of a monomial and a constant, or when the polynomial has rational roots.

scholarly journals A characterization of periodicity of bi-infinite words

2005 ◽  
Vol 347 (1-2) ◽  
pp. 419-422
Author(s):  
Tero Harju ◽  
Arto Lepistö ◽  
Dirk Nowotka
Keyword(s):  
Infinite Words
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