## THE FROBENIUS PROBLEM IN A FREE MONOID

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Citations: | 6 - 3 self |

### BibTeX

@MISC{Kao_thefrobenius,

author = {Jui-yi Kao and Jeffrey Shallit and Zhi Xu},

title = {THE FROBENIUS PROBLEM IN A FREE MONOID},

year = {}

}

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### Abstract

Abstract. The classical Frobenius problem over N is to compute the largest integer g not representable as a non-negative integer linear combination of non-negative integers x1, x2,..., xk, where gcd(x1, x2,..., xk) = 1. In this paper we consider novel generalizations of the Frobenius problem to the noncommutative setting of a free monoid. Unlike the commutative case, where the bound on g is quadratic, we are able to show exponential or subexponential behavior for several analogues of g, with the precise bound depending on the particular measure chosen. 1.

### Citations

107 |
Equations in free groups
- Lyndon
- 1960
(Show Context)
Citation Context ...w, x ∈ Σ + . Then sc({w, x} ∗ � ) ≤ |w| + |x|, d (g(|w|/d, |x|/d) + 1) + 2, if wx �= xw; if wx = xw and d = gcd(|w|, |x|) . Proof. If wx = xw, then by a classical theorem of Lyndon and Schützenberger =-=[20]-=-, we know there exists a word z and integers i, j ≥ 1 such that w = z i , x = z j . Thus {w, x} ∗ = {z i , z j } ∗ . Let e = gcd(i, j). Then L = {z i , z j } ∗ consists of all words of the form z ke f... |

100 |
H.S.: Uniqueness theorem for periodic functions
- Fine, Wilf
- 1965
(Show Context)
Citation Context ...e, as usual, g(x1, x2) denotes the Frobenius function introduced in Section 1. We need the following lemma, which is of independent interest and which generalizes a classical theorem of Fine and Wilf =-=[14]-=-. Lemma 5.1. Let w and x be nonempty words. Let y ∈ w{w, x} ω and z ∈ x{w, x} ω . Then the following conditions are equivalent: (a) y and z agree on a prefix of length |w| + |x| − gcd(|w|, |x|); (b) w... |

53 |
Salomaa: The state complexity of some basic operations on regular languages, Theoretical Computer Science 125
- Yu, Zhuang, et al.
- 1994
(Show Context)
Citation Context ...s section we study the measures S = sc(S ∗ ), N = nsc(S ∗ ), and S ′ = sc(x ∗ 1x ∗ 2 · · ·x ∗ k ). consider some results on state complexity. First we review previous results. Yu, Zhuang, and Salomaa =-=[32]-=- showed that if L is accepted by a DFA with n states, then L ∗ can be accepted by a DFA with at most 2 n−1 + 2 n−2 states. Furthermore, they showed this bound is realized, in the sense that for all n ... |

46 |
On a problem of partitions
- Brauer, Seelbinder
- 1954
(Show Context)
Citation Context ...ly large integer can be written as a non-negative integer linear combination of the xi if and only if gcd(x1, x2, . . . , xk) = 1. The famous Frobenius problem (so-called because, according to Brauer =-=[2]-=-, “Frobenius mentioned it occasionally in his lectures”) is the following: Given positive integers x1, x2, . . . , xk with gcd(x1, x2, . . . , xk) = 1, find the largest positive integer g(x1, x2, . . ... |

44 |
Lattice translates of a polytope and the Frobenius problem, Combinatorica 12
- Kannan
- 1992
(Show Context)
Citation Context ...compute g for two elements. For k = 3, efficient algorithms have been given by Greenberg [15] and Davison [10]; if x1 < x2 < x3, these algorithms run in time bounded by a polynomial in log x3. Kannan =-=[18]-=- gave a very complicated algorithm that runs in polynomial time in log xk if k is fixed, but is wildly exponential in k. However, Ramírez Alfonsín [22] proved that the general problem is NP-hard, unde... |

41 |
Regular expressions: new results and open problems. JALC
- Ellul, Krawetz, et al.
- 2004
(Show Context)
Citation Context ...≤ 2 m−k+1 . (c) If no xi is a prefix of any other xj, then sc({x1, x2, . . . , xk} ∗ ) ≤ m − k + 2. We now recall an example providing a lower bound for the state complexity of {x1, x2, . . . , xk} ∗ =-=[13]-=-. Let t be an integer ≥ 2, and define words as follows: y := 01 t−1 0 and xi := 1 t−i−1 01 i+1 for 0 ≤ i ≤ t − 2. Let St := {0, x0, x1, . . . , xt−2, y}. Theorem 4.2. S ∗ t has state complexity 3t2 t−... |

28 | Yu: State complexity of basic operations on finite language
- Campeanu, Culik, et al.
- 1999
(Show Context)
Citation Context ...e that for all n ≥ 2, there exists a DFA M with n states such that the minimal DFA accepting L(M) ∗ needs 2 n−1 + 2 n−2 states. This latter result was given previously by Maslov [21]. Câmpeanu et al. =-=[3, 5]-=- showed that if a DFA with n states accepts a finite language L, then L ∗ can be accepted by a DFA with at most 2 n−3 + 2 n−4 states for n ≥ 4. Furthermore, this bound is actually achieved for n > 4 f... |

27 | The Frobenius problem, rational polytopes, and FourierDedekind sums
- Beck, Diaz, et al.
- 2002
(Show Context)
Citation Context ...recent book by Ramírez Alfonsín [23] lists over 400 references on this problem. Applications to many different fields exist: to algebra [19]; the theory of matrices [11], counting points in polytopes =-=[1]-=-; the problem of efficient sorting using Shellsort [17], the theory of Petri nets [25]; the liveness of weighted circuits [8]; etc. Generally speaking, research on the Frobenius problem can be classif... |

18 |
Finite automata and unary languages. Theor
- Chrobak
- 1986
(Show Context)
Citation Context ...re ai ∈ N for 1 ≤ i ≤ k. The Frobenius problem is evidently linked to many problems over unary languages. It figures, for example, in estimating the size of the smallest DFA equivalent to a given NFA =-=[7]-=-. If L ⊆ Σ ∗ , by L we mean Σ ∗ − L, the complement of L. If L is a finite language, by |L| we mean the cardinality of L. Evidently we havesTHE FROBENIUS PROBLEM IN A FREE MONOID 3 Proposition 2.1. Su... |

18 |
On the linear Diophantine problem of Frobenius
- Davison
- 1994
(Show Context)
Citation Context ...vester [24], although he did not actually state it. Eq. (1.1) gives an efficient algorithm to compute g for two elements. For k = 3, efficient algorithms have been given by Greenberg [15] and Davison =-=[10]-=-; if x1 < x2 < x3, these algorithms run in time bounded by a polynomial in log x3. Kannan [18] gave a very complicated algorithm that runs in polynomial time in log xk if k is fixed, but is wildly exp... |

17 |
Improved upper bounds on Shellsort
- Incerpi, Sedgewick
(Show Context)
Citation Context ...ferences on this problem. Applications to many different fields exist: to algebra [19]; the theory of matrices [11], counting points in polytopes [1]; the problem of efficient sorting using Shellsort =-=[17]-=-, the theory of Petri nets [25]; the liveness of weighted circuits [8]; etc. Generally speaking, research on the Frobenius problem can be classified into three different areas: 2000 ACM Subject Classi... |

16 |
The value-semigroup of a one-dimensional Gorenstein ring
- Kunz
- 1973
(Show Context)
Citation Context ... subtle and intriguing aspects that continue to elicit study. A recent book by Ramírez Alfonsín [23] lists over 400 references on this problem. Applications to many different fields exist: to algebra =-=[19]-=-; the theory of matrices [11], counting points in polytopes [1]; the problem of efficient sorting using Shellsort [17], the theory of Petri nets [25]; the liveness of weighted circuits [8]; etc. Gener... |

13 |
Solution to a linear Diophantine equation for nonnegative integers
- Greenberg
- 1988
(Show Context)
Citation Context ...attributed to Sylvester [24], although he did not actually state it. Eq. (1.1) gives an efficient algorithm to compute g for two elements. For k = 3, efficient algorithms have been given by Greenberg =-=[15]-=- and Davison [10]; if x1 < x2 < x3, these algorithms run in time bounded by a polynomial in log x3. Kannan [18] gave a very complicated algorithm that runs in polynomial time in log xk if k is fixed, ... |

12 | M.: Nondeterministic descriptional complexity of regular languages
- Holzer, Kutrib
(Show Context)
Citation Context ...> 4 for an alphabet of size 3 or more. Unlike the examples we are concerned with in this section, however, the finite languages they construct contain exponentially many words in n. Holzer and Kutrib =-=[16]-=- examined the nondeterminstic state complexity of Kleene star. They showed that if an NFA M with n states accepts L, then L ∗ can be accepted by an NFA with n + 1 states, and this bound is tight. If L... |

12 |
The Diophantine Frobenius Problem
- Ramiréz-Alfonsín
- 2005
(Show Context)
Citation Context ...n time bounded by a polynomial in log x3. Kannan [18] gave a very complicated algorithm that runs in polynomial time in log xk if k is fixed, but is wildly exponential in k. However, Ramírez Alfonsín =-=[22]-=- proved that the general problem is NP-hard, under Turing reductions, by reducing from the integer knapsack problem. So it seems very likely that there is no simple formula for computing g(x1, x2, . .... |

11 |
Estimates of the number of states of finite automata
- Maslov
- 1970
(Show Context)
Citation Context ... realized, in the sense that for all n ≥ 2, there exists a DFA M with n states such that the minimal DFA accepting L(M) ∗ needs 2 n−1 + 2 n−2 states. This latter result was given previously by Maslov =-=[21]-=-. Câmpeanu et al. [3, 5] showed that if a DFA with n states accepts a finite language L, then L ∗ can be accepted by a DFA with at most 2 n−3 + 2 n−4 states for n ≥ 4. Furthermore, this bound is actua... |

9 |
On Weighted T-Systems
- Teruel, Chrzastowski-Wachtel, et al.
- 1992
(Show Context)
Citation Context ...any different fields exist: to algebra [19]; the theory of matrices [11], counting points in polytopes [1]; the problem of efficient sorting using Shellsort [16, 25, 30, 26]; the theory of Petri nets =-=[28]-=-; the liveness of weighted circuits [8]; etc. Generally speaking, research on the Frobenius problem can be classified into three different areas: • Formulas or algorithms for the exact computation of ... |

8 |
State complexity of regular languages: finite versus infinite
- Campeanu, Salomaa, et al.
- 2000
(Show Context)
Citation Context ...e that for all n ≥ 2, there exists a DFA M with n states such that the minimal DFA accepting L(M) ∗ needs 2 n−1 + 2 n−2 states. This latter result was given previously by Maslov [21]. Câmpeanu et al. =-=[3, 5]-=- showed that if a DFA with n states accepts a finite language L, then L ∗ can be accepted by a DFA with at most 2 n−3 + 2 n−4 states for n ≥ 4. Furthermore, this bound is actually achieved for n > 4 f... |

8 |
Gaps in the exponent set of primitive matrices
- Dulmage, Mendelson
- 1964
(Show Context)
Citation Context ...s that continue to elicit study. A recent book by Ramírez Alfonsín [23] lists over 400 references on this problem. Applications to many different fields exist: to algebra [19]; the theory of matrices =-=[11]-=-, counting points in polytopes [1]; the problem of efficient sorting using Shellsort [17], the theory of Petri nets [25]; the liveness of weighted circuits [8]; etc. Generally speaking, research on th... |

7 | Frobenius numbers by lattice point enumeration
- Einstein, Lichtblau, et al.
(Show Context)
Citation Context ...knapsack problem. So it seems very likely that there is no simple formula for computing g(x1, x2, . . . , xk) for arbitrary k. Nevertheless, recent work by Einstein, Lichtblau, Strzebonski, and Wagon =-=[12]-=- shows that in practice the Frobenius number can be computed relatively efficiently, even for very large numbers, at least for k ≤ 8. Another active area of interest is estimating how big g is in term... |

7 |
A circle-of-lights algorithm for the ”money-changing problem
- Wilf
- 1978
(Show Context)
Citation Context ...ctive area of interest is estimating how big g is in terms of x1, x2, . . .,xk for x1 < x2 < · · · < xk. It is known, for example, that g(x1, x2, . . .,xk) < x2 k . This follows from Wilf’s algorithm =-=[31]-=-. Many other bounds are known. One can also study variations on the Frobenius problem. For example, given positive integers x1, x2, . . .,xk with gcd(x1, x2, . . .,xk) = 1, what is the number f(x1, x2... |

6 |
The maximum state complexity for finite languages
- Câmpeanu, Ho
- 2004
(Show Context)
Citation Context ...f an NFA M with n states accepts L, then L ∗ can be accepted by an NFA with n + 1 states, and this bound is tight. If L is finite, then n − 1 states suffices, and this bound is tight. Câmpeanu and Ho =-=[4]-=- gave tight bounds for the number of states required to accept a finite language whose words are all bounded by length n.sTHE FROBENIUS PROBLEM IN A FREE MONOID 5 Proposition 4.1. (a) nsc({x1, x2, . .... |

5 |
Parsing with a finite dictionary
- Clément, Duval, et al.
- 2005
(Show Context)
Citation Context ...re the size of the input, conditions on the input, and measures of the size of the result. For an application of the noncommutative Frobenius problem, see Clément, Duval, Guaiana, Perrin, and Rindone =-=[9]-=-. In sections 2 and 3, we introduce the definition of the generalized Frobenius problem. In sections 4 and 5, we discuss the state complexity of this generalized problem. In sections 5 and 6, we will ... |

3 |
An algorithm to solve the Frobenius problem
- Owens
- 2003
(Show Context)
Citation Context ...he largest positive integer g(x1, x2, . . .,xk) which cannot be represented as a non-negative integer linear combination of the xi. Example 1. The Chicken McNuggets Problem ([29, pp. 19-20, 233–234], =-=[22]-=-). If Chicken McNuggets can be purchased at McDonald’s only in quantities of 6, 9, or 20 pieces, what is the largest number of McNuggets that cannot be purchased? The answer is g(6, 9, 20) = 43. 1sAlt... |

2 |
On weighted T -systems
- Teruel, Chrzastowski-Wachtel, et al.
- 1992
(Show Context)
Citation Context ...cations to many different fields exist: to algebra [19]; the theory of matrices [11], counting points in polytopes [1]; the problem of efficient sorting using Shellsort [17], the theory of Petri nets =-=[25]-=-; the liveness of weighted circuits [8]; etc. Generally speaking, research on the Frobenius problem can be classified into three different areas: 2000 ACM Subject Classification: F.4.3. Key words and ... |

2 |
Solution of the Frobenius problem
- Kannan
- 1989
(Show Context)
Citation Context ...compute g for two elements. For k = 3, efficient algorithms have been given by Greenberg [14] and Davison [10]; if x1 < x2 < x3, these algorithms run in time bounded by a polynomial in log x3. Kannan =-=[17, 18]-=- gave a very complicated algorithm that runs in polynomial time in log xk if k is fixed, but is wildly exponential in k. However, Ramírez Alfonsín [23] proved that the general problem is NP-hard, unde... |

2 |
On Shellsort and the Frobenius problem
- Selmer
- 1987
(Show Context)
Citation Context ...ferences on this problem. Applications to many different fields exist: to algebra [19]; the theory of matrices [11], counting points in polytopes [1]; the problem of efficient sorting using Shellsort =-=[16, 25, 30, 26]-=-; the theory of Petri nets [28]; the liveness of weighted circuits [8]; etc. Generally speaking, research on the Frobenius problem can be classified into three different areas: • Formulas or algorithm... |

1 |
Liveness of weighted circuits and the Diophantine problem of Frobenius
- Chrzastowski-Wachtel, Raczunas
- 1993
(Show Context)
Citation Context ...to algebra [19]; the theory of matrices [11], counting points in polytopes [1]; the problem of efficient sorting using Shellsort [17], the theory of Petri nets [25]; the liveness of weighted circuits =-=[8]-=-; etc. Generally speaking, research on the Frobenius problem can be classified into three different areas: 2000 ACM Subject Classification: F.4.3. Key words and phrases: combinatorics on words, Froben... |

1 |
Parsing with a finite dictioanry
- Clément, Duval, et al.
(Show Context)
Citation Context ...re the size of the input, conditions on the input, and measures of the size of the result. For an application of the noncommutative Frobenius problem, see Clément, Duval, Guaiana, Perrin, and Rindone =-=[9]-=-. 2sIn order to motivate our definitions, we consider the easiest case first: where Σ = {0}, a unary alphabet. 2 The unary case Suppose xi = 0 ai , for 1 ≤ i ≤ k. The Frobenius problem is evidently li... |

1 |
Shellsort and the Frobenius problem
- Weiss, Sedgewick, et al.
- 1988
(Show Context)
Citation Context ...ferences on this problem. Applications to many different fields exist: to algebra [19]; the theory of matrices [11], counting points in polytopes [1]; the problem of efficient sorting using Shellsort =-=[16, 25, 30, 26]-=-; the theory of Petri nets [28]; the liveness of weighted circuits [8]; etc. Generally speaking, research on the Frobenius problem can be classified into three different areas: • Formulas or algorithm... |