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47
Numeration systems, linear recurrences, and regular sets
 Inform. and Comput
, 1994
"... A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large mult ..."
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Cited by 36 (4 self)
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A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is orderpreserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an orderpreserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1
Alice’s Adventures in Wonderland
, 1916
"... "What is the use, " thought Alice, "of a book without pictures and ..."
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Cited by 30 (0 self)
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"What is the use, " thought Alice, "of a book without pictures and
A Simple AlphabetIndependent FMIndex
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
"... We design a succinct fulltext index based on the idea of Huffmancompressing the text and then applying the BurrowsWheeler transform over it. The resulting structure can be searched as an FMindex, with the benefit of removing the sharp dependence on the alphabet size, σ, present in that structu ..."
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Cited by 14 (7 self)
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We design a succinct fulltext index based on the idea of Huffmancompressing the text and then applying the BurrowsWheeler transform over it. The resulting structure can be searched as an FMindex, with the benefit of removing the sharp dependence on the alphabet size, σ, present in that structure. On a text of length n with zeroorder entropy H0, our index needs O(n(H0 + 1)) bits of space, without any significant dependence on σ. The average search time for a pattern of length m is O(m(H0 + 1)), under reasonable assumptions. Each position of a text occurrence can be located in worst case time O((H0 + 1)log n), while any text substring of length L can be retrieved in O((H0 + 1)L) average time in addition to the previous worst case time. Our index provides a relevant space/time tradeoff between existing succinct data structures, with the additional interest of being easy to implement. We also explore other coding variants alternative to Huffman and exploit their synchronization properties. Our experimental results on various types of texts show that our indexes are highly competitive in the space/time tradeoff map.
Automaticity II: Descriptional Complexity in the Unary Case
 Comput. Sci
, 1995
"... Let \Sigma and \Delta be finite alphabets, and let f be a map from \Sigma to \Delta. Then the deterministic automaticity of f , A f (n), is defined to be the size of the minimum finitestate machine that correctly computes f on all inputs of size n. A similar definition applies to languages L. We ..."
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Cited by 12 (6 self)
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Let \Sigma and \Delta be finite alphabets, and let f be a map from \Sigma to \Delta. Then the deterministic automaticity of f , A f (n), is defined to be the size of the minimum finitestate machine that correctly computes f on all inputs of size n. A similar definition applies to languages L. We denote the nondeterministic analogue (for languages L) of automaticity by NL (n). In a previous paper, J. Shallit and Y. Breitbart examined the properties of this measure of descriptional complexity in the case j\Sigmaj 2. In this paper, we continue the study of automaticity, focusing on the case where j\Sigmaj = 1. Research supported in part by DMS9206784. y Research supported in part by a grant from NSERC. Partial support under NSF Grant DCR 9208639 and the Wisconsin Alumni Research Foundation. We prove that A f (n) n + 1 \Gamma blog ` nc, where ` = j\Deltaj. We also prove that A f (n) ? n \Gamma 2 log ` n \Gamma 2 log ` log ` n for almost all functions f . In the nondetermi...
On cobweb posets’ most relevant codings
, 2009
"... One considers here acyclic digraphs named KoDAGs (****) which represent the outmost general chains of dibicliques denoting thus the outmost general chains of binary relations. Because of this fact KoDAGs start to become an outstanding concept of nowadays investigation. We propose here examples of ..."
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Cited by 12 (9 self)
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One considers here acyclic digraphs named KoDAGs (****) which represent the outmost general chains of dibicliques denoting thus the outmost general chains of binary relations. Because of this fact KoDAGs start to become an outstanding concept of nowadays investigation. We propose here examples of codings of KoDAGs looked upon as infinite hyperboxes as well as chains of rectangular hyperboxes in N ∞. Neither of KoDAGs ’ codings considered here is a poset isomorphism with Π = 〈P, ≤〉. Nevertheless every example of coding supplies a new view on possible investigation of KoDAGs properties. The codes proposed here down are by now recognized as most relevant codings for practical purposes including visualization. More than that. Employing quite arbitrary sequences F = {nF}n≥0 infinitely many new representations of natural numbers called an F base or baseF number system representations are introduced. These constitute mixed radixtype numeral systems. F base nonstandard positional numeral systems in which the numerical base varies from position to position have picturesque interpretation due to KoDAGs graphs and their correspondent posets which in turn are endowed on their own with combinatorial interpretation of uniquely assigned to KoDAGs F − nomial coefficients. The baseF number systems are used for KoDAGs ’ coding and are interpreted as chain coordinatization in KoDAGs pictures as well as systems of infinite number of boxes ’ sequences of Fvarying containers capacity of subsequent boxes. Needless to say how crucial is this baseF number system for KoDAGs hence consequently for arbitrary chains of binary relations. New Fbased numeral systems are umbral baseF number systems in a sense to be explained in what follows.
Robust Universal Complete Codes for Transmission and Compression
 Discrete Applied Mathematics
, 1996
"... Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variablelength codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is ..."
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Cited by 10 (4 self)
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Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variablelength codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is constant and need not be generated for every probability distribution. These codes can be used as alternatives to Huffman codes when the optimal compression of the latter is not required, and simplicity, faster processing and robustness are preferred. The codes are compared on several "reallife" examples. 1. Motivation and Introduction Let A = fA 1 ; A 2 ; \Delta \Delta \Delta ; An g be a finite set of elements, called cleartext elements, to be encoded by a static uniquely decipherable (UD) code. For notational ease, we use the term `code' as abbreviation for `set of codewords'; the corresponding encoding and decoding algorithms are always either given or clear from the context. A code i...
Fibonacci cubes—a class of selfsimilar graphs
 Fibonacci Quart
, 1993
"... The Fibonacci cube [6] is a new class of graphs that are inspired by the famous numbers. Because of the rich properties of the Fibonacci numbers [1], the graph also shows interesting properties. For a graph with AT nodes, it is known [6] that the diameter, the edge connectivity, and the node connect ..."
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Cited by 10 (0 self)
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The Fibonacci cube [6] is a new class of graphs that are inspired by the famous numbers. Because of the rich properties of the Fibonacci numbers [1], the graph also shows interesting properties. For a graph with AT nodes, it is known [6] that the diameter, the edge connectivity, and the node connectivity of the Fibonacci cube are in the order of 0(log N), which are similar to
Extensions and restrictions of Wythoff’s game preserving its P positions
 Journal of Combinatorial Theory, Series A
"... Abstract. We consider extensions and restrictions of Wythoff’s game having exactly the same set of P positions as the original game. No strict subset of rules give the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P po ..."
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Cited by 9 (5 self)
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Abstract. We consider extensions and restrictions of Wythoff’s game having exactly the same set of P positions as the original game. No strict subset of rules give the same set of P positions. On the other hand, we characterize all moves that can be adjoined while preserving the original set of P positions. Testing if a move belongs to such an extended set of rules is shown to be doable in polynomial time. Many arguments rely on the infinite Fibonacci word, automatic sequences and the corresponding number system. With these tools, we provide new twodimensional morphisms generating an infinite picture encoding respectively P positions of Wythoff’s game and moves that can be adjoined. 1.
The structure of complementary sets of integers: a 3shift theorem
 INTERNAT. J. PURE AND APPL. MATH
"... Let 0 < α < β be two irrational numbers satisfying 1/α +1/β = 1. Then the sequences a ′ n = ⌊nα⌋, b ′ n = ⌊nβ⌋, n ≥ 1, are complementary: 1 ≤ i < n}, n ≥ 1 over Z≥1, thus a ′ n satisfies: a ′ n = mex1{a ′ i, b ′ i (mex1(S), the smallest positive integer not in the set S). Suppose that c = β − α is a ..."
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Cited by 9 (7 self)
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Let 0 < α < β be two irrational numbers satisfying 1/α +1/β = 1. Then the sequences a ′ n = ⌊nα⌋, b ′ n = ⌊nβ⌋, n ≥ 1, are complementary: 1 ≤ i < n}, n ≥ 1 over Z≥1, thus a ′ n satisfies: a ′ n = mex1{a ′ i, b ′ i (mex1(S), the smallest positive integer not in the set S). Suppose that c = β − α is an integer. Then b ′ n = a ′ n + cn for all n ≥ 1. We define the following generalization of sequences a ′ n, b ′ n: Let c, n0 ∈ Z≥1, and let X ⊂ Z≥1 be an arbitrary finite set. Let an = mex1(X ∪{ai, bi: 1 ≤ i < n}), bn = an +cn, n ≥ n0. Let sn = a ′ n −an. We show that no matter how we pick c, n0 and X, from some point on the shift sequence sn assumes either one constant value or three successive values; and if the second case holds, it assumes these values in a very distinct fractallike pattern, which we describe. This work was motivated by a generalization of Wythoff’s game to N ≥ 3 piles.
On the Least Significant Digit of Zeckendorf Expansions." The Fibonacci Quarterly 34,2
, 1996
"... A wellknown digital expansion is the socalled Zeckendorf number system [7], where every positive integer n can be written as L k=Q ..."
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Cited by 6 (0 self)
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A wellknown digital expansion is the socalled Zeckendorf number system [7], where every positive integer n can be written as L k=Q