Jeffrey Shallit | University of Waterloo (original) (raw)
Papers by Jeffrey Shallit
We examine words w satisfying the following property: if x is a subword of w and |x| is at least ... more We examine words w satisfying the following property: if x is a subword of w and |x| is at least k for some fixed k, then the reversal of x is not a subword of w.
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Journal of Automata, Languages and Combinatorics, 2000
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The Electronic Journal of Combinatorics, 1998
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Theoretical Computer Science, 2003
In 1965, Fine and Wilf proved the following theorem: if (fn)n⩾0 and (gn)n⩾0 are periodic sequence... more In 1965, Fine and Wilf proved the following theorem: if (fn)n⩾0 and (gn)n⩾0 are periodic sequences of real numbers, of period lengths h and k, respectively, and fn=gn for 0⩽n<h+k−gcd(h,k), then fn=gn for all n⩾0. Furthermore, the constant h+k−gcd(h,k) is best possible. In this paper, we consider some variations on this theorem. In particular, we study the case where fn⩽gn instead of fn=gn. We also obtain generalizations to more than two periods.We apply our methods to a previously unsolved conjecture on iterated morphisms, the decreasing length conjecture: if is a morphism with |Σ|=n, and w is a word with |w|>|h(w)|>|h2(w)|>⋯>|hk(w)|, then k⩽n.
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Computing Research Repository, 2009
In this paper we examine decision problems associated with various classes of convex languages, s... more In this paper we examine decision problems associated with various classes of convex languages, studied by Ang and Brzozowski (under the name "continuous languages"). We show that we can decide whether a given language L is prefix-, suffix-, factor-, or subword-convex in polynomial time if L is represented by a DFA, but that the problem is PSPACE-hard if L is represented by an NFA. In the case that a regular language is not convex, we prove tight upper bounds on the length of the shortest words demonstrating this fact, in terms of the number of states of an accepting DFA. Similar results are proved for some subclasses of convex languages: the prefix-, suffix-, factor-, and subword-closed languages, and the prefix-, suffix-, factor-, and subword-free languages.
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Journal of Computer and System Sciences, 2002
It is well known that a context-free language defined over a one-letter alphabet is regular. This... more It is well known that a context-free language defined over a one-letter alphabet is regular. This implies that unary context-free grammars and unary pushdown automata can be transformed into equivalent finite automata. In this paper, we study these transformations from a descriptional complexity point of view. In particular, we give optimal upper bounds for the number of states of nondeterministic and deterministic finite automata equivalent to unary context-free grammars in Chomsky normal form. These bounds are functions of the number of variables of the given grammars. We also give upper bounds for the number of states of finite automata simulating unary pushdown automata. As a main consequence, we are able to prove a log log n lower bound for the workspace used by one-way auxiliary pushdown automata in order to accept nonregular unary languages. The notion of space we consider is the so-called weak space concept.
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International Journal of Foundations of Computer Science, 2005
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Theoretical Computer Science, 1999
Let Σ be a finite alphabet, and let h:Σ∗→Σ∗ be a morphism. Finite and infinite fixed points of mo... more Let Σ be a finite alphabet, and let h:Σ∗→Σ∗ be a morphism. Finite and infinite fixed points of morphisms—i.e., those words w such that h(w)=w—play an important role in formal language theory. Head characterized the finite fixed points of h, and later, Head and Lando characterized the one-sided infinite fixed points of h. Our paper has two main results. First, we complete the characterization of fixed points of morphisms by describing all two-sided infinite fixed points of h, for both the “pointed” and “unpointed” cases. Second, we completely characterize the solutions to the equation h(xy)=yx in finite words.
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In 1965, Fine & Wilf proved the following theorem: if (f n )n≥0 and (g n )n ≥0 are periodic seque... more In 1965, Fine & Wilf proved the following theorem: if (f n )n≥0 and (g n )n ≥0 are periodic sequences of real numbers, of periods h and k respectively, and f n = g n for 0 ≤n < h + k - gcd(h, k), then f n = g n for all n ≥0. Furthermore, the constant h + k - gcd(h, k) is best possible. In this paper we consider some variations on this theorem. In particular, we study the case where f n ≤ g n instead off n =g n . We also obtain a generalization to more than two periods.
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In this paper we define Sturmian graphs and we prove that all of them have a “counting” property.... more In this paper we define Sturmian graphs and we prove that all of them have a “counting” property. We show deep connections between this counting property and two conjectures, by Moser and by Zaremba, on the continued fraction expansion of real numbers. These graphs turn out to be the underlying graphs of CDAWGs of central Sturmian words. We show also that, analogously to the case of Sturmian words, these graphs converge to infinite ones.
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A numeration system based on a strictly increasing sequence of positive integers u 0=1, u 1u 2,..... more A numeration system based on a strictly increasing sequence of positive integers u 0=1, u 1u 2,... expresses a non-negative integer n as a sum n=∑ j=0iajuj. In this case we say the string a i a i −1 ...a1 a0 is a representation for n. 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 u 0=1, then the greedy representation, obtained via the greedy algorithm, is order-preserving. We prove that, subject to some technical assumptions, if the set of all representations in an order-preserving numeration system is regular, then the sequence u=(u j )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.
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Synthese, 2011
Intelligent design advocate William Dembski has introduced a measure of information called “compl... more Intelligent design advocate William Dembski has introduced a measure of information called “complex specified information”, or CSI. He claims that CSI is a reliable marker of design by intelligent agents. He puts forth a “Law of Conservation of Information” which states that chance and natural laws are incapable of generating CSI. In particular, CSI cannot be generated by evolutionary computation. Dembski asserts that CSI is present in intelligent causes and in the flagellum of Escherichia coli, and concludes that neither have natural explanations. In this paper, we examine Dembski’s claims, point out significant errors in his reasoning, and conclude that there is no reason to accept his assertions.
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Journal of Automata, Languages and Combinatorics, 2001
Abstract We define a new measure of complexity for finite strings, called automatic complexity an... more Abstract We define a new measure of complexity for finite strings, called automatic complexity and denoted A (x). Although A (x) is analogous to Kolmogorov-Chaitin complexity, it has the advantage of being computable. We give upper and lower bounds for A (x), and estimate it for some specific strings.
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Bookmarks Related papers MentionsView impact
Computing Research Repository, 2009
A famous theorem of Kuratowski states that, in a topological space, at most 14 distinct sets can ... more A famous theorem of Kuratowski states that, in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where closure is either Kleene closure or positive closure. We classify languages according to the structure of the algebra they generate under iterations of complement and closure. There are precisely 9 such algebras in the case of positive closure, and 12 in the case of Kleene closure. We study how the properties of being open and closed are preserved under concatenation. We investigate analogues, in formal languages, of the separation axioms in topological spaces; one of our main results is that there is a clopen partition separating two words if and only if the words do not commute. We can decide in quadratic time if the language specified by a DFA is closed, but if the language is specified by an NFA, the problem is PSPACE-complete.
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Theoretical Informatics and Applications, 1989
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Theory of Computing Systems / Mathematical Systems Theory, 1997
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Linear Algebra and Its Applications, 1999
Let A be an n × n matrix with non-negative entries and no entry in (0, 1). We prove that there ex... more Let A be an n × n matrix with non-negative entries and no entry in (0, 1). We prove that there exist integers r, s with 0 ⩽ r ⩽ s ⩽ 2n such that Ar ⩽ As. We prove that 2n cannot be replaced with e√n log n. We also give an application to the theory of formal languages.
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The Electronic Journal of Combinatorics, 2006
We characterize the squares occurring in infinite overlap-free binary words and construct various... more We characterize the squares occurring in infinite overlap-free binary words and construct various alpha power-free binary words containing infinitely many overlaps.
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Journal of Combinatorial Theory, 1989
In this paper we study the three-dimensional curves generated by the iterated bending of a piece ... more In this paper we study the three-dimensional curves generated by the iterated bending of a piece of wire, which are generalizations of the so-called “dragon curves” or “paper-folding sequences” previously studied by Davis and Knuth, Mendès France, and other writers. These “wire-bending sequences” have several surprising properties. We characterize the nth term of a wire-bending sequence in terms of the binary expansion of n. We prove that the curves traced out in R3 by many wire-bending sequences are actually bounded, although they are all aperiodic. Finally, we illustrate the close connection between wire-bending and the continued fractions for the transcendental numbers Σn ⩾ 0εng−2n, where εn = ± 1 and g ⩾ 3 is an integer.
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We examine words w satisfying the following property: if x is a subword of w and |x| is at least ... more We examine words w satisfying the following property: if x is a subword of w and |x| is at least k for some fixed k, then the reversal of x is not a subword of w.
Bookmarks Related papers MentionsView impact
Journal of Automata, Languages and Combinatorics, 2000
Bookmarks Related papers MentionsView impact
The Electronic Journal of Combinatorics, 1998
Bookmarks Related papers MentionsView impact
Theoretical Computer Science, 2003
In 1965, Fine and Wilf proved the following theorem: if (fn)n⩾0 and (gn)n⩾0 are periodic sequence... more In 1965, Fine and Wilf proved the following theorem: if (fn)n⩾0 and (gn)n⩾0 are periodic sequences of real numbers, of period lengths h and k, respectively, and fn=gn for 0⩽n<h+k−gcd(h,k), then fn=gn for all n⩾0. Furthermore, the constant h+k−gcd(h,k) is best possible. In this paper, we consider some variations on this theorem. In particular, we study the case where fn⩽gn instead of fn=gn. We also obtain generalizations to more than two periods.We apply our methods to a previously unsolved conjecture on iterated morphisms, the decreasing length conjecture: if is a morphism with |Σ|=n, and w is a word with |w|>|h(w)|>|h2(w)|>⋯>|hk(w)|, then k⩽n.
Bookmarks Related papers MentionsView impact
Computing Research Repository, 2009
In this paper we examine decision problems associated with various classes of convex languages, s... more In this paper we examine decision problems associated with various classes of convex languages, studied by Ang and Brzozowski (under the name "continuous languages"). We show that we can decide whether a given language L is prefix-, suffix-, factor-, or subword-convex in polynomial time if L is represented by a DFA, but that the problem is PSPACE-hard if L is represented by an NFA. In the case that a regular language is not convex, we prove tight upper bounds on the length of the shortest words demonstrating this fact, in terms of the number of states of an accepting DFA. Similar results are proved for some subclasses of convex languages: the prefix-, suffix-, factor-, and subword-closed languages, and the prefix-, suffix-, factor-, and subword-free languages.
Bookmarks Related papers MentionsView impact
Journal of Computer and System Sciences, 2002
It is well known that a context-free language defined over a one-letter alphabet is regular. This... more It is well known that a context-free language defined over a one-letter alphabet is regular. This implies that unary context-free grammars and unary pushdown automata can be transformed into equivalent finite automata. In this paper, we study these transformations from a descriptional complexity point of view. In particular, we give optimal upper bounds for the number of states of nondeterministic and deterministic finite automata equivalent to unary context-free grammars in Chomsky normal form. These bounds are functions of the number of variables of the given grammars. We also give upper bounds for the number of states of finite automata simulating unary pushdown automata. As a main consequence, we are able to prove a log log n lower bound for the workspace used by one-way auxiliary pushdown automata in order to accept nonregular unary languages. The notion of space we consider is the so-called weak space concept.
Bookmarks Related papers MentionsView impact
International Journal of Foundations of Computer Science, 2005
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Theoretical Computer Science, 1999
Let Σ be a finite alphabet, and let h:Σ∗→Σ∗ be a morphism. Finite and infinite fixed points of mo... more Let Σ be a finite alphabet, and let h:Σ∗→Σ∗ be a morphism. Finite and infinite fixed points of morphisms—i.e., those words w such that h(w)=w—play an important role in formal language theory. Head characterized the finite fixed points of h, and later, Head and Lando characterized the one-sided infinite fixed points of h. Our paper has two main results. First, we complete the characterization of fixed points of morphisms by describing all two-sided infinite fixed points of h, for both the “pointed” and “unpointed” cases. Second, we completely characterize the solutions to the equation h(xy)=yx in finite words.
Bookmarks Related papers MentionsView impact
In 1965, Fine & Wilf proved the following theorem: if (f n )n≥0 and (g n )n ≥0 are periodic seque... more In 1965, Fine & Wilf proved the following theorem: if (f n )n≥0 and (g n )n ≥0 are periodic sequences of real numbers, of periods h and k respectively, and f n = g n for 0 ≤n < h + k - gcd(h, k), then f n = g n for all n ≥0. Furthermore, the constant h + k - gcd(h, k) is best possible. In this paper we consider some variations on this theorem. In particular, we study the case where f n ≤ g n instead off n =g n . We also obtain a generalization to more than two periods.
Bookmarks Related papers MentionsView impact
In this paper we define Sturmian graphs and we prove that all of them have a “counting” property.... more In this paper we define Sturmian graphs and we prove that all of them have a “counting” property. We show deep connections between this counting property and two conjectures, by Moser and by Zaremba, on the continued fraction expansion of real numbers. These graphs turn out to be the underlying graphs of CDAWGs of central Sturmian words. We show also that, analogously to the case of Sturmian words, these graphs converge to infinite ones.
Bookmarks Related papers MentionsView impact
A numeration system based on a strictly increasing sequence of positive integers u 0=1, u 1u 2,..... more A numeration system based on a strictly increasing sequence of positive integers u 0=1, u 1u 2,... expresses a non-negative integer n as a sum n=∑ j=0iajuj. In this case we say the string a i a i −1 ...a1 a0 is a representation for n. 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 u 0=1, then the greedy representation, obtained via the greedy algorithm, is order-preserving. We prove that, subject to some technical assumptions, if the set of all representations in an order-preserving numeration system is regular, then the sequence u=(u j )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.
Bookmarks Related papers MentionsView impact
Synthese, 2011
Intelligent design advocate William Dembski has introduced a measure of information called “compl... more Intelligent design advocate William Dembski has introduced a measure of information called “complex specified information”, or CSI. He claims that CSI is a reliable marker of design by intelligent agents. He puts forth a “Law of Conservation of Information” which states that chance and natural laws are incapable of generating CSI. In particular, CSI cannot be generated by evolutionary computation. Dembski asserts that CSI is present in intelligent causes and in the flagellum of Escherichia coli, and concludes that neither have natural explanations. In this paper, we examine Dembski’s claims, point out significant errors in his reasoning, and conclude that there is no reason to accept his assertions.
Bookmarks Related papers MentionsView impact
Journal of Automata, Languages and Combinatorics, 2001
Abstract We define a new measure of complexity for finite strings, called automatic complexity an... more Abstract We define a new measure of complexity for finite strings, called automatic complexity and denoted A (x). Although A (x) is analogous to Kolmogorov-Chaitin complexity, it has the advantage of being computable. We give upper and lower bounds for A (x), and estimate it for some specific strings.
Bookmarks Related papers MentionsView impact
Bookmarks Related papers MentionsView impact
Computing Research Repository, 2009
A famous theorem of Kuratowski states that, in a topological space, at most 14 distinct sets can ... more A famous theorem of Kuratowski states that, in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where closure is either Kleene closure or positive closure. We classify languages according to the structure of the algebra they generate under iterations of complement and closure. There are precisely 9 such algebras in the case of positive closure, and 12 in the case of Kleene closure. We study how the properties of being open and closed are preserved under concatenation. We investigate analogues, in formal languages, of the separation axioms in topological spaces; one of our main results is that there is a clopen partition separating two words if and only if the words do not commute. We can decide in quadratic time if the language specified by a DFA is closed, but if the language is specified by an NFA, the problem is PSPACE-complete.
Bookmarks Related papers MentionsView impact
Theoretical Informatics and Applications, 1989
Bookmarks Related papers MentionsView impact
Theory of Computing Systems / Mathematical Systems Theory, 1997
Bookmarks Related papers MentionsView impact
Linear Algebra and Its Applications, 1999
Let A be an n × n matrix with non-negative entries and no entry in (0, 1). We prove that there ex... more Let A be an n × n matrix with non-negative entries and no entry in (0, 1). We prove that there exist integers r, s with 0 ⩽ r ⩽ s ⩽ 2n such that Ar ⩽ As. We prove that 2n cannot be replaced with e√n log n. We also give an application to the theory of formal languages.
Bookmarks Related papers MentionsView impact
The Electronic Journal of Combinatorics, 2006
We characterize the squares occurring in infinite overlap-free binary words and construct various... more We characterize the squares occurring in infinite overlap-free binary words and construct various alpha power-free binary words containing infinitely many overlaps.
Bookmarks Related papers MentionsView impact
Journal of Combinatorial Theory, 1989
In this paper we study the three-dimensional curves generated by the iterated bending of a piece ... more In this paper we study the three-dimensional curves generated by the iterated bending of a piece of wire, which are generalizations of the so-called “dragon curves” or “paper-folding sequences” previously studied by Davis and Knuth, Mendès France, and other writers. These “wire-bending sequences” have several surprising properties. We characterize the nth term of a wire-bending sequence in terms of the binary expansion of n. We prove that the curves traced out in R3 by many wire-bending sequences are actually bounded, although they are all aperiodic. Finally, we illustrate the close connection between wire-bending and the continued fractions for the transcendental numbers Σn ⩾ 0εng−2n, where εn = ± 1 and g ⩾ 3 is an integer.
Bookmarks Related papers MentionsView impact