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17
Two Nonholonomic Lattice walks in the Quarter Plane
, 2007
"... We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The nonholonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks w ..."
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Cited by 20 (3 self)
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We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The nonholonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions. The method also yields an asymptotic expression for the number of walks of length n.
The polynomial method for random matrices
, 2007
"... We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions a ..."
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Cited by 10 (1 self)
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We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the MarčenkoPastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability ” theory. We hope that the tools developed allow researchers to finally harness the power of the infinite random matrix theory.
Classical and intuitionistic logic are asymptotically identical
, 2007
"... This paper considers logical formulas built on the single binary connector of implication and a finite number of variables. When the number of variables becomes large, we prove the following quantitative results: asymptotically, all classical tautologies are simple tautologies. It follows that asymp ..."
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Cited by 6 (5 self)
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This paper considers logical formulas built on the single binary connector of implication and a finite number of variables. When the number of variables becomes large, we prove the following quantitative results: asymptotically, all classical tautologies are simple tautologies. It follows that asymptotically, all classical tautologies are intuitionistic.
The fraction of large random trees representing a given Boolean function in implicational logic
 Journal Random Structures and Algorithms
, 2011
"... We consider the logical system of Boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of Boolean functions expressible in this syste ..."
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Cited by 5 (3 self)
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We consider the logical system of Boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of Boolean functions expressible in this system. Then we show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f. We also prove that most expressions computing any given function in this system are “simple”, in a sense that we make precise. The probability of all readonce functions of a given complexity is also evaluated in this model. At last, using the same techniques, the relation between the probability of a function and its complexity is also obtained when random expressions are drawn according to a critical branching process.
Coinductive Counting With Weighted Automata
, 2002
"... A general methodology is developed to compute the solution of a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite weighted automaton; (2) the automaton is reduced by means of the quantitative notion of stream bisimulation; ..."
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Cited by 4 (0 self)
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A general methodology is developed to compute the solution of a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite weighted automaton; (2) the automaton is reduced by means of the quantitative notion of stream bisimulation; (3) the reduced automaton is used to compute an expression (in terms of stream constants and operators) that represents the stream of all counts.
Complexity and Limiting Ratio of Boolean Functions over Implication
"... We consider the logical system of boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of boolean functions expressible in this syst ..."
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Cited by 4 (3 self)
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We consider the logical system of boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of boolean functions expressible in this system. We then show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f. We also prove that most expressions computing any given function in this system are “simple”, in a sense that we make precise.
Local limit distributions in pattern statistics: beyond the Markovian models
 Proceedings 21st S.T.A.C.S., V. Diekert and M. Habib editors, Lecture Notes in Computer Science
, 2004
"... Abstract Motivated by problems of pattern statistics, we study the limit distribution of the random variable counting the number of occurrences of the symbol ¥ in a word of length ¦ chosen at random in § ¥©¨����� � , according to a probability distribution defined via a finite automaton equipped wit ..."
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Cited by 3 (2 self)
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Abstract Motivated by problems of pattern statistics, we study the limit distribution of the random variable counting the number of occurrences of the symbol ¥ in a word of length ¦ chosen at random in § ¥©¨����� � , according to a probability distribution defined via a finite automaton equipped with positive real weights. We determine the local limit distribution of such a quantity under the hypothesis that the transition matrix naturally associated with the finite automaton is primitive. Our probabilistic model extends the Markovian models traditionally used in the literature on pattern statistics. This result is obtained by introducing a notion of symbolperiodicity for irreducible matrices whose entries are polynomials in one variable over an arbitrary positive semiring. This notion and the related results we prove are of interest in their own right, since they extend classical properties of the Perron–Frobenius Theory for nonnegative real matrices.
And/Or trees revisited
 in « Combinatorics, Probability and Computing », To appear
"... We consider boolean functions over n variables. Any such function can be represented (and computed) by a complete binary tree with and or or in the internal nodes and a literal in the external nodes, and many different trees can represent the same function, so that a fundamental question is rela ..."
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Cited by 3 (1 self)
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We consider boolean functions over n variables. Any such function can be represented (and computed) by a complete binary tree with and or or in the internal nodes and a literal in the external nodes, and many different trees can represent the same function, so that a fundamental question is related to the socalled complexity of a boolean function: L(f) := minimal size of a tree computing f .
Coinductive counting: bisimulation in enumerative combinatorics (extended abstract). Report SENR0129, CWI, 2001. Available at URL http://www.cwi.nl. Also in
 L. Moss (Ed.), The Proc. CMCS’02, ENTCS, Vol. 65, Elsevier Science B.V
, 2002
"... Coinductive counting: bisimulation in enumerative combinatorics (extended abstract) ..."
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Cited by 1 (1 self)
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Coinductive counting: bisimulation in enumerative combinatorics (extended abstract)