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Mechanizing Coinduction and Corecursion in Higherorder Logic
 Journal of Logic and Computation
, 1997
"... A theory of recursive and corecursive definitions has been developed in higherorder logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresse ..."
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A theory of recursive and corecursive definitions has been developed in higherorder logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresses recursive functions over inductive data types; corecursion expresses functions that yield elements of coinductive data types. The theory rests on a traditional formalization of infinite trees. The theory is intended for use in specification and verification. It supports reasoning about a wide range of computable functions, but it does not formalize their operational semantics and can express noncomputable functions also. The theory is illustrated using finite and infinite lists. Corecursion expresses functions over infinite lists; coinduction reasons about such functions. Key words. Isabelle, higherorder logic, coinduction, corecursion Copyright c fl 1996 by Lawrence C. Paulson Content...
Sandpile Models and Lattices: A Comprehensive Survey
, 2001
"... Starting from some studies of (linear) integer partitions, we noticed that the lattice structure is strongly related to a large variety of discrete dynamical models, in particular sandpile models and chip firing games. After giving an historical survey of the main results which appeared about this, ..."
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Starting from some studies of (linear) integer partitions, we noticed that the lattice structure is strongly related to a large variety of discrete dynamical models, in particular sandpile models and chip firing games. After giving an historical survey of the main results which appeared about this, we propose a unified framework to explain the strong relationship between these models and lattices. In particular, we show that the apparent complexity of these models can be reduced, by showing the possibility of symplifying them, and we show how the known lattice properties can be deduced from this.
Generalized integer partitions, tilings of zonotopes and lattices
 Proceedings of the 12th international conference Formal Power Series and Algebraic Combinatorics (FPSAC'00
, 2000
"... Abstract: In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple dynamics, have the distributive lattice structure. Likewi ..."
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Abstract: In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of zonotopes, ordered with a simple and classical dynamics, is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical systems exist. These results give a better understanding of the behaviour of tilings of zonotopes with flips and dynamical systems involving partitions.
Structure of some sand piles model
, 1998
"... Abstract: spm (Sand Pile Model) is a simple discrete dynamical system used in physics to represent granular objects. It is deeply related to integer partitions, and many other combinatorics problems, such as tilings or rewriting systems. The evolution of the system started with n stacked grains gene ..."
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Abstract: spm (Sand Pile Model) is a simple discrete dynamical system used in physics to represent granular objects. It is deeply related to integer partitions, and many other combinatorics problems, such as tilings or rewriting systems. The evolution of the system started with n stacked grains generates a lattice, denoted by SPM(n). We study here the structure of this lattice. We first explain how it can be constructed, by showing its strong selfsimilarity property. Then, we define SPM(∞), a natural extension of spm when one starts with an infinite number of grains. Again, we give an efficient construction algorithm and a coding of this lattice using a selfsimilar tree. The two approaches give different recursive formulae for SPM(n).
The lattice structure of chip firing games
, 2000
"... Abstract: In this paper, we study a classical discrete dynamical system, the Chip Firing Game, used as a model in physics, economics and computer science. We use order theory and show that the set of reachable states (i.e. the configuration space) of such a system started in any configuration is a l ..."
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Abstract: In this paper, we study a classical discrete dynamical system, the Chip Firing Game, used as a model in physics, economics and computer science. We use order theory and show that the set of reachable states (i.e. the configuration space) of such a system started in any configuration is a lattice, which implies strong structural properties. The lattice structure of the configuration space of a dynamical system is of great interest since it implies convergence (and more) if the configuration space is finite. If it is infinite, this property implies another kind of convergence: all the configurations reachable from two given configurations are reachable from their infimum. In other words, there is a unique first configuration which is reachable from two given configurations. Moreover, the Chip Firing Game is a very general model, and we show how known models can be encoded as Chip Firing Games, and how some results about them can be deduced from this paper. Finally, we introduce a new model, which is a generalization of the Chip Firing Game, and about which many interesting questions arise.
Partitions of an Integer into Powers
, 2001
"... In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ..."
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Cited by 5 (2 self)
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In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer
The lattice of integer partitions and its infinite extension
, 1999
"... Abstract: In this paper, we use a simple discrete dynamical system to study integers partitions and their lattice. The set of reachable configurations equiped with the order induced by the transitions of the system is exactly the lattice of integer partitions equiped with the dominance ordering. We ..."
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Cited by 5 (3 self)
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Abstract: In this paper, we use a simple discrete dynamical system to study integers partitions and their lattice. The set of reachable configurations equiped with the order induced by the transitions of the system is exactly the lattice of integer partitions equiped with the dominance ordering. We first explain how this lattice can be constructed, by showing its selfsimilarity property. Then, we define a natural extension of the system to infinity. Using a selfsimilar tree, we obtain an efficient coding of the obtained lattice. This approach gives an interesting recursive formula for the number of partitions of an integer, where no closed formula have ever been found. It also gives informations on special sets of partitions, such as length bounded partitions. 1
Design systems: combinatorial characterizations of Delsarte Tdesigns via partially ordered sets
"... . A number of important designtheoretic structures, such as combinatorial block designs and orthogonal arrays, can be characterized as Delsarte T designs in cometric association schemes. Several recent papers extend this theory to more exotic types of designs; here, the relevant association sch ..."
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. A number of important designtheoretic structures, such as combinatorial block designs and orthogonal arrays, can be characterized as Delsarte T designs in cometric association schemes. Several recent papers extend this theory to more exotic types of designs; here, the relevant association schemes fail to be cometric. In all cases, however, the language of partially ordered sets can be used to describe the designs and to establish the connection to association schemes. In this paper, we introduce \design systems" as a tool for describing all of the known examples in a uniform fashion. A design system consists of an association scheme with a partial order on its eigenspaces together with a second partially ordered set having the vertices of the scheme as its maximal elements. The key axiom for these systems ties the incidence matrices of this partial order to the eigenspaces of the corresponding association scheme. The results of the paper are as follows. First, given a ...
Coding Distributive Lattices with Edge Firing Games
"... In this note, we show that any distributive lattice is isomorphic to the set of reachable configurations of an Edge Firing Game. Together with the result of James Propp, saying that the set of reachable configurations of any Edge Firing Game is always a distributive lattice, this shows that the two ..."
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In this note, we show that any distributive lattice is isomorphic to the set of reachable configurations of an Edge Firing Game. Together with the result of James Propp, saying that the set of reachable configurations of any Edge Firing Game is always a distributive lattice, this shows that the two concepts are equivalent.
Generalized Tilings with Height Functions
, 2001
"... In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be dened (for example, the famous tilings of polyominoes with dominoes). We show that many properties of these tilings can be seen as the consequences of pr ..."
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In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be dened (for example, the famous tilings of polyominoes with dominoes). We show that many properties of these tilings can be seen as the consequences of properties of the generalized tilings we introduce. In particular, we show that any tiling problem which can be modelized in our generalized framework has the following properties: the tilability of a region can be constructively decided in polynomial time, the number of connected components in the undirected ipaccessibility graph can be determined, and the directed ipaccessibility graph induces a distributive lattice structure. Finally, we give a few examples of known tiling problems which can be viewed as particular cases of the new notions we introduce.