Antimatroid
In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible states of such a process, or as a formal language modeling the different sequences in which elements may be included.
Dilworth was the first to study antimatroids, using yet another axiomatization based on lattice theory, and they have been frequently rediscovered in other contexts; see Korte et al. for a comprehensive survey of antimatroid theory with many additional references.
The axioms defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are defined by an exchange axiom, antimatroids are defined instead by an [|anti-exchange axiom], from which their name derives.
Antimatroids can be viewed as a special case of greedoids and of semimodular lattices, and as a generalization of partial orders and of distributive lattices.
Antimatroids are equivalent, by complementation, to [|convex geometries], a combinatorial abstraction of convex sets in geometry.
Antimatroids have been applied to model precedence constraints in scheduling problems, potential event sequences in simulations, task planning in artificial intelligence, and the states of knowledge of human learners.
Definitions
An antimatroid can be defined as a finite family F of sets, called feasible sets, with the following two properties:- The union of any two feasible sets is also feasible. That is, F is closed under unions.
- If S is a nonempty feasible set, then there exists some x in S such that S \ is also feasible. That is, F is an accessible set system.
- Every symbol of the alphabet occurs in at least one word of L.
- Each word of L contains at most one copy of any symbol.
- Every prefix of a string in L is also in L.
- If s and t are strings in L, and s contains at least one symbol that is not in t, then there is a symbol x in s such that tx is another string in L.
Examples
- A chain antimatroid has as its formal language the prefixes of a single word, and as its feasible sets the sets of symbols in these prefixes. For instance the chain antimatroid defined by the word "abcd" has as its formal language the strings and as its feasible sets the sets Ø,,,, and.
- A poset antimatroid has as its feasible sets the lower sets of a finite partially ordered set. By Birkhoff's representation theorem for distributive lattices, the feasible sets in a poset antimatroid form a distributive lattice, and any distributive lattice can be formed in this way. Thus, antimatroids can be seen as generalizations of distributive lattices. A chain antimatroid is the special case of a poset antimatroid for a total order.
- A shelling sequence of a finite set U of points in the Euclidean plane or a higher-dimensional Euclidean space is an ordering on the points such that, for each point p, there is a line that separates p from all later points in the sequence. Equivalently, p must be a vertex of the convex hull of it and all later points. The partial shelling sequences of a point set form an antimatroid, called a shelling antimatroid. The feasible sets of the shelling antimatroid are the intersections of U with the complement of a convex set. Every antimatroid is isomorphic to a shelling antimatroid of points in a sufficiently high-dimensional space.
- A perfect elimination ordering of a chordal graph is an ordering of its vertices such that, for each vertex v, the neighbors of v that occur later than v in the ordering form a clique. The prefixes of perfect elimination orderings of a chordal graph form an antimatroid. Antimatroids also describe some other kinds of vertex removal orderings in graphs, such as the dismantling orders of cop-win graphs.
Paths and basic words
For every feasible set S in the antimatroid, and every element x of S, one may find a path subset of S for which x is an endpoint: to do so, remove one at a time elements other than x until no such removal leaves a feasible subset. Therefore, each feasible set in an antimatroid is the union of its path subsets. If S is not a path, each subset in this union is a proper subset of S. But, if S is itself a path with endpoint x, each proper subset of S that belongs to the antimatroid excludes x. Therefore, the paths of an antimatroid are exactly the sets that do not equal the unions of their proper subsets in the antimatroid. Equivalently, a given family of sets P forms the set of paths of an antimatroid if and only if, for each S in P, the union of subsets of S in P has one fewer element than S itself. If so, F itself is the family of unions of subsets of P.
In the formal language formalization of an antimatroid we may also identify a subset of words that determine the whole language, the basic words.
The longest strings in L are called basic words; each basic word forms a permutation of the whole alphabet. For instance, the basic words of a poset antimatroid are the linear extensions of the given partial order. If B is the set of basic words, L can be defined from B as the set of prefixes of words in B. It is often convenient to define antimatroids from basic words in this way, but it is not straightforward to write an axiomatic definition of antimatroids in terms of their basic words.
Convex geometries
If F is the set system defining an antimatroid, with U equal to the union of the sets in F, then the family of setscomplementary to the sets in F is sometimes called a convex geometry and the sets in G are called convex sets. For instance, in a shelling antimatroid, the convex sets are intersections of U with convex subsets of the Euclidean space into which U is embedded.
Complementarily to the properties of set systems that define antimatroids, the set system defining a convex geometry should be closed under intersections, and for any set S in G that is not equal to U there must be an element x not in S that can be added to S to form another set in G.
A convex geometry can also be defined in terms of a closure operator τ that maps any subset of U to its minimal closed superset. To be a closure operator, τ should have the following properties:
- τ = ∅: the closure of the empty set is empty.
- Any set S is a subset of τ.
- If S is a subset of T, then τ must be a subset of τ.
- For any set S, τ = τ.
- If neither y nor z belong to τ, but z belongs to τ, then y does not belong to τ.
The undirected graphs in which the convex sets form a convex geometry are exactly the Ptolemaic graphs.
Join-distributive lattices
Any two sets in an antimatroid have a unique least upper bound and a unique greatest lower bound. Therefore, the sets of an antimatroid, partially ordered by set inclusion, form a lattice. Various important features of an antimatroid can be interpreted in lattice-theoretic terms; for instance the paths of an antimatroid are the join-irreducible elements of the corresponding lattice, and the basic words of the antimatroid correspond to maximal chains in the lattice. The lattices that arise from antimatroids in this way generalize the finite distributive lattices, and can be characterized in several different ways.- The description originally considered by concerns meet-irreducible elements of the lattice. For each element x of an antimatroid, there exists a unique maximal feasible set Sx that does not contain x. Sx is meet-irreducible, meaning that it is not the meet of any two larger lattice elements: any larger feasible set, and any intersection of larger feasible sets, contains x and so does not equal Sx. Any element of any lattice can be decomposed as a meet of meet-irreducible sets, often in multiple ways, but in the lattice corresponding to an antimatroid each element T has a unique minimal family of meet-irreducible sets Sx whose meet is T; this family consists of the sets Sx such that T ∪ belongs to the antimatroid. That is, the lattice has unique meet-irreducible decompositions.
- A second characterization concerns the intervals in the lattice, the sublattices defined by a pair of lattice elements x ≤ y and consisting of all lattice elements z with x ≤ z ≤ y. An interval is atomistic if every element in it is the join of atoms, and it is Boolean if it is isomorphic to the lattice of all subsets of a finite set. For an antimatroid, every interval that is atomistic is also boolean.
- Thirdly, the lattices arising from antimatroids are semimodular lattices, lattices that satisfy the upper semimodular law that for any two elements x and y, if y covers x ∧ y then x ∨ y covers x. Translating this condition into the sets of an antimatroid, if a set Y has only one element not belonging to X then that one element may be added to X to form another set in the antimatroid. Additionally, the lattice of an antimatroid has the meet-semidistributive property: for all lattice elements x, y, and z, if x ∧ y and x ∧ z are both equal then they also equal x ∧ . A semimodular and meet-semidistributive lattice is called a join-distributive lattice.
This representation of any finite join-distributive lattice as an accessible family of sets closed under unions may be viewed as an analogue of Birkhoff's representation theorem under which any finite distributive lattice has a representation as a family of sets closed under unions and intersections.
Supersolvable antimatroids
Motivated by a problem of defining partial orders on the elements of a Coxeter group, studied antimatroids which are also supersolvable lattices. A supersolvable antimatroid is defined by a totally ordered collection of elements, and a family of sets of these elements. The family must include the empty set. Additionally, it must have the property that if two sets A and B belong to the family, the set-theoretic difference B \ A is nonempty, and x is the smallest element of B \ A, then A ∪ also belongs to the family. As Armstrong observes, any family of sets of this type forms an antimatroid. Armstrong also provides a lattice-theoretic characterization of the antimatroids that this construction can form.Join operation and convex dimension
If A and B are two antimatroids, both described as a family of sets, and if the maximal sets in A and B are equal, we can form another antimatroid, the join of A and B, as follows:This is a different operation than the join considered in the lattice-theoretic characterizations of antimatroids: it combines two antimatroids to form another antimatroid, rather than combining two sets in an antimatroid to form another set.
The family of all antimatroids that have a given maximal set forms a semilattice with this join operation.
Joins are closely related to a closure operation that maps formal languages to antimatroids, where the closure of a language L is the intersection of all antimatroids containing L as a sublanguage. This closure has as its feasible sets the unions of prefixes of strings in L. In terms of this closure operation, the join is the closure of the union of the languages of A and B.
Every antimatroid can be represented as a join of a family of chain antimatroids, or equivalently as the closure of a set of basic words; the convex dimension of an antimatroid A is the minimum number of chain antimatroids in such a representation. If F is a family of chain antimatroids whose basic words all belong to A, then F generates A if and only if the feasible sets of F include all paths of A. The paths of A belonging to a single chain antimatroid must form a chain in the path poset of A, so the convex dimension of an antimatroid equals the minimum number of chains needed to cover the path poset, which by Dilworth's theorem equals the width of the path poset.
If one has a representation of an antimatroid as the closure of a set of d basic words, then this representation can be used to map the feasible sets of the antimatroid into d-dimensional Euclidean space: assign one coordinate per basic word w, and make the coordinate value of a feasible set S be the length of the longest prefix of w that is a subset of S. With this embedding, S is a subset of T if and only if the coordinates for S are all less than or equal to the corresponding coordinates of T. Therefore, the order dimension of the inclusion ordering of the feasible sets is at most equal to the convex dimension of the antimatroid. However, in general these two dimensions may be very different: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.
Enumeration
The number of possible antimatroids on a set of elements grows rapidly with the number of elements in the set. For sets of one, two, three, etc. elements, the number of distinct antimatroids isApplications
Both the precedence and release time constraints in the standard notation for theoretic scheduling problems may be modeled by antimatroids. use antimatroids to generalize a greedy algorithm of Eugene Lawler for optimally solving single-processor scheduling problems with precedence constraints in which the goal is to minimize the maximum penalty incurred by the late scheduling of a task.use antimatroids to model the ordering of events in discrete event simulation systems.
uses antimatroids to model progress towards a goal in artificial intelligence planning problems.
In Optimality Theory, grammars are logically equivalent to antimatroids.
In mathematical psychology, antimatroids have been used to describe feasible states of knowledge of a human learner. Each element of the antimatroid represents a concept that is to be understood by the learner, or a class of problems that he or she might be able to solve correctly, and the sets of elements that form the antimatroid represent possible sets of concepts that could be understood by a single person. The axioms defining an antimatroid may be phrased informally as stating that learning one concept can never prevent the learner from learning another concept, and that any feasible state of knowledge can be reached by learning a single concept at a time. The task of a knowledge assessment system is to infer the set of concepts known by a given learner by analyzing his or her responses to a small and well-chosen set of problems. In this context antimatroids have also been called "learning spaces" and "well-graded knowledge spaces".