A language L over an alphabet Σ is prefix-convex if, for any words x, y, z ∈ Σ * , whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages as special cases. We examine complexity properties of these special prefix-convex languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal, the size of the syntactic semigroup, and the quotient complexity of atoms. For binary operations we use arguments with different alphabets when appropriate; this leads to higher tight upper bounds than those obtained with equal alphabets. We exhibit right-ideal, prefix-closed, and prefix-free languages that meet the complexity bounds for all the measures listed above.

A language L over an alphabet Σ is prefix-convex if, for any words x, y, z ∈ Σ * , whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefixclosed, and prefix-free languages. We study complexity properties of prefix-convex regular languages. In particular, we find the quotient/state complexity of boolean operations, product (concatenation), star, and reversal, the size of the syntactic semigroup, and the quotient complexity of atoms. For binary operations we use arguments with different alphabets when appropriate; this leads to higher tight upper bounds than those obtained with equal alphabets. We exhibit most complex prefix-convex languages that meet the complexity bounds for all the measures listed above.Finally, if X = X ′ = ∅ and r ∈ Y ′ \ Y , then distinguish J X,Y and J X ′ ,Y ′ by a word that sends r → n − 2 and Q n \ {r} → {n − 1}. Hence, J X,Y and J X ′ ,Y ′ are distinct in all cases. Therefore, the quotients of A S counted in the upper bound are pairwise distinct and L n,k has maximal atomic complexity.

A language L over an alphabet Σ is suffix-convex if, for any words x, y, z ∈ Σ * , whenever z and xyz are in L, then so is yz. Suffixconvex languages include three special cases: left-ideal, suffix-closed, and suffix-free languages. We examine complexity properties of these three special classes of suffix-convex regular languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient complexity of their atoms.

We present a space-efficient data structure using O(n log n) bits that represents a distributive lattice on n elements and supports finding meets and joins in O(log n) time. Our data structure extends the ideal tree structure of Habib and Nourine which occupies O(n log n) bits of space and requires O(m) time to compute a meet or join, where m depends on the specific lattice and may be as large as n − 1. We also give an encoding of a distributive lattice using 10 7 n + O(log n) bits, which is very close to the information theoretic lower bound. This encoding can be created or decompressed in O(n log n) time.

We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space.