Enter the computing arboretum and you will find a variety of well-studied trees: AVL trees (Adel'son-Vel'skiĭ & Landis 1962), symmetric binary B-trees (Bayer 1972), Hopcroft's 2-3 trees (Aho et al. 1974), the bushy finger trees (Guibas et al. 1977) and the colourful red-black trees (Guibas & Sedgewick 1978). In this pearl, we look at a more exotic species of balanced search trees, 1-2 brother trees (Ottmann et al. 1979), which deserves to be better known. Brother trees lend themselves well to a functional implementation with deletion (Section 5) as straightforward as insertion (Section 3), both running in logarithmic time. Furthermore, brother trees can be constructed from ordered lists in linear time (Section 4). With some simple optimisations in place, this implementation of search trees is one of the fastest around. So, fasten your seat belts.

Experimental game theory is increasingly important for research in many fields. Unfortunately, it is poorly supported by computer tools. We have created Hagl, a domain-specific language embedded in Haskell, to reduce the development time of game-theoretic experiments and make the definition and exploration of games and strategies simple and fun.

A constraint programming system combines two essential components: a constraint solver and a search engine. The constraint solver reasons about satisfiability of conjunctions of constraints, and the search engine controls the search for solutions by iteratively exploring a disjunctive search tree defined by the constraint program. In this paper we give a monadic definition of constraint programming in which the solver is defined as a monad threaded through the monadic search tree. We are then able to define search and search strategies as first-class objects that can themselves be built or extended by composable search transformers. Search transformers give a powerful and unifying approach to viewing search in constraint programming, and the resulting constraint programming system is first class and extremely flexible.

We present natural semantics for acyclic as well as cyclic call-by-need lambda calculi, which are proved equivalent to the reduction semantics given by Ariola and Felleisen (J. Funct. Program., vol. 7, no. 3, 1997). The natural semantics are big-step and use global heaps, where evaluation is suspended and memorized. The reduction semantics are small-step, and evaluation is suspended and memorized locally in let-bindings. Thus two styles of formalization describe the call-by-need strategy from different angles. The natural semantics for the acyclic calculus is revised from the previous presentation by Maraist et al. (J. Funct. Program., vol. 8, no. 3, 1998), and its adequacy is ascribed to its correspondence with the reduction semantics, which has been proved equivalent to call-by-name by Ariola and Felleisen. The natural semantics for the cyclic calculus is inspired by that of Launchbury (1993) and Sestoft (1997), and we state its adequacy using a denotational semantics in the style of Launchbury; adequacy of the reduction semantics for the cyclic calculus is in turn ascribed to its correspondence with the natural semantics.