  With sound unification, Definite Clause Grammars and compact expression of
combinatorial generation algorithms, logic programming is shown to conveniently
host a declarative playground where interesting properties and behaviors emerge
from the interaction of heterogenous but deeply connected computational
objects.
  Compact combinatorial generation algorithms are given for several families of
lambda terms, including open, closed, simply typed and linear terms as well as
type inference and normal order reduction algorithms. We describe a
Prolog-based combined lambda term generator and type-inferrer for closed
well-typed terms of a given size, in de Bruijn notation.
  We introduce a compressed de Bruijn representation of lambda terms and define
its bijections to standard representations. Our compressed terms facilitate
derivation of size-proportionate ranking and unranking algorithms of lambda
terms and their inferred simple types.
  The S and K combinator expressions form a well-known Turing-complete subset
of the lambda calculus. We specify evaluation, type inference and combinatorial
generation algorithms for SK-combinator trees. In the process, we unravel
properties shedding new light on interesting aspects of their structure and
distribution.
  A uniform representation, as binary trees with empty leaves, is given to
expressions built with Rosser's X-combinator, natural numbers, lambda terms and
simple types. Using this shared representation, ranking/unranking algorithm of
lambda terms to tree-based natural numbers are described.
  Our algorithms, expressed as an incrementally developed literate Prolog
program, implement a declarative playground for exploration of representations,
encodings and computations with uniformly represented lambda terms, types,
combinators and tree-based arithmetic.
