Relational Programming in miniKanren: Techniques, Applications, and Implementations

Abstract

The promise of logic programming is that programs can be written

<italic>relationally</italic>, without distinguishing between input

and output arguments. Relational programs are remarkably

flexible&mdash;for example, a relational type-inferencer also performs

type checking and type inhabitation, while a relational theorem prover

generates theorems as well as proofs and can even be used as a simple

proof assistant.

Unfortunately, writing relational programs is difficult, and requires

many interesting and unusual tools and techniques. For example, a

relational interpreter for a subset of Scheme might use nominal

unification to support variable binding and scope, Constraint Logic

Programming over Finite Domains (CLP(FD)) to implement relational

arithmetic, and tabling to improve termination behavior.

In this dissertation I present <italic>miniKanren</italic>, a family

of languages specifically designed for relational programming, and

which supports a variety of relational idioms and techniques. I show

how miniKanren can be used to write interesting relational programs,

including an extremely flexible lean tableau theorem prover and a

novel constraint-free binary arithmetic system with strong termination

guarantees. I also present interesting and practical techniques used

to implement miniKanren, including a nominal unifier that uses

triangular rather than idempotent substitutions and a novel

&ldquo;walk&rdquo;-based algorithm for variable lookup in triangular

substitutions.

The result of this research is a family of languages that supports a

variety of relational idioms and techniques, making it feasible and

useful to write interesting programs as relations.