Search

Coinduction refers to both a technique for the definition of infinite streams, so-called codata, and a technique for proving the equality of coinductively specified codata. This article first reviews coinduction in declarative programming. Second, it reviews and slightly extends the formalism commonly used for specifying codata. Third, it generalizes the coinduction proof principle, which has been originally specified for the equality predicate only, to other predicates. This generalization makes the coinduction proof principle more intuitive and stresses its closeness with structural induction. The article finally suggests in its conclusion extensions of functional and logic programming with limited and decidable forms of the generalized coinduction proof principle.

Processing programs as data is one of the successes of functional and logic programming. Higher-order functions, as program-processing programs are called in functional programming, and meta-programs, as they are called in logic programming, are widespread declarative programming techniques. In logic programming, there is a gap between the meta-programming practice and its theory: The formalisations of meta-programming do not explicitly address its impredicativity and are not fully adequate. This article aims at overcoming this unsatisfactory situation by discussing the relevance of impredicativity to meta-programming, by revisiting former formalisations of meta-programming and by defining Reflective Predicate Logic, a conservative extension of first-order logic, which provides a simple formalisation of meta-programming., Comment: Under consideration for publication in Theory and Practice of Logic Programming

The model theory of a first-order logic called N^4 is introduced. N^4 does not eliminate double negations, as classical logic does, but instead reduces fourfold negations. N^4 is very close to classical logic: N^4 has two truth values; implications in N^4 are material, like in classical logic; and negation distributes over compound formulas in N^4 as it does in classical logic. Results suggest that the semantics of normal logic programs is conveniently formalized in N^4: Classical logic Herbrand interpretations generalize straightforwardly to N^4; the classical minimal Herbrand model of a positive logic program coincides with its unique minimal N^4 Herbrand model; the stable models of a normal logic program and its so-called complete minimal N^4 Herbrand models coincide., Comment: 16 pages. Originally published in proc. PCL 2002, a FLoC workshop; eds. Hendrik Decker, Dina Goldin, Jorgen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/)