Lambda the Ultimate

Logfun is a library of logic functors. A logic functor is a function that can be applied to zero, one or several logics so as to produce a new logic as a combination of argument logics. Each argument logic can itself be built by combination of logic functors. The signature of a logic is made of a parser and a printer of formulas, logical operations such as a theorem prover for entailment between formulas, and more specific operations required by Logical Information Systems (LIS). Logic functors can be concrete domains like integers, strings, or algebraic combinators like product or sum of logics.

Logic functors are coded as Objective Caml modules. A logic semantics is associated to each of these logic functors. This enables to define properties of logics like the consistency and completeness of the entailment prover, and to prove under which conditions a generated entailement prover satisfies these properties given the properties of argument logics.

Logic Functors form a framework for specifying new logics, and deriving automatically theorem provers and consistency/completeness diagnoses. Atomic functors are logics for manipulating symbols and concrete domains, while other functors are logic transformers that may add connectives or recursive structures, or may alter the semantics of a logic. The semantic structure of the framework is model theoretic as opposed to the verifunctional style often used in classical logic. This comes close to the semantics of description logics, and we show indeed that the logic ALC can be rebuilt using logic functors. This offers the immediate advantage that variants of ALC can be explored and implemented almost for free.

The use of OCaml functors here may be interesting even for those who aren't into logic languages. The system allows new logics to be created in OCaml itself, by simply composing OCaml functors. This is covered further in Implementation as ML Functors. The quickest way to get a feel for what this system does is to look at the examples.