[ Background | Reference ]

Infers is a library for deriving F# values from their types and, in a way, a direct application of the Curry-Howard correspondence aka Propositions as Types. Another way to describe Infers is as a logic programming language embedded in F#. Infers can be useful, for example, in situations where one might wish to use something like type classes or when one might want to do polytypic or datatype generic programming.

The basic idea of Infers is to view the types of static member functions as Horn clauses. Given a set of Rules, it is then possible to attempt to prove goals using a Prolog-style resolution algorithm. Infers invokes the rule functions during the resolution process to generate a value of the type given as the goal.

Aside from thinking of Infers as a library for generating values, another way to view Infers is as a specialized logic programming language embedded in F#. Applications such as solving logic puzzles that are typical for more general purpose logic programming languages are also possible with Infers. The logo, ⊃∧∨≡, lists the connectives, namely implication, conjunction, disjunction and equivalence, that are provided by the logic of Infers.

To support generation of F# values, the Infers resolution algorithm differs from general purpose logic programming languages in a number of ways:

• Infers prunes the search space so that when it encounters a goal to build a monomorphic value, it only tries to find one way, rather than all possible ways, to build it.

• Infers statically memoizes all final and intermediate values that it has built. When you invoke Infers twice with the same set of rules and the same type to generate, Infers returns the same (physical) value.

• Infers has special support, Rec<'t>, for building cyclic values. It is very common to need to build cyclic values to manipulate recursive types.

• Infers has special scoping rules such that when an antecedent value is built that contains Rules, those rules are added to the set of rules until the consequent has been built. This allows new rules, such as rules for viewing a type as a sum of products, to be generated dynamically.

Here is a toy example of a set of rules that can generate functions to arbitrarily reorder or flip the arguments of a given curried function:

type GFlip () =
  inherit Rules ()
  static member Id () = id
  static member First ab2yz = fun xb -> xb >> ab2yz
  static member Rest (ab2axc, ac2y) = fun ab ->
    let axc = ab2axc ab
    let xac = fun x a -> axc a x
    xac >> ac2y

To generate flipping functions we invoke generate:

let gflip f = generate<GFlip, (_ -> _) -> _ -> _> f

Now, for example, we could say:

gflip (sprintf "%s %d %c!") 2 'U' "Hello" = "Hello 2 U!"

There is a caveat: When you request Infers to generate a value, the value must have a monomorphic type.

Here is how you can get F# interactive running to try the above:

git clone https://github.com/Infers/Infers.git
cd Infers
./play.sh

namespace Infers

type Rules =
  inherit System.Attribute
  new: unit -> Rules

type Rec<'t> =
  new: unit -> Rec<'t>
  abstract Get: unit -> 't
  abstract Set: 't -> unit

[<AutoOpen>]
module Infers =
  val    generate<'r, 't when 'r :> Rules and 'r: (new: unit -> 'r)> : 't
  val generateDFS<'r, 't when 'r :> Rules and 'r: (new: unit -> 'r)> : 't

A type that inherits Rules is assumed to contain total static rule methods that are used by the resolution algorithm of Infers. Do not inherit from a class that inherits Rules. A rule class can specify dependencies to other rule classes as attributes. Specify any rule classes that you wish to include as attributes, e.g. type [<Rules1;...;RulesN>] MyRules.

A Rec<'t> is a proxy for a potentially cyclic value of type 't.

When the Infers resolution algorithm encounters a case where it needs to build a value in terms of itself, for example, when building a function manipulating a recursive union type, it automatically looks for a rule to create a proxy for the value. To support building cyclic values of type 't, a rule must be given to build a Rec<'t>.

The Get method must return a wrapper of type 't that corresponds to the value of the proxy. Note that Get may be called on a Rec proxy before Set is called.

The Set method must set the value of the proxy to close the resulting cyclic value.

The type-indexed functions of the Infers module, generate<'r, 't> and generateDFS<'r, 't> are the interface to the Infers resolution algorithm. They both try to generate a value of type 't by using the given set of rules 'r. If a value can be generated, it is statically memoized, so that subsequently the same value is returned instantly. An exception is raised in case Infers detects that there is no way to build the desired value with the given rules.

Uses IDDFS to try to find a way to generate a value of type 't using the rules 'r. IDDFS is slow, but works even in cases where the given rules allow infinite non-productive derivations. IDDFS also always finds a minimal solution in the sense that the depth of the derivation tree is minimal.

Uses DFS to try to find a way to generate a value of type 't using the rules 'r. DFS is fast, but requires that the given rules do not allow infinite non-productive derivations. DFS also does not necessarily find a minimal solution. Therefore, DFS should only be used when the rules are essentially deterministic, which basically means that there is only one way to generate a value of any given type using the rules.

In practice, one should almost always seek to create rules that work with DFS, but there are some valid applications where DFS will not work.