Modal programming in Haskell
Haskell 2010 extended with higher-rank polymorphism is approximately
System Fω and thus roughly corresponds to intuitionistic higher-order
propositional logic via a Curry–Howard isomorphism. GHC’s static
pointers extension, while having been conceived specifically for
distributed programming, makes it possible to turn Haskell into the
Curry–Howard correspondents of typical modal logics. Since the static
pointers extension only provides the general basis for modal
programming, specific features of languages corresponding to concrete
modal logics must be implemented as libraries. The modal-programming
package provides support for writing such libraries.
Modal logics and modal programming languages
A propositional modal logic usually extends a non-modal propositional logic with the following components:
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Modalities, which turn propositions into propositions
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Axioms that involve these modalities
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A rule, called N, which is of the form “if A is a theorem, then UA is a theorem”, where U is a dedicated modality
Therefore, we can build a modal programming language by adding the following constructs to a non-modal programming language:
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Modalities, which turn types into types
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Primitive values whose types involve these modalities
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A new form of expression, absolute e, that has type Universal τ, provided that e is a closed expression of type τ, where Universal is a dedicated modality
The crucial points in the above lists are the last ones. Let us look at these in more detail.
In a modal logic, one assumes the existence of different worlds. Truth of a proposition depends on the choice of a world, and a theorem is a proposition that is true in every world. Analogously, in a modal programming language, the set of inhabitants of a type depends on the choice of a world, and there is a notion of absolute values: values that exist in every world.
A proposition A is a theorem exactly if it is true unconditionally, that is, if ⊢ A. Therefore, rule N can be formulated, “if ⊢ A, then ⊢ UA”. Analogously, a value is absolute exactly if it arises as the result of a closed expression, that is, an expression e for which ⊢ e : τ for some type τ. Therefore, the typing rule for absolute expressions can be formulated, “if ⊢ e : τ, then ⊢ absolute e : Universal τ”.
While a closed expression is an expression that does not contain any free variables, we can permit the expressions e in expressions absolute e to mention global variables, because we can always replace global variables by the right-hand sides of their definitions. Therefore, we ultimately define an absolute value as the result of an expression whose free variables are all global.
Encoding of modal logics and modal programming languages
The modalities of a modal logic can be encoded in higher-order
propositional logic as predicates on propositions, and the axioms that
involve these modalities can be rephrased using the encodings of these
modalities. Analogously, the modalities of a modal programming language
can be encoded in System Fω as types of kind ∗ → ∗ and thus in Haskell
as types of kind Type -> Type, and the primitive values whose types
involve these modalities can be encoded as primitive values whose types
involve the encodings of these modalities instead.
Unfortunately, it is impossible to encode rule N in higher-order
propositional logic, and analogously it is impossible to encode
absolute expressions in System Fω. In Haskell, however, we can
encode absolute using the static construct that comes with GHC’s
static pointers extension.
The modal-programming package
The modal-programming package provides support for implementing
Curry–Howard correspondents of various modal logics. It comprises a
type class of universal modalities as well as various modal analogs of
type classes that reflect constructs from category theory, like
Functor and Monad.