BHK interpretation
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In mathematical logic, the BHK interpretation of intuitionistic logic was proposed by L. E. J. Brouwer, Arend Heyting and independently by Andrey Kolmogorov. It is also sometimes called the realizability interpretation, because of the connection with the realizability theory of Stephen Kleene.
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[edit] The interpretation
The interpretation states exactly what is to be considered a proof of a formula. This is specified by induction on the structure of the formula, presupposing that one begins with a) some notion of what a "primitive forumula" (simple statement) is, and b) some notion of what it means to be a proof of a primitive forumla.
- A proof of
is a pair <a,b> where a is a proof of P and b is a proof of Q. - A proof of
is a pair <a,b> where a is 0 and b is a proof of P, or a is 1 and b is a proof of Q. - A proof of
is a function f which will convert any proof of P into a proof of Q. - A proof of
is a pair <a,b> where a is an element of S, and b is a proof of φ(a). - A proof of
is a function f which will convert any element a of S into a proof of φ(a). 
is absurdity, which is normally assumed to have no proof (e.g., could stand for "0=1").- The formula
is defined as 
, so a proof of it is a function f which converts a proof of P into a proof of .
What is considered a "function" is open for interpretation, however, a typical conceptual example would be a "proof fragment" that could be appended to a proof of P to obtain a proof of Q (as for 3), or a "proof template" with blanks into which a particular element a could be substituted to obtain a proof of φ(a) (as for 5).
The interpretation of a primitive formula is supposed to be known from context.
[edit] Examples
The identity function is a proof of the formula 
, no matter what P is.
The law of non-contradiction 
expands to 
:
- A proof of is a function f which converts a proof of
to a proof of . - A proof of is a pair of proofs <a,b>, where a is a proof of P, and b is a proof of .
- A proof of is a function which converts a proof of P to a proof of .
Putting it all together, a proof of is a function f which converts a pair <a,b> -- where a is a proof of P, and b is a function which converts a proof of P to a proof of -- into a proof of . The function 
fits the bill, proving the law of non-contradiction, no matter what P is.
On the other hand, the law of excluded middle 
expands to 
, and in general has no proof. According to the interpretation, a proof of is a pair <a,b> where a is 0 and b is a proof of P, or a is 1 and b is a proof of . Thus if neither P nor is provable then neither is .
[edit] What is absurdity?
It is not possible for a logical system to have a formal negation operator such that there is a proof of "not" P exactly when there isn't a proof of P (in general -- see Gödel's incompleteness theorems). The BHK interpretation instead takes "not" P to mean that P leads to absurdity, designated , so that a proof of ¬P is a function converting a proof of P into a proof of absurdity.
A standard example of absurdity is found in dealing with arithmetic. Assume that 0=1, and proceed by mathematical induction: 0=0 by the axiom of equality. Now (induction hypothesis), if 0 were equal to a certain natural number n, then 1 would be equal to n+1, (Peano axiom: Sm = Sn if and only if m = n), but since 0=1, therefore 0 would also be equal to n+1. By induction, 0 is equal to all numbers, and therefore any two natural numbers become equal.
Therefore, there is a way to go from a proof of 0=1 to a proof of any basic arithmetic equality, and thus to a proof of any complex arithmetic proposition. Furthermore, to get this result it was not necessary to invoke the Peano axiom which states that 0 is "not" the successor of any natural number. This makes 0=1 suitable as in Heyting arithmetic (and the Peano axiom is rewritten 0=Sn → 0=S0). This use of 0=1 validates the principle of explosion.
[edit] What is a function?
The BHK interpretation will depend on the view taken about what constitutes a function which converts one proof to another, or which converts an element of a domain to a proof. Different versions of constructivism will diverge on this point.
Kleene's realizability theory identifies the functions with the computable functions. It deals with Heyting arithmetic, where the domain of quantification is the natural numbers and the primitive propositions are of the form x=y. A proof of x=y is simply the trivial algorithm if x evaluates to the same number that y does (which is always decidable for natural numbers), otherwise there is no proof. These are then built up by induction into more complex algorithms.
