Krivine machine

A redex[1] (one says also β-redex) is a term of the lambda calculus of the form (λ x. t) u. If a term has the shape (λ x. t) u₁ ... u_n it is said to be a head redex. A head normal form is a term of the lambda calculus which is not a head redex.[lower-alpha 1] A head reduction is a (non empty) sequence of contractions of a term which contracts head redexes. A head reduction of a term t (which is supposed not to be in head normal form) is a head reduction which starts from a term t and ends on a head normal form. From an abstract point of view, head reduction is the way a program computes when it evaluates a recursive sub-program. To understand how such a reduction can be implemented is important. One of the aims of the Krivine machine is to propose a process to reduct a term in head normal form and to describe formally this process. Like Turing used an abstract machine to describe formally the notion of algorithm, Krivine used an abstract machine to describe formally the notion of head normal form reduction.

The term ((λ 0) (λ 0)) (λ 0) (which corresponds, if one uses explicit variables, to the term (λx.x) (λy.y) (λz.z)) is not in head normal form because (λ 0) (λ 0) contracts in (λ 0) yielding the head redex (λ 0) (λ 0) which contracts in (λ 0) and which is therefore the head normal form of ((λ 0) (λ 0)) (λ 0). Said otherwise the head normal form contraction is:

((λ 0) (λ 0)) (λ 0) ➝ (λ 0) (λ 0) ➝ λ 0,

which corresponds to :

(λx.x) (λy.y) (λz.z) ➝ (λy.y) (λz.z) ➝ λz.z.

We will see further how the Krivine machine reduces the term ((λ 0) (λ 0)) (λ 0).

To implement the head reduction of a term u v which is an application, but which is not a redex, one must reduce the body u to exhibit an abstraction and therefore create a redex with v. When a redex appears, one reduces it. To reduce always the body of an application first is called call by name.[lower-alpha 2] The Krivine machine implements call by name.

The state has three components[2]

1. a term,
2. a stack,
3. an environment.

The term is a λ-term with de Bruijn indices. The stack and the environment belong to the same recursive data structure. More precisely, the environment and the stack are lists of pairs <term, environment>, that are called closures. In what follows, the insertion as the head of a list ℓ (stack or environment) of an element a is written a:ℓ, whereas the empty list is written □. The stack is the location where the machine stores the closures that must be evaluated furthermore, whereas the environment is the association between the indices and the closures at a given time during the evaluation. The first element of the environment is the closure associated with the index 0, the second element corresponds to the closure associated with index 1 etc. If the machine has to evaluate an index, it fetches there the pair <term, environment> the closure that yields the term to be evaluated and the environment in which this term must be evaluated.[lower-alpha 3] This intuitive explanations allow understanding the operating rules of the machine. If one writes t for term, p for stack,[lower-alpha 4] and e for environment, the states associated with these three entities will be written t, p, e. The rules explain how the machine transforms a state into another state, after identifying the patterns among the states.

The initial state aims to evaluate a term t, it is the state t,□,□, in which the term is t and the stack and the environment are empty. The final state (in absence of error) is of the form λ t, □, e, in other words, the resulting terms is an abstraction together with its environment and an empty stack.

The Krivine machine[2] has four transitions : App, Abs, Zero, Succ.

The transition App removes the parameter of an application and put it on the stack for further evaluation. The transition Abs removes the λ of the term and pop up the closure from the top of the stack and put it on the top of the environment. This closure corresponds to the de Bruijn index 0 in the new environment. The transition Zero takes the first closure of the environment. The term of this closure becomes the current term and the environment of this closure becomes the current environment. The transition Succ removes the first closure of the environment list and decreases the value of the index.

Let us evaluate the term (λ 0 0) (λ 0) which corresponds to the term (λ x. x x) (λ x. x). Let us start with the state (λ 0 0) (λ 0), □, □.

The conclusion is that the head normal form of the term (λ 0 0) (λ 0) is λ 0. This translates with variables in: the head normal form of the term (λ x. x x) (λ x. x) is λ x. x.

Let us evaluate the term ((λ 0) (λ 0)) (λ 0) as shown belown:

This confirms the above fact that the normal form of the term ((λ 0) (λ 0)) (λ 0) is (λ 0).