Refocusing (semantics)


In computer science, refocusing is a program transformation used to implement a reduction semantics—i.e., a small-step operational semantics with an explicit representation of the reduction context—more efficiently. It is a step towards implementing a deterministic semantics as a deterministic abstract machine.
A small-step operational semantics defines the meaning of a given program as a sequence of one-step reductions that starts with and continues with a sequence of reducts, where :
A one-step reduction from to is achieved by
  1. locating the smallest potentially reducible term using a given reduction strategy, if this potential redex exists in ; and
  2. contracting this potential redex using given contraction rules if it is an actual one.
A reduction semantics is a small-step operational semantics with an explicit representation of the context of each potential redex.
Writing for such context, the sequence of one-step reductions above reads:
where
  1. is first decomposed into the context and a potential redex,
  2. is contracted into the contractum,
  3. is then recomposed around and then decomposed into the context and a potential redex,
  4. etc.
This succession of decompositions, contractions, and recomposition is depicted as follows:

 contract contract contract 
 o--------->o o--------->o o--------->o 
 / \ / \ / \ 
 / recompose \ / recompose \ / recompose \ 
 / \ / \ / \ 
 / decompose \ / decompose \ / decompose \ 
 / \ / \ / \ 
 o--------------------->o--------------------->o--------------------->o
 reduce reduce reduce

Refocusing is a deforestation of the successive reducts:

 contract refocus contract refocus contract 
 o--------->o---------->o--------->o---------->o--------->o------
 / \ / \ / \ 
 / recompose \ / recompose \ / recompose \ 
 / \ / \ / \ 
 / decompose \ / decompose \ / decompose \ 
 / \ / \ / \ 
 o o o o

After the initial decomposition, the succession of contractions and refocusings has the structure of a deterministic abstract machine.

Background

The semantics of a programming language defines the meaning of the programs written in this programming language.
Plotkin's Structural Operational Semantics is a small-step semantics where the meaning of a program is defined step by step and where each step is an elementary operation that is carried out with contraction rules.

Example

Consider the following deterministic language of arithmetic expressions over integers with additions and quotients, in the manner of Hutton's razor.
In OCaml:

type operator = Add | Quo;;
type expression = Lit of int | Opr of expression * operator * expression;;

So
is parsed as Opr
and
is parsed as Opr .

type value = Int of int;;
let expression_of_value : expression = match v with Int n -> Lit n;;

The smallest potentially reducible expressions are operations over values, and they are carried out with a contraction function that maps an actual redex to an expression and otherwise yields an error message:

type potential_redex = PR of value * operator * value;;
type contractum_or_error = Contractum of expression | Error of string;;
let contract : contractum_or_error =
match pr with
PR ->
Contractum
| PR ->
if n2 = 0
then Error
else Contractum ;;

The addition of two integers is an actual redex, and so is the quotient of an integer and a nonzero integer.
So for example, the expression
Opr ,
i.e.,
,
reduces to
Opr ,
i.e.,
the expression
Opr ,
i.e.,
reduces to
Lit 5,
i.e.,
since,
and
the expression
Opr ,
i.e.,
,
reduces to
Error "1 / 0".
Say that the reduction strategy is leftmost-innermost, as captured by the following congruence rules:

 e1 -> e1'
 ---------------------------------------
 Opr -> Opr 
 e2 -> e2'
 -----------------------------------------------
 Opr -> Opr

The following one-step reduction function implements this strategy:

let rec reduce_d : value_or_expression_or_stuck =
match e with
Lit n ->
Value
| Opr ->
match reduce_d e1 with
Value v1 ->

| Expression e2' ->
Expression
| Stuck s ->
Stuck s)
| Expression e1' ->
Expression
| Stuck s ->
Stuck s;;

In words:
  • a literal reduces to a value;
  • if the expression e1 is stuck, then so is the expression Opr , for any expression e2;
  • if the expression e1 reduces to an expression e1', then for any expression e2, Opr reduces to Opr ;
  • if the expression e1 reduces to a value v1, then
  • * if the expression e2 is stuck, then so is the expression Opr ;
  • * if the expression e2 reduces to an expression e2', then Opr reduces to Opr ;
  • * if the expression e2 reduces to a value v2, then Opr is a potential redex:
  • ** if this potential redex is an actual one, then contracting it yields an expression; Opr reduces to this expression;
  • ** if this potential redex is not an actual one, then Opr is stuck.
Evaluation is achieved by iterated reduction.
It yields either a value or an error message:

type result = Normal_form of value | Wrong of string;;
let rec normalize_d : result =
match reduce_d e with
Value v ->
Normal_form v
| Expression e' ->
normalize_d e'
| Stuck s ->
Wrong s;;

This one-step reduction function is structurally recursive.
It implements a Structural Operational Semantics for this minimalistic language of arithmetic expressions with errors.
For example, the reduction function implicitly constructs the following proof tree to carry out the reduction step

 ------------
 5 + 5 -> 10
 ---------------------
 1 - -> 1 - 10
 -----------------------------------------------
 - -> -

Reformatting this proof tree to emphasize the implicit decomposition yields:

 -------------------------------------------------->
 ^ contraction |
 | ----------------------------- |
 | 5 + 5 -> 10 |
 implicit | ---------------------------------- | implicit
 decomposition | 1 - -> 1 - 10 | recomposition
 | ----------------------------------------------- |
 | - -> - v

A reduction semantics is a small-step operational semantics where the implicit context of a potential redex is made explicit.
So one reduction step gives rise to
  1. constructing the context of the redex,
  2. contracting this redex, and
  3. recomposing the context around the contractum to yield the reduct:

 - \
 | explicit
 1 - ] - ] | decomposition
 1 - /
 contraction
 1 - \
 1 - 10 ] - ] | explicit
 | recomposition
 - /

And pictorially, an arithmetic expression is evaluated in successive steps:

 contract contract contract 
 o--------->o o--------->o o--------->o 
 / \ / \ / \ 
 / recompose \ / recompose \ / recompose \ 
 / \ / \ / \ 
 / decompose \ / decompose \ / decompose \ 
 / \ / \ / \ 
 o--------------------->o--------------------->o--------------------->o
 reduce reduce reduce

Transforming the one-step reduction function in Continuation-Passing Style, delimiting the continuation from type value_or_expression_or_stuck -> 'a to type value_or_expression_or_stuck -> value_or_expression_or_stuck,
and splitting this delimited continuation into two makes it simple to implement the corresponding normalization function:

type value_or_decomposition_cc =
Val_cc of value
let rec decompose_expression_cc : value_or_decomposition_cc =
match e with
Lit n ->
kd
| Opr ->
decompose_expression_cc
e1

)
;;
let decompose_cc : value_or_decomposition_cc =
decompose_expression_cc e ;;
let rec iterate_cc_rb : result =
match vod with
Val_cc v ->
Normal_form v
| Dec_cc ->

Wrong s);;
let normalize_cc_rb : result =
iterate_cc_rb ;;

In the underlined code, the contractum is recomposed and the result is decomposed.
This normalization function is said to be reduction-based because it enumerates all the reducts in the reduction sequence.

Refocusing

Extensionally, the refocusing thesis is that there is no need to reconstruct the next reduct in order to decompose it in the next reduction step.
In other words, these intermediate reducts can be deforested.
Pictorially:

 contract refocus contract refocus contract 
 o--------->o---------->o--------->o---------->o--------->o------
 / \ / \ / \ 
 / recompose \ / recompose \ / recompose \ 
 / \ / \ / \ 
 / decompose \ / decompose \ / decompose \ 
 / \ / \ / \ 
 o o o o

Intensionally, the refocusing thesis is that this deforestation is achieved by continuing the decomposition over the contractum in the current context.

let rec iterate_cc_rf : result =
match vod with
Val_cc v ->
Normal_form v
| Dec_cc ->

| Error s ->
Wrong s);;
let normalize_cc_rf : result =
iterate_cc_rf ;;

In the underlined code, the decomposition is continued.
This normalization function is said to be reduction-free because it enumerates none of the reducts in the reduction sequence.
In practice, the two continuations are defunctionalized into traditional first-order, inside-out contexts, which yields an implementation of Felleisen and Hieb's reduction semantics,
a small-step semantics that was designed independently of continuations and defunctionalization,
but whose representation—as illustrated here—can be obtained by CPS-transforming and defunctionalizing the representation of a Structural Operational Semantics.
The construction sketched above is completely formalized using the Rocq Proof Assistant.

Applications

Over the years, refocusing has been used for inter-deriving calculi and abstract machines.
Besides the CEK Machine, the Krivine machine, and the SECD machine, examples also include the chemical abstract machine and abstract machines for JavaScript.
Bach Poulsen and Mosses have also used refocusing for implementing Structural Operational Semantics and Modular Structural Operational Semantics.
More broadly, refocusing has been used for deriving type systems and implementations for coroutines,
for going from type checking via reduction to type checking via evaluation,
for deriving a classical call-by-need sequent calculus,
for deriving interpretations of the gradually-typed lambda calculus,
and for full reduction.

Correctness

Danvy and Nielsen stated conditions for refocusing and proved them informally.
Sieczkowski, Biernacka, and Biernacki formalized refocusing using the Rocq Proof Assistant.
Bach Poulsen proved the correctness of refocusing for XSOS using rule induction.
Biernacka, Charatonik, and Zielińska generalized refocusing using the Rocq Proof Assistant.
Using Agda, Swiestra proved the refocusing step as part of his formalization of the syntactic correspondence between the calculus with a normal-order reduction strategy and the Krivine machine.
Also using Agda, Rozowski proved the refocusing step as part of his formalization of the syntactic correspondence between the calculus with an applicative-order reduction strategy and the CEK Machine.