Traversal Combinators¶
There are many ways to traverse a tree. For example, a bottomup traversal, visits the subterms of a node before it visits the node itself, while a topdown traversal visits nodes before it visits children. Onepass traversals traverse the tree one time, while fixedpoint traversals, such as innermost, repeatedly traverse a term until a normal form is reached.
Rather than provide builtin implementations for all traversals needed in transformations, Stratego defines traversals in terms of the primitive ingredients of traversal. For example, a topdown, onepass traversal strategy will first visit a node, and then descend to the children of a node in order to recursively traverse all subterms. Similarly, the bottomup, fixedpoint traversal strategy innermost, will first descend to the children of a node in order to recursively traverse all subterms, then visit the node itself, and possibly recursively reapply the strategy.
Traversal in Stratego is based on the observation^{1} that a full term traversal is a recursive closure of a onestep descent, that is, an operation that applies a strategy to one or more direct subterms of the subject term. By separating this onestep descent operator from recursion, and making it a firstclass operation, many different traversals can be defined.
Here we explore the ways in which Stratego supports the definition of traversal strategies. We start with explicitly programmed traversals using recursive traversal rules. Next, congruences operators provide a more concise notation for such datatype specific traversal rules. Finally, generic traversal operators support data type independent definitions of traversals, which can be reused for any data type. Given these basic mechanisms, we conclude with an exploration of idioms for traversal and standard traversal strategies in the Stratego Library.
Congruence Operators¶
Congruence operators provide a convenient abbreviation of traversal with rewrite rules.
A congruence operator applies a strategy to each direct subterm of a specific constructor.
For each nary constructor c
declared in a signature, there is a corresponding congruence operator c(s1 , ..., sn)
, which applies to terms of the form c(t1 , ..., tn)
by applying the argument strategies to the corresponding argument terms.
A congruence fails if the application of one the argument strategies fails or if constructor of the operator and that of the term do not match.
For example, consider the following signature of expressions:
module expressions
signature
sorts Exp
constructors
Int : String > Exp
Var : String > Exp
Plus : Exp * Exp > Exp
Times : Exp * Exp > Exp
The following applications apply the congruence operators Plus
and Times
to a term:
<Plus(!Var("a"), id)> Plus(Int("14"),Int("3")) => Plus(Var("a"),Int("3"))
<Times(id, !Int("42"))> Plus(Var("a"),Int("3")) // fails
The first application shows how a congruence transforms a specific subterm, that is the strategy applied can be different for each subterm. The second application shows that a congruence only succeeds for terms constructed with the same constructor.
Defining Traversals with Congruences¶
Since congruence operators define a onestep traversal for a specific constructor, they capture the pattern of traversal rules. That is, a traversal rule such as
proptr(s) : And(x, y) > And(<s>x, <s>y)
can be written by the congruence And(s,s)
.
Applying this to the propdnf
program we can replace the traversal rules by congruences as follows:
module propdnf10
imports proprules
strategies
proptr(s) = Not(s) <+ And(s, s) <+ Or(s, s) <+ Impl(s, s) <+ Eq(s, s)
propbu(s) = try(proptr(propbu(s))); s
strategies
dnf = propbu(try(DN <+ (DefI <+ DefE <+ DMA <+ DMO <+ DAOL <+ DAOR); dnf))
cnf = propbu(try(DN <+ (DefI <+ DefE <+ DMA <+ DMO <+ DOAL <+ DOAR); cnf))
Traversing Tuples and Lists¶
Congruences can also be applied to tuples, (s1,s2,...,sn)
, and lists, [s1,s2,...,sn]
.
A special list congruence is []
which ‘visits’ the empty list.
As an example, consider again the definition of map(s)
using recursive traversal rules:
map(s) : [] > []
map(s) : [x  xs] > [<s> x  <map(s)> xs]
Using list congruences we can define this strategy as:
map(s) = [] <+ [s  map(s)]
The []
congruence matches an empty list.
The [s  map(s)]
congruence matches a nonempty list, and applies s
to the head of the list and map(s)
to the tail.
Thus, map(s)
applies s
to each element of a list:
<map(inc)> [1,2,3] => [2,3,4]
Note that map(s)
only succeeds if s
succeeds for each element of the list.
The fetch
and filter
strategies are variations on map that use the failure of s
to list elements.
fetch(s) = [s  id] <+ [id  fetch(s)]
The fetch
strategy traverses a list until it finds a element for which s
succeeds and then stops.
That element is the only one that is transformed.
filter(s) = [] + ([s  filter(s)] <+ ?[ <id>]; filter(s))
The filter
strategy applies s
to each element of a list, but only keeps the elements for which it succeeds.
even = where(<eq>(<mod>(<id>,2),0))
<filter(even)> [1,2,3,4,5,6,7,8] => [2,4,6,8]
Format Checking¶
Another application of congruences is in the definition of format checkers. A format checker describes a subset of a term language using a recursive pattern. This can be used to verify input or output of a transformation, and for documentation purposes. Format checkers defined with congruences can check subsets of signatures or regular tree grammars. For example, the subset of terms of a signature in a some normal form.
As an example, consider checking the output of the dnf
and cnf
transformations.
conj(s) = And(conj(s), conj(s)) <+ s
disj(s) = Or (disj(s), disj(s)) <+ s
// Conjunctive normal form
conjnf = conj(disj(Not(Atom(id)) <+ Atom(id)))
// Disjunctive normal form
disjnf = disj(conj(Not(Atom(id)) <+ Atom(id)))
The strategies conj(s)
and disj(s)
check that the subject term is a conjunct or a disjunct, respectively, with terms satisfying s at the leaves.
The strategies conjnf
and disjnf
check that the subject term is in conjunctive or disjunctive normal form, respectively.
Generic Traversal¶
Using congruence operators we constructed a generic, i.e. transformation independent, bottomup traversal for proposition terms.
The same can be done for other data types.
However, since the sets of constructors of abstract syntax trees of typical programming languages can be quite large, this may still amount to quite a bit of work that is not reusable across data types; even though a strategy such as bottomup traversal, is basically datatype independent.
Thus, Stratego provides generic traversal by means of several generic onestep descent operators.
The operator all
, applies a strategy to all direct subterms.
The operator one
, applies a strategy to one direct subterm, and the operator some
, applies a strategy to as many direct subterms as possible, and at least one.
Visiting All Subterms¶
The all(s)
strategy transforms a constructor application by applying the parameter strategy s
to each direct subterm.
An application of all(s)
fails if the application to one of the subterms fails.
The following example shows how all (1) applies to any term, and (2) applies its argument strategy uniformly to all direct subterms.
That is, it is not possible to do something special for a particular subterm (that’s what congruences are for).
<all(!Var("a"))>
Plus(Int("14"), Int("3")) => Plus(Var("a"), Var("a"))
<all(!Var("z"))>
Times(Var("b"), Int("3")) => Times(Var("z"), Var("z"))
Defining Traversals with All¶
The all(s)
operator is really the ultimate replacement for the traversal with rules idiom.
Instead of specifying a rule or congruence for each constructor, the single application of the all
operator takes care of traversing all constructors.
Thus, we can replace the propbu
strategy by a completely generic definition of bottomup traversal.
Consider again the last definition of propbu
:
proptr(s) = Not(s) <+ And(s, s) <+ Or(s, s) <+ Impl(s, s) <+ Eq(s, s)
propbu(s) = try(proptr(propbu(s))); s
The role of proptr(s)
in this definition can be replaced by all(s)
, since that achieves exactly the same, namely applying s
to the direct subterms of constructors:
propbu(s) = all(propbu(s)); s
Moreover, all
succeeds on any constructor in any signature, so we can also drop the try
as well, which was there only because proptr
fails on the Atom(...)
, True()
, and False()
nodes at the leaves.
However, the strategy now is completely generic, i.e. independent of the particular structure it is applied to.
In the Stratego Library this strategy is called bottomup(s)
, and defined as follows:
bottomup(s) = all(bottomup(s)); s
It first recursively transforms the subterms of the subject term and then applies s
to the result.
Using this definition, the normalization of propositions now reduces to the following module, which is only concerned with the selection and composition of rewrite rules:
module propdnf11
imports proprules
strategies
dnf = bottomup(try(DN <+ (DefI <+ DefE <+ DMA <+ DMO <+ DAOL <+ DAOR); dnf))
cnf = bottomup(try(DN <+ (DefI <+ DefE <+ DMA <+ DMO <+ DOAL <+ DOAR); cnf))
In fact, these definitions still contain a reusable pattern. With a little squinting we see that the definitions match the following pattern:
dnf = bottomup(try(dnfrules; dnf))
cnf = bottomup(try(cnfrules; cnf))
In which we can recognize the definition of innermost reduction, which the Stratego Library defines as:
innermost(s) = bottomup(try(s; innermost(s)))
The innermost
strategy performs a bottomup traversal of a term.
After transforming the subterms of a term it tries to apply the transformation s
.
If successful the result is recursively transformed with an application of innermost
.
This brings us to the final form for the proposition normalizations:
module propdnf12
imports proprules
strategies
dnf = innermost(DN <+ DefI <+ DefE <+ DMA <+ DMO <+ DAOL <+ DAOR)
cnf = innermost(DN <+ DefI <+ DefE <+ DMA <+ DMO <+ DOAL <+ DOAR)
Different transformations can be achieved by using a selection of rules and a strategy, which is generic, yet defined in Stratego itself using strategy combinators.
Visiting One Subterm¶
The one(s)
strategy transforms a constructor application by applying the parameter strategy s
to exactly one direct subterm.
An application of one(s)
fails if the application to all of the subterms fails.
The following applications illustrate the behavior of the combinator:
<one(!Var("a"))>
Plus(Int("14"), Int("3")) => Plus(Var("a"), Int("3"))
<one(\ Int(x) > Int(<addS>(x,"1")) \ )>
Plus(Var("a"), Int("3")) => Plus(Var("a"), Int("4"))
<one(?Plus(_,_))>
Plus(Var("a"), Int("4")) // fails
Defining Traversals with One¶
A frequently used application of one
is the oncetd(s)
traversal, which performs a left to right depth first search/transformation that stops as soon as s has been successfully applied.
oncetd(s) = s <+ one(oncetd(s))
Thus, s
is first applied to the root of the subject term.
If that fails, its direct subterms are searched one by one (from left to right), with a recursive call to oncetd(s)
.
An application of oncetd
is the contains(t)
strategy, which checks whether the subject term contains a subterm that is equal to t.
contains(t) = oncetd(?t)
Through the depth first search of oncetd
, either an occurrence of t
is found, or all subterms are verified to be unequal to t
.
Here are some other onepass traversals using the one combinator:
oncebu(s) = one(oncebu(s)) <+ s
spinetd(s) = s; try(one(spinetd(s)))
spinebu(s) = try(one(spinebu(s))); s
Here are some fixepoint traversals, i.e., traversals that apply their argument transformation exhaustively to the subject term.
reduce(s) = repeat(rec x(one(x) + s))
outermost(s) = repeat(oncetd(s))
innermostI(s) = repeat(oncebu(s))
The difference is the subterm selection strategy.
Visiting Some Subterms¶
The some(s)
strategy transforms a constructor application by applying the parameter strategy s
to as many direct subterms as possible and at least one. An application of some(s)
fails if the application to all of the subterms fails.
Some onepass traversals based on some:
sometd(s) = s <+ some(sometd(s))
somebu(s) = some(somebu(s)) <+ s
A fixedpoint traversal with some:
reducepar(s) = repeat(rec x(some(x) + s))
References¶

Bas Luttik and Eelco Visser. Specification of rewriting strategies. In M. P. A. Sellink, editor, 2^{nd} International Workshop on the Theory and Practice of Algebraic Specifications (ASF+SDF 1997), Electronic Workshops in Computing. Berlin, November 1997. SpringerVerlag. ↩
Created: May 17, 2023