Pattern to match only “children” of certain elements

Question

I would like to be able to have a pattern that matches only expressions that are (alternately: are not) children of certain other elements.

For example, a pattern to

Answer 1

I will propose a solution based on expression pre-processing and soft redefinitions of operations using rules, rather than rules themselves. Here is the code:

ClearAll[matchChildren, exceptChildren];
Module[{h, preprocess},
  preprocess[expr_, parentPtrn_, lhs_, match : (True | False)] :=
     Module[{pos, ptrnPos, lhsPos},
       ptrnPos = Position[expr, parentPtrn];
       lhsPos = Position[expr, lhs];
       pos = Cases[lhsPos, {Alternatives @@ PatternSequence @@@ ptrnPos, __}];
       If[! match,pos = Complement[Position[expr, _, Infinity, Heads -> False], pos]];
       MapAt[h, expr, pos]];

  matchChildren /: 
    fun_[expr_, matchChildren[parentPtrn_, lhs : Except[_Rule | _RuleDelayed]],
    args___] :=
       fun[preprocess[expr, parentPtrn, lhs, True], h[lhs], args] //. 
           h[x_] :> x;

  matchChildren /: 
    fun_[expr_, matchChildren[parentPtrn_, lhs_ :> rhs_], args___] :=
       fun[preprocess[expr, parentPtrn, lhs, True], h[lhs] :> rhs, args] //. 
           h[x_] :> x;

  exceptChildren /: 
   fun_[expr_,exceptChildren[parentPtrn_, lhs : Except[_Rule | _RuleDelayed]], 
   args___] :=
       fun[preprocess[expr, parentPtrn, lhs, False], h[lhs], args] //. 
           h[x_] :> x;

  exceptChildren /: 
   fun_[expr_, exceptChildren[parentPtrn_, lhs_ :> rhs_], args___] :=
       fun[preprocess[expr, parentPtrn, lhs, False], h[lhs] :> rhs, args] //. 
          h[x_] :> x;
]

A few details on implementation ideas, and how it works. The idea is that, in order to restrict the pattern that should match, we may wrap this pattern in some head (say h), and also wrap all elements matching the original pattern but also being (or not being) within some other element (matching the "parent" pattern) in the same head h. This can be done for generic "child" pattern. Technically, one thing that makes it possible is the intrusive nature of rule application (and function parameter-passing, which have the same semantics in this respect). This allows one to take the rule like x_List:>f[x], matched by generic pattern lhs_:>rhs_, and change it to h[x_List]:>f[x], generically by using h[lhs]:>rhs. This is non-trivial because RuleDelayed is a scoping construct, and only the intrusiveness of another RuleDelayed (or, function parameter-passing) allows us to do the necessary scope surgery. In a way, this is an example of constructive use of the same effect that leads to the leaky functional abstraction in Mathematica. Another technical detail here is the use of UpValues to overload functions that use rules (Cases, ReplaceAll, etc) in the "soft" way, without adding any rules to them. At the same time, UpValues here allow the code to be universal - one code serves many functions that use patterns and rules. Finally, I am using the Module variables as a mechanism for encapsulation, to hide the auxiliary head h and function preprocess. This is a generally very handy way to achieve encapsulation of both functions and data on the scale smaller than a package but larger than a single function.

Here are some examples:

In[171]:= expr = {{1,2,3},Graphics[Line[{{1,2},{3,4}}]]};

In[168]:= expr/.matchChildren[_Graphics,x_List:>f[x]]//FullForm
Out[168]//FullForm= List[List[1,2,3],Graphics[Line[f[List[List[1,2],List[3,4]]]]]]

In[172]:= expr/.matchChildren[_Graphics,x:{__Integer}:>f[x]]//FullForm
Out[172]//FullForm= List[List[1,2,3],Graphics[Line[List[f[List[1,2]],f[List[3,4]]]]]]

In[173]:= expr/.exceptChildren[_Graphics,x_List:>f[x]]//FullForm
Out[173]//FullForm= List[f[List[1,2,3]],Graphics[Line[List[List[1,2],List[3,4]]]]]

In[174]:= expr = (Tan[p]*Cot[p+q])*(Sin[Pi n]+Cos[Pi m])*(Tan[q]+Cot[q]);

In[175]:= expr/.matchChildren[_Plus,x_Tan:>f[x]]
Out[175]= Cot[p+q] (Cot[q]+f[Tan[q]]) (Cos[m \[Pi]]+Sin[n \[Pi]]) Tan[p]

In[176]:= expr/.exceptChildren[_Plus,x_Tan:>f[x]]
Out[176]= Cot[p+q] f[Tan[p]] (Cos[m \[Pi]]+Sin[n \[Pi]]) (Cot[q]+Tan[q])

In[177]:= Cases[expr,matchChildren[_Plus,x_Tan:>f[x]],Infinity]
Out[177]= {f[Tan[q]]}

In[178]:= Cases[expr,exceptChildren[_Plus,x_Tan:>f[x]],Infinity]
Out[178]= {f[Tan[p]]}

In[179]:= Cases[expr,matchChildren[_Plus,x_Tan],Infinity]
Out[179]= {Tan[q]}

In[180]:= Cases[expr,matchChildren[_Plus,x_Tan],Infinity]
Out[180]= {Tan[q]}

It is expected to work with most functions which have the format fun[expr_,rule_,otherArgs___]. In particular, those include Cases,DeleteCases, Replace, ReplaceAll,ReplaceRepeated. I did not generalize to lists of rules, but this should be easy to do. It may not work properly in some subtle cases, e.g. with non-trivial heads and pattern-matching on heads.

Answer 2

According to your explanation in the comment to the acl's answer:

Actually I'd like it to work at any level in the expression <...>. <...> what I need is replacement: replace all expression that match this "pattern", and leave the rest unchanged. I guess the simplest possible solution is finding the positions of elements, then using ReplacePart. But this can also get quite complicated in the end.

I think it could be done in one pass with ReplaceAll. We can rely here on the documented feature of the ReplaceAll: it does not look at the parts of the original expression which were already replaced even if they are replaced by themselves! Citing the Documentation: "ReplaceAll looks at each part of expr, tries all the rules on it, and then goes on to the next part of expr. The first rule that applies to a particular part is used; no further rules are tried on that part, or on any of its subparts."

Here is my solution (whatIwant is what you want to do with matched parts):

replaceNonChildren[lst_List] := 
 ReplaceAll[#, {x_List :> whatIwant[x], y_ :> y}] & /@ lst

Here is your test case:

replaceNonChildren[{{1, 2, 3}, Graphics[Line[{{1, 2}, {3, 4}}]]}] // InputForm

=> {whatIwant[{1, 2, 3}], Graphics[Line[{{1, 2}, {3, 4}}]]}

Here is a function that replaces only inside certain head (Graphics in this example):

replaceChildren[lst_List] := 
 ReplaceAll[#, {y : Graphics[__] :> (y /. x_List :> whatIwant[x])}] & /@ lst

Here is a test case:

replaceChildren[{{1, 2, 3}, Graphics[Line[{{1, 2}, {3, 4}}]]}] // InputForm

=> {{1, 2, 3}, Graphics[Line[whatIwant[{{1, 2}, {3, 4}}]]]}

Answer 3

I am probably misunderstanding you, but, if I do understand correctly you want to match all expressions with head List which have the property that, going upwards in the expression tree, we'll never meet a Graphics. I m not sure how to do this in one pass, but if you are willing to match twice, you can do something like

lst = {randhead[5], {1, 2, {3, 5}}, Graphics[Line[{{1, 2}, {3, 4}}]]};
Cases[#, _List] &@Cases[#, Except@Graphics[___]] &@lst
(*
----> {{1, 2, {3, 5}}}
*)

which first selects elements so that the Head isn't Graphics (this is done by Cases[#, Except@Graphics[___]] &, which returns {randhead[5], {1, 2, {3, 5}}}), then selects those with Head List from the returned list. Note that I've added some more stuff to lst.

But presumably you knew this and were after a single pattern to do the job?

Answer 4

You might write a recursive function that descends an expression tree and acts on the types of expression you want only if inside the right type of sub-expression, while leaving everything else alone. Patterns would be used heavily in the definition of the function.

Consider, for example, the following expression.

test = {{1, 2}, Graphics[{
  Point[{{-1, 0}, {1, 0}}],
  Line[{{-1, 0}, {1, 0}}]},
 Frame -> True, 
 PlotRange -> {{-1, 1}, {-0.5, 0.5}}]};

Let's suppose that we want to rotate every ordered pair that we see in the first argument of Graphics about the origin through the angle Pi/4, while leaving other points alone. The following function does this.

Clear[f];
f[{x_?NumericQ, y_?NumericQ}] := If[flag === True,
  RotationMatrix[Pi/4].{x, y}, {x, y}];
f[Graphics[primitives_, rest___]] := Block[{flag = True},
  Graphics[f[primitives], rest]];
f[x_?AtomQ] := x;
f[x_] := f /@ x;

Now we check

f[test]