Pattern-matching with Flat Attribute: Different from Mathematica

Question

Please consider

SetAttributes[eqv, Flat]
eqv[p, q, q, p] /. eqv[x_, y_] :> {x, y}

which produces

{q, eqv[q, q, p]}

in mathics, and

{eqv[q], eqv[q, q, p]}

in Mathematica. This has ramifications in symbolic manipulations with custom associative operators because the missing head eqv is observable in semantics layers.

该提问来源于开源项目：mathics/Mathics

Answer 1

Looks like mathics assumes OneIdentity. Let's consider Mathematica's built-in Or, first. Its relevant attributes are just Flat, OneIdentity:

In[97]:= Attributes[Or]
Out[97]= {Flat, HoldAll, OneIdentity, Protected}

It also has no Default:

In[99]:= Default[Or]
Out[99]= Default[Or]

That produces the following behavior:

In[107]:= Or[p, p, p] /. Or[a_, b_] :> {a, b}
Or[p, p, p] /. Or[a_., b_.] :> {a, b}

Out[107]= {p, p || p}
Out[108]= {p, p} || p

In[109]:= Or[p, p, p]
Out[109]= p || p || p

In[112]:= Or[p, q] === Or[q, p]
Out[112]= False

This is similar to, but not exactly the same, as mathics; Pattern produces a nodef message by side effect when presented with Optional[Blank[]] patterns (syntax: _., underscore-dot) and reduces as if the patterns were not optional.

In[9]:= Or[p, p, p] /. Or[a_, b_] :> {a, b}
Out[9]= {p, p || p}

In[10]:= Or[p, p, p] /. Or[a_., b_.] :> {a, b}
Pattern::nodef: No default setting found for Or in position 1 when length is 2.
Out[10]= {p, p || p}

In[11]:= Or[p, p, p]
Out[11]= p || p || p

In[12]:= Or[p, q] === Or[q, p]
Out[12]= False

We want our or to be symmetric, so we will add Orderless. First, here is what Mathematica says, in stages, adding the attributes one at a time:

In[43]:= ClearAll[or]; SetAttributes[or, {Flat}]
or[p, p, p] /. or[a_, b_] :> {a, b}
or[p, p, p] /. or[a_., b_.] :> {a, b}

Out[44]= {or[p], or[p, p]}     {a, b}
or[p, p, p] /. or[a_., b_.] :> {a, b}

Out[41]= {p, or[p, p]}     {a, b}
or[p, p, p] /. or[a_., b_.] :> {a, b}

Out[47]= {p, or[p, p]}    

Mathics disagrees right away:

In[13]:= ClearAll[or]; SetAttributes[or, {Flat}]
In[14]:= or[p, p, p] /. or[a_, b_] :> {a, b}
Out[14]= {p, or[p, p]}

In[15]:= or[p, p, p] /. or[a_., b_.] :> {a, b}
Pattern::nodef: No default setting found for or in position 1 when length is 2.
Out[15]= {p, or[p, p]}

In[16]:= ClearAll[or]; SetAttributes[or, {Flat, OneIdentity}]
In[17]:= or[p, p, p] /. or[a_, b_] :> {a, b}
Out[17]= {p, or[p, p]}

In[18]:= or[p, p, p] /. or[a_., b_.] :> {a, b}
Pattern::nodef: No default setting found for or in position 1 when length is 2.
Out[18]= {p, or[p, p]}

In[19]:= ClearAll[or]; SetAttributes[or, {Flat, Orderless, OneIdentity}]
In[20]:= or[p, p, p] /. or[a_, b_] :> {a, b}
Out[20]= {p, or[p, p]}

In[21]:= or[p, p, p] /. or[a_., b_.] :> {a, b}
Out[21]= {p, or[p, p]}