symbol

© 2005,2007,2008 John Abbott
GNU Free Documentation License, Version 1.2



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User documentation for symbol

symbol is short for "Symbolic Name". A value of type symbol represents a "variable name" possibly with some integer indices attached. Its primary use is for input and output of polynomials: the name of each indeterminate in a polynomial ring is a symbol, similarly for a PPMonoid.

A symbol value has two components: its head which is a string comprising letters and underscores (but the first character must be a letter), and its indices which are a vector of integers (indices may be negative). Examples of symbols are: (in standard printed forms)

   x, X, alpha, z_alpha, x[2], gamma[-2,3,-9]

Constructors

symbol(head) where head is a std::string
this produces a symbol with no indices
symbol(head, ind) where head is a std::string and ind a long
this produces a symbol with a single index
symbol(head, ind1, ind2) where head is a std::string and ind1, ind2 longs
this produces a symbol with a two indexes
symbol(head, inds) where head is a std::string and
inds is a std::vector<long> this produces a symbol with indices

Operations on a symbol Let sym, sym1, and sym2 be objects of type symbol

   head(sym)        head of sym as a const ref to ``std::string``
   NumIndices(sym)  number of indices sym has
                    (gives 0 if sym has no indices)
   index(sym, n)    gives n-th index of sym
   cmp(sym1, sym2)  <0, =0, >0 according as sym1 < = > sym2
             (using some total ordering: currently lex on heads, then lex on index vectors)
   
   sym1 < sym2      comparisons defined in terms of ``cmp``
   sym1 <= sym2    
   sym1 > sym2     
   sym1 >= sym2    
   sym1 == sym2    
   sym1 != sym2    
  
   out << sym       print sym on out
   in >> sym        read a symbol into sym (NB see "bugs" section)
                    (expected format is x, y[1], z[2,3], etc.)

Several polynomial ring pseudo-constructors expect a vector of symbols to specify the names of the indeterminates. There are several "convenience" functions for constructing commonly used collections of symbols.

   symbols(hd1)             create vector of length 1 containing symbol(hd1)
   symbols(hd1,hd2)         ... length 2...
   symbols(hd1,hd2,hd3)     ... length 3...
   symbols(hd1,hd2,hd3,hd4) ... length 4...
  
   SymbolRange(hd, lo, hi)      create vector of hd[lo], hd[lo+1], ... hd[hi]
   SymbolRange(sym1, sym2)      create vector of "cartesian product" of the indices,
                                e.g. given x[1,3] and x[2,4] produces
                                     x[1,3], x[1,4], x[2,3], x[2,4]
  
   AreDistinct(vecsyms)         true iff all symbols are distinct
   AreArityConsistent(vecsyms)  true iff all symbols with the same head have the same arity

Maintainer documentation for symbol

The implementation is extremely simple. Efficiency does not seem to be important (e.g. symbols and SymbolRange copy the vector upon returning). Implementation of SymbolRange is mildly delicate when we have to make checks to avoid integer overflow -- see comments in the code.

The total ordering could be useful, for instance, if someone wants to make a std::map using symbols. "Lex on the heads then lex on the index vectors" seemed simple, and probably fast enough.

The function symbol::myInput is a stop-gap implementation.

Bugs, Shortcomings and other ideas

The member function myInput handles white space wrongly. For CoCoALib whitespace is space, TAB, or backslash-newline; newline without backslash is not considered white space.

It might be nice to have a function which returns the vector of indices of a name.

Decided not to permit big integers as indices; I don't see when it could ever be useful.

I wonder what sending a symbol on an OpenMath channel would mean (given that OpenMath is supposed to preserve semantics, and a symbolic name is by definition devoid of semantics).