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   Copyright (c)  2005 John Abbott
   Permission is granted to copy, distribute and/or modify this document
   under the terms of the GNU Free Documentation License, Version 1.2;
   with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
   A copy of the licence is included in the file COPYING in this directory.

User documentation for the classes PPMonoid, PPMonoidElem and PPMonoidBase

The classes [PPMonoid] and [PPMonoidElem] are analogous to [ring] and [RingElem]. A [PPMonoid] represents a (multiplicative) power product monoid with grading and compatible total arithmetic ordering; a [PPMonoidElem] represents an element of a [PPMonoid], i.e. a power product.

Functions for [PPMonoid]s

Recall that every PP monoid is graded, and has a degree-compatible total arithmetical ordering; the grading and ordering are specified when the [PPMonoid] is created. For convenient input and output, also the names of the indeterminates generating the monoid must be specified when the monoid is created.

There are not many operations which apply directly to a [PPMonoid]. The [PPMonoid] serves primarily to specify the "type" of a [PPMonoidElem], and so appears as argument in constructors of [PPMonoidElem].

To create a PPMonoid use the function NewPPMonoid. To create a [PPMonoid] object of a specific type use one of the pseudo-constructors related to the concrete monoid classes: [PPMonoidEvOv], [PPMonoidEv], and [PPMonoidOv].

cout << PPM          -- print PPM on cout
NumIndets(PPM)       -- number of indeterminates
ordering(PPM)        -- the PP ordering inherent in PPM
GradingDim(PPM)      -- the dimension of the grading (zero if ungraded)
symbols(PPM)         -- std::vector of the symbols in PPM (i.e. names of the indets)
PPM1 == PPM2         -- true iff PPM1 and PPM2 are identical (same addr)
PPM1 != PPM2         -- true unless PPM1 and PPM2 are identical

These pseudo-constructors are described in the section about [PPMonoidElem]s

indet(PPM, var)
IndetPower(PPM, var, exp)

Summary of functions for [PPMonoidElem]s

See also some example programs in the CoCoALib/examples/ directory.

When a new object of type [PPMonoidElem] is created the monoid to which it belongs must be specified (either explicitly as a constructor argument, or implicitly as the monoid associated with some constructor argument). Once the [PPMonoidElem] object has been created it is not possible to make it belong to any another monoid. Arithmetic between objects of type [PPMonoidElem] is permitted only if they belong to the same identical monoid.

NOTE: when writing a function which has an argument of type [PPMonoidElem], you should specify the argument type as [ConstRefPPMonoidElem], or [RefPPMonoidElem] if you want to modify its value.

Let PPM be a PPMonoid; for convenience, in comments we shall use x[i] to refer to the i-th indeterminate in PPM. Let pp be a non-const PPMonoidElem, and pp1 and pp2 const PPMonoidElems (all belonging PPM).

PPMonoidElem t(PPM);       -- create new PP in PPM, value is 1
PPMonoidElem t(PPM, expv); -- create new PP in PPM, value is product x[i]^expv[i]
PPMonoidElem t(pp1);       -- create a new copy of pp1, belongs to same PPMonoid as pp1
one(PPM)                   -- the 1 belonging to PPM
indet(PPM, v)              -- create a new copy of x[v] the v-th indeterminate of PPM
IndetPower(PPM, v, n)      -- create x[v]^n, n-th power of v-th indeterminate of PPM  

owner(pp1)                 -- returns the PPMonoid to which pp1 belongs

pp1 == pp2                 -- the six standard comparison operators...
pp1 != pp2                 --   ...
pp1 < pp2                  --   ... (inequalities use the ordering inherent in PPM)
pp1 <= pp2                 --   ...
pp1 > pp2                  --   ...
pp1 >= pp2                 --   ... 

pp1 * pp2                  -- product of pp1 and pp2
pp1 / pp2                  -- quotient of pp1 by pp2, quotient MUST be exact
                              (see the function IsDivisible below)
colon(pp1, pp2)            -- "colon" quotient of pp1 by pp2; equal to pp1/gcd(pp1,pp2)
gcd(pp1, pp2)              -- gcd of pp1 and pp2
lcm(pp1, pp2)              -- lcm of pp1 and pp2
power(pp1, n)              -- n-th power of pp1
IsCoprime(pp1, pp2)        -- tests whether pp1 and pp2 are coprime
IsDivisible(pp1, pp2)      -- tests whether pp1 is divisible by pp2
pp = pp1                   -- assignment (pp and pp1 must belong to same PPMonoid)
pp *= pp1                  -- same as pp = pp * pp1
pp /= pp1                  -- same as pp = pp / pp1
deg(pp1)                   -- degree of pp1 (using specified grading); result is of type [degree] 
CmpDeg(pp1, pp2)           -- <0 =0 >0 according as deg(pp1) < = > deg(pp2)
log(pp1, v)                -- power to which x[v] appears in pp1
cout << pp1                -- print out the value of pp1

exponents(expv, pp)        -- fill expv so that expv[i] = log(pp, i)
swap(pp, pp0)              -- swaps the values of pp and pp0 
AssignOne(pp)              -- set pp equal to 1

Library Contributor Documentation

This section comprises two parts: the first is about creating a new type of PP monoid; the second comments about calling the member functions of [PPMonoidBase] directly.

To add a new type of concrete PPMonoid class

My first suggestion is to look at the code implementing [PPMonoidEv]. This is a simple PP monoid implementation: the values are represented as C arrays of exponents. Initially you should ignore the class [CmpBase] and those derived from it; they are simply to permit fast comparison of PPs in certain special cases.

First, a note about "philosophy". As far as we can tell the programming language C++ does not have a built-in type system sufficiently flexible (and efficient) for our needs, consequently we have to build our own type system on top of what C++ offers. The way we have chosen to do this is as follows (note that the overall scheme used here is similar to that used for rings and their elements).

To fit into CoCoALib your new class must be derived from [PPMonoidBase]. Remember that any operation on elements of your PP monoid will be effected by calling a member function of your new monoid class.

The monoid must be a cartesian power of N, the natural numbers, with the monoid operation (called "multiplication") being vector addition -- the vector should be thought of as the vector of exponents in a power product. The monoid must have a total arithmetic ordering; often this will be specified when the monoid is created. The class [PPOrdering] represents the possible orderings.

Here is a summary of the member functions which must be implemented. All the functions may be called for a "const" PPMonoid (for brevity the const qualifier is omitted). I use two abbreviations:

 RawPP       is short for  PPMonoidElemRawPtr
 ConstRawPP  is short for  PPMonoidElemConstRawPtr

NOTE: all arithmetic function must tolerate argument aliasing (i.e. any pair of arguments may be identical).


these all allocate memory which must eventually be freed (by calling myDelete); the result is a pointer to the memory allocated.

PPMonoidElemRawPtr PPMonoidBase::myNew()                     // initialize pp to the identity
PPMonoidElemRawPtr PPMonoidBase::myNew(const vector<int>& v) // initialize pp from  exponent vector v
PPMonoidElemRawPtr PPMonoidBase::myNew(const RawPP& pp1)     // initialize pp from pp1


there is only one of these, its argument must be initialized

void PPMonoidBase::myDelete(PPMonoidElemRawPtr pp)           // destroy pp, frees memory

Assignment etc

void PPMonoidBase::mySwap(RawPP pp1, RawPP pp2)             // swap the values of pp1 and pp2
void PPMonoidBase::myAssign(RawPP pp, ConstRawPP pp1)       // assign the value of pp1 to pp
void PPMonoidBase::myAssign(RawPP pp, const vector<int>& v) // assign to pp the PP with exponent vector v


  • in all cases the first arg is where the answer is placed,
  • aliasing is permitted (i.e. arguments need not be distinct);
  • div result is undefined if the quotient does not exist!
const PPMonoidElem& myOne()                          // reference to 1 in the monoid
void myMul(RawPP pp, ConstRawPP pp1, ConstRawPP pp2) // pp = pp1*pp2
void myMulIndetPower(RawPtr pp, size_t var, int exp) // pp *= indet(var)^exp
void myDiv(RawPP pp, ConstRawPP pp1, ConstRawPP pp2) // pp = pp1/pp2 (if it exists)
void myColon(RawPP pp, ConstRawPP pp1, Const RawPP pp2)// pp = pp1/gcd(pp1,pp2)
void myGcd(RawPP pp, ConstRawPP pp1, ConstRawPP pp2) // pp = gcd(pp1, pp2)
void myLcm(RawPP pp, ConstRawPP pp1, ConstRawPP pp2) // pp = lcm(pp1, pp2)
void myPower(RawPP pp, ConstRawPP pp1, int exp)      // pp = pp1^exp

Comparison and testing

Each PPMonoid has associated with it a "term ordering", a total ordering which respects the monoid operation (multiplication)

bool myIsCoprime(ConstRawPP pp1, ConstRawPP pp2)     // true iff gcd(pp1, pp2) is 1
bool myIsDivisible(ConstRawPP t1, ConstRawPP t2)     // true iff t1 is divisible by t2
int myCmp(ConstRawPP t1, ConstRawPP t2)              // result is <0, =0, >0 according as t1 <,=,> t2
int myHomogCmp(ConstRawPP t1, ConstRawPP t2)         // as cmp, but assumes t1 and t2 have the same degree


degree myDeg(ConstRawPP t)                         // total degree
int myLog(ConstRawPP t, unsigned int var)          // degree in var
void myExponents(vector<int>& v, ConstRawPP t)     // get exponents
ostream& myOutput(ostream& out, const RawPP& t)    // prints t on out; default defn in PPMonoid.C

Query functions

size_t myNumIndets()                               // number of indeterminates generating the monoid
const string& myIndetName(size_t var)              // name of indet with index var

To add a new member function to PPMonoidBase

You will have to edit PPMonoid.H and possibly PPMonoid.C (e.g. if there is to be a default definition). Arguments representing PPs should be of type [RawPP] if they may be modified, or of type [ConstRawPP] if they should never be modified. See also the Coding Conventions about names of member functions.

If you do add a new pure virtual member function, you will have to add definitions to all the existing concrete PP monoid classes (otherwise they will become uninstantiable). Don't forget to update the documentation too!

Calculating directly with raw PPs

Values of type [PPMonoidElem] are intended to be simple and safe to use but with some performance penalty. There is also a "fast, ugly, unsafe" option which we shall describe here.

The most important fact to heed is that a [PPMonoidElemRawPtr] value is NOT a C++ object -- it does not generally know enough about itself even to destroy itself. This places a considerable responsibility on the programmer, and probably makes it difficult to write exception clean code. You really must view the performance issue as paramount if you plan to use raw PPs! In any case the gain in speed will likely be only slight.

The model for creation/destruction and use of raw PPs is as follows: [NB see "bug" section about exception-safety]

  1. an uninitialized raw PP is acquired from the system;
  2. the raw PP is "initialized" by calling an "init" function -- this will generally acquire further resources;
  3. now the RawPP may be used for i/o, arithmetic, and so forth;
  4. finally, when the value is no longer required the extra resources acquired during initialization should be released by calling the "kill" function -- failure to call "kill" will probably result in a memory leak.

Here is some pseudo C++ code to give an idea

const PPMonoid& M;     // A PPMonoid from somewhere

PPMonoidElemRawPtr t;  // A wrapped opaque pointer; initially points into hyperspace.

t = M->myNew();        // Allocate resources for a new PP belonging to M;
                        // there are two other myNew functions.

.... operations on t; always via a member function of the monoid M ...

M->myDelete(t);        // "destroy" the value t held; t points into hyperspace again.

NOTE: the only functions which take a pointer into hyperspace are PPMonoidBase::myNew; many functions, e.g. PPMonoidBase::myMul, write their result into the first argument and require that that first argument be already allocated/initialized.

NOTE: if an exception is thrown after M->myNew and before M->myDelete then there will be a memory leak. If t is just to hold a temporary local value then it is better to create a full PPMonoidElem and then let t be its "RawPtr"; this should avoid memory leaks.

Maintainer documentation for PPMonoid, PPMonoidElem, and PPMonoidBase

The general structure here mirrors that of rings and their elements, so you may find it helpful to read ring.txt if the following seems too opaque. At first sight the design may seem complex (because it comprises several classes), but there's no need to be afraid.

The class [PPMonoid] is a reference counting smart pointer to an object derived from [PPMonoidBase]. This means that making copies of a [PPMonoid] is very cheap, and that it is easy to tell if two [PPMonoid]s are identical. Assignment of [PPMonoid]s is disabled because I am not sure whether it is useful/meaningful. [operator->] allows member functions of [PPMonoidBase] to be called using a simple syntax.

The class [PPMonoidBase] is what specifies the class interface for each concrete PP monoid implementation, i.e. the operations that it must offer. It includes an intrusive reference count for compatibility with [PPMonoid]. Since it is inconceivable to have a PP monoid without an ordering, there is a data member for memorizing the inherent [PPOrdering]. This data member is [protected] so that it is accessible only to friends and derived classes.

The function [PPMonoidBase::myOutput] for printing PPs has a reasonable default definition.

The situation for elements of a PP monoid could easily appear horrendously complicated. The basic idea is that a PP monoid element comprises two components: one indicating the [PPMonoid] to which the value belongs, and the other indicating the actual value. This allows the user to employ a notationally convenient syntax for many operations -- the emphasis is on notational convenience rather than ultimate run-time efficiency.

For an element of a PP monoid, the owning [PPMonoid] is specified during creation and remains fixed throughout the life of the object; in contrast the value may be varied (if C++ const rules permit). The value is indicated by an opaque pointer (essentially a wrapped [void*]): only the owning [PPMonoid] knows how to interpret the data pointed to, and so all operations on the value are effected by member functions of the owning [PPMonoid].

I do not like the idea of having naked [void*] values in programs: it is too easy to get confused about what is pointing to what. Since the value part of a [PPMonoidElem] is an opaque pointer (morally a [void*]), I chose to wrap it in a lightweight class; actually there are two classes depending on whether the pointed to value is [const] or not. These classes are [PPMonoidElemRawPtr] and [PPMonoidElemConstRawPtr]; they are opaque pointers pointing to a value belonging to some concrete PP monoid (someone else must keep track of precisely which PP monoid is the owner).

The constructors for [PPMonoidElemRawPtr] and [PPMonoidElemConstRawPtr] are [explicit] to avoid potentially risky automatic conversion of any old pointer into one of these types. The naked pointer may be accessed via the member functions [myRawPtr]. Only implementors of new PP monoid classes are likely to find these two opaque pointer classes useful.

I now return to the classes for representing fully qualified PPs. There are three very similar yet distinct classes for elements of PP monoids; the distinction is to keep track of constness and ownership. I have used inheritance to allow natural automatic conversion among these three classes (analogously to [RingElem], [RefRingElem] etc.)

  • A [PPMonoidElem] is the owner of its value; the value will be deleted when the object ceases to exist.
  • A [RefPPMonoidElem] is not the owner of its value, but the value may be changed (and the owner of the value will see the change too).
  • A [ConstRefPPMonoidElem] is not the owner of its value, and its value may not be changed (through this reference).

The data layout is determined in [ConstRefPPMonoidElem], and the more permissive classes inherit the data members. I have deliberately used a non-constant [PPMonoidElemRawPtr] for the value pointer as it is easier for the class [ConstRefPPMonoidElem] to add in constness appropriately than it is for the other two classes to remove it.

By convention the member functions of [PPMonoidBase] which operate on raw PP values assume that the values are valid (e.g. belong to the same PP monoid, division is exact in [myDiv]). The validity of the arguments is checked by the syntactically nice equivalent operations (see the code in PPMonoid.C). This permits a programmer to choose between safe clean code (with nice syntax) or faster unsafe code (albeit with uglier syntax).

Bugs, Shortcomings and other ideas

The section on "Advanced Use" is a bit out of date and too long.

(1) Should more operations on PPMonoidElems be inlined? [perhaps not, since speed is not so important for PPMonoidElems]

(2) We would like a way of performing divisibility tests faster when there are few indeterminates and relatively high degrees. In this case the DivMask is useless. The "gonnet" example is slow because it entails many divisibility tests. One suggestion would be to maintain a "randomly weighted" degree and use that as a simple heuristic for deciding quickly some cases.

(3) I've fixed the various arithmetic functions for PPMonoidElems so that they are obviously exception safe, BUT they now make an extra copy of the computed value (as it is returned from a local variable to the caller). Here is an idea for avoiding that extra copy. Create a new type (say PPMonoidElem_local) which offers just raw(..) and a function export(..) which allows the return mechanism to create a full PPMonoidElem (just by copying pointers) and empty out the PPMonoidElem_local. If the PPMonoidElem_local is not empty then it can destroy the value held within it. By not attempting to make PPMonoidElem_locals behave like full PPMonoidElems I save a lot of "useless" function definitions. Indeed the "export" function need not exist: an implicit ctor for a PPMonoidElem from a PPMonoidElem_local could do all the work. I'll wait to see profiling information before considering implementing.

(4) Is assignment for PPMonoids likely to be useful to anyone? I prefer to forbid it, as I suspect a program needing to use it is really suffering from poor design...

(5) I have chosen not to use operator^ for computing powers because of a significant risk of misunderstanding between programmer and compiler. The syntax/grammar of C++ cannot be changed, and operator^ binds less tightly than (binary) operator*, so any expression of the form a*b^c will be parsed as (a*b)^c; this is almost certainly not what the programmer intended. To avoid such problems of misunderstanding I have preferred not to define operator^; it seems too dangerous.