Abstract

This document motivates the concept of “disordered vector” using coefficients of multivariate polynomials as represented by associative maps such as the STL map class as used by the mvp package. Values and keys of a map are stored in an implementation-specific way so certain extraction and replacement operations should be forbidden: for example, the “first” value is implementation specific…but the maximum value has a well-defined result. The disordR package makes forbidden operations impossible while allowing transparent R idiom for permitted operations. An illustrative R session is given in which the disordR package is used abstractly, without reference to any particular application, and then shows how the disordR package is used in the mvp package. disordR is used in the clifford, freealg, hyper2, mvp, spray, stokes, and weyl packages.

Introduction

Accessing elements of an mvp object is problematic because the order of the terms of an mvp object is not well-defined. This is because the map class of the STL does not specify an order for the key-value pairs (and indeed the actual order in which they are stored may be implementation dependent). The situation is applicable to the hyper2, spray, clifford and freealg packages which use the STL in a similar way.

A disord object is a vector of coefficients of a mvp object. But it is not a conventional vector; in a conventional vector, we can identify the first element unambiguously, and the second, and so on. An mvp is a map from terms to coefficients, and a map has no intrinsic ordering: the maps

x -> 1, y -> 3, xy -> 3, xy^3 -> 4

and

xy^3 -> 4, xy -> 3, x -> 1, y -> 3

are the same map and correspond to the same multinomial (symbolically, \(x+3y+3xy+4xy^3=4xy^3+3xy+x+3y\)). Thus the coefficients of the multinomial might be c(1,3,3,4) or c(4,3,1,3), or indeed any ordering. Internally, the elements are stored in some order but the order used is implementation-specific. Quite often, I am interested in the coefficients per se, without consideration of their meaning in the context of a multivariate polynomial. I might ask:

These are reasonable and mathematically meaningful questions. Compare a meaningless question: “what is the second coefficient?”. This is meaningless because of the order ambiguity discussed above: the answer is at best implementation-specific, but fundamentally it is a question that one should not be allowed to ask.

To deal with the coefficients in isolation in R, one might be tempted to use a multiset. However, this approach does not allow one to link the coefficients with the terms. Suppose I coerce the coefficients to a multiset object (as per the sets package, for example): then it is impossible to extract the terms with coefficient greater than 2 (which would be the polynomial \(3y+3xy+4xy^3\)) because the link between the coefficients and the terms is not included in the multiset object. Sensible questions involving this aspect of mvp objects might be:

and these questions cannot be answered if the the coefficients are stored as a multiset (compare inadmissible questions such as “give me the first three terms”). Further note that replacement methods are mathematically meaningful, for example:

Again these operations are reasonable but precluded by multiset formalism (compare inadmissible replacements: “replace the first two terms with zero”, or “double the last term” would be inadmissible).

What we need is a system that forbids stupid questions and stupid operations, while permitting sensible questions and operations

The disord class of the disordR package is specificially designed for this situation. This class of object has a slot for the coefficients in the form of a numeric R vector, but also another slot which uses hash codes to prevent users from misusing the ordering of the numeric vector.

For example, a multinomial x+2y+3z might have coefficients c(1,2,3) or c(3,1,2). Package idiom to extract the coefficients of a multivariate polynomial a is coeffs(a); but this cannot return a standard numeric vector. If stored as a numeric vector, the user might ask “what is the first element?” and this question should not be asked [and certainly not answered!], because the elements are stored in an implementation-specific order. The disordR package uses disord objects which are designed to return an error if such inadmissible questions are asked. But disord objects can answer admissible questions and perform admissible operations.

Suppose we have two multivariate polynomials, a as defined as above with a=x+2y+3z and b=x+3y+4z. Even though the sum a+b is well-defined algebraically, idiom such as coeffs(a) + coeffs(b) is not defined because there is no guarantee that the coefficients of the two multivariate polynomials are stored in the same order. We might have c(1,2,3)+c(1,3,4)=c(2,5,7) or c(1,2,3)+c(1,4,3)=c(2,6,6), with neither being more “correct” than the other. In the package, this ambiguity is rendered void: coeffs(a) + coeffs(b) will return an error. Note carefully that coeffs(a+b) is perfectly well defined, although the result is subject to the same ambiguity as coeffs(a).

In the same way, coeffs(a) + 1:3 is not defined and will return an error. Further, idiom such as coeffs(a) <- 1:3and coeffs(a) <- coeffs(b) are not defined and will also return an error. However, note that

coeffs(a) + coeffs(a)
coeffs(a) + coeffs(a)^2
coeffs(a) <- coeffs(a)^2
coeffs(a) <- coeffs(a)^2 + 7

are perfectly well defined, with package idiom behaving as expected.

Idiom such as disord(a) <- disord(a)^2 is OK: one does not need to know the order of the coefficients on either side, so long as the order is the same on both sides. The idiomatic English equivalent would be: “the coefficient of each term of a becomes its square”; note that this operation is insensitive to the order of coefficients. The whole shebang is intended to make idiom such as coeffs(a) <- coeffs(a)%%2 possible, so we can manipulate polynomials over finite rings, here \(Z/2Z\).

The replacement methods are defined so that an expression like coeffs(a)[coeffs(a) < 5] <- 0 works as expected; the English idiom would be “replace any coefficient less than 5 with 0”.

To fix ideas, consider a fixed small mvp object:

library("mvp")
## Loading required package: magrittr
## 
## Attaching package: 'mvp'
## The following object is masked from 'package:base':
## 
##     trunc
a <- as.mvp("5 a c^3 + a^2 d^2 f^2 + 4 a^3 b e^3 + 3 b c f + 2 b^2 e^3")
a
## mvp object algebraically equal to
## 5 a c^3  +  a^2 d^2 f^2  +  4 a^3 b e^3  +  3 b c f  +  2 b^2 e^3

Extraction presents issues; consider coeffs(a)<3. This object has Boolean elements but has the same ordering ambiguity as coeffs(a). One might expect that we could use this to extract elements of coeffs(a): specifically, those elements less than 5. We may use replace methods for coefficients if this makes sense. Idiom such as

coeffs(a)[coeffs(a)<5] <- 4 + coeffs(a)[coeffs(a)<5]
coeffs(a) <- pmax(a,3)

is algebraically meaningful and allowed in the package. Idiomatically: “Add 4 to any element less than 5”; “coefficients become the parallel maximum of themselves and 3” respectively. Further note that coeffs(a) <- rev(coeffs(a)) is disallowed (although coeffs(a) <- rev(rev(coeffs(a))) is meaningful and admissible).

So the output of coeffs(x) is defined only up to an unknown rearrangement. The same considerations apply to the output of vars(), which returns a list of character vectors in an undefined order, and the output of powers(), which returns a numeric list whose elements are in an undefined order. However, even though the order of these three objects is undefined individually, their ordering is jointly consistent in the sense that the first element of coeffs(x) corresponds to the first element of vars(x) and the first element of powers(x). The identity of this element is not defined—but whatever it is, the first element of all three accessor methods refers to it.

Note also that a single term (something like 4a3*b*c^6) has the same issue: the variables are not stored in a well-defined order. This does not matter because the algebraic value of the term does not depend on the order in which the variables appear and this term would be equivalent to 4bc^6*a^3.

An R session with the disordR package

We will use the disordR package to show how the idiom works.

library("disordR")
## Loading required package: Matrix
set.seed(0)
a <- rdis()
a
## A disord object with hash 5b7279f3c05d00cf1e8f999a755151e0451c56ec and elements
## [1] 9 4 7 1 2 6 3 8 5
## (in some order)

Object a is a disord object but it behaves similarly to a regular numeric vector in many ways:

a^2
## A disord object with hash 5b7279f3c05d00cf1e8f999a755151e0451c56ec and elements
## [1] 81 16 49  1  4 36  9 64 25
## (in some order)
a+1/a
## A disord object with hash 5b7279f3c05d00cf1e8f999a755151e0451c56ec and elements
## [1] 9.111111 4.250000 7.142857 2.000000 2.500000 6.166667 3.333333 8.125000
## [9] 5.200000
## (in some order)

Above, note how the result has the same hash code as a. Other operations that make sense are max() and sort():

max(a)
## [1] 9
sort(a)
## [1] 1 2 3 4 5 6 7 8 9

Above, see how the result is a standard numeric vector. However, inadmissible operations give an error:

a[1]  # asking for the first element is inadmissible
## Error in .local(x, i, j = j, ..., drop): if using a regular index to extract, must extract each element once and once only (or none of them)
a[1] <- 1000 # also cannot replace the first element
## Error in .local(x, i, j = j, ..., value): if using a regular index to replace, must specify each element once and once only

Standard R semantics generally work as expected:

x <- a + 1/a
x
## A disord object with hash 5b7279f3c05d00cf1e8f999a755151e0451c56ec and elements
## [1] 9.111111 4.250000 7.142857 2.000000 2.500000 6.166667 3.333333 8.125000
## [9] 5.200000
## (in some order)
y <- a*2-9
y
## A disord object with hash 5b7279f3c05d00cf1e8f999a755151e0451c56ec and elements
## [1]  9 -1  5 -7 -5  3 -3  7  1
## (in some order)
x+y
## A disord object with hash 5b7279f3c05d00cf1e8f999a755151e0451c56ec and elements
## [1] 18.1111111  3.2500000 12.1428571 -5.0000000 -2.5000000  9.1666667  0.3333333
## [8] 15.1250000  6.2000000
## (in some order)

Above, observe that objects a, x and y have the same hash code: they are “compatible”, in disordR idiom. However, if we try to combine object a with another object with different hash, we get errors:

b <- rdis()
b
## A disord object with hash 488e1c6f4e2c062379d47b5511730a9785661318 and elements
## [1] 2 3 8 1 5 6 9 7 4
## (in some order)
a
## A disord object with hash 5b7279f3c05d00cf1e8f999a755151e0451c56ec and elements
## [1] 9 4 7 1 2 6 3 8 5
## (in some order)
a+b
## Error in a + b: consistent(e1, e2) is not TRUE

The error is given because objects a and b are stored in an implementation-specific order (we say that a and b are incompatible). In the package, many extract and replace methods are implemented whenever this is admissible:

a[a<0.5] <- 0  # round down
a
## A disord object with hash 5b7279f3c05d00cf1e8f999a755151e0451c56ec and elements
## [1] 9 4 7 1 2 6 3 8 5
## (in some order)
b[b>0.6] <- b[b>0.6] + 3  # add 3 to every element greater than 0.6
b
## A disord object with hash 488e1c6f4e2c062379d47b5511730a9785661318 and elements
## [1]  5  6 11  4  8  9 12 10  7
## (in some order)

Usual semantics follow, provided one is careful to maintain the hash code:

d <- disord(1:10)
d
## A disord object with hash 65e11d78de79b7f584068ad856749e3748cb837c and elements
##  [1]  1  2  3  4  5  6  7  8  9 10
## (in some order)
e <- 10 + 3*d - d^2
e
## A disord object with hash 65e11d78de79b7f584068ad856749e3748cb837c and elements
##  [1]  12  12  10   6   0  -8 -18 -30 -44 -60
## (in some order)
e<4
## A disord object with hash 65e11d78de79b7f584068ad856749e3748cb837c and elements
##  [1] FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
## (in some order)
d[e<4] <- e[e<4]
d
## A disord object with hash 65e11d78de79b7f584068ad856749e3748cb837c and elements
##  [1]   1   2   3   4   0  -8 -18 -30 -44 -60
## (in some order)

Above, the replacement command works because d and e and e<4 [which is a Boolean disord object] all have the same hash code.

An R session with the mvp package

The mvp package implements multivariate polynomials using the STL map class. Following commands only work as intended here with mvp >= 1.0-12. Below we see how disordR idiom allows mathematically meaningful operation while suppressing inadmissible ones:

library("mvp")
set.seed(0)
a <- rmvp()
b <- rmvp()
a
## mvp object algebraically equal to
## 3 a b^9 e^4 f  +  7 a^2 b^4 d^6 e f^4  +  4 a^4 b^6 c^5 d^11 f^4  +  6 a^6 b^3 c^14 f^2  +  5 a^11 e^6 f^6  +  b^8 e^7 f^12  +  2 b^10 d^10 f^4
b
## mvp object algebraically equal to
## 5 a c^2 e^8 f^7  +  4 a^2 b^5 c^6 e^3  +  7 a^2 b^7 c^4 d e^2  +  a^4 d^6 e^5 f  +  6 a^6 d^6 f^6  +  3 b^7 c^7 e^5  +  2 b^10 c^3 f^7

Observe that standard multivariate polynomial algebra works:

a + 2*b
## mvp object algebraically equal to
## 3 a b^9 e^4 f  +  10 a c^2 e^8 f^7  +  7 a^2 b^4 d^6 e f^4  +  8 a^2 b^5 c^6 e^3  +  14 a^2 b^7 c^4 d e^2  +  4 a^4 b^6 c^5 d^11 f^4  +  2 a^4 d^6 e^5 f  +  6 a^6 b^3 c^14 f^2  +  12 a^6 d^6 f^6  +  5 a^11 e^6 f^6  +  6 b^7 c^7 e^5  +  b^8 e^7 f^12  +  4 b^10 c^3 f^7  +  2 b^10 d^10 f^4
(a+b)*(a-b) == a^2-b^2   # should be TRUE (expression is quite long)
## [1] TRUE

We can extract the coefficients of these polynomials using the coeffs() function:

coeffs(a)
## A disord object with hash 76b070e3d27bf2e3a548b56a02678d79881de0ce and elements
## [1] 3 7 4 6 5 1 2
## (in some order)
coeffs(b)
## A disord object with hash 40b9beff42bebe889cb596f78d096c90ef279834 and elements
## [1] 5 4 7 1 6 3 2
## (in some order)

observe that the coefficients are returned as a disord object. We may manipulate the coefficients of a polynomial in many ways. We may do the following things:

coeffs(a)[coeffs(a) < 4] <- 0   # set any coefficient of a that is <4 to zero
a
## mvp object algebraically equal to
## 7 a^2 b^4 d^6 e f^4  +  4 a^4 b^6 c^5 d^11 f^4  +  6 a^6 b^3 c^14 f^2  +  5 a^11 e^6 f^6
coeffs(b) <- coeffs(b)%%2       # consider coefficients of b modulo 2
b
## mvp object algebraically equal to
## a c^2 e^8 f^7  +  a^2 b^7 c^4 d e^2  +  a^4 d^6 e^5 f  +  b^7 c^7 e^5

However, many operations which have reasonable idiom are in fact meaningless and are implicitly prohibited. For example:

x <- rmvp()     # set up new mvp objects x and y
y <- rmvp()

Then the following should all produce errors:

coeffs(x) + coeffs(y)  # order implementation specific
## Error in coeffs(x) + coeffs(y): consistent(e1, e2) is not TRUE
coeffs(x) <- coeffs(y) # ditto
## Error in `coeffs<-.mvp`(`*tmp*`, value = new("disord", .Data = c(5, 2, : consistent(vars(x), value) is not TRUE
coeffs(x) <- 1:2       # replacement value not length 1
## Error in .local(x, i = i, j = j, ..., value): consistent(x, value) is not TRUE
coeffs(x)[coeffs(x) < 3] <- coeffs(x)[coeffs(y) < 3]
## Error in .local(x, i, j = j, ..., drop): consistent(x, i) is not TRUE

vars() and powers() return disord objects

The disord() function takes a list argument, and this is useful for working with mvp objects:

(a <- as.mvp("x^2 + 4 - 3*x*y*z"))
## mvp object algebraically equal to
## 4  -  3 x y z  +  x^2
vars(a)
## A disord object with hash 9395842c6e67dfb0be871e04b3a8964a1c9b9bd5 and elements
## [[1]]
## character(0)
## 
## [[2]]
## [1] "x" "y" "z"
## 
## [[3]]
## [1] "x"
## 
## (in some order)
powers(a)
## A disord object with hash 9395842c6e67dfb0be871e04b3a8964a1c9b9bd5 and elements
## [[1]]
## integer(0)
## 
## [[2]]
## [1] 1 1 1
## 
## [[3]]
## [1] 2
## 
## (in some order)
coeffs(a)
## A disord object with hash 9395842c6e67dfb0be871e04b3a8964a1c9b9bd5 and elements
## [1]  4 -3  1
## (in some order)

Note that the hash of all three objects is identical, generated from the polynomial itself (not just the relevant element of the three-element list that is an mvp object). This allows us to do some rather interesting things:

double <- function(x){2*x}
(a <- rmvp())
## mvp object algebraically equal to
## a^2 c^10 d^2 f^2  +  7 a^3 d^5 e^14  +  6 a^5 c^7 d^4 e^2  +  4 a^8 c d^5 e^6  +  2 b^2 c^4 d^10 e^5 f  +  5 b^2 c^6 d^2 e^7 f^6  +  3 c^6 d^4 e^2 f^6
pa <- powers(a)
va <- vars(a)
ca <- coeffs(a)
pa[ca<4] <- sapply(pa,double)[ca<4]
mvp(va,pa,ca)
## mvp object algebraically equal to
## 7 a^3 d^5 e^14  +  a^4 c^20 d^4 f^4  +  6 a^5 c^7 d^4 e^2  +  4 a^8 c d^5 e^6  +  5 b^2 c^6 d^2 e^7 f^6  +  2 b^4 c^8 d^20 e^10 f^2  +  3 c^12 d^8 e^4 f^12

Above, a was a multivariate polynomial and we doubled the powers of all variables in terms with coefficients less than 4. Or even:

a <- as.mvp("3 + 5*a*b - 7*a*b*x^2 + 2*a*b^2*c*d*x*y -6*x*y + 8*a*b*c*d*x")
a
## mvp object algebraically equal to
## 3  +  5 a b  +  8 a b c d x  -  7 a b x^2  +  2 a b^2 c d x y  -  6 x y
pa <- powers(a)
va <- vars(a)
ca <- coeffs(a)
va[sapply(pa,length) > 4] <- sapply(va,toupper)[sapply(pa,length) > 4]
mvp(va,pa,ca)
## mvp object algebraically equal to
## 3  +  8 A B C D X  +  2 A B^2 C D X Y  +  5 a b  -  7 a b x^2  -  6 x y

Above, we took multivariate polynomial a and replaced the variable names in every term with more than four variables with their uppercase equivalents. The mvp package does not include powers<-() or vars<-() functions, but these would have made the idiom nicer (I will get round to writing them when the disordR package is more stable).