Basic
http://www.openmath.org/OpenMath/CDs/Basic
31/Dec/1997
official
The basic semantic dictionary that every OM compliant
implementation should accept.
Pi
The mathematical constant Pi, approximately 3.14159
Constant
() -> Intersection( real, Non(rational),
Non(algebraic) )
Pi = 3.1415926535897932385, "20-digit approximation"
Pi = 4*arctan(1)
Pi = 16*arctan(1/5)-4*arctan(1/239), "Machin's formula"
\pi = 16 \arctan(1/5) - 4 \arctan (1/239)
Pi
16 1 5
4 1 239
true
The boolean value true
Constant
false
The boolean value false
Constant
infinity
A value greater than any computable value.
Infinity characterizes the positive real infinity.
Negative infinity is represented as -infinity.
Constant
a < infinity, "for any a which is not infinity itself"
a <= infinity, "for any a which is not infinity itself"
NaN
The result of an ill-posed floating computation.
OpenMath borrows the definition of NaN from the IEEE
standard for floating point representations.
Constant
0/0 = NaN
NaN+a = NaN, "for any a which is not infinity"
NaN*a = NaN, "for any a which is not infinity or 0"
NaN-a = NaN, "for any a which is not infinity"
NaN/a = NaN, "for any a which is not 0"
ImagUnit
The imaginary unit, the square root of -1.
The number (0,1) in the pair-representation of complex
numbers.
The name "i" has been avoided, as it causes too many
clashes with the variable "i" and thus renaming would
be often required.
Constant
(-1)^(1/2) = ImagUnit
ImagUnit^2+1=0
exp(ImagUnit*Pi)+1=0
i = \sqrt{-1}
ImagUnit
-1 1 2
i^2+1 = 0
ImagUnit 2 1
0
e^{i\pi}+1 = 0
ImagUnit Pi 1
0
e
The base of the natural logarithms.
Also called Euler's constant.
Also called the base of the Napierian logarithms.
Although e^x = exp(x), for mathematical purposes
the functional form is preferred, while for displaying
purposes, the exponential form is preferred.
Constant
e=exp(1)
e^x = exp(x)
e = limit( (1+1/n)^n, n, infinity )
e = 2.71828182845904523536028747135,
"30-digit approximation"
e^x = \lim_{n \rightarrow \infty} (1+x/n)^n
x
1 1 n n
n
infinity
+ plus
The addition operator of any group
Binary, Operator
Associative, Commutative,
Error( "invalid cancellation of infinity" )
(complex complex) -> complex
(real real) -> real
(rational rational) -> rational
(integer integer) -> integer
(symbolic symbolic) -> symbolic
a+b=b+a, commutativity
a+(b+c)=(a+b)+c, associativity
a+0=a, identity
0+a=a, identity
a+(b+c)=(a+b)+c
a
b c
a b
c
+ plus
The unary addition operator. It behaves
like the identity operator.
Unary, Operator
complex -> complex
real -> real
rational -> rational
integer -> integer
symbolic -> symbolic
+a=a
* times
The multiplication operator of any commutative
ring
Binary, Operator
Associative, Commutative
(complex complex) -> complex
(real real) -> real
(rational rational) -> rational
(integer integer) -> integer
(symbolic symbolic) -> symbolic
a*b=b*a, commutativity
a*(b*c)=(a*b)*c, associativity
a*1=a, identity
1*a=a, identity
a*0=0
0*a=0
- minus
The subtraction operator of any group,
defined as a-b = a + inverse(b)
Binary, Operator
Error( "invalid cancellation of infinity" )
(complex complex) -> complex
(real real) -> real
(rational rational) -> rational
(integer integer) -> integer
(symbolic symbolic) -> symbolic
a-(b+c)=a-b-c
- minus
The unary minus operator. The inverse operator
of a group.
Unary, Operator
complex -> complex
real -> real
rational -> rational
integer -> integer
symbolic -> symbolic
a+(-a)=0
/ over
The division operator of any commutative
field, defined as the multiplication by the
multiplicative inverse of the second argument.
Binary, Operator
Error( "division by zero" ),
Error( "invalid cancellation of infinity" )
(complex complex) -> complex
(real real) -> real
(rational rational) -> rational
(integer integer) -> rational
(symbolic symbolic) -> symbolic
^ power
The powering operator
Binary, Operator
Error( "division by zero" ),
Error( "0^0 is undefined" )
(complex complex) -> complex
(real real) -> complex
(rational rational) -> complex
(rational integer) -> rational
(integer integer) -> rational
(symbolic symbolic) -> symbolic
a^0=1
a^1=a
1^a=1
apply
Explicit use of function application.
Apply a function to a sequence of (0 or more) arguments.
The first argument of apply is taken as a function, and
this function is applied over the second, third, etc.
arguments of apply. Symbolically:
apply( f, a, b, c, .... ) == f( a, b, c, ... )
Function application is provided by the encoding, so
in general it is not necessary to use apply. For example,
sin(x) is encoded as in SGML and
as (sin x) in the lisp-like functional encoding.
Apply is needed in two special cases:
When the name of the function clashes with some name
used by the encoding or by OpenMath, or
when the function to be applied is not a symbol, and
hence the SGML encoding will not be able to produce
valid tags.
Nary
apply( f ) = f( )
apply( f, a ) = f( a )
apply( f, a, b ) = f( a, b )
apply( f, a, b, c ) = f( a, b, c )
* NC A B
lambda
The operation of lambda calculus that makes a
function from an expression and a variable. The definition
at this level uses only one variable. Lambda is a binary
function, where the first argument is the variable and
the second argument is a the expression.
Lambda( x, F ) is written as \lambda x [F] in the lambda
calculus literature.
The lambda function can be viewed as the inverse of function
application.
Although the expression F may contain x, the lambda expression
is interpreted to be free of x. That is, the x variable is
a variable local to the environment of the definition of
the function or operator. Formally, lambda(x,F) is free of
x, and any substitutions, evaluations or tests for x in
lambda(x,F) should not happen.
A lambda expression on an arbitrary function applied to a
simple argument is equivalent to the arbitrary function.
E.g. lambda(x, f(x)) == f. This is a common shortcut.
Binary
apply( lambda(x, F), x ) = F,
"for any expression F and symbol x"
lambda(x, f(x)) = f
x x
from
Interpretation of a name (typically a function,
constant or operator name) from a specific CD. This
is necessary only in special circumstances, when a name
is defined in more than one CD.
When a name is used in an OpenMath expression, the
usage is associated with a hierarchy of CD definitions.
The interpretation of an unqualified name is that of
the first CD in which appears.
For example, suppose that the multiplication operator is
used. Assume that we are using the "Basic" CD and on
top of this, the "NC" (non-commutative) CD. Both define
the operator "*". An unqualified reference to "*" will
refer to the definition of "Basic". To refer to the NC
"*" the form: from("*",Basic) should be used.
Binary, Function
(symbol symbol) -> constructor
and
The boolean conjunction
Binary, Function
(boolean boolean) -> boolean
(symbolic symbolic) -> symbolic
and(true, true) = true
and(true, false) = false
and(false, true) = false
and(false, false) = false
or
The boolean disjunction
Binary, Function
(boolean boolean) -> boolean
(symbolic symbolic) -> symbolic
or(true, true) = true
or(true, false) = true
or(false, true) = true
or(false, false) = false
xor
The boolean exclusive disjunction
Binary, Function
(boolean boolean) -> boolean
(symbolic symbolic) -> symbolic
xor(true, true) = false
xor(true, false) = true
xor(false, true) = true
xor(false, false) = false
not
The boolean negation
Unary, Function
(boolean) -> integer
(symbolic) -> symbolic
not(true) = false
not(false) = false
union
The union of two sets
Binary, Function
(set set) -> set
intersection
The intersection of two sets
Binary, Function
(set set) -> set
setdiff
The difference of two sets i.e. the set of the members
of the first that are not members of the second.
Binary, Function
(set set) -> set
member
The membership testing operation (also commonly
called "in" or "including"). Returns true if the first
argument is part of the second argument. The second
argument must be a set.
Binary, Function
(symbolic set) -> boolean
IntegerQuotient
Integer quotient, the result of integer
division. For arguments a and b, it returns q,
where a = b*q+r, |r| < |b| and a*r ≥ 0 (or
the sign of r is the same as the sign of a).
Binary, Function
Error( "division by zero" )
(integer integer) -> integer
(symbolic symbolic) -> symbolic
a = b*IntegerQuotient(a,b) + IntegerRemainder(a,b)
IntegerRemainder
Integer remainder, the result of integer
division. For arguments a and b, it returns r,
where a = b*q+r, |r| < |b| and a*r ≥ 0 (the
sign of r is the same as the sign of a when both are
non-zero).
Binary, Function
Error( "division by zero" )
(integer integer) -> integer
(symbolic symbolic) -> symbolic
a = b*IntegerQuotient(a,b) + IntegerRemainder(a,b)
IntegerGcd
The integer greatest common divisor of
all its arguments. The function is extended to work
with 0 arguments (returns 0) and with one argument
(returns the argument).
Nary, Function
Associative, Commutative
Sequence(integer) -> integer
Sequence(symbolic) -> symbolic
IntegerGcd() = 0
IntegerGcd(a) = a
IntegerGcd(a,b) = IntegerGcd(b,a)
IntegerGcd(a,b,c) = IntegerGcd(a,IntegerGcd(b,c))
exp
The exponential function
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.2]
Unary, Function
real -> real
symbolic -> symbolic
exp(0) = 1
exp(x) = e^x
exp(x) = limit( (1+x/n)^n, n, infinity )
ln
The logarithmic function. Also called
the natural logarithm.
The inverse of the exponential function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.1]
Unary, Function
Error( "logarithm has a singularity at 0" )
Intersect(real,positive) -> real
symbolic -> symbolic
ln(1) = 0
ln(exp(x)) = x, "for real x"
exp(ln(x)) = x, always
sin
The circular trigonometric function sine
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.3]
Unary, Function
real -> real
symbolic -> symbolic
sin(0) = 0
sin(integer*Pi) = 0
sin((Z+1/2)*Pi) = (-1)^Z, "for integer Z"
-1 <= sin(real)
sin(real) <= 1
sin(3*x)=-4*sin(x)^3+3*sin(x), "triple angle formula"
ditto, [4.3.27]
cos
The cosine function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.3]
Unary, Function
real -> real
symbolic -> symbolic
cos(0) = 1
cos(integer*Pi+Pi/2) = 0
cos(Z*Pi) = (-1)^Z, "for integer Z"
-1 <= cos(real)
cos(real) <= 1
tan
The tangent function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.3]
Unary, Function
real -> real
symbolic -> symbolic
tan(integer*Pi) = 0
tan(x) = sin(x)/cos(x)
cot
The cotangent function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.3]
Unary, Function
real -> real
symbolic -> symbolic
cot(integer*Pi+Pi/2) = 0
cot(x) = cos(x)/sin(x)
sec
The secant function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.3]
Unary, Function
real -> real
symbolic -> symbolic
sec(x) = 1/cos(x)
csc
The cosecant function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.3]
Unary, Function
real -> real
symbolic -> symbolic
csc(x) = 1/sin(x)
arcsin
The inverse of the sine function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.4]
Unary, Function
real -> real
symbolic -> symbolic
sin(arcsin(x)) = x
arcsin(sin(x)) = x, "for x between -Pi/2 and Pi/2"
arccos
The inverse of the cosine function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.4]
Unary, Function
real -> real
symbolic -> symbolic
cos(arccos(x)) = x
arccos(cos(x)) = x, "for x between 0 and Pi"
arctan
The inverse of the tangent function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.4]
Unary, Function
real -> real
symbolic -> symbolic
tan(arctan(x)) = x
arctan(tan(x)) = x, "for x between -Pi/2 and Pi/2"
arccot
The inverse of the cotangent function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.4]
Unary, Function
real -> real
symbolic -> symbolic
cot(arccot(x)) = x
arcsec
The inverse of the secant function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.4]
Unary, Function
real -> real
symbolic -> symbolic
sec(arcsec(x)) = x
arccsc
The inverse of the cosecant function.
M. Abramowitz and I. Stegun, Handbook of Mathematical
Functions, [4.4]
Unary, Function
real -> real
symbolic -> symbolic
csc(arccsc(x)) = x
ExpandedForm
The expanded form of a mathematical object. The formal
definition of an expanded form is that every occurrence
of a*(b+c) is replaced by a*b+a*c. This is expansion
with respect to arithmetic operators (including integer
powers).
ExpandedForm defines a mathematical object with certain
properties, it does not necessarily imply performing an
operation over its argument.
Unary, Function
symbolic -> symbolic
ExpandedForm( a*(b+c) ) = a*b+a*c
ExpandedForm( (x+y)^2 ) = x^2 + 2*x*y + y^2
diff
The derivative of its first argument evaluated at the second
argument. The first argument is a unary function or a unary
operator. A lambda expression is naturally a valid first
argument. The result of the differentiation is an operator,
which is then immediately applied to the second argument.
This is necessary to be able to express concepts like, for
example, the derivative of an unknown function at 0.
diff( sin, x ) = cos(x)
diff( lambda(x, sin(x)), x ) = cos(x)
diff( lambda(x, x^2+3), x ) = 2*x
diff( lambda(x, x^2+3), a ) = 2*a
diff( lambda(x, x^2+3), 1 ) = 2
The following common expressions in the math literature are
encoded as follows (using a latex notation for mathematics)
\frac{\partial f}{\partial x} == diff( f, x )
f^{\prime}(x) == diff( f, x )
f^{\prime}(0) == diff( f, 0 )
f^{\prime} == lambda( x, diff(f,x) )
Binary, Function
(symbolic symbol) -> symbolic
diff
The unary differentiation function. This is identical to
the diff function with two arguments, except that it does
not evaluate the resulting expression at the point, it
returns the operator. It is ideally used for repeated
differentiation or when we are interested in the differentiated
operator.
The following relation defines the unary diff formally:
diff( f ) == lambda( x, diff(f,x) )
Examples:
diff( sin ) = cos
diff( lambda(x, sin(x)) ) = cos
diff( lambda(x, x^2+3) ) = lambda(x, 2*x)
The following common expressions in the math literature are
encoded as follows (using a latex notation for mathematics)
f^{\prime} == diff(f)
f^{\prime \prime} == diff(diff(f))
f^{\prime \prime}x == diff(diff(f),x)
f^{\prime \prime \prime} == diff(diff(diff(f)))
Unary, Function
symbolic -> symbolic
integral
The indefinite integral of a unary operator or function
(first argument) evaluated at the second argument.
integral( cos, x ) = sin(x)
integral( lambda(x, cos(x)), x ) = sin(x)
integral( lambda(x, 2*x^2+3), x ) = x^3+3*x
integral( lambda(x, 2*x^2+3), y ) = y^3+3*y
integral( lambda(x, 2*x^2+3), a+b ) = (a+b)^3+3*(a+b)
The following common expressions in the math literature are
encoded as follows (using a latex notation for mathematics)
\int x dx == integral(lambda(x,x),x)
Binary , Function
(symbolic symbol) -> symbolic
integral
The unary integration function. This is identical to
the integral function with two arguments, except that it does
not evaluate the resulting expression at the point, it
returns the operator. It is ideally used for repeated
integration or when we are interested in the integrated
operator.
The following relation defines the unary integral formally:
integral( f ) == lambda( x, integral(f,x) )
integral( cos ) = sin
integral( lambda(x, cos(x)) ) = sin
integral( lambda(x, 2*x^2+3) ) = lambda(x, x^3+3*x)
The following common expressions in the math literature are
encoded as follows (using a latex notation for mathematics)
\int f == integral(f)
\int \int f == integral(integral(f))
Unary , Function
symbolic -> symbolic
defint
The definite integral of a unary operator or function
(first argument) evaluated between the second and third
argument.
integral( cos, 0, Pi ) = sin(Pi)-sin(0) = 0
integral( lambda(x, cos(x)), 0, x ) = sin(x)
integral( lambda(x, 2*x^2+3), 0, x ) = x^3+3*x
integral( lambda(x, 2*x^2+3), 0, 1 ) = 4
The following common expressions in the math literature are
encoded as follows (using a latex notation for mathematics)
\int_0^a x^2 dx == defint( lambda(x, x^2), 0, a )
\int_0^x x^2 dx == defint( lambda(x, x^2), 0, x )
\int_0^x t^2 dt == defint( lambda(t, t^2), 0, x )
\int_0^x \sin == defint( sin, 0, x )
Binary , Function
= equal
The equality operator.
Binary, Operator
Commutative
(symbolic symbolic) -> boolean
<> unequal
The in-equality operator.
Binary, Operator
Commutative
(symbolic symbolic) -> boolean
< less
The less-than operator
Binary, Operator
(symbolic symbolic) -> boolean
<= lessequal
The less-or-equal operator
Binary, Operator
(symbolic symbolic) -> boolean
> greater
The greater-than operator
Binary, Operator
(symbolic symbolic) -> boolean
>= greaterequal
The greater-or-equal operator
Binary, Operator
(symbolic symbolic) -> boolean
list
An ordered list of elements. The list structure
does not impose any restriction on the number or type of
the elements. Just the ordering and number of elements
is preserved. Many systems will use alternative notations
to denote lists, for example [a,b,c...], or (a,b,c,...) or
{a,b,c,....}.
Nicklaus Wirth, Algorithms + Data Structures =
Programs, Prentice-Hall, [4.3]
The list may be homogeneous (all elements of the same type)
or inhomogeneous (elements can be of different types).
Nary, Constructor
2 3 5 7 11 13 17 19 23 29

set
An unordered list of elements. The set structure
does not impose any restriction on the number or type of
the elements. Repeated elements are discarded from a set.
Many systems will use alternative notations to denote sets,
for example {a,b,c,....}.
The set may be homogeneous (all elements of the same type)
or inhomogeneous (elements can be of different types).
Nary, Constructor
1 2 3
3 3 2 1 2 3
array
The computer science notion of an array, that
is a data structure with n dimensions, each dimension
with its own range of valid values where elements can
be stored and retrieved.
Intuitively, a one dimensional array, e.g.
array( segment(1,n), [....] ) is equivalent to a vector.
A two-dimensional array, e.g.
array( segment(1,m), segment(1,n), [ [...], [....], ...] )
is equivalent to a matrix.
An n-dimensional array has n+1 arguments. The first
n arguments are integer segments or ranges which define
the lower and upper bound of each dimension. The last
argument is a list or a list of list or an n-iterated
list of lists containing the values stored in the array.
if the last argument is missing, it is assumed that the
array entries are undefined.
Nary, Constructor
1 10
2 3 5 7 11 13 17 19 23 29

1 100 1 100
segment
A segment (or range or interval) of values,
defined by two limiting values, the lowest and the highest.
A segment is typically used in the definition of arrays,
definite integration and summation, and every time that
a range of values needs to be defined. Some programming
languages use the notation a .. b for segments.
Segments are closed, i.e. the end points are included in
the segment.
Binary, Constructor
select
The selection operation of an array, list,
set or any other non-primitive object.
Select has at least two arguments, the first argument
is the object to be selected from, and the rest are the
integer selectors.
Select with n arguments (n>2) can only be applied to
arrays with n-1 dimensions.
The normal representation of select is not in the form
of a function, but as a subscript (e.g. a_i in latex)
or in the form of an index (a[i] in many programming
languages). E.g.
select(a,i,j,...) == a[i,j,...]
The second, third, etc arguments of select are called
the selectors. When the selector is a negative integer,
it is interpreted as its complement counted from the left
to the right. Hence if a is a list, a[-1] is the last
element of a. a[-3] is the third element from left to
right. In general, a[-k] == a[ length(a)+1-k ].
For the binary case, when the selector is 0, select
has the value of the function or operator at the top
of the expression tree.
Nary
(anything integer) -> anything
(array integer integer) -> anything
(array integer integer integer) -> anything
select(a,k) = a[k]
select(a,i,j) = a[i,j]
select(a+b,1) = a
length
The length of a structure, typically a list
set or array. Every OpenMath object has a length, and
in general, the length is the arity of the function/
operator at the top of the expression tree.
The length of symbols and strings is the number of
characters in their encoding.
The length of integers and floating point numbers is
not defined here.
Unary
symbolic -> integer
structure -> integer
length(list(1,2,3)) = 3
length(list()) = 0
length(a+b) = 2
length("abc") = 3
sin(x)^2+cos(x)^2-1=0, "Invariant relation followed by sin
and cos" M. Abramowitz and I. Stegun, Handbook of
Mathematical Functions [4.3.10]
sin(Z*Pi)=0, "for any integer Z"
exp(ImagUnit*Pi) + 1 = 0, "Identity linking ImagUnit, Pi, 0 and 1"
ditto, [4.2.26]
sin(2*x) = 2*sin(x)*cos(x), "sine duplicating formula"
csc(x) = 1/sin(x)
sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x)
cot(x) = 1/tan(x)