ivy

Overview ¶

Ivy is an interpreter for an APL-like language. It is a plaything and a work in progress.

Unlike APL, the input is ASCII and the results are exact (but see the next paragraph). It uses exact rational arithmetic so it can handle arbitrary precision. Values to be input may be integers (3, -1), rationals (1/3, -45/67) or floating point values (1e3, -1.5 (representing 1000 and -3/2)).

Some functions such as sqrt are irrational. When ivy evaluates an irrational function, the result is stored in a high-precision floating-point number (default 256 bits of mantissa). Thus when using irrational functions, the values have high precision but are not exact.

Unlike in most other languages, operators always have the same precedence and expressions are evaluated in right-associative order. That is, unary operators apply to everything to the right, and binary operators apply to the operand immediately to the left and to everything to the right. Thus, 3*4+5 is 27 (it groups as 3*(4+5)) and iota 3+2 is 1 2 3 4 5 while 3+iota 2 is 4 5. A vector is a single operand, so 1 2 3 + 3 + 3 4 5 is (1 2 3) + 3 + (3 4 5), or 7 9 11.

As a special but important case, note that 1/3, with no intervening spaces, is a single rational number, not the expression 1 divided by 3. This can affect precedence: 3/6*4 is 2 while 3 / 6*4 is 1/8 since the spacing turns the / into a division operator. Use parentheses or spaces to disambiguate: 3/(6*4) or 3 /6*4.

Ivy has complex numbers, which are constructed using the unary or binary j operator. As with rationals, the token 1j2 (the representation of 1+2i) is a single token. The individual parts can be rational, so 1/2j-3/2 is the complex number 0.5-1.5i and scans as a single value.

Indexing uses [] notation: x[1], x[1; 2], and so on. Indexing by a vector selects multiple elements: x[1 2] creates a new item from x[1] and x[2]. An empty index slot is a shorthand for all the elements along that dimension, so x[] is equivalent to x, and x[;3] gives the third column of two-dimensional array x.

Only a subset of APL's functionality is implemented, but all numerical operations are supported.

Semicolons separate multiple statements on a line. Variables are alphanumeric and are assigned with the = operator. Assignment is an expression.

After each successful expression evaluation, the result is stored in the variable called _ (underscore) so it can be used in the next expression.

The APL operators, adapted from https://en.wikipedia.org/wiki/APL_syntax_and_symbols, and their correspondence are listed here. The correspondence is incomplete and inexact.

Unary operators

Name              APL   Ivy     Meaning
Roll              ?B    ?       One integer selected randomly from the first B integers
Ceiling           ⌈B    ceil    Least integer greater than or equal to B
                                If B is complex, the complex ceiling, as defined by McDonnell
Floor             ⌊B    floor   Greatest integer less than or equal to B
                                If B is complex, the complex floor, as defined by McDonnell
Shape             ⍴B    rho     Vector of number of components in each dimension of B
Count             ≢B    count   Scalar number of elements at top level of B
Flatten           ∊B    flatten Vector of all the scalar elements within B
Not               ∼B    not     Logical: not 1 is 0, not 0 is 1
Absolute value    ∣B    abs     Magnitude of B
Index generator   ⍳B    iota    Vector of the first B integers
                                If B is a vector, matrix of coordinates
Unique            ∪B    unique  Remove all duplicate elements from B
Enclose           ⊂B    box     Wrap B in one level of nesting
Disclose          ⊃B    first   First element of B in ravel order
Split             ↓B    split   Create vector of nested elements from matrix B; inverse of mix
Mix               ↑B    mix     Create matrix from elements of vector B; inverse of split
Exponential       ⋆B    **      e to the B power
Negation          −B    -       Change sign of B
Identity          +B    +       No change to B
Signum            ×B    sgn     -1 if B<0; 0 if B=0; 1 if B>0. More generally: B/abs B if B!=0
Reciprocal        ÷B    /       1 divided by B
Ravel             ,B    ,       Reshapes B into a vector
Matrix inverse    ⌹B    inv     Inverse of B; for vector (conj v)/v+.*conj v
Pi times          ○B            Multiply by π
Logarithm         ⍟B    log     Natural logarithm of B
Reversal          ⌽B    rot     Reverse elements of B along last axis
Reversal          ⊖B    flip    Reverse elements of B along first axis
Grade up          ⍋B    up      Indices of B which will arrange B in ascending order
Grade down        ⍒B    down    Indices of B which will arrange B in descending order
Execute           ⍎B    ivy     Execute an APL (ivy) expression
Monadic format    ⍕B    text    A character representation of B
Monadic transpose ⍉B    transp  Reverse the axes of B
Factorial         !B    !       Product of integers 1 to B
Bitwise not             ^       Bitwise complement of B (integer only)
Square root       B⋆.5  sqrt    Square root of B.
Sine                    sin     sin(A); APL uses binary ○ (see below)
Cosine                  cos     cos(A); ditto
Tangent                 tan     tan(A); ditto
Arcsine                 asin    arcsin(B)
Arccosine               acos    arccos(B)
Arctangent              atan    arctan(B)
Hyperbolic sine         sinh    sinh(B)
Hyperbolic cosine       cosh    cosh(B)
Hyperbolic tangent      tanh    tanh(B)
Hyperbolic arcsine      asinh   arcsinh(B)
Hyperbolic arccosine    acosh   arccosh(B)
Hyperbolic arctangent   atanh   arctanh(B)
Rotation by 90°         j       Multiplication by sqrt(-1)
Real part               real    Real component of the value
Imaginary part          imag    Imaginary component of the value
Phase                   phase   Phase of the value in the complex plane (-π to π)
Conjugate         +B    conj    Complex conjugate of the value
System functions  ⎕     sys     Argument is a string; run "sys 'help'" for details

Binary operators

Name                  APL   Ivy       Meaning
Add                   A+B   +         Sum of A and B
Subtract              A−B   -         A minus B
Multiply              A×B   *         A multiplied by B
Divide                A÷B   /         A divided by B (exact rational division)
                            div       A divided by B (Euclidean)
                            idiv      A divided by B (Go)
Exponentiation        A⋆B   **        A raised to the B power
Circle                A○B             Trigonometric functions of B selected by A
                                      A=1: sin(B) A=2: cos(B) A=3: tan(B); ¯A for inverse
                            sin       sin(B); ivy uses traditional name.
                            cos       cos(B); ivy uses traditional name.
                            tan       tan(B); ivy uses traditional name.
Deal                  A?B   ?         A distinct integers selected randomly from the first B integers
Membership            A∈B   in        1 for elements of A present in B; 0 where not.
Intersection          A∩B   intersect A with all elements that are also in B removed
Union                 A∪B   union     A followed by all members of B not already in A
Maximum               A⌈B   max       The greater value of A or B
Minimum               A⌊B   min       The smaller value of A or B
Reshape               A⍴B   rho       Array of shape A with data B
Take                  A↑B   take      Select the first (or last) A elements of B according to sgn A
Drop                  A↓B   drop      Remove the first (or last) A elements of B according to sgn A
Decode                A⊥B   decode    Value of a polynomial whose coefficients are B at A
                                      'T' decode B creates a seconds value from the time vector B
Encode                A⊤B   encode    Base-A representation of the value of B
                                      'T' encode B creates a time vector from the seconds value B
Residue               A∣B              B modulo A
                            mod       A modulo B (Euclidean)
                            imod      A modulo B (Go)
Catenation            A,B   ,         Elements of B appended to the elements of A along last axis
Catenation            A,B   ,%        Elements of B appended to the elements of A along first axis
Expansion             A\B   fill      Insert zeros (or blanks) in B corresponding to zeros in A
                                      In ivy: abs(A) gives count, A <= 0 inserts zero (or blank)
Compression           A/B   sel       Select elements in B corresponding to ones in A
                                      In ivy: abs(A) gives count, A <= 0 inserts zero
Index of              A⍳B   iota      The location (index) of B in A; 1+⌈/⍳⍴A if not found
                                      In ivy: origin-1 if not found (i.e. 0 if one-indexed)
Matrix divide         A⌹B   mdiv      Solution to system of linear equations Ax = B
                                      For real vectors, the magnitude of A projected on B
Rotation              A⌽B   rot       The elements of B are rotated A positions left
Rotation              A⊖B   flip      The elements of B are rotated A positions along the first axis
Logarithm             A⍟B   log       Logarithm of B to base A
Dyadic format         A⍕B   text      Format B into a character matrix according to A
                                      A is the textual format (see format special command);
                                      otherwise result depends on length of A:
                                      1 gives decimal count, 2 gives width and decimal count,
                                      3 gives width, decimal count, and style ('d', 'e', 'f', etc.).
                                      'T' text B formats seconds value B as a Unix date
General transpose     A⍉B   transp    The axes of B are ordered by A
Combinations          A!B   !         Number of combinations of B taken A at a time
Less than             A<B   <         Comparison: 1 if true, 0 if false
Less than or equal    A≤B   <=        Comparison: 1 if true, 0 if false
Equal                 A=B   ==        Comparison: 1 if true, 0 if false
Greater than or equal A≥B   >=        Comparison: 1 if true, 0 if false
Greater than          A>B   >         Comparison: 1 if true, 0 if false
Not equal             A≠B   !=        Comparison: 1 if true, 0 if false
Or                    A∨B   or        Logic: 0 if A and B are 0; 1 otherwise
And                   A∧B   and       Logic: 1 if A and B are 1; 0 otherwise
Nor                   A⍱B   nor       Logic: 1 if both A and B are 0; otherwise 0
Nand                  A⍲B   nand      Logic: 0 if both A and B are 1; otherwise 1
Xor                         xor       Logic: 1 if A != B; otherwise 0
Bitwise and                 &         Bitwise A and B (integer only)
Bitwise or                  |         Bitwise A or B (integer only)
Bitwise xor                 ^         Bitwise A exclusive or B (integer only)
Left shift                  <<        A shifted left B bits (integer only)
Right Shift                 >>        A shifted right B bits (integer only)
Complex construction        j         The complex number A+Bi

Operators and axis indicator

Name                APL  Ivy  APL Example  Ivy Example  Meaning (of example)
Reduce (last axis)  /    /    +/B          +/B          Sum across B
Reduce (first axis) ⌿    /%   +⌿B                       Sum down B
Scan (last axis)    \    \    +\B          +\B          Running sum across B
Scan (first axis)   ⍀    \%   +⍀B                       Running sum down B
Inner product       .    .    A+.×B        A +.* B      Matrix product of A and B
Outer product       ∘.   o.   A∘.×B        A o.* B      Outer product of A and B
                                                       (lower case o; may need preceding space)

Type-converting operations

Name              APL   Ivy     Meaning
Code                    code B  The integer Unicode value of char B
Char                    char B  The character with integer Unicode value B
Float                   float B The floating-point representation of B;
                                for complex numbers, the result is
                                (float A)j(float B)

Pre-defined constants ¶

The constants e (base of natural logarithms) and pi (π) are pre-defined to high precision, about 3000 decimal digits truncated according to the floating point precision setting.

Character data ¶

Strings are vectors of "chars", which are Unicode code points (not bytes). Syntactically, string literals are very similar to those in Go, with back-quoted raw strings and double-quoted interpreted strings. Unlike Go, single-quoted strings are equivalent to double-quoted, a nod to APL syntax. A string with a single char is just a singleton char value; all others are vectors. Thus “, "", and ” are empty vectors, `a`, "a", and 'a' are equivalent representations of a single char, and `ab`, `a` `b`, "ab", "a" "b", 'ab', and 'a' 'b' are equivalent representations of a two-char vector.

Unlike in Go, a string in ivy comprises code points, not bytes; as such it can contain only valid Unicode values. Thus in ivy "\x80" is illegal, although it is a legal one-byte string in Go.

Strings can be printed. If a vector contains only chars, it is printed without spaces between them.

Chars have restricted operations. Printing, comparison, indexing and so on are legal but arithmetic is not, and chars cannot be converted automatically into other singleton values (ints, floats, and so on). The unary operators char and code enable transcoding between integer and char values.

User-defined operators ¶

Users can define unary and binary operators, which then behave just like built-in operators. Both a unary and a binary operator may be defined for the same name.

The syntax of a definition is the 'op' keyword, the operator and formal arguments, an equals sign, and then the body. The names of the operator and its arguments must be identifiers. For unary operators, write "op name arg"; for binary write "op leftarg name rightarg". The final expression in the body is the return value. Operators may have recursive definitions; see the paragraph about conditional execution for an example.

The body may be a single line (possibly containing semicolons) on the same line as the 'op', or it can be multiple lines. For a multiline entry, there is a newline after the '=' and the definition ends at the first blank line (ignoring spaces).

Conditional execution is done with the ":" binary conditional return operator, which is valid only within the code for a user-defined operator. The left operand must be a scalar. If it is non-zero, the right operand is returned as the value of the function. Otherwise, execution continues normally. The ":" operator has a lower precedence than any other operator; in effect it breaks the line into two separate expressions.

Example: average of a vector (unary):

op avg x = (+/x)/rho x
avg iota 11
result: 6

Example: n largest entries in a vector (binary):

op n largest x = n take x[down x]
3 largest 7 1 3 24 1 5 12 5 51
result: 51 24 12

Example: multiline operator definition (binary):

op a sum b =
	a = a+b
	a

iota 3 sum 4
result: 1 2 3 4 5 6 7

Example: primes less than N (unary):

op primes N = (not T in T o.* T) sel T = 1 drop iota N
primes 50
result: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

Example: greatest common divisor (binary):

op a gcd b =
	a == b: a
	a > b: b gcd a-b
	a gcd b-a

1562 gcd !11
result: 22

On mobile platforms only, due to I/O restrictions, user-defined operators must be presented on a single line. Use semicolons to separate expressions:

op a gcd b = a == b: a; a > b: b gcd a-b; a gcd b-a

To declare an operator but not define it, omit the equals sign and what follows.

op foo x
op bar x = foo x
op foo x = -x
bar 3
result: -3
op foo x = /x
bar 3
result: 1/3

Within a user-defined operator body, identifiers are local to the invocation if they are assigned before being read, and global if read before being written. To write to a global without reading it first, insert an unused read.

total = 0
last = 0
op save x =
	total = total + x  # total is global because total is read before written
	last; last = x     # unused read makes last global

save 9; save 3
total last
result: 12 3

To remove the definition of a unary or binary user-defined operator,

opdelete foo x
opdelete a gcd b

Special commands ¶

Ivy accepts a number of special commands, introduced by a right paren at the beginning of the line. Most report the current value if a new value is not specified. For these commands, numbers are always read and printed base 10 and must be non-negative on input.

) help
	Describe the special commands. Run )help <topic> to learn more
	about a topic, )help <op> to learn more about an operator.
) base 0
	Set the number base for input and output. The commands ibase and
	obase control setting of the base for input and output alone,
	respectively.  Base 0 allows C-style input: decimal, with 037 being
	octal and 0x10 being hexadecimal. Bases above 16 are disallowed.
	To output large integers and rationals, base must be one of
	0 2 8 10 16. Floats are always printed base 10.
) cpu
	Print the duration of the last interactive calculation.
) debug name 0|1
	Toggle or set the named debugging flag. With no argument, lists
	the settings.
) demo
	Run a line-by-line interactive demo. On mobile platforms,
	use the Demo menu option instead.
) format ""
	Set the format for printing values. If empty, the output is printed
	using the output base. If non-empty, the format determines the
	base used in printing. The format is in the style of golang.org/pkg/fmt.
	For floating-point formats, flags and width are ignored.
) get "save.ivy"
	Read input from the named file; return to interactive execution
	afterwards. If no file is specified, read from "save.ivy".
	(Unimplemented on mobile.)
) maxbits 1e6
	To avoid consuming too much memory, if an integer result would
	require more than this many bits to store, abort the calculation.
	If maxbits is 0, there is no limit; the default is 1e6.
) maxdigits 1e4
	To avoid overwhelming amounts of output, if an integer has more
	than this many digits, print it using the defined floating-point
	format. If maxdigits is 0, integers are always printed as integers.
) maxstack 1e5
	To avoid using too much stack, the number of nested active calls to
	user-defined operators is limited to maxstack.
) op X
	If X is absent, list all user-defined operators. Otherwise,
	show the definition of the user-defined operator X. Inside the
	definition, numbers are always shown base 10, ignoring the ibase
	and obase.
) origin 1
	Set the origin for indexing a vector or matrix. Must be non-negative.
) prec 256
	Set the precision (mantissa length) for floating-point values.
	The value is in bits. The exponent always has 32 bits.
) prompt ""
	Set the interactive prompt.
) save "save.ivy"
	Write definitions of user-defined operators and variables to the
	named file, as ivy textual source. If no file is specified, save to
	"save.ivy".
	(Unimplemented on mobile.)
) seed 0
	Set the seed for the ? operator.
) timezone "Local"
	Set the time zone to be used for display. If the argument is
	missing, print the name and zone offset in seconds east.
) var X
	If X is absent, list all defined variables. Otherwise, show the
	definition of the variable X in a form that can be evaluated
	to recreate the value.