picos.glyphs

String templates used to print (algebraic) expressions.

PICOS internally uses this module to produce string representations for the algebraic expressions that you create. The function-like objects that are used to build such strings are called “glyphs” and are instanciated by this module following the singleton pattern. As a result, you can modify the glyph objects listed below to influence how PICOS will format future strings, for example to disable use of unicode symbols that your console does not suppport or to adapt PICOS’ output to the rest of your application.

Here’s an example of first swapping the entire character set to display expressions using only Latin-1 characters, and then modifying a single glyph to our liking:

>>> import picos
>>> X = picos.SymmetricVariable("X", (2,2))
>>> print(X >> 0)
X ≽ 0
>>> picos.glyphs.latin1()
>>> print(X >> 0)
X » 0
>>> picos.glyphs.psdge.template = "{} - {} is psd"
>>> print(X >> 0)
X - 0 is psd

Note that glyphs understand some algebraic rules such as operator precedence and associativity. This is possible because strings produced by glyphs remember how they were created.

>>> one_plus_two = picos.glyphs.add(1, 2)
>>> one_plus_two
'1 + 2'
>>> one_plus_two.glyph.template, one_plus_two.operands
('{} + {}', (1, 2))
>>> picos.glyphs.add(one_plus_two, 3)
'1 + 2 + 3'
>>> picos.glyphs.sub(0, one_plus_two)
'0 - (1 + 2)'

The positive semidefinite glyph above does not yet know how to properly handle arguments with respect to the - symbol involved, but we can modify it further:

>>> print(X + X >> X + X)
X + X - X + X is psd
>>> # Use the same operator binding strength as regular substraction.
>>> picos.glyphs.psdge.order = picos.glyphs.sub.order
>>> print(X + X >> X + X)
X + X - (X + X) is psd

You can reset all glyphs to their initial state as follows:

>>> picos.glyphs.default()

Classes

class picos.glyphs.Am(glyph)[source]

Bases: Op

A math atom glyph.

__init__(glyph)[source]

Construct an Am glyph.

Parameters

glyph (str) – The glyph’s format string template.

class picos.glyphs.Br(glyph)[source]

Bases: Op

A math operator glyph with enclosing brackets.

__init__(glyph)[source]

Construct a Br glyph.

Parameters

glyph (str) – The glyph’s format string template.

class picos.glyphs.Fn(glyph)[source]

Bases: Op

A math operator glyph in function form.

__init__(glyph)[source]

Construct a Fn glyph.

Parameters

glyph (str) – The glyph’s format string template.

class picos.glyphs.Gl(glyph)[source]

Bases: object

The basic “glyph”, an (algebraic) string formatting template.

Sublcasses are supposed to extend formatting routines, going beyond of what Python string formatting is capabale of. In particular, glyphs can be used to craft unambiguous algebraic expressions with the minimum amount of parenthesis.

__init__(glyph)[source]

Construct a glyph.

Parameters

glyph (str) – The glyph’s format string template.

rebuild()[source]

If the template was created using other glyphs, rebuild it.

Returns

True if the template has changed.

reset()[source]

Reset the glyph to its initial formatting template.

update(new)[source]

Change the glyph’s formatting template.

class picos.glyphs.GlStr(string, glyph, operands)[source]

Bases: str

A string created from a glyph.

It has an additional glyph field pointing to the glyph that created it, and a operands field containing the values used to create it.

__init__(string, glyph, operands)[source]

Augment the Python string with metadata on its origin.

static __new__(cls, string, glyph, operands)[source]

Create a regular Python string.

reglyphed(replace={})[source]

Returns a rebuilt version of the string using current glyphs.

Parameters

replace (dict) – Replace leaf-node (non GlStr) strings with new strings. This can be used, for instance, to change the names of varaibles.

glyph

The glyph used to create the string.

operands

The operands used to create the string.

class picos.glyphs.Op(glyph, order, assoc=False, closed=False)[source]

Bases: Gl

The basic math operator glyph.

__init__(glyph, order, assoc=False, closed=False)[source]

Construct a math operator glyph.

Parameters
  • glyph (str) – The glyph’s format string template.

  • order (int) – The operator’s position in the binding strength hierarchy. Operators with lower numbersbind more strongly.

  • assoc (bool) – If this is True, then the operator is associative, so that parenthesis are always omitted around operands with an equal outer operator. Otherwise, (1) parenthesis are used around the right hand side operand of a binary operation of same binding strength and (2) around all operands of non-binary operations of same binding strength.

  • closed (bool or list(bool)) – If True, the operator already encloses the operands in some sort of brackets, so that no additional parenthesis are needed. For glyphs where only some operands are enclosed, this can be specified per operand in the form of a list.

reset()[source]

Reset the glyph to its initial behavior.

update(new)[source]

Change the glyph’s behavior.

class picos.glyphs.OpStr(string, glyph, operands)[source]

Bases: GlStr

A string created from a math operator glyph.

class picos.glyphs.Rl(glyph)[source]

Bases: Op

A math relation glyph.

__init__(glyph)[source]

Construct a Rl glyph.

Parameters

glyph (str) – The glyph’s format string template.

class picos.glyphs.Tr(glyph)[source]

Bases: Op

A math glyph in superscript/trailer form.

__init__(glyph)[source]

Construct a Tr glyph.

Parameters

glyph (str) – The glyph’s format string template.

Functions

picos.glyphs.ascii()[source]

Let PICOS create future strings using only ASCII characters.

picos.glyphs.clever_add(left, right)[source]

Describe the addition of two values in a clever way.

A wrapper around add that resorts to sub if the second operand was created by neg or is a negative number (string). In both cases the second operand is adjusted accordingly.

Example

>>> from picos.glyphs import neg, add, clever_add, matrix
>>> add("x", neg("y"))
'x + -y'
>>> clever_add("x", neg("y"))
'x - y'
>>> add("X", matrix(neg("y")))
'X + [-y]'
>>> clever_add("X", matrix(neg("y")))
'X - [y]'
>>> clever_add("X", matrix(-1.5))
'X - [1.5]'
picos.glyphs.clever_div(left, right)[source]

Describe the division of one value by another in a clever way.

A wrapper around div that factors out identity factors.

picos.glyphs.clever_dotp(left, right, complexLHS, scalar=False)[source]

Describe an inner product in a clever way.

Parameters

complexRHS (bool) – Whether the right hand side is complex.

picos.glyphs.clever_mul(left, right)[source]

Describe the multiplocation of two values in a clever way.

A wrapper around mul that factors out identity factors.

picos.glyphs.clever_neg(value)[source]

Describe the negation of a value in a clever way.

A wrapper around neg that resorts to unnegating an already negated value.

Example

>>> from picos.glyphs import neg, clever_neg, matrix
>>> neg("x")
'-x'
>>> neg(neg("x"))
'-(-x)'
>>> clever_neg(neg("x"))
'x'
>>> neg(matrix(-1))
'-[-1]'
>>> clever_neg(matrix(-1))
'[1]'
picos.glyphs.clever_sub(left, right)[source]

Describe the substraction of a value from another in a clever way.

A wrapper around sub that resorts to add if the second operand was created by neg or is a negative number(string). In both cases the second operand is adjusted accordingly.

Example

>>> from picos.glyphs import neg, sub, clever_sub, matrix
>>> sub("x", neg("y"))
'x - -y'
>>> clever_sub("x", neg("y"))
'x + y'
>>> sub("X", matrix(neg("y")))
'X - [-y]'
>>> clever_sub("X", matrix(neg("y")))
'X + [y]'
>>> clever_sub("X", matrix(-1.5))
'X + [1.5]'
picos.glyphs.col_vectorize(*entries)[source]

Describe a column vector with the given symbolic entries.

picos.glyphs.default(rebuildDerivedGlyphs=True)

Let PICOS create future strings using only unicode characters.

picos.glyphs.free_var_name(string)[source]

Return a variable name not present in the given string.

picos.glyphs.from_glyph(string, theGlyph)[source]

Whether the given string was created by the given glyph.

picos.glyphs.is_negated(value)[source]

Check if a value can be unnegated by unnegate.

picos.glyphs.latin1(rebuildDerivedGlyphs=True)[source]

Let PICOS create future strings using only ISO 8859-1 characters.

picos.glyphs.make_function(*names)[source]

Create an ad-hoc composite function glyphs.

Example

>>> from picos.glyphs import make_function
>>> make_function("log", "sum", "exp")("x")
'log∘sum∘exp(x)'
picos.glyphs.matrix_cat(left, right, horizontal=True)[source]

Describe matrix concatenation in a clever way.

A wrapper around matrix, horicat and vertcat.

Example

>>> from picos.glyphs import matrix_cat
>>> Z = matrix_cat("X", "Y")
>>> Z
'[X, Y]'
>>> matrix_cat(Z, Z)
'[X, Y, X, Y]'
>>> matrix_cat(Z, Z, horizontal = False)
'[X, Y; X, Y]'
picos.glyphs.rebuild()[source]

Update glyphs that are based upon other glyphs.

picos.glyphs.row_vectorize(*entries)[source]

Describe a row vector with the given symbolic entries.

picos.glyphs.scalar(value)[source]

Format a scalar value.

This function mimics an operator glyph, but it returns a normal string (as opposed to an OpStr) for nonnegative numbers.

This is not realized as an atomic operator glyph to not increase the recursion depth of is_negated and unnegate unnecessarily.

Example

>>> from picos.glyphs import scalar
>>> str(1.0)
'1.0'
>>> scalar(1.0)
'1'
picos.glyphs.shape(theShape)[source]

Describe a matrix shape that can contain wildcards.

A wrapper around size that takes just one argument (the shape) that may contain wildcards which are printed as '?'.

picos.glyphs.show(*args)[source]

Show output from all glyphs.

Parameters

args (list(str)) – Strings to use as glyph operands.

picos.glyphs.unicode(rebuildDerivedGlyphs=True)[source]

Let PICOS create future strings using only unicode characters.

picos.glyphs.unnegate(value)[source]

Unnegate a value, usually a glyph-created string, in a sensible way.

Unnegates a operator glyph created string or other value in a sensible way, more precisely by recursing through a sequence of glyphs used to create the value and for which we can factor out negation, and negating the underlaying (numeric or string) value.

Raises

ValueError – When is_negated returns False.

Objects

picos.glyphs.CAN_FACTOR_OUT_NEGATION

Operator glyphs for which negation may be factored out.

Default value
(<picos.glyphs.Br object at 0x7dc21772d010>,
 <picos.glyphs.Fn object at 0x7dc217705bd0>,
 <picos.glyphs.Fn object at 0x7dc214b884b0>,
 <picos.glyphs.Fn object at 0x7dc21773a360>,
 <picos.glyphs.Fn object at 0x7dc214b88590>,
 <picos.glyphs.Fn object...
picos.glyphs.abs

Absolute value glyph.

Default value
<picos.glyphs.Br object at 0x7dc217706850>
picos.glyphs.add

Addition glyph.

Default value
<picos.glyphs.Op object at 0x7dc214b73250>
picos.glyphs.affpart

Affine part glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b4ad50>
picos.glyphs.and_

Logical and glyph.

Default value
<picos.glyphs.Gl object at 0x7dc214d78dd0>
picos.glyphs.bcasted

Broadcasted glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b2bd20>
picos.glyphs.blinpart

Bilinear part glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b3f150>
picos.glyphs.closure

Set closure glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21772dbe0>
picos.glyphs.comma

Seperated by comma glyph.

Default value
<picos.glyphs.Gl object at 0x7dc217731370>
picos.glyphs.compose

Function composition glyph.

Default value
<picos.glyphs.Gl object at 0x7dc21773a140>
picos.glyphs.compsep

Compact seperator glyph.

Default value
<picos.glyphs.Gl object at 0x7dc21774c180>
picos.glyphs.conj

Complex conugate glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214bf15a0>
picos.glyphs.cstpart

Constant part glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b2b7e0>
picos.glyphs.cubed

Cubed value glyph.

Default value
<picos.glyphs.Tr object at 0x7dc217707750>
picos.glyphs.desvec

Symmetric de-vectorization glyph.

Default value
<picos.glyphs.Fn object at 0x7dc215532b70>
picos.glyphs.det

Determinant glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214d79010>
picos.glyphs.diag

Diagonal matrix glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b88590>
picos.glyphs.div

Division glyph.

Default value
<picos.glyphs.Op object at 0x7dc214b94120>
picos.glyphs.dotp

Scalar product glyph.

Default value
<picos.glyphs.Br object at 0x7dc217706490>
picos.glyphs.element

Set element glyph.

Default value
<picos.glyphs.Rl object at 0x7dc21772e120>
picos.glyphs.eq

Equality glyph.

Default value
<picos.glyphs.Rl object at 0x7dc2177060d0>
picos.glyphs.exp

Exponentiation glyph.

Default value
<picos.glyphs.Fn object at 0x7dc217732210>
picos.glyphs.exparg

Expected value glyph.

Default value
<picos.glyphs.Op object at 0x7dc214b73150>
picos.glyphs.expected

Epected value glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21452c1a0>
picos.glyphs.forall

Universal quantification glyph.

Default value
<picos.glyphs.Gl object at 0x7dc214b883d0>
picos.glyphs.fromto

Range glyph.

Default value
<picos.glyphs.Gl object at 0x7dc2155327b0>
picos.glyphs.frozen

Frozen mutables glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b2b850>
picos.glyphs.ge

Greater or equal glyph.

Default value
<picos.glyphs.Rl object at 0x7dc214530050>
picos.glyphs.geomean

Matrix geometric mean glyph.

Default value
<picos.glyphs.Op object at 0x7dc214bf1650>
picos.glyphs.gt

Greater than glyph.

Default value
<picos.glyphs.Rl object at 0x7dc21774d0f0>
picos.glyphs.hadamard

Hadamard product glyph.

Default value
<picos.glyphs.Op object at 0x7dc215532d50>
picos.glyphs.horicat

Horizontal concatenation glyph.

Default value
<picos.glyphs.Op object at 0x7dc214b4adf0>
picos.glyphs.htransp

Matrix hermitian transposition glyph.

Default value
<picos.glyphs.Tr object at 0x7dc21773a690>
picos.glyphs.idmatrix

Identity matrix glyph.

Default value
<picos.glyphs.Am object at 0x7dc217707110>
picos.glyphs.imag

Imaginary part glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214bf14f0>
picos.glyphs.index

Index glyph.

Default value
<picos.glyphs.Tr object at 0x7dc21773a7a0>
picos.glyphs.infty

Infinity glyph.

Default value
<picos.glyphs.Am object at 0x7dc21772dd30>
picos.glyphs.interval

Interval glyph.

Default value
<picos.glyphs.Gl object at 0x7dc214b72e50>
picos.glyphs.intrange

Integer range glyph.

Default value
<picos.glyphs.Gl object at 0x7dc215532990>
picos.glyphs.inverse

Matrix inverse glyph.

Default value
<picos.glyphs.Tr object at 0x7dc21774ce90>
picos.glyphs.kron

Kronecker product glyph.

Default value
<picos.glyphs.Op object at 0x7dc214b88670>
picos.glyphs.lambda_

Lambda symbol glyph.

Default value
<picos.glyphs.Am object at 0x7dc2177065d0>
picos.glyphs.le

Lesser or equal glyph.

Default value
<picos.glyphs.Rl object at 0x7dc21774d220>
picos.glyphs.leadsto

Successorship glyph.

Default value
<picos.glyphs.Gl object at 0x7dc214ff3790>
picos.glyphs.linpart

Linear part glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b259a0>
picos.glyphs.log

Logarithm glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21773a250>
picos.glyphs.lt

Lesser than glyph.

Default value
<picos.glyphs.Rl object at 0x7dc2177329f0>
picos.glyphs.maindiag

Main diagonal glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b94050>
picos.glyphs.matrix

Matrix glyph.

Default value
<picos.glyphs.Br object at 0x7dc21772d010>
picos.glyphs.max

Maximum glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21774c510>
picos.glyphs.maxarg

The basic math operator glyph.

Default value
<picos.glyphs.Op object at 0x7dc214b25b50>
picos.glyphs.min

Minimum glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21774c8a0>
picos.glyphs.minarg

The basic math operator glyph.

Default value
<picos.glyphs.Op object at 0x7dc214b3f250>
picos.glyphs.mul

Multiplication glyph.

Default value
<picos.glyphs.Op object at 0x7dc214b88750>
picos.glyphs.ncnorm

Nuclear Norm glyph.

Default value
<picos.glyphs.Op object at 0x7dc21774c9d0>
picos.glyphs.ncstpart

Nonconstant part glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b3f1d0>
picos.glyphs.neg

Negation glyph.

Default value
<picos.glyphs.Op object at 0x7dc214d79190>
picos.glyphs.norm

Norm glyph.

Default value
<picos.glyphs.Br object at 0x7dc21774c770>
picos.glyphs.or_

Logical or glyph.

Default value
<picos.glyphs.Gl object at 0x7dc214d78710>
picos.glyphs.parenth

Parenthesis glyph.

Default value
<picos.glyphs.Gl object at 0x7dc217706710>
picos.glyphs.plsmns

Plus/Minus glyph.

Default value
<picos.glyphs.Op object at 0x7dc214d79250>
picos.glyphs.pnorm

p-Norm glyph.

Default value
<picos.glyphs.Op object at 0x7dc21772de80>
picos.glyphs.power

Power glyph.

Default value
<picos.glyphs.Tr object at 0x7dc21772dfd0>
picos.glyphs.pqnorm

pq-Norm glyph.

Default value
<picos.glyphs.Op object at 0x7dc217707890>
picos.glyphs.probdist

Probability distribution glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21452c130>
picos.glyphs.prod

Product glyph.

Default value
<picos.glyphs.Fn object at 0x7dc217705e50>
picos.glyphs.psdge

Lesser or equal w.r.t. the p.s.d. cone glyph.

Default value
<picos.glyphs.Rl object at 0x7dc21773a8b0>
picos.glyphs.psdle

Greater or equal w.r.t. the p.s.d. cone glyph.

Default value
<picos.glyphs.Rl object at 0x7dc21773a9c0>
picos.glyphs.ptrace

Matrix p-Trace glyph.

Default value
<picos.glyphs.Op object at 0x7dc21774cd60>
picos.glyphs.ptrace_

Matrix partial trace glyph.

Default value
<picos.glyphs.Op object at 0x7dc21773a580>
picos.glyphs.ptransp

Matrix partial transposition glyph.

Default value
<picos.glyphs.Tr object at 0x7dc2177327b0>
picos.glyphs.ptransp_

Matrix partial transposition glyph.

Default value
<picos.glyphs.Op object at 0x7dc21773a470>
picos.glyphs.qe

Quantum entropy glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21452c210>
picos.glyphs.qre

Quantum relative entropy glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21452c280>
picos.glyphs.quadpart

Quadratic part glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b4acb0>
picos.glyphs.real

Real part glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214d790d0>
picos.glyphs.renyi

Renyi entropy glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21452c2f0>
picos.glyphs.repr1

Representation glyph.

Default value
<picos.glyphs.Gl object at 0x7dc21772da90>
picos.glyphs.repr2

Long representation glyph.

Default value
<picos.glyphs.Gl object at 0x7dc217705d10>
picos.glyphs.reshaped

Column-major (Fortran) reshaped glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b2b9a0>
picos.glyphs.reshaprm

Row-major (C-order) reshaped glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b2ba10>
picos.glyphs.sep

Seperator glyph.

Default value
<picos.glyphs.Gl object at 0x7dc21774c640>
picos.glyphs.set

Set glyph.

Default value
<picos.glyphs.Gl object at 0x7dc214b72d50>
picos.glyphs.shortint

Shortened interval glyph.

Default value
<picos.glyphs.Gl object at 0x7dc214b882f0>
picos.glyphs.shuffled

Matrix reshuffling glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21452c0c0>
picos.glyphs.size

Matrix size/shape glyph.

Default value
<picos.glyphs.Gl object at 0x7dc21773a030>
picos.glyphs.slice

Expression slicing glyph.

Default value
<picos.glyphs.Op object at 0x7dc217732330>
picos.glyphs.spnorm

Spectral Norm glyph.

Default value
<picos.glyphs.Op object at 0x7dc217705f90>
picos.glyphs.sqrt

Square root glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21452c050>
picos.glyphs.squared

Squared value glyph.

Default value
<picos.glyphs.Tr object at 0x7dc217707610>
picos.glyphs.sub

Substraction glyph.

Default value
<picos.glyphs.Op object at 0x7dc215532c60>
picos.glyphs.sum

Summation glyph.

Default value
<picos.glyphs.Fn object at 0x7dc217705bd0>
picos.glyphs.svec

Symmetric vectorization glyph.

Default value
<picos.glyphs.Fn object at 0x7dc215532a80>
picos.glyphs.trace

Matrix trace glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b884b0>
picos.glyphs.transp

Matrix transposition glyph.

Default value
<picos.glyphs.Tr object at 0x7dc21774cfc0>
picos.glyphs.trilvec

Lower triangular vectorization glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b72f50>
picos.glyphs.triuvec

Upper triangular vectorization glyph.

Default value
<picos.glyphs.Fn object at 0x7dc214b73050>
picos.glyphs.vec

Vectorization glyph.

Default value
<picos.glyphs.Fn object at 0x7dc21773a360>
picos.glyphs.vertcat

Vertical concatenation glyph.

Default value
<picos.glyphs.Op object at 0x7dc214b4ae90>
picos.glyphs.wgeomean

Weighted matrix geometric mean glyph.

Default value
<picos.glyphs.Op object at 0x7dc214bf1700>