The language#

A spec is one YAML file that states an optimisation problem without its data. The file declares the dimensions that index the problem, the data the spec expects, the decisions the solver makes, and the rules those decisions obey. The pages in this section list every key such a file accepts, and the rules such a file must meet. The glossary defines each word these pages use, such as entry, coordinate and absent, in one fixed sense.

File shape lists the top-level keys of a spec file, and how mathspec reads the YAML.
Given, parameters, variables, objective and constraints lists the keys of the data, the decisions, the goal and the rules, and of the given: section.
Dimensions lists the keys of a dimension, the set of labels that entries are indexed over.
Relations and maps lists the keys of a relation, a set of combinations of labels, and of a map, which gives each row one label, and how each operator uses them.
Expressions defines the grammar of arithmetic and of conditions, and how dimensions combine.
Named expressions, conditions and macros lists the keys of the three kinds of named piece of an expression.
Operators lists every operator, such as sum and shift, and the math each one prints.
Piecewise curves and SOS lists the keys of a piecewise-linear curve and of a special ordered set (SOS).
Assumptions lists the keys that state what a spec expects of its data.
Absence and where describes which rows do not exist, and how absence spreads through an expression.
Errors and advice lists what mathspec refuses at load, and what it advises without refusing.

A whole spec#

The file below is a complete spec, which states the least-cost dispatch of a fleet of generators against an hourly load.

dispatch.yaml

description: Least-cost dispatch of a generator fleet against an hourly load.

dimensions:
  snapshot: { dtype: int }
  generator: { dtype: str }

parameters:
  load: { dims: [snapshot] }
  cost: { dims: [generator] }
  capacity: { dims: [generator] }

variables:
  dispatch:
    dims: [snapshot, generator]
    where: "capacity > 0"
    bounds: { lower: 0, upper: capacity }

constraints:
  power_balance:
    dims: [snapshot]
    expression: sum(dispatch, over=generator) == load

objective:
  sense: minimize
  expression: sum(dispatch * cost)

python -m mathspec markdown dispatch.yaml --no-legend prints the math that the file states. It prints one heading for the variable domains, one for the objective and one for the constraints, and under them these three equations:

\[ 0 \le \mathit{dispatch}_{g,t} \le \mathrm{capacity}_{g} \qquad \forall\, g \in \mathcal{G},\ t \in \mathcal{T} \,:\, \mathrm{capacity}_{g} > 0 \]

\[ \min \sum_{g \in \mathcal{G},\ t \in \mathcal{T}} \mathit{dispatch}_{g,t} \cdot \mathrm{cost}_{g} \]

\[ \sum_{g \in \mathcal{G}} \mathit{dispatch}_{g,t} = \mathrm{load}_{t} \qquad \forall\, t \in \mathcal{T} \]

The variable dispatch exists only where capacity > 0, and the objective sums over both dimensions. power_balance has one row per snapshot, because the sum removes generator.

The eleven rules#

A file obeys these eleven rules. Where a rule needs no data, mathspec refuses the file when to_spec reads it, before any data exists (at load). In the message, mathspec suggests the rewrite.

#	Rule	Stated in full
1	A file has fourteen section keys, plus `version` and `description`. mathspec refuses an unknown key, and suggests the nearest valid key.	File shape
2	`to_spec` checks at load every rule that needs no data.	Errors and advice
3	Every name is declared once. mathspec refuses a parameter and a dimension that are both called `snapshot`.	Names
4	The kind of a name sets where the name may appear. A dimension follows `over=` or `along=`, and is never multiplied.	Names
5	`a + b` is over the dimensions of `a` and of `b` together. The expression of a constraint is over exactly its `dims`, and the objective is over none.	How dimensions combine
6	At the coordinates it excludes, a `where:` on a variable makes the variable absent. A `where:` on a constraint makes the row absent.	Absence
7	A row that uses an absent variable is absent. Inside a `sum`, an absent term is left out, and the row stays.	Absence
8	A tool refuses a row missing from a data table, unless the `missing:` of the entry sets the row to absent or to a value.	Absence
9	Two variables may be multiplied in the objective and in a constraint, up to degree two, and not in a bound or a `piecewise:` link. `x / y` and `a ** b` need their divisor, their base and their exponent free of variables.	Expressions
10	The operators are `sum`, `sum_back`, `at` and `shift`. A file cannot add an operator.	Operators
11	A construct that steps or counts along a dimension needs the dimension declared `ordered: true`. These are `shift` in an expression or a condition, `sum_back`, `position`, `piecewise` with `over:`, and `sos` of `type: 2`.	`ordered`