Category:Fōrmulæ
This programming language may be used to instruct a computer to perform a task.
Official website 

Execution method:  Interpreted 

Garbage collected:  Yes 
Parameter passing methods:  By value 
Typing:  Untyped 
See Also: 

If you know Fōrmulæ, please write code for some of the tasks not implemented in Fōrmulæ.
Contents
What is Fōrmulæ ?[edit]
Fōrmulæ is a framework to do symbolic computation.
Symbolic computation works by the repeatedly application of rules of transformation —also called rewriting rules — on symbolic expressions, until no rule can be applied.
As a very simple example, consider the following sequence of rule application:
Symbolic expression  Step 

2 + (3 × 5)  Starting expression 
2 + 15  Numeric multiplication rule 
17  Numeric addition rule 
17  Final result, no rule can be applied 
Mathematics is not the only field where symbolic computation can be used.
Traditional symbolic software, such as Mathematica, Maple or Matlab are provided with a considerable but fixed set of rewriting rules.
The Fōrmulæ project works in a different way. We do not only create rewriting rules, It provides an specification with which anybody can write —in a regular programming language— rewriting rules that can be also published. It converts the process of creating rewriting rules to a colaborative one.
Fōrmulæ dictionary[edit]
As you can see, the number of rules are always increasing and they need to be documented. The Fōrmulæ dictionary is the official mean to do it.
The dictionary is organized on expression tags. An expression tag is a unique name given to a kind of symbolic expression, i.e. the mathematical addition operation receives the expression tag Math.Arithmetic.Addition. The Math.Arithmetic.Addition dictionary entry enumerates and describes every known rule applied to this expression tag.
Expression visualization[edit]
The Fōrmulæ framework is not only a specification to create rewriting rules. It also defines a specification of how to visualize symbolic expressions, and how to create methods for editing symbolic expressions.
It means that anybody can write code to define how an expression is to be shown. It is called a visualization rule. Moreover, there can be multiple visualization rules for a specific expression, although only one can be used at the same time.
It is very common that these visualization rules show expression as humans do. It is usually called prettyprint. Moreover, different visualization rules can also written to match science field, localization (see Expressions with localized implementations), or even personal preferences.
Entries in the dictionary also describe the known visualization rules for a specific expression tag.
Expression edition[edit]
The Fōrmulæ framework also defines a specification to edit (create or modify) symbolic expressions, called edition rules.
Entries in the dictionary also describe the known edition rules for a specific expression tag.
Fōrmulæ packages[edit]
A package is a combination of zero or more rewriting rules, zero or more visualization rules and zero or more edition rules, packed as a single file, ready to be published. These rules are generally related to the same field.
There are several packages developed by the Fōrmulæ team, called the standard packages, but anyone can pack rules he/she creates.
The following are some of the standard packages:
Standard package  Description 

Arithmetic  Number definition, arbitrary precision calculations, rational numbers, imaginary unit, infinity 
Relation  Comparisons, threeway comparisons, min/max value in set/list, set/list membership (∈) 
Logic  Logic literals (true, false), logic operations, first order quantifiers, predicates 
String  String definition, strings operations 
Symbolic  Symbols, functions, global and local scope of symbols, lambda expressions 
Expression  Subexpression retrieval, expression cardinality, tag management, ask/show expressions 
List  List definition, dynamic creation of lists, matrices, addition/multiplication of matrices, cartesian product, transposing, determinants 
Color  Color definition, dynamic creation of color 
Programming  Sequence control (blocks), decision control (if, ifelse) and iteration control (while, for, foreach), recursion 
There are standard packages that are being developed, and they will be available soon:
Standard package  Description 

Diagram  Several kind of diagrams 
Chart  Bar, line, pie charts 
Plot  Mathematic plots, 2D/3D functions, parametric, polar, contour line, vector, sufaces 
Time  Date datatype definition, timezones 
Localization  Locale definition, localized operations such as number, string or date representation 
Object oriented programming  Class definition, object instantiation, inheritance, polymorphism 
Logic programming  Conversion to Horn clauses, backtracking 
Theorem proving  High order and equational logic, conversion to conjunctive normal form, resolution tree 
Quantum programming (simulated)  Qbit definition, quantum logic gates, reversibility, measurement 
Fōrmulæ frontends[edit]
There must be a program where symbolic expressions are created, visualized, edited and where to apply the rewriting rules. It is a frontend. Anyone can write such a program, according to the Fōrmulæ specification.
There is, however a standard program, called the Desktop. It has the following features:
 It can be used to install, reinstall, disable or remove Fōrmulæ packages.
 It can be used to choose the visualization rule for a tag, when it has multiple of them.
 It can be used to disable specific edition or rewriting rules.
 It works in readevalprintloop mode. It is, the user creates an expression (a question), then the program invokes the process of rule application, returning the result (the answer), and the process continues again, creating a list of questionanswers (a script), like a commandline mode. Unlike a commandline interface, a user can modify and reevaluate any of the previous questions.
 The script can be saved to a file, to be retrieved later.
 The script can be printed out.
 It supports cut/copy/paste operations between expressions and subexpressions.
 It supports undo/redo operations.
Fōrmulæ as a programming language[edit]
There are rewriting rules to do what can be done with traditional programming languages, so it can be used as a programming language too. It is called the Fōrmulæ programming language and it is Turingcomplete, so it can compute everything that is computable.
Given that rewriting rules, visualization rules and edition rules are continuously created or improved, it produces radical consecuences never seen before in traditional programming languages:
A dynamically defined programming language[edit]
The term "dynamically defined programming language" is defined here as a language which its features and capabilities are defined and added to the language over time. It is a language able to evolve and change in time.
Traditionally, programming languages are first fully designed. A grammar is created in order to construct tools, such as compilers. Once it is done, it is very hard or impossible to make changes on the language structure. Adding features to a language usually leads in a new and different language, i.e. the C and the C++ languages.
Nonunique visualization of programs[edit]
Because there can be multiple ways symbolic expressions look, there are several forms of visualization for the same program.
The next example shows the definition of a function that generates a Farey sequence of a given order, using different visualization rules —for some of its subexpressions—.
Flowchart style  Traditional, indented code style 

File:FormulaeExampleFlowchart.png  File:FormulaeExampleCode.png 
Official sites[edit]
Site  Description 

www.formulae.org  Main page. Official repository of the library, Desktop program, packages and source code. 
dictionary.formulae.org  Reference of expressions and implementations 
Pages in category "Fōrmulæ"
The following 19 pages are in this category, out of 19 total.