Programming Paradigms via Mathematica

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Programming Paradigms via Mathematica

• Dr Dejan Toic, ETF Beograd tosic_at_etf.bg.ac.yu
  http//kondor.etf.bg.ac.yu/tosic/

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Mathematica

• Mathematica is the world's only fully integrated
  environment for technical computing.
• The visionary concept of Mathematica was to
  create once and for all a single system that
  could handle all the various aspects of
  technical computing in a coherent and unified
  way.
• The key intellectual advance that made this
  possible was the invention of a new kind of
  symbolic computer language that could for the
  first time manipulate the very wide range of
  objects involved in technical computing using
  only a fairly small number of basic primitives.

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Mathematica combines ideas from many different
sources

• LISP Mathematica can manipulate lists using the
  data structures of the programming language LISP.
  
• PROLOG Mathematica uses pattern matching
  capabilities taken from PROLOG to express
  mathematical rules in a simple way.
• APL Mathematica's simple manipulation of
  structured data has been pioneered by APL.
• SIMULA, SMALLTALK, and C Object-oriented
  elements make code development easier.
• PASCAL, FORTRAN, and C Traditional procedural
  programming style.
• Other Features Modularization and Built-In
  Functions.

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Software development cycle
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Square brackets , Curly braces , and
Parentheses ( )

• Parentheses ( ) are used in Mathematica for
  grouping in arithmetic.
• Square brackets are used for enclosing the
  arguments of functions.
• Curly braces are used for entering lists.
• Using square brackets to enclose function
  arguments avoids the ambiguity in traditional
  mathematical notation between parentheses that
  enclose function arguments and parentheses that
  are used for grouping arithmetic operations.

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Normal Expression

• A normal expression is an expression that has a
  head and elements (arguments).
• The head of the normal expression fa,b, for
  example, is f, and the elements are a and b.
• The head and the elements in a normal expression
  are also expressions, and can, for example, be
  other normal expressions.

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Atoms

• Expressions that are not normal expressions are
  called atoms.
• Atoms are either
• symbols,
• numbers, or
• strings.
• All expressions are constructed from atoms.

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Patterns

• A pattern is an expression that is used to
  specify a set of expressions.
• Any expression in the set of expressions
  specified by a pattern is said to match the
  pattern.
• The simplest non-trivial pattern is Blank,
  which matches any expression. Blank is usually
  entered as a single underscore character _.
• For example, the pattern _,_ matches any list
  with two elements.

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Canonical example
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Built-in Functions
One way to program a function that computes the
factorial of a non-negative integer is to simply
use the built-in function Factorial.
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Composition of Functions Anonymous (pure)
function
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Procedural FORTRANesque programming Do
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Straight-forward implementation Product
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Functional programming
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Range generates a list of consecutive integers
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Times multiplies together its arguments
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Rule-based programming
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Transformation rules

• Rules are a mechanism for replacing a variable
  with a value in an expression.
• A rule can transform one expression, lhs, into
  another expression, rhs lhs -gt rhs

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Rule-based programming style

• The transformations are described by rules which
  separate the description of the sub-object to be
  replaced (the pattern) from the calculation of
  the replacement.
• Optionally, rules can have further conditions
  that restrict their applicability.
• The transformations are controlled by explicit or
  implicit strategies.

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Mixed programming Definitions, Rules, Patterns
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Dynamically Stored Results
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Information on objects ? command
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Recursive programming
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C-style programming For
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Iterative programming While
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Traditional form Product
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Definition using pure function
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Benefits

• Visionary concept of advanced computing
• Fully integrated mathematical environment
• Symbolic computer language
• Various programming paradigms
• Short software development cycle