Automata

Class NumberSystem

java.lang.Object

Automata.NumberSystem

public class NumberSystem
extends java.lang.Object

The class NumberSystem represents a number system.
A number system consists of the following four:
- a rule to represent non-negative numbers, and hence an alphabet
- a rule for addition
- a rule for comparison, and by comparison we mean equality and less than testing.
- a flag that determines whether numbers are represented in most significant digit order or least significant digit.
For example msd_3 is a number system. It is the number system in which
- numbers are represented in base 3 in most significant digit first. Hence the alphabet is {0,1,2}.
- the rule for addition, is a simple two states automaton, that gets 3-tuples (a,b,c) in base three and accepts iff c=a+b.
- the rule for comparison is a simple two state automaton, that gets 2-tuples (a,b) in base three and accepts iff a < b.
- we already mentioned that numbers are in most significant digit first (msd) in this number system.
We mandate that 0 and 1 belong to the alphabet of every number systems. In addition, we require that 0* represent the additive identity in all number systems. We also mandate that either 0*1 or 10* (depending on msd/lsd) represent multiplicative identity in all number systems.
From here on by rule we mean a finite automaton. If the users want to create a new number system, they, at least, have to provide the rule for addition (in the Custom Bases directory). They can further provide the rule for less than. If less than rule is not provided, we use the lexicographic ordering on integers to create an automaton for less than testing. So for example if the alphabet is {-2,0,7} then in lexicographic order, we have -270 < 0-270-2.
Rule for equality is always the rule for word equalities, i.e., two numbers are equal if the words representing them are equal.
Rules for base n already exist in the system for every n>1. However the user can override them.

VERY IMPORTANT: ALL PRIVATE METHODS WHICH RETURN AUTOMATON MUST BE USED WITH CAUTION. THEIR RETURNED AUTOMATON SHOULD NOT BE ALTERED. IF YOU WANT TO ALTER THEIR RETURNED VALUE, THEN YOU HAVE GOT TO MAKE A CLONE AND DO THE MODIFICATION ON THE CLONE.

Field Summary

Fields

Modifier and Type	Field and Description
Automaton	addition Automata for addition, lessThan, and equal -addition has three inputs, and it accepts iff the third is the sum of the first two.
Automaton	allRepresentations Automata for addition, lessThan, and equal -addition has three inputs, and it accepts iff the third is the sum of the first two.
Automaton	equality Automata for addition, lessThan, and equal -addition has three inputs, and it accepts iff the third is the sum of the first two.
Automaton	lessThan Automata for addition, lessThan, and equal -addition has three inputs, and it accepts iff the third is the sum of the first two.

Constructor Summary

Constructors

Constructor and Description
NumberSystem(java.lang.String name)

Method Summary

Methods

Modifier and Type	Method and Description
Automaton	arithmetic(int a, java.lang.String b, java.lang.String c, java.lang.String arithmeticOperator)
Automaton	arithmetic(java.lang.String a, int b, java.lang.String c, java.lang.String arithmeticOperator)
Automaton	arithmetic(java.lang.String a, java.lang.String b, java.lang.String c, java.lang.String arithmeticOperator)
Automaton	comparison(int a, java.lang.String b, java.lang.String comparisonOperator)
Automaton	comparison(java.lang.String a, int b, java.lang.String comparisonOperator)
Automaton	comparison(java.lang.String a, java.lang.String b, java.lang.String comparisonOperator)
Automaton	get(int n)
Automaton	getAllRepresentations()
java.util.List<java.lang.Integer>	getAlphabet()
Automaton	getDivision(int n)
Automaton	getMultiplication(int n)
boolean	isMsd()
boolean	should_we_use_allRepresentations()
java.lang.String	toString()

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Field Detail

addition

public Automaton addition

Automata for addition, lessThan, and equal
-addition has three inputs, and it accepts iff the third is the sum of the first two. So the input is ordered!
-lessThan has two inputs, and it accepts iff the first one is less than the second one. So the input is ordered!
-equal has two inputs, and it accepts iff they are equal.

lessThan

public Automaton lessThan

Automata for addition, lessThan, and equal
-addition has three inputs, and it accepts iff the third is the sum of the first two. So the input is ordered!
-lessThan has two inputs, and it accepts iff the first one is less than the second one. So the input is ordered!
-equal has two inputs, and it accepts iff they are equal.

equality

public Automaton equality

Automata for addition, lessThan, and equal
-addition has three inputs, and it accepts iff the third is the sum of the first two. So the input is ordered!
-lessThan has two inputs, and it accepts iff the first one is less than the second one. So the input is ordered!
-equal has two inputs, and it accepts iff they are equal.

allRepresentations

public Automaton allRepresentations

Automata for addition, lessThan, and equal
-addition has three inputs, and it accepts iff the third is the sum of the first two. So the input is ordered!
-lessThan has two inputs, and it accepts iff the first one is less than the second one. So the input is ordered!
-equal has two inputs, and it accepts iff they are equal.

Constructor Detail

NumberSystem

public NumberSystem(java.lang.String name)
             throws java.lang.Exception

Throws:

java.lang.Exception

Method Detail

isMsd

public boolean isMsd()

should_we_use_allRepresentations

public boolean should_we_use_allRepresentations()

getAlphabet

public java.util.List<java.lang.Integer> getAlphabet()

getAllRepresentations

public Automaton getAllRepresentations()

get

public Automaton get(int n)
              throws java.lang.Exception

Parameters:

n -

Returns:

an automaton that accepts only n.

Throws:

java.lang.Exception

getDivision

public Automaton getDivision(int n)
                      throws java.lang.Exception

Throws:

java.lang.Exception

getMultiplication

public Automaton getMultiplication(int n)
                            throws java.lang.Exception

Throws:

java.lang.Exception

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object

comparison

public Automaton comparison(java.lang.String a,
                   java.lang.String b,
                   java.lang.String comparisonOperator)
                     throws java.lang.Exception

Parameters:

a -

b -

comparisonOperator - can be any of "<",">","<=",">=","=","!="

Returns:

an Automaton with two inputs, with labels a and b. It accepts iff a comparisonOperator b. Note that the order of inputs, in the resulting automaton, is not guaranteed to be either (a,b) or (b,a). So the input is not ordered!

Throws:

java.lang.Exception

comparison

public Automaton comparison(java.lang.String a,
                   int b,
                   java.lang.String comparisonOperator)
                     throws java.lang.Exception

Parameters:

a -

b - a non negative integer

comparisonOperator - can be any of "<",">","<=",">=","=","!="

Returns:

an Automaton with single input, with label = [a]. It accepts iff a comparisonOperator b.

Throws:

java.lang.Exception

comparison

public Automaton comparison(int a,
                   java.lang.String b,
                   java.lang.String comparisonOperator)
                     throws java.lang.Exception

Parameters:

a - a non negative integer

b -

comparisonOperator - can be any of "<",">","<=",">=","=","!="

Returns:

an Automaton with single input, with label = [b]. It accepts iff a comparisonOperator b.

Throws:

java.lang.Exception

arithmetic

public Automaton arithmetic(java.lang.String a,
                   java.lang.String b,
                   java.lang.String c,
                   java.lang.String arithmeticOperator)
                     throws java.lang.Exception

Parameters:

a -

b -

c -

arithmeticOperator - can be any of "+", "-","*","/"

Returns:

an Automaton with three inputs with labels a,b,c. It accepts iff c = a arithmeticOperator b. Note that the order of inputs, in the resulting automaton, is not guaranteed to be in any fixed order like (a,b,c) or (c,b,a) ... So the input is not ordered!

Throws:

java.lang.Exception

arithmetic

public Automaton arithmetic(java.lang.String a,
                   int b,
                   java.lang.String c,
                   java.lang.String arithmeticOperator)
                     throws java.lang.Exception

Parameters:

a -

b - a non-negative integer

c -

arithmeticOperator - can be any of "+","-","*","/"

Returns:

an Automaton with two inputs, with labels a and c. It accepts iff c = a arithmeticOperator b. Note that the order of inputs, in the resulting automaton, is not guaranteed to be in any fixed order like [a,c] or [c,a]. So the input is not ordered!

Throws:

java.lang.Exception

arithmetic

public Automaton arithmetic(int a,
                   java.lang.String b,
                   java.lang.String c,
                   java.lang.String arithmeticOperator)
                     throws java.lang.Exception

Parameters:

a - a non-negative integer

b -

c -

arithmeticOperator - can be any of "+","-","*","/"

Returns:

an Automaton with two inputs, with labels b and c. It accepts iff c = a arithmeticOperator b. Note that the order of inputs, in the resulting automaton, is not guaranteed to be in any fixed order like [b,c] or [c,b]. So the input is not ordered!

Throws:

java.lang.Exception