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This manual is for NI version 0.2.3.

Copyright © 2006 Michael Jumper

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

1. Introduction

1.1 What is NI?

NI (an acronym for not-intractable, pronounced as the English "knee") is a handy boolean algebra program which has been released as free software under the GNU General Public License. It is capable of computing and manipulating truth tables for boolean functions involving hundreds (or thousands) of variables, even though your computer doesn't even remotely have enough memory or speed. How does it do this? Simple: it cheats.

1.2 What do you mean there are only 2^{250 particles in the universe?

Although truth tables are usually a convenient way to solve boolean logic problems, they rapidly become inconvenient as the number of variables increases (a truth table for $n$ variables is 2^n in size). This is particularly bad for problems involving 250 or more variables, since not only is your computer not fast enough to finish such a calculation before the world ends, but the universe doesn't even contain enough particles to store the result. That's okay, though, because NI cheats: it compresses the result. In fact, the entire truth table is produced and manipulated in a compressed form. This is good for a number of reasons:

• The compressed form will fit in your computers memory.
• Manipulating a small piece of data is quicker than manipulating a large piece of data; your calculations will finish in a sane amount of time, rather than in billions of years.

1.3 But compression is a myth!

It is. You cannot losslessly make data smaller in some cases without making it larger in others -- it simply isn't possible. NI will solve many problems in a much smaller amount of time, but, by the laws of information theory, there must be cases where NI's compressed form is equal to or larger than the original uncompressed truth table. In such a case, NI would probably be slower than a simple brute-force truth table method, but I have yet to find a function involving many variables that demonstrates this condition.

1.4 So what are NI's limitations?

NI makes use of a giant memory-speed tradeoff, so it is definitely possible to find a function which uses up all available memory on your computer -- this is an unavoidable physical limitation of any computer program. Just because NI is capable of performing huge calculations quickly, doesn't mean that your computer has enough memory to perform these calculations; often, for complicated expressions involving hundreds of variables, the amount of memory required is several gigabytes. Keep in mind, however, that for hundreds of variables, a couple gigabytes of memory is nothing compared to the size of the uncompressed truth table.

2. NI Terminology

2.1 General Variables

General variables represent a single binary bit (0 or 1) of information and vary independently from each other. A general variable in NI essentially represents an unknown value, or all possible values of a single bit. All general variables are predefined in NI, and the syntax of their names is simple: the first general variable is g0, the second is g1, the third is g2, etc. For convenience, the lowercase letters a through z can be used in place of the first 26 general variables (g0 through g25). Using N general variables together in an expression is logically equivalent (but not algorithmically equivalent) to calculating the entire truth table for that expression.

2.2 Truth Tables

NI uses a specialized form of truth table internally, but it can easily convert this form into a more human-readable, standard truth table using the tt= command.

NI> a & b & c
NI> tt=

 a b c | result
-------+-------
 0 0 0 | 0
 1 0 0 | 0
 0 1 0 | 0
 1 1 0 | 0
 0 0 1 | 0
 1 0 1 | 0
 0 1 1 | 0
 1 1 1 | 1

NI>

Note that the variable a (g0) changes its value every 2^0 bits, b (g1) changes its value every 2^1 bits, and c (g2) changes its value every 2^2 bits. This is true of all general variables: the general variable gN changes its value every 2^N bits.

2.3 Linear Truth Tables

Because the values of general variables follow a simple, reproducable pattern, the left side of a truth table (the side which lists the values of each variable involved in the function) can be left out while still preserving the content of the truth table. This form of a truth table is close to what NI uses internally, and is displayed on only one line, hence the name. The ltt= command can be used to display the linear truth table representing the previous result.

NI> a & b & c
NI> ltt=
00000001
NI>

2.4 Abbreviated Linear Truth Tables

Abbreviated linear truth tables (ALTTs) are the internal representation of truth tables used in NI. They are substantially harder to read than a standard truth table, but they take up much less memory. In an ALTT, repeating sections of a linear truth table are split into pieces with lengths that are a power of two (called "segments"). The notation of a segment is simple, either 0:n, 1:n, or (ALTT):n, where ALTT is any abbreviated linear truth table of length n or less, and n is an integer representing the exponent of a power of two. The | character is placed between two segments to denote that the segments are joined together. The = command is used to print the ALTT of the previous result.

NI> a & b & c
NI> =
0:2|(0:1|(0:0|1:0):1):2
NI>

The usefulness of ALTTs is more easily demonstrated with larger truth tables.

NI> g100 & g200
NI> =
0:200|(0:100|1:100):200
NI>

The truth table of the above calculation is beyond the capabilities of any computer to print, since it involves 2^{200 bits. If you attempt to print such a truth table, NI refuses with an error.

NI> g100 & g200
NI> =
0:200|(0:100|1:100):200
NI> tt=
ERROR: Physical result of operation is too large.
tt=
^
NI>

3. Examples

3.1 Validating and Invalidating Argument Forms

NI's built-in assume and verify commands can be used not only to apply hypothetical conditions, but also to validate arguments. In this section, we will verify three valid argument forms (modus ponens, modus tollens, and disjunctive syllogism) and verify the invalidity of two logical fallacies (affirming the consequent, and denying the antecedent).

3.1.1 Modus Ponens

The argument form of Modus Ponens is as follows:

If P, then Q.
P.
Therefore, Q.

These can be translated into NI expressions quite easily:

p implies q
p
q

Now, to verify the validity, we need to use the assume command on each premise, and the verify command on the conclusion.

        NI (Not Intractable) version 0.2.3 (almost-stable)
        http://www.nongnu.org/not-intractable
        Distributed under the GNU General Public License.
        See the file COPYING for more information.
        Copyright (C) 2006 Michael Jumper

        Type quit to exit.

NI> assume (p implies q)
NI> assume (p)
NI> verify (q)
True.

NI agrees that this is a valid argument.

3.1.2 Modus Tollens

The argument form of Modus Tollens is as follows:

If P, then Q.
Q is false.
Therefore, P is false.

The corresponding equivalents in NI are:

p implies q
q == 0
p == 0

And now for the verification:

NI> reset
NI> assume (p implies q)
NI> assume (q == 0)
NI> verify (p == 0)
True.

NI agrees that this argument form is valid. Because there were conditions left over from the previous argument form verification (the modus ponens example), we needed to use the reset command to remove those conditions.

3.1.3 Disjunctive Syllogism (Modus Tollendo Ponens)

The argument form of Disjunctive Syllogism is as follows:

P or Q.
Not P.
Therefore, Q.

Translated into NI expressions:

p | q
!p
q

And now for the verification:

NI> reset
NI> assume (p | q)
NI> assume (!p)
NI> verify (q)
True.

Once again, NI agrees with the validity of this argument form.

3.1.4 Affirming the Consequent

The form of the logical fallacy Affirming the Consequent is as follows:

If P, then Q.
Q.
Therefore, P.

The NI expressions:

p implies q
q
p

And now for the actual test:

NI> reset
NI> assume (p implies q)
NI> assume (q)
NI> verify (p)
False.

As you can see, NI is also capable of showing argument forms to be invalid. This particular logical fallacy is fallacious because it assumes that the "if ... then ..." goes both ways, when it does not. The argument would be valid if if-and-only-if were used instead (as this is a bidirectional if):

NI> reset
NI> assume (p iff q)
NI> assume (q)
NI> verify (p)
True.

3.1.5 Denying the Antecedent

The form of the logical fallacy Denying the Antecedent is as follows:

If P, then Q.
P is false.
Therefore, Q is false.

The NI expressions:

p implies q
p == 0
q == 0

And now for the actual test:

NI> reset
NI> assume (p implies q)
NI> assume (p == 0)
NI> verify (q == 0)
False.

Indeed, this argument form is false.

4. Optimization

4.1 The Concept

NI is faster than other truth-table based algorithms because it solves a common problem in a different way. The same can be said about certain expressions given to NI; one particular expression can likely be rewritten to solve the same problem, yet run faster. NI takes care of a large amount of the optimization necessary to solve problems with large numbers of variables, but still some effort must be made to lessen NI's workload wherever possible.

4.2 An Example

Lets say we want to XOR together 256 variables, and calculate the corresponding truth table. In NI, this is as simple as typing the expression (or having another program generate it for you). Because of the properties of XOR, it doesn't matter logically what order the variables are placed within the expression, but just for the sake of experiment, lets benchmark NI on the same expression in two different orders.

4.2.1 256 XORs: Order 1

First, we'll try XORing the variables together with the numbers of each general variable decreasing from g255 to g0 (the entire expression will not fit nicely on one page, so only a piece is shown below):

g255 ^ g255 ^ g254 ^ g253 ^ g252 ^ g251 ^ g250 ^ g249 ^ ...

Timing how long it takes NI to calculate the result shows that, for this particular problem, NI needs 0.250 seconds.

4.2.2 256 XORs: Order 2

Now, lets try reversing the order, such that the general variable numbers are increasing from g0 to g255:

g0 ^ g1 ^ g2 ^ g3 ^ g4 ^ g5 ^ g6 ^ g7 ^ ... 

Timimg this reveals that NI requires only 0.006 seconds to solve the exact same problem when the order is reversed.

4.2.3 Whats going on?

Because of how NI solves problems, having the general variables decrease in number from left to right forces NI to move further recursively into the ALTT than it would have to if the variables were increasing in number from left to right. For most expressions, this is true; therefore, in general, if you do not care what general variable is used to represent which variable of your function, try to keep the general variable numbers increasing from left to right. For example, use a | (b & c) rather than c | (b & a) or (a | b) & (a | c). To be more specific: try to keep complicated results small in length, and simple results or values (like general variables) larger in length. Doing so will reduce the amount of work that NI must perform to accomplish your task.

A. Syntax Reference

A.1 Comments

Comments begin with a # symbol and are effective until the end of the current line.

A.2 Operators

Operators take two arguments: the first argument is simply the previous result, the second is the value which follows the operator. The overall syntax is OPERATOR VALUE.

!&

NAND

|

OR (implemented through NAND)

&

AND (implemented through NAND)

^

XOR (faster than the NAND version)

^^

XOR (implemented through NAND)

=>

Store result in the following named variable.

undef

Delete (undefine) the named variable.

==

Compares two values. Result has 1's where equal, 0's where not equal. Equivalent to the inverse of an XOR.

!=

Compares two values. Result has 0's where equal, 1's where not equal. Equivalent to XOR.

implies

Logical implication / if ... then

iff

if-and-only-if (implemented with non-NAND XOR)

iff2

iff (implemented through NAND)

A.3 Values

Values are the arguments of operators, but can exist on their own as the entire expression.

(EXPRESSION)

A parenthesized expression.

!VALUE

The inverse of VALUE. Something like !!a is legal but useless.

gN

General variable. Replace N with an integer >= 0. General variables are taken to be unknown and independant from each other.

a

Any lowercase letter is an alias for g0 through g25 (where a = g0, b = g1, c = g2, etc.)

$VARNAME

Named variable.

0

Binary 0 / Contradiction

1

Binary 1 / Tautology

?

An unknown value which may be 0 or 1.

$ANS

The last result (ANSwer from the previous line).

$MASK

The mask created by direct assignment or through the assume command.

A.4 Commands

Commands are like operators but they do not operate on the last result.

info

Print statistical information.

verify

Takes one parameter. If the parameter is universally true or true because of an assume statement, prints "True". Otherwise prints "False". Returns nothing.

assume

Takes one parameter. The parameter is used as a hypothetical statement, and all calculations and verifications thereafter are restricted by the parameter. Returns nothing.

reset

Removes all previous assumptions. Equivalent to undef $MASK.

simplify

Returns a simplified version of the previous result based on $MASK.

A.5 Printing Results

NI provides several ways of printing the last result; each prints the result in a different format.

=

Print the last result as an ALTT(1).

tt=

Print the last result as a truth table

ltt=

Print the last result as an LTT

id=

Print a unique number representing the result (a memory address, actually). Equivalent results will have equivalent ids.

f=

Print a function which has a result that is logically equivalent to the previous result.

B. Invoking NI

B.1 Syntax

ni [OPTIONS] [FILE [PARAMETERS(2)]]

B.2 Options

-v

Increases the verbosity level by one. Repeating this option increases the verbosty even further.

-sN

Set the simplification to N, where N is an integer from 0 to 2. This only affects the simplify and f= commands. The current default is -s1. -s2 is experimental and fails for some expressions.

-h, --help

Display information on usage.

--print-large

By default, NI will refuse to print truth tables for functions involving more than 26 variables, even though the calculations have been performed, returning an error referring to the size of the result. This option overrides that error, and forces NI to print such truth tables anyway. Use with caution. description

C. Errors

If you happen to get an error, it is usually related to syntax or spelling. Double check the syntax reference (included in the appendices). If you believe you have found a bug, see the section entitled Reporting Bugs.

ERROR: Variable name expected.

You typed an operator which takes a named variable as an argument (like $my_variable), but did not provide the named variable.

ERROR: Value expected.

You typed an operator, but did not provide the required argument. A misspelled command or operator will also cause this error.

ERROR: Variable undefined.

You typed a named variable, but that variable hasn't been defined yet.

ERROR: Physical result of operation is too large.

You attempted to print the last result as a truth table or linear truth table, but the last result is too large to be printed this way. Try the info command to see how large the result is, or restart NI with the --print-large command line option to override this error (if necessary).

ERROR: Could not open input file: FILENAME

The file provided on the command line does not exist; check the spelling, check the current working directory, etc.

ERROR: [undefined error number... this shouldn't be happening]

If you ever get this error, report it immediately! You've found a bug!

C.1 Reporting Bugs

If you believe you have found a bug, send a descriptive e-mail (and a copy of the script that reproduces the bug, if possible) to not-intractable@nongnu.org.

D. Copying This Manual

D.1 GNU Free Documentation License

Version 1.2, November 2002

Copyright © 2000,2001,2002 Free Software Foundation, Inc.
51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA

Everyone is permitted to copy and distribute verbatim copies
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    The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.

    Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

D.1.1 ADDENDUM: How to use this License for your documents

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

  Copyright (C)  year  your name.
  Permission is granted to copy, distribute and/or modify this document
  under the terms of the GNU Free Documentation License, Version 1.2
  or any later version published by the Free Software Foundation;
  with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
  Texts.  A copy of the license is included in the section entitled ``GNU
  Free Documentation License''.

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this:

    with the Invariant Sections being list their titles, with
    the Front-Cover Texts being list, and with the Back-Cover Texts
    being list.

If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

Index

Command and Operator Index

Footnotes

(1)

Results which involve many variables may only be able to be printed as ALTTs because of sheer size

(2)

If a file is specified, all remaining arguments on the command line will be passed to the file itself as the variables $1, $2, $3, ... etc.

Table of Contents

About This Document

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