1. Introduction to qepmax

Qepmax is a Qepcad B - Maxima interfacing module written in Maxima language and Common Lisp. As Qepcad B provides so called quantifier elimination in the real domain, qepmax adds this functionality to Maxima computer algebra system.

With using quantifier elimination (Q.E. in short), a variety of problems can be expressed and solved. Optimization problems with multi-variate polynomials with equality and inequality constraints can be exactly(!) and symbolically(!!) solved.

Qepmax requires Qepcad B, developed at and distributed from the United States Naval Academy. Qepmax users are suggested to visit Qepcad B homepage to become familier with Qepcad B, to obtain the distribution package, and to know how to install it.

Lincese terms of Qepmax can be found in each source files. Qepmax package is in GPL v2.

Qepmax package is written by Yasuaki Honda, then Prof. Dr. Reinhard Oldenburg of Goethe-Universität Frankfurt has contributed the improvements for allowing the use of logical operators and rational functions as well as better function interface.

Further, ehito-san has contributed the improvement of the treatment of rational functions, nested qe() call, and strict rational function treatment.

Hereafter is the quick introduction of:

• What is Quantifier Elimination?
• What does Qepcad B do?
• What is qepmax package and how to use it?

What is Quantifier Elimination

A large size of problems can be expressed as a first order predicate logic with the following properties:

• variables are in real number domain
• predicates are =, <, >, <=, >=, #(not equal)
• terms are polinomials or rational functions

A logical formula with some of its variables quntified is called a Tarski sentence. Example: a circle x^2+y^2=1 and a line y=x/2+1/3 crosses with each other. Tarski sentence for this is: E x, E y, x^2+y^2=1 and y=x/2+1/3 where E denotes the existential quantifier.

The quntifier elimination is a technique to obtain the equivarent condition to the given Tarski sentence. The resulted condition does not contain any quantified variables.

By using the quantifier elimination technique, we can obtain an equivarent condition: true meaning that, yes, there is a point (x,y) which satisfies both circle and line formulas.

On the other hand, from E x, E y, x^2+y^2=1 and y=x/2+5 we will obtain false since there is no such intersection point (x,y).

Further more, considering the following Tarski sentence with free variable a: E x, E y, x^2+y^2=1 and y=x/2+a

Using the quantifier elimination, we obtain a condition on a: 4*a^2-5<=0

So, while a is in this range, the circle and the line cross, but they don't cross if a is not in the range above.

What does Qepcad B do?

Qepcad B is a software that implements a quantifier elimination algorithm called Cylindrical Algebraic Decomposition (CAD). The software is developed by Prof. Chris Brown of US Naval Academy. The homepage of the software is: http://www.usna.edu/CS/~qepcad/B/QEPCAD.html

On starting the program Qepcad B, the interactive session starts. You give a list of variables, number of free variables, a list of quntified variables, and a Tarski sentence with Qepcad B syntax during the session. Alternatively, you can supply all such information in a file and Qepcad B processes it. Then the program computes the equivarent condition to the specified Tarski sentence, only including the free variables.

It is a fun to use Qepcad B directly from its command line interface, as it provides many commands and options to use during the interactive sessions.

What is qepmax package and how to use it?

Qepmax is a Maxima package that calls Qepcad B internally using system() function to process Tarski sentence represented using Maxima syntax.

Assuming it is correctly installed according to INSTALL, you can load the package by:

(%i1) load("qepmax.mac");

Qepmax package provides a function qe(QFList, aLogicalExp). QFList is a list of quantified variables. aLogicalExp is an expression in the real domain. You can use =,<,>,<=,>=,# as the predicates, polynomials and rational functions as terms, and %and, %or, %not, %implies as the logical operators. In polimomials and rational functions, you can use rational numbers and variables as coefficients.

The function qe() generates a temporary file that can be used for the input of Qepcad B. Then Qepcad B is invoked with the file contents supplied from standard input. Qepcad generates an output file in which quantifier elimination result is written.

Then qe() read the output file and translate the result into Maxima formula and return it from qe(). You can of course use it for the futher computation using Maxima.

@opencatbox @category{Symbolic methods} @category{Optimization} @category{Share packages} @category{Package qepmax} @closecatbox

2. Functions and Variables for qepmax

Function: qe (QFVList, LogicalExp)

Compute a formula equivarent to LogicalExp with quantified variables specified in QFVList. The computed formula does NOT contain any quantified variables in QFVList.

Before giving preciese syntax definition, here goes the examples appeared in the Quantifier Elimination explanation.

(%i1) qe([[E,x],[E,y]], x^2+y^2=1 %and y=x/2+1/3);
(%o1) true
(%i2) qe([[E,x],[E,y]],x^2+y^2=1 %and y=x/2+5);
(%o2) false
[ qe1 ]

QFVList is a list of pairs (two element list) of a quantifier and a variable. A quantifier is one of A,E,F,G,C,and Xk, where k is a positive integer. Variables are ordinal Maxima variables, however, indexed variables are not supported. Quantifier A is the universal quantifier (for all). E is the existential quantifier (there exists). F,G,C are described in Qepcad B homepage. Xk is similar to E, but the number of existence is exactly k.

An example of QFVList is [[A,x],[A,y]], meaning "For all x and for all y in R". Another example is [[E,x],[A,y],[A,z]], meaning "There exists x in R such that for all y and for all z in R". Variables appeared in QVList are called quantified variables.

LogicalExp is a boolean formula in which you can use %and, %or, %implies, and %not as boolean operators. Predicates in atomic formulas appeared in the LogicalExp are restricted to <, <=, >, >=, =, #. Terms in the atomic formulas are restricted to multivariate polynomials or rational functions with coefficients being rational numbers. Atomic formula may also be a maxima function which returns a boolean formula.

Qepmax supports the Extended Tarski Formula, the atomic formula of the form var op etf_root(k, polynomial). Detailed semantics of this atomic formula is difficult explain here. Instead you should refer to the following URL http://www.usna.edu/CS/~qepcad/B/user/ETF.html.

Here I give you a very typical example and its meaning. In the Q.E., you may want to express some conditions on one of roots of an equation. Extended Tarski Formula can be used for this purpose. For an example, you can specify that x is a second smallest root of x^2=2 by the atomic formula x=etf_root(2,x^2-2). This is a shortcut of expressing that x is sqrt(2).

Any quantified variables specified in the QVList can be used in the LogicalExp. Variables that are not specified in the QVList can also be used. They are called free variables.

Couple of examples of LogicalExp are:

x^2>=0
y=x/2+m %and y<x+1 %and y<-x+1

qe returns a boolean formula which only contains free variables in the LogicalExp, still the returned formula is equivalent to LogicalExp quantified by QVList.

Option variable: qepcad_installed_dir

Default value: "~/qepmax/qesource"

By extracting Qepcad B source tarball, all the files are placed in the directory qesource. This variable needs to be set up to point that directory.

Option variable: qepcad_input_file

Default value: sconcat(maxima_tempdir, "/qepcad_input.txt")

Qepmax creates temporary files during its execution. One is the input file to the qepcad command. This variable can be used to specify the file path and the file name of the input file.

The default value is the concatenation of maxima_tempdir and "/qepcad_input.txt". Note that if you change the value of this variable, you should also consider to change the value of the variable qepcad_file_pattern.

Option variable: qepcad_input_file

Default value: sconcat(maxima_tempdir, "/qepcad_output.txt")

Qepmax creates temporary files during its execution. One is the output file to the qepcad command. This variable can be used to specify the file path and the file name of the output file.

The default value is the concatenation of maxima_tempdir and "/qepcad_output.txt". Note that if you change the value of this variable, you should also consider to change the value of the variable qepcad_file_pattern.

Option variable: qepcad_file_pattern

Default value: sconcat(maxima_tempdir,"/qepcad_*.txt")

Qepmax creates temporary files during its execution. Prior to the creation, if exists, qepmax deletes such temporary files using rm command using system() function. This variable is used to specify the files to be deleted.

The default value is the concatenation of maxima_tempdir and "/qepcad_*.txt", which are good for default file names of the input and output files. If you change the value of qepcad_input_file and / or qepcad_output_file, you should also consider to change the value of this variable.

Option variable: qepcad_option

Default value: " +N20000000 +L10000 "

Qepcad B accepts several options. This variable is used to specify such options to be passed.

The default value is " +N20000000 +L10000 ", which allocates more memory to the computation library used in qepcad.

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