IAsolver 0.1beta1: 10/97

The Brandeis Interval Arithmetic Constraint Solver (10/97)

PUSH THE "Show IAsolver" BUTTON BELOW TO START THE IASOLVER APPLET

You need a java-enabled web brower to run the applet.

Featured Applet for the week of 15-22 August 1997 in the Gamelan Java Archive

Source Code is now Available

The source code for IAsolver is available for browsing in this directory, or you can download this gzipped tar file (414K) directly. This source code is being released as three libraries: ia_math, ia_parser, and ia_solver. All of these are currently being distributed using the GNU LGPL copyleft for libraries.

Portability

This applet has reportedly run successfully on the following configurations:

Silicon Graphics workstations
- Netscape Navigator 3.01 on an SGI Indy running IRIX 5.3
- Netscape Navigator 4.02 on an SGI Indy running IRIX 5.3
Macintosh
- Netscape Navigator 3.01 a Mac Performa 637 running MacOS 7.5
- Netscape Communicator Pro 4.01 on a Power Mac 7300/200 Mac OS 7.5.5
- Netscape 3.02 on a Power Mac running Mac OS 7.6.1
- Netscape 4.04 on a Power Mac running MkLinux
PC's
- Netscape 4.0 on a PC running Windows NT 4.0
- Netscape 4.04 on a Gateway2000 running (Red Hat) Linux 2.0.30

There have been some reports of problems under some Microsoft and Linux operating systems.

Netscape 4.0beta on x86 running Linux 2.0.31 -- It seems always to resolve variables to [-1, 0], and narrowing fails.

If you do have a problem, I would greatly appreciate it if you could send me email at tim@cs.brandeis.edu describing the machine, browser, operating system and problem, and I'll post it here and try to work on it. Thanks.

Please be patient. It may take a few minutes to download this applet over the net. You can read the instructions and examples below while waiting for it to load!

Instructions

When the applet has been fully loaded onto your machine, a button labelled Show IAsolver will appear at the top left of this page. Pushing this button will open a new window on which you can enter and solve systems of arithmetic constraints using narrowing. If there are two or more variables in the constraint set, then you can use the Interval Arithmetic plotting feature to simultaneously view one or more projections of the solution set.

The main window of the solver contains an area for entering constraints, an area for viewing the current bounds on the variables, and several control buttons, as shown below:

NOTE: THE FOLLOWING IS JUST A GIF IMAGE. THE REAL APPLET IS AT THE TOP LEFT OF THIS PAGE.

Narrow This button starts a simple narrowing iteration loop in which each constraint is used to narrow the intervals of the variables it contains. This processes is repeated a fixed number of times (initially 10), and you can change this parameter using the text field at the bottom of the window. The narrowed intervals for the variables appearing in the constraints are shown in the textarea labelled "variables" and you can watch the intervals shrinking while the narrowing is going on. In principle the narrowing operation has the property that it shrinks the intervals associated to each variable without removing any of the solution set. In practice, the current applet relies on the underlying math library which is not guaranteed to be accurate to the last bit, so this ideal property is currently not guaranteed to hold.
Absolve This button applies another form of narrowing which attempts to trim the upper and lower bounds of each variable using a proof by contradiction approach. More precisely, it trims an upper bound of a variable by first binding that variable to the region it wishes to trim away. If narrowing fails, then the region is removed; otherwise, the region remains. The current system first tries to trim away 1/2 of the interval, then it trims away 1/16.
Clear Variables This button sets the bounds of all variables to [-infty,infty], but has no effect on the set of constraints.
Clear Constraints This clears the constraints window, but leaves the variable bindings alone.
Hide This hides the window and stops the applet.
Stop This stops the current narrowing operation.
Plot This button, draws a 2d plot of the projection of the constraint onto two (user-specified) variables. You can view several projections simultaneously. Also, you can use the mouse to zoom in on any interesting section of a projection. This has the effect of adding additional constraints to the constraint set (limiting the projected variables), and hence all other projections will reflect the new "pruned" view. You can also "unzoom". This feature allows one to examine several discrete solutions individually.
- depth This choice selector allows you to control the level of detail which the plotter presents. A level of k means that the x and y axes are divided into 2^k segments and each of the 4^k boxes is then narrowed. If narrowing fails the box is painted white, otherwise it is painted blue.
- Zoom/Unzoom This allows you to use the mouse to zoom in on any segment of the plot. It has the effect of temporarily adding new constraints to the constraint system and this will therefore affect all other plotting windows that you have opened. Unzooming reverts to the previous view as usual.
- Size to Fit This increases the magnification so that the solution set fills the window and then it narrows to depth 3. By repeatedly pressing this button you can often effectively narrow in on an isolated solution point.
- max_narrows This is normally set to 1. It is the number of times that he constraintns are narrowed for each small subbox that is being plotted.
- Shade boundary When this is selected it draws a 10 pixel red boundary around the blue solution set. This is useful when the solution consists of 1x1 blue pixels which can easily be overlooked (especially if your screen is dirty!)

Examples of allowable constraints

The constraints must each end with a semicolon and are constructed from variable names and simple decimal numbers using the standard operators listed below. Some examples of valid constraints are:

```
  sin(x*cos(y))=cos(y*sin(x)); x= [-6.3,6.3]; y = [0,10];
```
This constraint has a complex (and beautiful) solution set which you can view using the PLOT function.
The region which is displayed properly contains the solution set. It was plotted with the DEPTH parameter set to 8. By selecting a greater DEPTH parameter for the plot you can view more detail.

```
    y^2 = x*(x^2-1)+z^2; x*y*z=1;
    x = [-2,2];     z = [-2,2];                                            
```
The solution set of this constraint is a 1 dimensional curve in 3-space. Plotting its (x,y) projection we get the following curve:
which has four components. If we add the additional constraint sin(x*cos(y))=cos(y*sin(x)), the system has four solutions which can be isolated using the Plot, Zoom and UnZoom facilities.

   x^5 + 3*x^2 +x+50 = 7*exp(x^2) ; x = [-100,0];

has one solution in the following box:

    x = [-1.3960403710880638,   -1.396040371088062]

x**x = 5; x = [1,4] ;

This has one solution in the following interval:

    x = [2.129372482760152,   2.1293724827601612]

x^2 + y^2 = 25; (x-5)^4 + y^4 = 25; y > 0;

This has one solution in the following box:

    x = [4.472998922610509,    4.47299892261052]
    y = [2.2343412090200485,   2.2343412090200516]

```
      x = [0.88,0.89]; 
      v := MP(x); 
      y := v - (v^4 - 12*v^3 + 47*v^2 - 60*v + 24)/
              (4*x^3 - 36*x^2 + 94*x - 60);
      y=x;
```
This example shows how the interval newton method can be implemented using IAsolver 0.0. The "y:=E" operator evaluates its right hand side E using interval arithmetic and assigns that value to the variable y on the left. The MP(x) function returns the midpoint of the interval x, when evaluated using interval arithmetic, but it is a nop when narrowed. We could also have used the left endpoint (LP(x)) or the right endpoint (RP(x)), or some point midway (0.25*LP(x)+0.75*RP(x)). This constraint yields the following solution:
```
    x = [0.8883057790717404,   0.8883057790717662]
    v = [0.8883057790717532,   0.8883057790717532]
    y = [0.8883057790717404,   0.8883057790717662]
```

SYNTAX

A Constraint is one of the following relations, where A,B are expressions: A=B, A<B, A>B, A!=B, A<=B, A>=B, V:=E The last is the assignment operator. It evaluates E using interval arithmetic, and binds the resulting interval to the variable V. The allowable expressions are defined below:

An Expression E is formed from variables, numbers, operators, and subexpressions A and B as follows:
- Decimal numbers (e.g. 3.141592 or -2.718281828459045
- infty and -infty: the IEEE 754 infinite numbers.
- variables A variable is a string of letters and/or digits, starting with a letter.
- [A,B] the interval notation. This is equivalent to a variable X with the additional constraints that A<=X and X <= B. Here A and B must be numbers, they cannot be variables or compound expressions.
- A+B, A-B, A*B, A/B the arithmetic operators.
- A^n the integer power operator, where n must evaluate to an integer.
- A**B the real power operator, where A must be positive.
- exp(A), log(A) the exponential function and the logarithm.
- MP(x), LP(x), RP(x) The midpoint function MP(x) returns the midpoint of the interval x when evaluated using interval arithmetic (i.e. in the RHS of an assignment statement v := E), but which is the identity function when narrowed (i.e. when it occurs in an equation F=E). The left and right endpoint functions (LP(x), RP(x)) are also available and behave similarly.
- sin(A), cos(A), tan(A), asin(A), acos(A), atan(A) The trigonometric functions and their inverses.

Commercial Interval Arithmetic Constraint Systems

If you need a more powerful interval arithmetic constraint solver you may be interested in the ALS Prolog implementation of CLP(BNR) which combines a powerful (and efficient) interval arithmetic constraint solver with a high quality Prolog compiler and it runs on almost every platform.

Cautions and Provisos

This software is experimental and you should use at your own risk. In particular, the current version is NOT sound since it relies on Java's underlying math library which is not proven to be accurate to the last bit. Please send any bug reports, suggestions, or comments to tim@cs.brandeis.edu The web page http://tigereye.cs.brandeis.edu/Applets/IAsolver2.html is a stripped down version of this page which has only the applet and a few comments.

The development of this applet was partially supported by NSF grant CCR-9403427.

IAsolver

Copyright (C) 1997 Timothy J. Hickey This library is free software; you can redistribute it and/or modify it under the terms of the GNU Library General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Library General Public License for more details.