To see the mathematical expressions in this document, begin by carrying out the instructions in Section 2.

Double-starred items (**) can be ignored by the general user.

There are computer algebra systems (Macsyma, Maple, Reduce, Scratchpad, SMP), and there are high quality document formatting systems that support mathematical expressions (Eqn-Troff, TEX, INFOR, VORTEX). There is still no adequate support for the working scientist who may have databases of math to browse, and who needs "living" notebooks and technical papers. Such a worker typically wants to examine a document or database, edit or algebraically manipulate expressions therein, and save new expressions. Not only should editing and algebra serve databases and documents, but conversely: one would like high-quality logs of algebra sessions.

CaminoReal is a prototype system for integration of documents, editing, and algebra. It supports interactive, syntax-directed, two-dimensional, WYSIWYG editing of mathematical expressions, placing/fetching such expressions in/from Tioga documents, and algebraic manipulation of expressions. The algebraic manipulation is based on an object-oriented paradigm that supports generic procedures (the AlgebraStructures package). Thus CaminoReal also provides client interfaces for generic operations in structured domains (e.g. vectors, matrices, polynomials) over such ground domains as Mesa INTs, Mesa REALs, Complexes, and BigRats. For example, with either the user environment or the interfaces, one can easily read and do simple arithmetic on matrices of polynomials with complex number coefficients, or matrices of such matrices, etc.

The expression editor is essentially Carl Waldspurger's MEDDLE package, extended to dovetail with the algebraic manipulation facilities (MEDDLE remains available as a separate CedarChest package).

It is perhaps most natural to view the algebra system as resident in the editor, i.e. algebraic manipulation of expressions is just a kind of editing of expressions. The current algebra functionality is a small Cedar package. [In the future we hope to offer access to other algebra systems running in either Unix or Lisp. This may be done either by extending the current user interface paradigm to incorporate functionality provided by the foreign system, or by providing new user interfaces for algebraic manipulation that allow the (necessarily knowledgeable) user to work with the foreign system in some facsimile of its own user interface paradigm] .

The new user should get some acquaintance with the terminology of Section 3. Sections 4-8 deal with editing of expressions, and moving expressions back and forth between CaminoReal and Tioga documents. Sections 9-12 deal with doing algebra. Section 14 pertains to some rather esoteric data structures pertaining to algebraic curves and surfaces. Section 15 describes how a client would use the algebra interfaces.

Bringover -p /Cyan/Imaging/Top/CaminoReal.df. Type "CaminoReal" to a command tool. You will get a viewer named "CaminoRealItem" opened in the left column [and in the future another iconified viewer named "Cad2DViewer"].

A Item is just another name for a CaminoReal viewer tool. An Item has a name, displayed in the viewer banner. You can change an items name by typing a new one in the ScratchPad and clicking the SetName menu button. You can get a new Item at any time by bugging the "NewItem" button in an existing Item, or typing "Camino" to a command tool.

An Expression can be displayed and edited, but is not capable of having algebraic manipulations performed on it. Every Expression is attached to some Item; every Item has one and only one (possibly null) Current Expression, which, at any moment, is what you see.

Strictly speaking, a Domain is just a set. Typically the Domains of interest are also algebraic structures, e.g. the integers, 2 x 2 matrices of complex numbers. Associated with any Domain is a certain collection of operations, such as conversion of Domain elements from Ropes or elements of other Domains, and arithmetic.

Abstractly, an Object is an element of a Domain. CaminoReal Objects possess Display Expressions, which we look at to see what Object we've got. Given an arbitrary Expression and an arbitrary Domain, the Expression may or may not be interpretable as an element of that Domain (obviously a Display Expression can always be interpreted as an element of its Object's Domain). In other words, from a given (Expression, Domain) pair we may or may not be able to construct an Object.

Every Object is attached to some Item, and every Item has one and only one (possibly empty) Current Object. The Domain to which an Item's Object belongs is the Item's Object Domain. Items also have a Working Domain, which can be edited at will. A common action is to set an Item's Current Object by interpreting its Current Expression as an element of its Working Domain; this is done by bugging SetObject with the Current Expression being the primary selection ("primary selection" is defined below).

When an Item has a nontrivial Object, it will usually be the case that the Item's Current Expression is the Object's Display Expression, but this is not necessarily so. We may edit the Current Expression, and the changes have no affect on the Current Object until we explicitly request that a new Current Object be established (e.g. by bugging SetObject with the appropriate primary selection (see below) in effect).

PlaceHolder:

Replace:

Wrap:

In terms of the definitions given above, we can state the basic fact of life for users doing algebraic manipulation: one Object, one Viewer. Every time an operation (e.g. addition of two Objects) that creates a new Object is performed, a new Item is created to which that Object is attached. Thus during a typical algebra session, Viewers accumulate quickly. A better way, e.g. logging new Objects in a continuously growing Tioga document, will come.

Before doing any algebra, we first have to have Domains to do it in, which is the subject of the next section.

Bug the

button in the control panel. You'll get a pop-up menu of choices. Some of these are Ground Domains, and others are Domain Structuring Operations. The current Ground Domains likely to be of interest to the general user are:

You must initially select one of these; a short name for it (e.g. Z for Integers) will appear in the text viewer next to the Working Domain button.

These are rings of square matrices of specified size, whose entries are elements of the current Working Domain. When you select Matrices, you get a pop-up menu to select the size.

Creating an Object can be done in two ways: with the SetObject button described above, and by reading an Algebra-format Rope format rope for an object, as described in the next section.

Once you have a Current Object, bug the OperateObject button to get a list of available operations. The Current Object is the implicit first operand to any of these.

The reader may find it a useful exercise, for each of these examples, to create an Item, set its WorkingDomain appropriately, Tioga-select the Algebra-format Rope given below for a sample element of that Domain, bug the OperateWorkingDomain button, and choose the FromLinearRope operation. In each instance, you should get a new Current Object and corresponding Display Expression.

You can always see what CaminoReal thinks is an appropriate Algebra-format Rope for the Primary Selection by bugging the "ToASRope" button.

The RealAlgebraicNumbers constructor is applied to a domain of univariate polynomials over the Rationals, e.g. Q[x]. Having set the Working Domain of an Item to this Domain, a particular real algebraic number can be created from two Objects (which must be attached to other Items): an element of the appropriate domain of univariate polynomials (this should be an irreducible polynomial), and an isolating interval for a particular real root of that polynomial. For example, the polynomial might be:

and the isolating interval

The latter can be constructed as an element of the domain of Points of length 2 over the Rationals. Primary select the polynomial and secondary select the interval. Then in the Viewer whose Working Domain is Real Algebraic Numbers, bug OperateWorkingDomain, and you will see the algebraic number displayed. For the above example, the result would be:

The ExtensionField constructor can only be applied in an Item whose Current Object is a Real Algebraic Number, constructed as just described. Just select ExtensionField from the Working Domain menu, and the domain constructed is the extension of the Rationals by that Algebraic Number. Elements of this domain are now univariate polynomials in the appropriate variable reduced mod the minimal polynomial. Thus in the above example, if we create a Current Expression of

primary select it, and bug SetObject, after reduction mod the minimal polynomial of X, we will get:

[The FormulaOperators and Cads ground Domains, and the Formulas, RealAlgebraicNumbers, ExtensionField, SamplePoints, CoveringSets, and Cells structuring operations are used, for algebraic curves and surfaces. Cad2DViewer is used for displaying algebraic curves. Exercising this stuff involves a detour through the Vax currently; consult Arnon if interested.]

To seek out the interfaces you want, look at AlgebraStructures.df. The class record slots, which are the same for all Domains, are defined in AlgebraClasses.mesa.

Thanks to Rick Beach and Michael Plass for enlightenment on Tioga and other matters. Thanks to Ken Pier for the "point and stuff" part of the CaminoReal-Tioga interface.