DIRECTORY
SafeStorage,
IO,
Atom,
Rope,
Basics,
SymTab,
Imager,
MathExpr;

MathObjects: CEDAR DEFINITIONS
 
= BEGIN

ROPE: TYPE = Rope.ROPE;
STREAM: TYPE = IO.STREAM;
Object: TYPE = REF ObjectRep;

ObjectRep: TYPE;  -- internal concrete rep

Method: TYPE = REF MethodRep;

MethodRep: TYPE;

MethodDictionary: TYPE = REF MethodDictionaryRep;
MethodDictionaryRep: TYPE;

SystemObjects: Object;

SystemObjectsData: TYPE ~ REF SystemObjectsDataRep; --  probably could be like a StructureDataRep;  includes a MethodDictionary (including an $eltFlavor method for SystemObject's), and a LIST of all current SystemObject's
SystemObjectsDataRep: TYPE;

EnumerateSystemObjects: PROC RETURNS [LIST OF Object]; -- maybe unneeded

SystemObjectsToRope: ToRopeOp;
ApplyMethod: PROC [method: Method, argList: LIST OF REF] RETURNS[value: REF];
StaticTypes: Object; -- SystemObject

StaticTypesData: TYPE ~ REF StaticTypesDataRep;
StaticTypesDataRep: TYPE; -- probably a RefTab of method static type names and associated applicator procs; user augmentable

MethodType: TYPE = {Value, SideEffect, TrueNullaryOp, NullaryOp, UnaryOp, BinaryOp, BinaryMixedOp, TernaryOp, TernaryMixedOp, QuaternaryOp, LegalFirstCharOp, ReadOp, FromRopeOp, ToRopeOp, ToExprOp, FromExprOp, FromBOOLOp, FromINTOp, UnaryPredicate, BinaryPredicate, CompareToZeroOp, BinaryCompareOp, StructuredToGroundOp, ElementRankOp, UnaryImbedOp, BinaryImbedOp, ListImbedOp, MatrixImbedOp, StructureFromSetConstructor, VectorStructureConstructor, SequenceStructureConstructor, MatrixStructureConstructor, PolynomialStructureConstructor};
SideEffect: TYPE = PROC [] RETURNS [];

TrueNullaryOp: TYPE = PROC [] RETURNS [result: Object];

UnaryOp: TYPE = PROC [arg: Object] RETURNS [result: Object];

UnaryToRefOp: TYPE = PROC [arg: Object] RETURNS [result: REF];

UnaryPredicate: TYPE = PROC [arg: Object] RETURNS [BOOL];
UnaryInPlaceOp: TYPE = PROC [arg: Object];

UnaryToListOp: TYPE = PROC [arg: Object] RETURNS [result: LIST OF Object];

BinaryOp: TYPE = PROC [firstArg, secondArg: Object] RETURNS [result: Object];

BinaryPredicate: TYPE = PROC [firstArg, secondArg: Object] RETURNS [BOOL];

EqualityOp: TYPE = PROC [firstArg, secondArg: Object] RETURNS [BOOL]; -- redundant

BinaryInPlaceOp: TYPE = PROC [firstArg: Object, secondArg: REF];

BinaryMixedOp: TYPE = PROC [firstArg: Object, secondArg: REF] RETURNS [result: Object];

BinaryToPairOp: TYPE = PROC [firstArg: Object, secondArg: REF] RETURNS [firstResult, secondResult: Object];

TernaryOp: TYPE = PROC [firstArg, secondArg, thirdArg: Object] RETURNS [result: Object];

TernaryMixedOp: TYPE = PROC [firstArg, secondArg: Object, thirdArg: REF] RETURNS [result: Object];

QuaternaryOp: TYPE = PROC [firstArg, secondArg, thirdArg, fourthArg: Object] RETURNS [result: Object];
MakeMethod: PROC [type: MethodType, operator: BOOL, value: REF, desiredArgStructures: REF UnaryToListOp, doc: ROPE] RETURNS[Method];

DesiredArgStructures: PROC [methodSelector: ATOM, structure: Object] RETURNS[LIST OF Object];

DefaultDesiredArgStructures: UnaryToListOp;
GetMethodAndRecastArgs: PROC [methodSelector: ATOM, structure: Object, inArgs: LIST OF Object] RETURNS [ok: BOOL, method: Method, outArgs: LIST OF Object _ NIL];
RecastArgs: PROC [method: Method, structure: Object, inArgs: LIST OF Object] RETURNS [ok: BOOL, outArgs: LIST OF Object _ NIL];

ApplyLegalFirstCharMethod: PROC [method: Method, char: CHAR, structure: Object _ NIL] RETURNS[BOOL];

ApplyFromRopeMethod: PROC [method: Method, in: ROPE, structure: Object _ NIL] RETURNS[Object];

ApplyReadMethod: PROC [method: Method, in: STREAM, structure: Object _ NIL] RETURNS[Object];

ApplyFromExprMethod: PROC [method: Method, in: EXPR, structure: Object] RETURNS[Object];

ApplyCompareToZeroMethod: PROC [method: Method, arg: Object] RETURNS[Basics.Comparison];

ApplyBinaryCompareMethod: PROC [method: Method, firstArg, secondArg: Object] RETURNS[Basics.Comparison];

ApplyBinaryImbedMethod: PROC [method: Method, data1: Object, data2: REF, structure: Object] RETURNS[Object];

ApplyMixedMethod: PROC [method: Method, objectArgs: LIST OF Object, refArg: REF] RETURNS[Object];

ApplyPredNoLkpNoRecast: PROC [method: Method, argList: LIST OF Object] RETURNS[BOOL];

ApplyPredNoLkpRecast: PROC [method: Method, structure: Object, argList: LIST OF Object] RETURNS[BOOL];

ApplyPredLkpNoRecast: PROC [methodSelector: ATOM, structure: Object, argList: LIST OF Object] RETURNS[BOOL];

ApplyPredLkpRecast: PROC [methodSelector: ATOM, structure: Object, argList: LIST OF Object] RETURNS[BOOL];

ApplyNoLkpNoRecastRef: PROC [method: Method, argList: LIST OF Object] RETURNS[value: REF];

ApplyNoLkpRecastRef: PROC [method: Method, structure: Object, argList: LIST OF Object] RETURNS[value: REF];

ApplyLkpNoRecastRef: PROC [methodSelector: ATOM, structure: Object, argList: LIST OF Object] RETURNS[value: REF];

ApplyLkpRecastRef: PROC [methodSelector: ATOM, structure: Object, argList: LIST OF Object] RETURNS[value: REF];

ApplyNoLkpNoRecastObject: PROC [method: Method, argList: LIST OF Object] RETURNS[value: Object];

ApplyNoLkpRecastObject: PROC [method: Method, structure: Object, argList: LIST OF Object] RETURNS[value: Object];

ApplyLkpNoRecastObject: PROC [methodSelector: ATOM, structure: Object, argList: LIST OF Object] RETURNS[value: Object];
ApplyLkpRecastObject: PROC [methodSelector: ATOM, structure: Object, argList: LIST OF Object] RETURNS[value: Object];

ApplyNoLkpNoRecastExpr: PROC [method: Method, argList: LIST OF Object] RETURNS[value: EXPR];
ApplyNoLkpRecastExpr: PROC [method: Method, structure: Object, argList: LIST OF Object] RETURNS[value: EXPR];

ApplyLkpRecastExpr: PROC [methodSelector: ATOM, structure: Object, argList: LIST OF Object] RETURNS[value: EXPR];
ApplyLkpNoRecastExpr: PROC [methodSelector: ATOM, structure: Object, argList: LIST OF Object] RETURNS[value: EXPR];

MakeClass: PROC [name: Rope.ROPE, superClass: Object, methodDictionary: MethodDictionary _ NIL] RETURNS[class: Object];

AddMethodToClass: PROC [methodSelector: ATOM, method: Method, class: Object];

SetSuperClass: PROC [object: Object, superClass: Object];
LookupMethodInClass: PROC [methodSelector: ATOM, class: Object] RETURNS[method: Method];
BuildClassOperators: PROC [class: Object] RETURNS[opNames: LIST OF ROPE, operators: LIST OF Method];
Category: TYPE ~  {set, lattice, group, ring, field, module, vectorSpace, algebra, divisionAlgebra};
PrintNameProc: TYPE = PROC [structure: Object] RETURNS [Rope.ROPE]; -- redundant with ToRopeOp

StructureToExprOp: TYPE = PROC [structure: Object] RETURNS [out: EXPR];

StructureEqualityTest: TYPE = PROC [thisStructure, otherStructure: Object] RETURNS [BOOL];

StructureRankOp: TYPE = PROC [structure: Object] RETURNS [CARDINAL];
ReportOpsProc: TYPE = PROC [structure: Object] RETURNS [opNames: LIST OF Rope.ROPE, refOps: LIST OF REF];
BinaryStructureLUBOp: TYPE = PROC [firstStructure, secondStructure: Object] RETURNS [LUBStructure: Object];
StructureFromSetConstructor: TYPE = PROC [set: Object] RETURNS [structure: Object];

VectorStructureConstructor: TYPE = PROC [coordinateStructure: Object, dimension: NAT, row: BOOL _ TRUE] RETURNS [vectorStructure: Object];

SequenceStructureConstructor: TYPE = PROC [elementStructure: Object, row: BOOL _ TRUE] RETURNS [sequenceStructure: Object];

MatrixStructureConstructor: TYPE = PROC [elementStructure: Object, nRows, nCols: NAT] RETURNS [matrixStructure: Object];

PolynomialStructureConstructor: TYPE = PROC [coeffRing, variableSeq: Object] RETURNS [polynomialStructure: Object];
IsCategory: PROC [structure: Object, category: Category] RETURNS [BOOL];

HasProperty: PROC [structure: Object, property: ATOM] RETURNS [BOOL];

MakeStructure: PROC [name: Rope.ROPE, class: Object, instanceData: REF] RETURNS[structure: Object];

StructureEqual: PROC [structure1, structure2: Object] RETURNS [BOOL];
LookupMethodInStructure: PROC [methodSelector: ATOM, structure: Object] RETURNS[Method];
LookupMethodInAllStructures: PROC [methodSelector: ATOM] RETURNS[method: Method, structure: Object];
ResetStructureRegistry: PROC[];

InstallStructure: PROC[structure: Object];

LookupStructure: PROC[name: ROPE] RETURNS[structure: Object];
KillStructure: PROC[name: ROPE];

LegalFirstCharOp: TYPE = PROC [char: CHAR, structure: Object] RETURNS [BOOL];
ReadOp: TYPE = PROC [in: IO.STREAM, structure: Object _ NIL] RETURNS [out: Object];

FromRopeOp: TYPE = PROC [in: Rope.ROPE, structure: Object _ NIL] RETURNS [out: Object];

ToRopeOp: TYPE = PROC [in: Object] RETURNS [out: Rope.ROPE];

WriteOp: TYPE = PROC [stream: IO.STREAM, in: Object];

FromExprOp: TYPE = PROC [in: EXPR, structure: Object] RETURNS [out: Object];

ToExprOp: TYPE = PROC [in: Object] RETURNS [out: EXPR];

FromBOOLOp: TYPE = PROC [in: BOOL, structure: Object _ NIL] RETURNS [out: Object];

FromINTOp: TYPE = PROC [in: INT, structure: Object _ NIL] RETURNS [out: Object];

NullaryOp: TYPE = PROC[structure: Object] RETURNS [result: Object];

CompareToZeroOp: TYPE = PROC [arg: Object] RETURNS [Basics.Comparison];

BinaryCompareOp: TYPE = PROC [firstArg, secondArg: Object] RETURNS [Basics.Comparison];

StructuredToGroundOp: TYPE = PROC [structuredElt: Object] RETURNS [groundElement: Object];
ElementRankOp: TYPE = PROC [arg: Object] RETURNS [CARDINAL];
Display2DOp: TYPE = PROC [object: Object, context: Imager.Context, dotWidth, segmentWidth: REAL];
Copy: UnaryOp;

ElementOf: PROC [object: Object, structure: Object] RETURNS [BOOL];
UnaryImbedOp: TYPE = PROC [in: Object, structure: Object] RETURNS [out: Object];
BinaryImbedOp: TYPE = PROC [data1: Object, data2: REF, structure: Object] RETURNS [out: Object];
ListImbedOp: TYPE = PROC [data: LIST OF Object, structure: Object] RETURNS [out: Object];
MatrixImbedOp: TYPE = PROC [elements: LIST OF Object, structure: Object] RETURNS [out: Object];

END.

.v
MathObjects.mesa
Copyright c 1989 by Xerox Corporation.  All rights reserved.
Arnon, August 28, 1989 1:42:36 pm PDT

Types From Referenced Interfaces
Object Type Definitions
Conceptually, a MethodDictionary is a data structure containing [key, Method] pairs, on which some search algorithm is defined that, given a key, either returns a corresponding Method, or reports that none can be found.
It is not required that there be at most one [key, Method] pair for a given key in a MethodDictionary.  In fact, multiple [key, Method] pairs with the same key can be stored so as to be found in a particular order.  It is the search proc implementor's job to determine how to handle the possibility of multiple [key, Method] pairs with the same key.

We have in mind multiple possible reps, e.g. Atom.PropList, or more structured, non-flat, e.g. tree of methods, or a tree of Atom.PropList
Should MethodDictionary's, or some portion of the info in them, be "externalizable" , or Objects, so that a domain or category can report info about itself?
A method's methodKey, i.e. methodSelector, should serve as the method's "name" for any external purpose, e.g. user menu entry, or "logging" of a method application;  should be identical to what some parser expects to be able to construct a MethodApplication object.
Note that we need to plan NOW for universal names, so we don't get into $sum, $Sum, $add confusions.
System (Global) Objects
SystemObjects is the .class for all SystemObject's.  Its .data records all currently known SystemObject's.  Its .class is some special Object SystemObjectsClass, which we insist is never manipulated in any way, and whose .class is NIL) and its flavor is $SystemRoot.  Its usefulness is that it provides a single, comprehensive access point to system state (which indeed should be definable as the set of all SystemObject's), which can thus be externalized as the composition (e.g. list) of the externalizations of the individual SystemObject's.
SystemObjects' .methods provides methods for access to the constituents of its Elements (i.e. SystemObject's), e.g. search, a $flavor method (returns $SystemObject), and an $eltDisplay method.

Thus we mandate an architecture that imposes some limitation of generality on class relationships.  We assert that there is no useful value to be had by explicitly creating a class of all Objects.  Instead we assert that every (Math)Object has a certain flavor, which (currently) is one of:
$Element
$Domain
$View
$Category
$Environment
$Package
$SystemObject
$SystemRoot
For the first ($Element), we need know no more than what Structure it belongs to.  For the next five ($Domain, $View, $Category, $Environment, or $Package), any Object of that flavor is an element of some corresponding SystemObject, and that is the most we need to know.  Any $SystemObject Object is an element of SystemObjects, and that is all we need to know.  SystemObjects is the unique Object of flavor $SystemRoot.  The only other (kind of) Object that does, or ever can, exist in the system is SystemObjectsClass
Method Application
It is assumed that argList conforms to method's runtimeType (no check). 
Note that args and return value, if not Objects, must be at least REF's.  I.e. the ability of this method apply procedure to handle Cedar types is limited to REF's.  Thus all methods to be invoked by the general method apply machinery must meet this constraint.
Static Types
Need to declare aplicator procs in the right places (e.g. FromINTOp in Ints), for each of the following
Operations on Objects -  Static Types

Method Operations
Does method lookup
RETURN[ LIST[structure] ];  (arg is expected to be a Structure)

Lookup method in structure, and recast args

Try to recast args
LookupMethodForStructure, RETURN[NARROW[ApplyNoLkpNoRecastRef]].
Error if anything at all goes wrong.
Apply method from structure.  It is assumed that method exists, and that correct number and types of args are supplied.
LookupMethodForStructure, RETURN[NARROW[ApplyNoLkpNoRecastRef]].
Error if anything at all goes wrong.
ApplyNoLkpNoRecastRef[GetMethodAndRecastArgs];  It is assumed that method exists, that correct number of args are supplied, and that args are recastable to desiredArgStructure. Error if anything at all goes wrong.
ERROR if method output type is not imbeddable as an Object
LookupMethodForStructure, RETURN[NARROW[ApplyNoLkpNoRecastRef]].
Error if anything at all goes wrong.


ApplyNoLkpNoRecastObject[GetMethodAndRecastArgs];  It is assumed that method exists, that correct number of args are supplied, and that args are recastable to desiredArgStructure. Error if anything at all goes wrong.
RETURN[NARROW[ApplyNoLkpNoRecastRef]].
Error if anything at all goes wrong.

LookupMethodForStructure, RETURN[NARROW[ApplyNoLkpNoRecastRef]].
Error if anything at all goes wrong.

LookupMethodForStructure, RETURN[NARROW[ApplyNoLkpNoRecastRef]].
Error if anything at all goes wrong.
Class Operations
Typically a class is created with MethodDictionary = NIL, then MethodDictionary loaded with AddMethodToClass
Existing method, if any (generally there shouldn't be), is replaced
Sets class field of a Class, or class.class field of a Structure

Look up method in this class and its superclasses.

Structure Categories
Structure Operation Types
Identify the particular structure to the world.
Test whether some other structure is the same algebraic domain as this one
For things like ring characteristic (smallest positive integer k such that k*1 = 0;  0 if k infinite) and vector space dimension.

Report structure operations to outside world, e.g. a user interface
Should be made a concrete proc to be applied to method dictionaries

If either Structure NIL, return other
Structure Constructor Types
Make a Structure whose (finite) underlying set is the given argument.
A particular vector structure is defined by its coordinateStructure, which can be any Structure, its dimension, and whether its elements are displayed as rows or columns.
A particular sequence structure is defined by its elementStructure, which can be any Structure, and whether its elements are displayed as rows or columns.
A particular matrix structure is defined by its elementStructure and its nRows, nCols. elementStructure can be a ring, field, algebra, or divisionAlgebra.
A particular polynomial structure is defined by its coeffRing and its variables.
coeffRing can be a ring, field, algebra, or divisionAlgebra.
variableSeq is a sequence of any length, and the right thing happens.
Structure Operations
RETURN[structure1.name = structure2.name]

LookupMethodInClass[structure.class]

Linear search of StructureRegistry, LookupMethodForStructure until found or EOF.  If found, then structure is Structure where found.  Returns [NIL, NIL] if not found.
Structure Registry
Structures have nonNIL names. Mathematically different Structures have different names.

A central registry of Structures is kept, and we never have more than one instance of a given mathematical structure in existence. 

effects:  Installs structure in global StructureRegistry database. Replaces existing occurrence of a Structure with same name, if any, and otherwise places structure at the tail of StructureRegistry.   
effects:  Returns the Structure Object associated with name.
          returns NIL if not found

effects:  delete structure from global StructureRegistry DataBase, if present. Only needed when want to delete a given Structure name entirely, not if just want to replace.
StructureElement Operation Types
  True if char is legal first char in an external rep of a structure element.

Since no object arg, need a hook to structure
For things like determinant and coefficient extraction

For things like degree functions in euclidean domains, rank of matrices


Display an object in a 2D context.
StructureElement Operations
  True if object is an element of structure.  (check equality of its structure's name with this structure's name)

StructureElement Creation Types
Imbed in as an element of structure

Construct an element of structure from data1 and data2.
Sample use: constructing a term of a polynomial.

Construct an element of a point, sequence, or vector Structure from data

Construct an element of a matrix Structure from data.  Assumes that correct number of elements supplied, in correct order, and that each element supplied belongs to structure.elementStructure
*Old* Structure Class Record Defn
Category: TYPE = {set, lattice, group, ring, field, module, vectorSpace, algebra, divisionAlgebra};

ClassData: TYPE = REF ClassDataRec;
ClassDataRec: TYPE = RECORD [

Structure properties and operations
category: Category,

flavor: Rope.ROPE _ NIL,
Flavor identifies outermost constructor (if any) used to build objects of this structure.  Is a rope to allow expansion without interface changes. Examples:
"Ints"
"BigRats"
"Points"
"Polynomials"
"Matrices"
"Quotients"
"Structures"

printName: PrintNameProc,
longNameGenerator: PrintNameProc,
shortPrintName: PrintNameProc,  -- if none, should be set to printName
shortNameGenerator: PrintNameProc,  -- if none, should be set to printName
code: Rope.ROPE, -- for quick checks of structure equality
structureEqual: StructureEqualityTest, -- should use codes
characteristic: StructureRankOp _ NIL, -- rings, fields, modules, vector spaces, algebras, divisionAlgebras only
dimension: StructureRankOp _ NIL,-- modules, vector spaces, algebras, divisionAlgebras only
reportOps: ReportOpsProc _ NIL,


Test whether a given item belongs to this structure (run-time typechecking): check equality of its structure's code with this structure's code.
isElementOf: ElementOfProc,

I/O and Conversion of Structure elements
legalFirstChar: LegalFirstCharOp,
read: ReadOp,
fromRope: FromRopeOp,
toRope: ToRopeOp,
write: WriteOp,
fromExpr: FromExprOp _ NIL,
toExpr: ToExprOp _ NIL,
toIndexRope: ToRopeOp, -- print short object key (for use in sequences of objects)  
recast: UnaryImbedOp _ NIL, -- convert an element of a structure that conforms to this one (should be a noop on elements of this structure). Should use fastest available determination of structure, e.g. hashCode or shortPrintName.

Addition (lattices, rings, fields, vectorSpaces, algebras, divisionAlgebras only)
add: BinaryOp _ NIL,
negate: UnaryOp _ NIL,
subtract: BinaryOp _ NIL,
zero: NullaryOp _ NIL,

Multiplication (lattices, groups, rings, fields, algebras, divisionAlgebras only)
multiply: BinaryOp _ NIL, -- assumed associative; commutative or noncommutative ok
commutative: BOOL _ TRUE,
invert: UnaryOp _ NIL,
divide: BinaryOp _ NIL, -- (lattices only) set difference operation goes here if appropriate
one: NullaryOp,

Equality testing 
equal: EqualityOp _ NIL,  -- need not be present in sets and lattices

Module ops
rightModule: BOOL _ FALSE,  -- modules are left modules by default

Scalar multiplication (modules, vectorSpaces, algebras, divisionAlgebras only)
scalarMultiply: BinaryOp _ NIL, -- unless rightModule, this is left multiplication, i.e. scalar * vector. The underlying field of a vector space may be commutative or noncommutative.

Lattice ops
booleanAlgebra: BOOL _ FALSE,
complement: UnaryOp _ NIL,

Ordered structure ops (rings, fields, modules, vectorSpaces, algebras, divisionAlgebras only)
ordered: BOOL _ FALSE,
sign: CompareToZeroOp _ NIL,
abs: UnaryOp _ NIL,
compare: BinaryCompareOp _ NIL,

Common varieties of rings (and algebras) and their ops
integralDomain: BOOL _ FALSE,
gcdDomain: BOOL _ FALSE,
gcd: BinaryOp _ NIL, -- gcdDomains only
euclideanDomain: BOOL _ FALSE,
remainder: BinaryOp _ NIL, -- euclideanDomains only
degree: ElementRankOp _ NIL, -- euclideanDomains only

Common varieties of fields (and divisionAlgebras) and their ops
completeField: BOOL _ FALSE,
realField: BOOL _ FALSE,
realClosedField: BOOL _ FALSE,
algebraicallyClosedField: BOOL _ FALSE,

propList: Atom.PropList _ NIL
];
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