NAME

Language::MuldisD::Core - Muldis D core data types and operators

VERSION

This document is Language::MuldisD::Core version 0.9.1.

PREFACE

This document is part of the Muldis D language specification, whose root document is Language::MuldisD; you should read that root document before you read this one, which provides subservient details.

DESCRIPTION

Muldis D has a mandatory core set of system-defined (eternally available) entities, which is referred to as the Muldis D core or the core; they are the minimal entities that all Muldis D implementations need to provide; they are mutually self-describing and are used to bootstrap the language; any entities outside the core, called Muldis D extensions, are non-mandatory and are defined in terms of the core or each other, but the reverse isn't true.

This current Core document features the boolean, order, integer, rational, bit string, and character string types and operators, plus the tuple and relation type constructors (and quasi- variants) and operators, plus the type system minimal and maximal types, plus the special types used to define the system catalog, and the polymorphic operators that all types, or some types including core types, have defined over them.

Extensions are in these other documents: Language::MuldisD::Ext::Temporal, Language::MuldisD::Ext::Spatial.

TYPE SUMMARY

Following are all the data types and data type factories described in this document, arranged in a type graph according to their proper sub|supertype relationships:

sys.Core.Universal.Universal

    sys.Core.Universal.Empty

    sys.Core.Scalar.Scalar
        sys.Core.Ordered.Ordered

            # The following are all regular ordered scalar types.

            sys.Core.Bool.Bool
            sys.Core.Order.Order
            sys.Core.Int.Int
                sys.Core.Int.UInt
                    sys.Core.Int.PInt
            sys.Core.Rat.Rat
                sys.Core.Rat.URat
                    sys.Core.Rat.PRat
                sys.Core.Rat.BRat
                sys.Core.Rat.DRat
            sys.Core.Blob.Blob
                sys.Core.Blob.NEBlob
            sys.Core.Text.Text
                sys.Core.Text.NEText

    # The following are all nonscalar type factories.

    sys.Core.Tuple.Tuple
        sys.Core.Tuple.Database

    sys.Core.Relation.Relation
        sys.Core.Relation.Set
            sys.Core.Relation.Maybe
        sys.Core.Relation.Seq
        sys.Core.Relation.Bag

    # The following are all quasi-nonscalar type factories.

    sys.Core.QuasiTuple.QuasiTuple

    sys.Core.QuasiRelation.QuasiRelation
        sys.Core.QuasiRelation.QuasiSet
            sys.Core.QuasiRelation.QuasiMaybe
        sys.Core.QuasiRelation.QuasiSeq
        sys.Core.QuasiRelation.QuasiBag

Note that sys.Core.Universal.Empty is a proper subtype of all of the other types in this graph, but every other type has only one immediate supertype shown, and hence the graph of them is a simple hierarchy.

These system-defined subtypes are specific to defining the system catalog, more or less:

sys.Core.Universal.Universal

    sys.Core.Universal.Empty

    sys.Core.Scalar.Scalar

        # The following is actually a union over many scalar types.

        sys.Core.Cat.ScalarLiteral

        sys.Core.Ordered.Ordered

            # The following are all regular ordered scalar types.

            sys.Core.Cat.Name
            sys.Core.Cat.NameChain
            sys.Core.Cat.FlattenedNameChain

        # The following are all regular non-ordered scalar types.

        sys.Core.Cat.E_TK
        sys.Core.Cat.E_TDM
        sys.Core.Cat.E_EK

    sys.Core.Tuple.Tuple

        # The following are all regular tuple types.

        sys.Core.Cat.Type
        sys.Core.Cat.Expr
        sys.Core.Cat.Exception

    sys.Core.Relation.Relation

        # The following are all regular relation types.

        sys.Core.Cat.NameTypeMap
        sys.Core.Cat.NameMap
        sys.Core.Cat.BiDiNameMap

        sys.Core.Relation.Set

            # The following are all regular set types.

            sys.Core.Cat.SetOfName
            sys.Core.Cat.SetOfSetOfName
            sys.Core.Cat.SetOfNameMap
            sys.Core.Cat.SetOfNameChain

        sys.Core.Relation.Seq

            # The following are all regular sequence types.

            sys.Core.Cat.NESeqOfName

These system-defined subtypes have also been defined for convenience, as they are anticipated to be frequently used; in fact, most of them are used as the declared parameter types of various core operators:

sys.Core.Universal.Universal

    sys.Core.Universal.Empty

    sys.Core.Scalar.Scalar
        sys.Core.Ordered.Ordered
            sys.Core.Int.Int
                sys.Core.Int.UInt
                    sys.Core.Int.PInt

                        # The following are all finite integer types.

                        sys.Core.Spec.PInt1_4
                        sys.Core.Spec.PInt2_36

    sys.Core.Relation.Relation
        sys.Core.Relation.Set

            # The following are all regular set types.

            sys.Core.Spec.SetOfBool
            sys.Core.Spec.SetOfInt
            sys.Core.Spec.SetOfRat
            sys.Core.Spec.SetOfBlob
            sys.Core.Spec.SetOfText

            # The following are all nonscalar type factories.

            sys.Core.Spec.SetOfRelation

            sys.Core.Relation.Maybe

                # The following are all regular maybe types.

                sys.Core.Spec.MaybeOfBool
                sys.Core.Spec.MaybeOfInt
                sys.Core.Spec.MaybeOfRat
                sys.Core.Spec.MaybeOfBlob
                sys.Core.Spec.MaybeOfText

        sys.Core.Relation.Seq

            # The following are all regular sequence types.

            sys.Core.Spec.SeqOfBool
            sys.Core.Spec.SeqOfInt
            sys.Core.Spec.SeqOfRat
            sys.Core.Spec.SeqOfBlob
            sys.Core.Spec.SeqOfText

        sys.Core.Relation.Bag

            # The following are all regular bag types.

            sys.Core.Spec.BagOfBool
            sys.Core.Spec.BagOfInt
            sys.Core.Spec.BagOfRat
            sys.Core.Spec.BagOfBlob
            sys.Core.Spec.BagOfText

            # The following are all nonscalar type factories.

            sys.Core.Spec.BagOfRelation

    sys.Core.QuasiRelation.QuasiRelation
        sys.Core.QuasiRelation.QuasiSet

            # The following are all quasi-nonscalar type factories.

            sys.Core.Spec.QuasiSetOfRelation

Note that, in later operator documentation, if you see something like Foo{Bar} as a declared type, it corresponds to sys.Core.Spec.FooOfBar.

SYSTEM-DEFINED CORE MAXIMAL AND MINIMAL DATA TYPES

These core data types are special and are the only Muldis D types that are neither scalar nor nonscalar nor quasi-nonscalar types. They are all system-defined and it is impossible for users to define more types of this nature.

sys.Core.Universal.Universal

The Universal type is the maximal type of the entire Muldis D type system, and contains every value that can possibly exist. Every other type is implicitly a proper subtype of Universal, and Universal is implicitly a union type over all other types. Its default value is Bool:false. The cardinality of this type is infinity.

sys.Core.Universal.Empty

The Empty type is the minimal type of the entire Muldis D type system, and is the only type that contains exactly zero values. Every other type is implicitly a proper supertype of Empty and Empty is implicitly an intersection type over all other types. It has no default value. The cardinality of this type is zero.

SYSTEM-DEFINED CORE PSEUDO-TYPES

These are pseudo-types (that is, they aren't types at all) which represent generic contexts that could accept a multiplicity of types, but are not the same as contexts named after some actual types. For example, they are used as the declared parameter types of some dyadic polymorphic operators to refer to, per instance, 2 given types that need to be compatible, but the compatibility isn't simply defined by a "are any 2 subtypes of" such as is true with most operators. They are all system-defined and it is impossible for users to define more types of this nature.

sys.Core.Some.Universal

Contexts defined by the Some.Universal pseudo-type accept values of potentially any type of Universal.

sys.Core.Some.Ordered

Contexts defined by the Some.Ordered pseudo-type accept values of potentially any type of Ordered.

SYSTEM-DEFINED CORE SCALAR DATA TYPES

These core scalar data types are the most fundamental Muldis D types. Each one has zero possreps, and hence has no named components that can be referenced. Concrete Muldis D provides a specific syntax per type to select a value of one of these types, which does not look like a routine invocation, but rather like a scalar literal in a typical programming language; details of that syntax are not given here, but in Language::MuldisD::Grammar. Abstract Muldis D as hosted in another language will essentially use literals of corresponding host language types, whatever they use for eg booleans and integers and character strings, but tagged with extra meta-data if the host language is more weakly typed or lacks one-to-one type correspondence; see Language::MuldisD::PerlHosted for a Perl-based example.

sys.Core.Scalar.Scalar

The Scalar type is the maximal type of all Muldis D scalar types, and contains every scalar value that can possibly exist. Every other scalar type is implicitly a proper subtype of Scalar, and Scalar is implicitly a union type over all other scalar types. Its default value is Bool:false. The cardinality of this type is infinity.

sys.Core.Ordered.Ordered

The Ordered type is a proper subtype of Scalar that is a proper supertype of all scalar types that are considered ordered. Any scalar types which consider themselves ordered, which happens to include every system-defined core scalar root type, will explicitly declare themselves as subtypes of Ordered in their type definitions; the definition of Ordered does not specify what other types it is a union over. Its default value is Bool:false. The cardinality of this type is infinity.

sys.Core.Bool.Bool

A Bool is an enumeration consisting of the 2 values Bool:false and Bool:true. It represents a truth value, and is the result type of any is_equal or is_not_equal routine; it is the only essential scalar data type of a D language. Its default and minimum value is Bool:false; its maximum value is Bool:true. The cardinality of this type is 2.

sys.Core.Order.Order

An Order is an enumeration consisting of the 3 values Order:increase, Order:same, Order:decrease. It is the result type of any compare routine that is used on 2 values of a sys.Core.Ordered.Ordered subtype. Its default value is Order:same; its minimum and maximum values are, respectively, Order:increase and Order:decrease. The cardinality of this type is 3.

sys.Core.Int.Int

An Int is a single exact integral number of any magnitude. Its default value is zero; its minimum and maximum values are conceptually infinities and practically impossible. The cardinality of this type is infinity; to define a most-generalized finite Int subtype, you must specify the 2 integer end-points of the inclusive range that all its values are in.

sys.Core.Int.UInt

A UInt (unsigned / non-negative integer) is a proper subtype of Int where all member values are greater than or equal to zero. Its minimum value is zero.

sys.Core.Int.PInt

A PInt (positive integer) is a proper subtype of UInt where all member values are greater than zero. Its default and minimum value is 1.

sys.Core.Rat.Rat

A Rat is a single exact rational number of any magnitude. It is conceptually a numerator (Int) divided by a denominator (PInt). Its default value is zero; its minimum and maximum values are conceptually infinities and practically impossible. The cardinality of this type is infinity; to define a most-generalized finite Rat subtype, you must specify the greatest magnitude value denominator, plus the the 2 integer end-points of the inclusive range of the value numerator. Common subtypes specify that all denominators are positive powers of a particular radix (PInt), where the radix is usually either 2 or 10; types such as these will easily map exactly to common human or physical numeric representations, so they tend to perform better.

sys.Core.Rat.URat

A URat (unsigned / non-negative rational) is a proper subtype of Rat where all member values are greater than or equal to zero. Its minimum value is zero.

sys.Core.Rat.PRat

A PRat (positive integer) is a proper subtype of URat where all member values are greater than zero. Its default value is 1.

sys.Core.Rat.BRat

A BRat (binary rational) is a proper subtype of Rat where the radix is 2; it is the best option to exactly represent rational numbers that are conceptually binary or octal or hexadecimal.

sys.Core.Rat.DRat

A DRat (decimal rational) is a proper subtype of Rat where the radix is 10; it is the best option to exactly represent rational numbers that are conceptually the decimal numbers that humans typically work with.

sys.Core.Blob.Blob

A Blob is an undifferentiated string of bits. Its default and minimum value is the empty string; its maximum value is an infinite-length string and practically impossible. The cardinality of this type is infinity; to define a most-generalized finite Blob subtype, you must specify a maximum length in bits that the subtype's strings are.

sys.Core.Blob.NEBlob

A NEBlob (non-empty blob) is a proper subtype of Blob where its length in bits must be at least 1; it can be any Blob except for the empty string. Its default and minimum value is a single zero bit.

sys.Core.Text.Text

A Text is a string of characters. Its default and minimum value is the empty string; its maximum value is an infinite-length string and practically impossible. Note that there is only one system-defined character repertoire for Text types, which is the newest Unicode repertoire (5.0.0). The cardinality of this type is infinity; to define a most-generalized finite Text subtype, you must specify a maximum length in characters (that is, eg, in NFC graphemes) that the subtype's strings are.

sys.Core.Text.NEText

A NEText (non-empty text) is a proper subtype of Text where its length in characters must be more than zero; it can be any Text except for the empty string. Its default value is a single "space" character; its minimum value has one character, but which character that is depends on the default or current collation.

SYSTEM-DEFINED CORE NONSCALAR PARAMETERIZED DATA TYPES

These are only called nonscalar data types in a loose sense, because by themselves they are incomplete type definitions. Actual nonscalar data type definitions are derived from these by supplying the balance of the type definitions, such as what their attributes are and/or what their attribute types are. Associated with these incomplete type definitions are a set of system-defined routines that can be applied to values of any actual nonscalar types derived from them; such are called generic nonscalar operators. In the Muldis D type system, these incomplete nonscalar types are defined as union types over all actual types derived from them, and are proper supertypes of said.

Some actual nonscalar data types are system-defined, for use in defining the Muldis D system catalog / meta-model (see further below in the current document), and some others are system-defined for convenience since they are the types of many core operators (see further below in the current document), but all other actual nonscalar data types are user-defined. Users can also define their own incomplete nonscalar data types that are tuple or relation types.

sys.Core.Tuple.Tuple

The Tuple type is the maximal type of all Muldis D tuple (nonscalar) types, and contains every tuple value that could possibly exist. A Tuple is an unordered heterogeneous collection of 0..N named attributes (the count of attributes being its degree), where all attribute names are mutually distinct, and each attribute may be of distinct types; the mapping of a tuple's attribute names and their declared data types is called the tuple's heading. Its default value is the sole value of the sole tuple data type that has zero attributes. The cardinality of this type is equal to the product of the number of permutations drawable from the values of each of its attributes' declared data types; for a Tuple subtype to be finite, all of its attribute types must be.

sys.Core.Tuple.Database

A Database is a proper subtype of Tuple where all of its attributes are each of relation types or of database types (the leaves of this recursion are all relation types); it is otherwise the same.

sys.Core.Relation.Relation

The Relation type is the maximal type of all Muldis D relation (nonscalar) types, and contains every relation value that could possibly exist. A Relation is analogous to a set of 0..N tuples where all tuples have the same heading (the degrees match and all attribute names and corresponding declared data types match), but that a Relation data type still has its own corresponding heading (attribute names and declared data types) even when it consists of zero tuples. Its default value is the zero-tuple value of the sole relation data type that has zero attributes. Matters of its cardinality are generally the same as for Tuple. A relation data type can also have (unique) keys each defined over a subset of its attributes, which constrain its set of values relative to there being no explicit keys, but having the keys won't turn an infinite relation type into a finite one.

sys.Core.Relation.Set

A Set is a proper subtype of Relation that has 1 attribute, and its name is value; it can be of any declared type. A Set subtype is normally used by any system-defined N-ary operators where the order of their argument elements or result is not significant, and that duplicate values are not significant. Its default value has zero tuples.

sys.Core.Relation.Maybe

A Maybe is a proper subtype of Set that may have at most one element; that is, it is a unary Relation with a nullary key. Operators that work specifically with Maybe subtypes can provide a syntactic shorthand for working with sparse data; so Muldis D has something which is conceptually close to SQL's nullable types without actually having 3-valued logic; it would probably be convenient for code that round-trips SQL by way of Muldis D to use the Maybe type. Its default value has zero tuples.

sys.Core.Relation.Seq

An Seq is a proper subtype of Relation that has 2 attributes, and their names are index and value, where index is a unary key and its declared type is an UInt subtype (value can be non-unique and of any declared type). A Seq is considered dense, and all index values in one are numbered consecutively from 0 to 1 less than the count of tuples, like array indices in typical programming languages. A Seq subtype is normally used by any system-defined N-ary operators where the order of their argument elements or result is significant (and duplicate values are significant); specifically, index defines an explicit ordering for values. Its default value has zero tuples.

sys.Core.Relation.Bag

A Bag is a proper subtype of Relation that has 2 attributes, and their names are value and count, where value is a unary key (that can have any declared type) and count is a PInt subtype. A Bag subtype is normally used by any system-defined N-ary operators where the order of their argument elements or result is not significant, but that duplicate values are significant; specifically, count defines an explicit count of occurrences for values. Its default value has zero tuples.

SYSTEM-DEFINED CORE QUASI-NONSCALAR PARAMETERIZED DATA TYPES

These quasi-nonscalar incomplete data type definitions correspond to their similarly-named (differing only by the Quasi) nonscalar data types, and their use is intended to be limited to the few situations where the corresponding nonscalar data types can't be used. It should be noted in particular that there is no "QuasiDatabase" type, since all normal data or catalog databases should be composed of normal relations only; but all of the other nonscalar incomplete types have counterparts here.

sys.Core.QuasiTuple.QuasiTuple

A QuasiTuple is like a Tuple but that the declared types of its attributes can be anything at all. Its cardinality is infinite.

sys.Core.QuasiRelation.QuasiRelation

A QuasiRelation is like a Relation but that the declared types of its attributes can be anything at all. Its cardinality is infinite.

sys.Core.QuasiRelation.QuasiSet

A QuasiSet is a proper subtype of QuasiRelation in the corresponding manner to Set being a proper subtype of Relation. Its cardinality is infinite.

sys.Core.QuasiRelation.QuasiMaybe

A QuasiMaybe is a proper subtype of QuasiRelation in the corresponding manner to Maybe being a proper subtype of Relation. Its cardinality is infinite.

sys.Core.QuasiRelation.QuasiSeq

A QuasiSeq is a proper subtype of QuasiRelation in the corresponding manner to Seq being a proper subtype of Relation. Its cardinality is infinite.

sys.Core.QuasiRelation.QuasiBag

A QuasiBag is a proper subtype of QuasiRelation in the corresponding manner to Bag being a proper subtype of Relation. Its cardinality is infinite.

SYSTEM-DEFINED CORE CATALOG SCALAR DATA TYPES

These core scalar data types are more special-purpose in nature and are intended for use in defining or working with the system catalog, which is mainly composed of nonscalar types built using these.

sys.Core.Cat.ScalarLiteral

A Cat.ScalarLiteral is a union type over all the system-defined scalar types that are allowed to be used directly as hard-coded literal values in Muldis D expressions; 'directly' meaning not by way of explicitly invoking a selector function. Generally speaking, this union type includes all of the core scalar types that aren't themselves defined as union types. The full list that ScalarLiteral unions is: Bool, Order, Int, Rat, Blob, Text, Cat.Name, Cat.NameChain, Cat.E_TK, Cat.E_TDM, Cat.E_EK, ...

sys.Core.Cat.Name

A Cat.Name is a canonical short name for any kind of DBMS entity (or named component) when declaring it; this short name is sufficient to identify the entity within its immediate namespace. Similarly, a DBMS entity can often be invoked or referred to using just its Cat.Name, depending on the context; other times, a Cat.NameChain must be used instead to also qualify the reference with a namespace. Cat.Name is the same as NEText in all ways but that it is specifically intended for use in naming DBMS entities rather than being normal data.

sys.Core.Cat.NameChain

A Cat.NameChain is a canonical long name for invoking or referring to a DBMS entity, when its name needs to be qualified with a namespace. A Cat.NameChain has 2 possreps; one possrep is a sequence of 1..N Cat.Name (represented by a Cat.NESeqOfName), the 1..N elements being ordered from parent-most to child-most component name; the other possrep is a character string (represented by a Cat.FlattenedNameChain) like when the elements of the first possrep are catenated (in order with the first element at the start of the string), with a period (.) between the parts, and each part escaped such that backslashes, single-quotes, and periods are escaped as \b, \q and \p respectively.

sys.Core.Cat.FlattenedNameChain

The Cat.FlattenedNameChain type is used as the definition of the character string possrep of a Cat.NameChain (see that type for details); while being a character string like Cat.Name, the two are disjoint.

sys.Core.Cat.E_TK

A Cat.E_TK (type kind) is an enumeration consisting of the 7 values Cat.E_TK:special (mainly for system-defined implicit supertypes), Cat.E_TK:scalar, Cat.E_TK:tuple, Cat.E_TK:relation, Cat.E_TK:quasi_tuple, Cat.E_TK:quasi_relation, Cat.E_TK:remnant.

sys.Core.Cat.E_TDM

A Cat.E_TDM (type definition method) is an enumeration consisting of the 9 values Cat.E_TDM:special (for some system-defined types that don't best fit in other categories, and all parameterized types; users can not define the latter for now), Cat.E_TDM:root (complete root type defined in terms of explicit attribute collection; all are fully defined, not parameterized), Cat.E_TDM:restrict (defined as "explicit other-type where condition"), Cat.E_TDM:alias (so the same one type can have multiple names), Cat.E_TDM:(union|intersection|exclusion) (defined as explicit union|intersection|exclusion of other types), Cat.E_TDM:difference (defined as explicit difference of 2 other types), Cat.E_TDM:negation (defined as explicit negation of another type).

sys.Core.Cat.E_EK

A Cat.E_EK (expression kind) is an enumeration consisting of the 8 values Cat.E_EK:default (default value of expression's type), Cat.E_EK:(scalar|tuple|relation|quasi_tuple|quasi_relation) (hard-coded literal scalar|tuple|relation|quasi-tuple|quasi-relation value), Cat.E_EK:param (value of expression-containing function parameter), Cat.E_EK:func (result of function invocation, or inlining of function body). Its default value is Cat.E_EK:scalar.

SYSTEM-DEFINED CORE CATALOG TUPLE DATA TYPES

These tuple data types, essentially all of the system-defined tuple types are special-purpose in nature and are intended for use in defining or working with the system catalog. They are all completely defined types.

Note that many of these types might conceptually have name attributes, but those would actually be provided by any larger types in which they are embedded, rather than by these types themselves.

Note that whenever an attribute of one of these types isn't significant, given the context (determined by other attributes of the same type), and should be ignored, its value is the default for its type.

To keep things simpler for now, most constraint definitions for these types are missing, or just defined informally.

sys.Core.Cat.Type

A Cat.Type is a Tuple. It defines a data type, which can either be (sometimes) invoked directly for values, or be invoked by or embedded into other type definitions. Cat.Type is used in the catalogs for defining both system and user types (just the interfaces in the former case).

A Cat.Type has these 7 attributes:

tk - Cat.E_TK: This broadly declares what kind of type this is (eg, scalar|tuple|relation); it partially says how to interpret the rest of the attributes and/or which of them have significant values.
tdm - Cat.E_TDM: This is the type definition method for the type; it partially says how to interpret the rest of the attributes and/or which of them have significant values. If tk is Cat.E_TK:special, then tdm must be Cat.E_TDM:special; tdm can be special at other times too.
types - Cat.SetOfNameChain: For some values of tdm, the data type is defined at least partially in terms of other data types not by way of attribute definitions, and then types lists all/most of those types. Iff tdm is Cat.E_TDM:alias, then the data type is just a symbolic reference for some other data type (as far as the type system is concerned, they are the same data type, invokable by an extra name), and types has one element that is the name of that type. Iff tdm is Cat.E_TDM:restrict, then the data type is defined as an explicit subtype of another type by way of an explicit further type constraint applied to it, and types has one element that is the name of that supertype. Iff tdm is Cat.E_TDM:(union|intersection|exclusion), then the data type is defined as an explicit union|intersection|exclusion of N other types, and types lists their names, one per element; 2+ elements is the norm; just 1 element means that the data type is a simple alias for the named element; zero elements means the data type is a simple alias for, respectively, the type Empty, Universal, or Empty. Iff tdm is Cat.E_TDM.difference, then the data type is defined as the difference of 2 other types, and types has one element that is the name of the minuend type. Iff tdm is Cat.E_TDM.negation, then the data type is defined as the negation of some other type, and types has one element that is the name of that type.
subtr_type - Cat.NameChain: Iff tdm is Cat.E_TDM.difference, then the data type is defined as the difference of 2 other types, and subtr_type is the name of the subtrahend type.
attrs - Cat.NameTypeMap: Iff tdm is Cat.E_TDM:root, then the data type is defined fundamentally in terms of an explicit attribute collection, and attrs defines the names and declared types of those attributes. Iff additionally tk is Cat.E_TK:scalar, then attrs specifically defines the attributes of just the core/initial/only possrep; otherwise, attrs defines the heading of the tuple or relation etc type. It is valid to have zero attributes; in this case, the type consists of exactly one value.
keys - Cat.SetOfSetOfName: Iff tk is Cat.E_TK:(relation|quasi_relation) and tdm is not Cat.E_TDM:special, then the data type is or resembles a relation type and can have explicit keys (duality of unique key constraints and terser unique identifiers for the q/relation's member q/tuples) defined over its attributes, and keys defines those keys in the canonical simplest form (in contrast with using constraint instead). Each element of keys defines one key of the q/relation, and that element is a set of the attribute names comprising that key. For q/relation types, if no keys are explicitly defined, then it implicitly has a single key comprising all of its attributes. If any explicit keys are defined, then every one must be over a distinct proper subset of the type's attributes, and moreover no key's attributes may be a proper subset of any other key's attributes; if 2 such candidates appear, just use the one that has the subset. It is valid for a key to consist of zero attributes; in this case, that key is the only key of the q/relation type, and values of the type may each consist of no more than one tuple.
constraint - Cat.TypeConstraint ... default one always ret True: Iff tdm is Cat.E_TDM:root, then the data type is defined fundamentally in terms of an explicit attribute collection, and constraint defines/names a generalized type constraint that validates the collection as a whole. Iff tdm is Cat.E_TDM:restrict, then the data type is defined as an explicit subtype of another type by way of an explicit further type constraint applied to it, and constraint defines/names that further constraint.

The default value of Cat.Type is an alias of Empty.

sys.Core.Cat.Expr

A Cat.Expr is a Tuple. It specifies a named expression node, which is the majority component of functional Muldis D code. All arbitrarily complex Muldis D expression trees, including relational queries, are composed of just Cat.Expr, either directly, or indirectly by way of function invocations, as each function body is itself composed entirely of a single expression tree. Only functions may contain Cat.Expr, so for any procedures that would conceptually include them, those portions of the procedures need to be separated out and encapsulated by named functions.

A Cat.Expr has these 8 attributes:

name - Cat.Name: This is the declared name of the expression node; other Muldis D code that is composing this expression node into their own definitions would reference it with this name.
type - Cat.NameChain: This is the declared result data type of the expression node. If the expression node represents a literal value, this is often necessary for properly interpreting the details of that literal, particularly for literals that are not scalar. Otherwise, although the result type of an expression can sometimes be inferred from context, such as from the result type of a function it is an invocation of, this redundancy helps make static analysis of Muldis D code easier, so it is easier to validate and optimize in isolation from the other routines it interacts with. Moreover, this type declaration provides a terse way to use the treat function frequently in code, so that the programmer and compiler knows that some generic routines are actually supposed to be returning a subtype of their normal result types.
kind - Cat.E_EK: This is the declared kind of expression that this node is; it says how to interpret the rest of the attributes and/or which of them have significant values.
scal_lit - Cat.ScalarLiteral: Iff kind is Cat.E_EK:scalar, then the expression represents a hard-coded scalar literal of one of a certain collection of system-defined core scalar types (or subtype thereof), and this is that literal value.
coll_lit - Cat.SetOfNameMap: Iff kind is Cat.E_EK:(|quasi_)(tuple|relation), then the expression represents a collection literal, and these are the values of its components. Each element defines one tuple, and each sub-element of that element defines one attribute value for one tuple, with the sub-element key matching the attribute name, and the sub-element value naming another local Cat.Expr which defines the value. The value of coll_lit defines exactly one tuple when kind specifies a tuple or quasi-tuple, and it defines 0..N tuples when kind specifies a relation or quasi-relation.
param - Cat.Name: Iff kind is Cat.E_EK:param, then the expression represents the value of the containing function's parameter which this names.
func - Cat.NameChain: Iff kind is Cat.E_EK:func, then the expression represents the result of invoking a named function with specific arguments, and this the name of that function.
func_args - Cat.NameMap: Iff func is used, then these are the arguments for the function invocation. Each element defines one argument value, with the element key matching the parameter name, and the element value naming another local Expr which defines the value.

The default value of Cat.Expr represents the literal scalar value Bool:false.

sys.Core.Cat.Exception

TODO.

SYSTEM-DEFINED CORE CATALOG RELATION DATA TYPES

These relation data types, essentially all of the system-defined relation types are special-purpose in nature and are intended for use in defining or working with the system catalog. They are all completely defined types.

To keep things simpler for now, most constraint definitions for these types are missing, or just defined informally.

sys.Core.Cat.NameTypeMap

A Cat.NameTypeMap is a Relation. It defines a basic component list, meaning a set of names, with a declared data type for each. It forms the foundation for most componentized type definitions, including all tuple and relation types (for which it is named heading), and it is used also for the components list of a scalar possrep. It is also used to define parameter lists for routines. A Cat.NameTypeMap has 2 attributes, name (a Cat.Name) and type (a Cat.NameChain); the name is the declared name of the attribute or parameter, and comprises a unary key; the type is the declared data type of the attribute or parameter. Its default value has zero tuples.

sys.Core.Cat.NameMap

A Cat.NameMap specifies a map of short entity names to other short entity names. It is a binary Relation whose 2 attributes are named key and value, and both attributes have declared types of Cat.Name; the key attribute is a unary key. Its default value has zero tuples.

sys.Core.Cat.BiDiNameMap

A Cat.BiDiNameMap is a proper subtype of Cat.NameMap where its value attribute is also a unary key. It is used as a specification for how to rename attributes of a relation.

sys.Core.Cat.SetOfName

A Cat.SetOfName is a (Set) whose value attribute has a declared type of Cat.Name.

sys.Core.Cat.SetOfSetOfName

A Cat.SetOfSetOfName is a (Set) whose value attribute has a declared type of Cat.SetOfName.

sys.Core.Cat.SetOfNameMap

A Cat.SetOfNameMap is a (Set) whose value attribute has a declared type of Cat.NameMap.

sys.Core.Cat.SetOfNameChain

A Cat.SetOfNameChain is a (Set) whose value attribute has a declared type of Cat.NameChain.

sys.Core.Cat.NESeqOfName

A Cat.NESeqOfName is a (Seq) whose value attribute has a declared type of Cat.Name and that must have at least 1 element; this type is used as the definition of the sequence possrep of a Cat.NameChain (see that type for details).

SYSTEM-DEFINED CORE SPECIALIZED SUBTYPES

These types are proper subtypes of other core types, and they are system-defined for convenience, as they are anticipated to be frequently used; in fact, most of them are used as the declared parameter types of various core operators.

sys.Core.Spec.PInt1_4: A PInt1_4 is a proper subtype of PInt where all member values are between 1 and 4. Its maximum value is 4. The cardinality of this type is 4.
sys.Core.Spec.PInt2_36: A PInt2_36 is a proper subtype of PInt where all member values are between 2 and 36. (The significance of the number 36 is 10 digits plus 26 letters.) Its default and minimum value is 2; its maximum value is 36. The cardinality of this type is 35.
sys.Core.Spec.(Set|Maybe|Seq|Bag)Of(Bool|Int|Rat|Blob|Text): A (Set|Maybe|Seq|Bag)Of(Bool|Int|Rat|Blob|Text) is a completely defined proper subtype of (Set|Maybe|Seq|Bag) whose value attribute has a declared type of a (Bool|Int|Rat|Blob|Text) subtype.
sys.Core.Spec.(Set|Bag)OfRelation: A (Set|Bag)OfRelation is an incompletely defined proper subtype of Set|Bag whose value attribute has a declared type of a Relation subtype.
sys.Core.Spec.QuasiSetOfRelation: A QuasiSetOfRelation is an incompletely defined proper subtype of QuasiSet whose value attribute has a declared type of Relation; this is the parameter type of the N-ary relational join operator.

SYSTEM-DEFINED GENERIC UNIVERSAL FUNCTIONS

These functions are applicable to values of any data type at all.

sys.Core.Universal.is_equal of Bool (Some.Universal $v1, Some.Universal $v2): This function results in Bool:true iff its 2 arguments are exactly the same value, and Bool:false otherwise. This function's arguments must be of compatible declared types; in this case, 2 declared types are compatible iff at least one of the following is true: 1. they are both subtypes of a common scalar root type; 2. they are both subtypes of a common non-incomplete tuple or relation type, that is they essentially have the same headings; 3. at least one type is a generic (eg-Universal) or incomplete (eg-Seq) type, and it is a supertype of the other. This function is commutative.
sys.Core.Universal.is_not_equal of Bool (Some.Universal $v1, Some.Universal $v2): This function is exactly the same as sys.Core.Universal.is_equal except that it results in the opposite boolean value when given the same arguments.
sys.Core.Universal.is_value_of_type of Bool (Cat.NameChain $type, Universal $v): This function results in Bool:true iff the value of its $v argument is a member of the data type whose name is given in the $type argument, and Bool:false otherwise. As trivial cases, this function always results in Bool:true if the named type is Universal, and Bool:false if it is Empty. This function will fail if the named type doesn't exist in the virtual machine.
sys.Core.Universal.treat of Some.Universal (Cat.NameChain $as, Some.Universal $v): This function results in the value of its $v argument, but that the declared type of the result is the not-Empty data type whose name is given in the $as argument. This function will fail if the named type doesn't exist in the virtual machine, or if $v isn't a member of the named type. The purpose of treat is to permit taking values from a context having a more generic declared type, and using them in a context having a more specific declared type; such an action would otherwise be blocked at compile time due to a type-mismatch error; treat causes the type-mismatch validation, and possible failure, to happen at runtime instead, on the actual value rather than declared value. For example, if you are storing an Int value in a Scalar-typed variable, using treat will cause the compiler to let you use that variable as an argument to Int.sum, which it otherwise wouldn't.
sys.Core.Universal.default of Some.Universal (Cat.NameChain $of): This function results in the default value of the not-Empty data type whose name is given in the $of argument, and the declared type of the result is that same type. This function will fail if the named type doesn't exist in the virtual machine, either at compile or runtime depending whether the type is in the system or user namespace. This function is conceptually implicitly used to provide default values for variables, so they always hold valid values of their declared type.

SYSTEM-DEFINED GENERIC ORDERED FUNCTIONS

These functions are applicable to values of any data type which is a subtype of Ordered. They provide a common syntax for sort-related functionality, though technically every type having these functions is re-implementing its own version. If values of an ordered data type can conceivably be sorted using multiple criteria (such as different text collations), then these functions just represent the default criteria; any additional criteria are represented by additional functions declared for just the types they apply to.

sys.Core.Ordered.compare of Order (Some.Ordered $v1, Some.Ordered $v2): This function results in Order:same iff its 2 arguments are exactly the same value, and otherwise it results in Order:increase if the value of the $v2 argument is considered to be an increase (as defined by the type) over the value of the $v1 argument, and otherwise it results in Order:decrease as the reverse of the last condition would be true. This function's arguments must be of compatible declared types; in this case, 2 declared types are compatible iff they are both subtypes of a common scalar type that declares itself an Ordered subtype. Note that compare is considered the only fundamental ordered-specific operator, and all others are defined over it.
sys.Core.Ordered.reverse_compare of Order (Some.Ordered $v1, Some.Ordered $v2): This function is exactly the same as sys.Core.Ordered.compare except that it results in the reverse value when given the same arguments. It is a short-hand for applying sys.Core.Order.reverse to the result of sys.Core.Ordered.compare with the same arguments.
sys.Core.Ordered.is_increase of Bool (Some.Ordered $v1, Some.Ordered $v2): This function results in Bool:true iff sys.Core.Ordered.compare would result in Order:increase when given the same arguments, and Bool:false otherwise.
sys.Core.Ordered.is_decrease of Bool (Some.Ordered $v1, Some.Ordered $v2): This function results in Bool:true iff sys.Core.Ordered.compare would result in Order:decrease when given the same arguments, and Bool:false otherwise.
sys.Core.Ordered.min of Some.Ordered (Set{Some.Ordered} $topic): This function is a reduction operator that recursively takes each pair of its N input element values and picks the minimum of the 2 (a process which is both commutative and associative) until just one is left, which is the function's result. If $topic has zero values, then min results in the result type's concept of positive infinity, which is the identity value for min. This function will fail on a $topic of zero values if the result type's concept of positive infinity is impossible or impractically large to represent, such as with the infinite Text type.
sys.Core.Ordered.max of Some.Ordered (Set{Some.Ordered} $topic): This function is exactly the same as sys.Core.Ordered.min except that it results in the maximum input element value rather than the minimum one, and its identity value is the result type's concept of negative infinity.
sys.Core.Ordered.maybe_min of Maybe{Some.Ordered} (Set{Some.Ordered} $topic): This function is exactly the same as sys.Core.Ordered.min except that it results in a Maybe of what is otherwise the result type, and that result has zero elements if the argument has zero elements.
sys.Core.Ordered.maybe_max of Maybe{Some.Ordered} (Set{Some.Ordered} $topic): This function is to sys.Core.Ordered.max as sys.Core.Ordered.maybe_min is to sys.Core.Ordered.min.

SYSTEM-DEFINED CORE SCALAR FUNCTIONS

These functions are applicable to just one or more specific system-defined core scalar data type.

Functions for sys.Core.Bool.Bool

These functions select values of the Bool enumeration.

sys.Core.Bool.(false|true) of Bool (): This selector function results in the Bool:(false|true) value.

These functions implement commonly used boolean operations.

sys.Core.Bool.not of Bool (Bool $topic): This function results in the logical not of its argument.
sys.Core.Bool.and of Bool (Set{Bool} $topic): This function is a reduction operator that recursively takes each pair of its N input element values and does a logical and (which is both commutative and associative) on them until just one is left, which is the function's result. If $topic has zero values, then and results in Bool:true, which is the identity value for logical and.
sys.Core.Bool.or of Bool (Set{Bool} $topic): This function is a reduction operator that recursively takes each pair of its N input element values and does a logical inclusive-or (which is both commutative and associative) on them until just one is left, which is the function's result. If $topic has zero values, then or results in Bool:false, which is the identity value for logical inclusive-or.
sys.Core.Bool.xor of Bool (Bag{Bool} $topic): This function is a reduction operator that recursively takes each pair of its N input element values and does a logical exclusive-or (which is both commutative and associative) on them until just one is left, which is the function's result. If $topic has zero values, then xor results in Bool:false, which is the identity value for logical exclusive-or.

Functions for sys.Core.Order.Order

These functions select values of the Order enumeration.

sys.Core.Order.(increase|same|decrease) of Order (): This selector function results in the Order:(increase|same|decrease) value.

These functions implement commonly used order-enumeration operations.

sys.Core.Order.reverse of Order (Order $topic): This function results in the reverse value of its argument; an Order:increase or Order:decrease argument results in the other one of the two; an Order:same argument results in Order:same.

Functions for sys.Core.Int.Int

These functions implement commonly used integer numeric operations.

sys.Core.Int.abs of UInt (Int $topic): This function results in the absolute value of its argument.
sys.Core.Int.sum of Int (Bag{Int} $addends): This function results in the sum of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and adds (which is both commutative and associative) them together until just one is left, which is the result. If $addends has zero values, then sum results in the integer zero, which is the identity value for addition.
sys.Core.Int.difference of Int (Int $minuend, Int $subtrahend): This function results in the difference when its $subtrahend argument is subtracted from its $minuend argument.
sys.Core.Int.product of Int (Bag{Int} $factors): This function results in the product of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and multiplies (which is both commutative and associative) them together until just one is left, which is the result. If $factors has zero values, then product results in the integer 1, which is the identity value for multiplication.
sys.Core.Int.quotient of Int (Int $dividend, Int $divisor): This function results in the quotient when its $dividend argument is divided by its $divisor argument using integer division. This function will fail if $divisor is zero.
sys.Core.Int.remainder of UInt (Int $dividend, Int $divisor): This function results in the remainder when its $dividend argument is divided by its $divisor argument using integer division. This function will fail if $divisor is zero.
sys.Core.Int.maybe_quotient of Maybe{Int} (Int $dividend, Int $divisor): This function is exactly the same as sys.Core.Int.quotient except that it results in a Maybe of what is otherwise the result, and that result has zero elements if $divisor is zero.
sys.Core.Int.maybe_remainder of Maybe{UInt} (Int $dividend, Int $divisor): This function is exactly the same as sys.Core.Int.remainder except that it results in a Maybe of what is otherwise the result, and that result has zero elements if $divisor is zero.
sys.Core.Int.range of Int (Set{Int} $topic): This function results in the difference between the lowest and highest element values of its argument. If $topic has zero values, then range results in the integer zero.
sys.Core.Int.median of Set{Int} (Bag{Int} $topic): This function results in the 1 or 2 median values of the N element values of its argument; they are returned as a set. It is equivalent to first arranging the input values from least to greatest, and then taking the single middle value, if the count of input values is odd, or taking the 2 middle values, if the count of input values is even (but if the 2 middle values are the same value, the output has one element). If $topic has zero values, then the result set is empty.
sys.Core.Int.mode of Set{Int} (Bag{Int} $topic): This function results in the mode of the N element values of its argument; it is the set of values that appear the most often as input elements, and all have the same count of occurrances. As a trivial case, if all input elements have the same count of occurrances, then they will all be in the output. If $topic has zero values, then the result set is empty.
sys.Core.Int.power of Int (Int $radix, UInt $exponent): This function results in its $radix argument taken to the power of its (unsigned integer) $exponent argument. This function will fail if $radix and $exponent are both zero.

These functions convert between Int values and canonically formatted representations of integers as character strings.

sys.Core.Int.Int_from_NEText of Int (NEText $text, PInt2_36 $radix): This selector function results in the Int value that its $text argument maps to when the whole character string is evaluated as a base-$radix integer. Extending the typical formats of [base-2, base-8, base-10, base-16], this function supports base-2 through base-36; to get the latter, the characters 0-9 and A-Z represent values in 0-35. This function will fail if $text can't be mapped as specified.
sys.Core.Int.NEText_from_Int of NEText (Int $int, PInt2_36 $radix): This selector function results in the NEText value where its $int argument is formatted as a base-$radix integer.

These functions convert between Int values and canonically formatted representations of integers as binary strings. Conjecture: These may not actually be useful, and perhaps only operators that take an argument specifying a fixed-length field size, with big and little endian versions, would be appropriate instead. Or maybe both kinds are necessary.

sys.Core.Int.Int_from_Blob_S_VBE of Int (NEBlob $blob): This selector function results in the Int value that its $blob argument maps to when the whole bit string is treated literally as a variable-length binary (two's complement) signed integer of 1 or more bits in length. The first bit is taken as the sign bit, and any other bits provide greater precision than the -1 thru 0 range. The bit string is assumed to be big-endian, since it may not be possible to use little-endian in situations where the bit length isn't a multiple of 8.
sys.Core.Int.Blob_S_VBE_from_Int of NEBlob (Int $int): This selector function results in the Blob value where its $int argument is formatted as a variable-length binary (two's complement) signed integer of 1 or more bits in length; the smallest number of bits necessary to store $int is used.
sys.Core.Int.Int_from_Blob_U_VBE of UInt (NEBlob $blob): This function is the same as sys.Core.Int.Int_from_Blob_S_VBE but that it does unsigned integers.
sys.Core.Int.Blob_U_VBE_from_Int of UInt (NEBlob $blob): This function is the same as sys.Core.Int.Blob_S_VBE_from_Int but that it does unsigned integers.

Functions for sys.Core.Rat.Rat

These functions are essentially selectors and attribute extractors for the conceptual possrep of a rational in terms of an integral numerator plus denominator.

sys.Core.Rat.Rat_from_Int_pair of Rat (Int $numerator, PInt $denominator): This selector function results in the Rat value that its $numerator and $denominator arguments map to when collectively interpreted in the appropriate fashion, as if there were a possrep for the Rat type that was composed of such 2 attributes, but keeping in mind that multiple distinct argument pairs can map to each same Rat value, since the arguments don't have to be a canonical pair.
sys.Core.Rat.numerator of Int (Rat $topic): This function results in the canonical numerator of its argument, which assumes said rational is represented with the pair of smallest possible integers.
sys.Core.Rat.denominator of PInt (Rat $topic): This function results in the canonical denominator of its argument, which assumes said rational is represented with the pair of smallest possible integers.

These functions implement commonly used rational numeric operations.

sys.Core.Rat.abs of URat (Rat $topic): This function results in the absolute value of its argument.
sys.Core.Rat.sum of Rat (Bag{Rat} $addends): This function results in the sum of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and adds (which is both commutative and associative) them together until just one is left, which is the result. If $addends has zero values, then sum results in the rational zero, which is the identity value for addition.
sys.Core.Rat.difference of Rat (Rat $minuend, Rat $subtrahend): This function results in the difference when its $subtrahend argument is subtracted from its $minuend argument.
sys.Core.Rat.product of Rat (Bag{Rat} $factors): This function results in the product of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and multiplies (which is both commutative and associative) them together until just one is left, which is the result. If $factors has zero values, then product results in the rational 1, which is the identity value for multiplication.
sys.Core.Rat.quotient of Rat (Rat $dividend, Rat $divisor): This function results in the quotient when its $dividend argument is divided by its $divisor argument using rational division. This function will fail if $divisor is zero.
sys.Core.Rat.maybe_quotient of Maybe{Rat} (Rat $dividend, Rat $divisor): This function is exactly the same as sys.Core.Rat.quotient except that it results in a Maybe of what is otherwise the result, and that result has zero elements if $divisor is zero.
sys.Core.Rat.range of Rat (Set{Rat} $topic): This function results in the difference between the lowest and highest element values of its argument. If $topic has zero values, then range results in the rational zero.
sys.Core.Rat.mean of Rat (Bag{Rat} $topic): This function results in the mean or arithmetic average of the N element values of its argument. It is equivalent to first taking the sum of the input values, and dividing that sum by the count of the input values. If $topic has zero values, then this function will fail.
sys.Core.Rat.maybe_mean of Maybe{Rat} (Bag{Rat} $topic): This function is exactly the same as sys.Core.Rat.mean except that it results in a Maybe of what is otherwise the result, and that result has zero elements if $topic has zero values.
sys.Core.Rat.median of Set{Rat} (Bag{Rat} $topic): This function results in the 1 or 2 median values of the N element values of its argument; they are returned as a set. It is equivalent to first arranging the input values from least to greatest, and then taking the single middle value, if the count of input values is odd, or taking the 2 middle values, if the count of input values is even (but if the 2 middle values are the same value, the output has one element). If $topic has zero values, then the result set is empty.
sys.Core.Rat.mean_of_median of Rat (Bag{Rat} $topic): This function is a wrapper over sys.Core.Rat.median that will return the mean of its result elements; it will fail if there are zero elements.
sys.Core.Rat.mode of Set{Rat} (Bag{Rat} $topic): This function results in the mode of the N element values of its argument; it is the set of values that appear the most often as input elements, and all have the same count of occurrances. As a trivial case, if all input elements have the same count of occurrances, then they will all be in the output. If $topic has zero values, then the result set is empty.
sys.Core.Rat.power of PRat (PRat $radix, Rat $exponent): This function results in its (positive rational) $radix argument taken to the power of its $exponent argument. Note that, while this function might conceptually have multiple real number results for some negative $exponent, it will always only return the one that is positive.
sys.Core.Rat.log of Rat (PRat $topic, PRat $radix, PInt $rebase_radix, PInt $rebase_max_denom, Cat.RoundMeth $rebase_round): This function results in the logarithm of its $topic argument to the base given in its (positive rational) $radix argument. Since the result would be an irrational number in the general case, the additional 3 $rebase_\w+ parameters specify how to coerce the conceptual result into a rational number that is the actual result; see also the similarly named parameters of the sys.Core.Rat.rebase function.
sys.Core.Rat.natural_power of PRat (Rat $exponent, PInt $rebase_radix, PInt $rebase_max_denom, Cat.RoundMeth $rebase_round): This function results in the special mathematical constant e (which is the base of the natural logarithm) taken to the power of its $exponent argument. The 3 $rebase_\w+ parameters are as per log.
sys.Core.Rat.natural_log of Rat (PRat $topic, PInt $rebase_radix, PInt $rebase_max_denom, Cat.RoundMeth $rebase_round): This function results in the natural logarithm of its $topic argument. The 3 $rebase_\w+ parameters are as per log.

These functions convert between Rat values and canonically formatted representations of rationals as character strings.

sys.Core.Rat.Rat_from_NEText of Rat (NEText $text, PInt2_36 $radix): This selector function results in the Rat value that its $text argument maps to when the whole character string is evaluated as a base-$radix rational. Extending the typical formats of [base-2, base-8, base-10, base-16], this function supports base-2 through base-36; to get the latter, the characters 0-9 and A-Z represent values in 0-35. This function will fail if $text can't be mapped as specified.
sys.Core.Rat.NEText_from_Rat of NEText (Rat $rat, PInt2_36 $radix): This selector function results in the NEText value where its $rat argument is formatted as a base-$radix rational.

These functions convert between Rat values and equal or nearly equal Int values.

sys.Core.Rat.Rat_from_Int of Rat (Int $int): This selector function results in the Rat value that is conceptually equal to its Int argument.
sys.Core.Rat.round_half_up of Int (Rat $topic): This rounding function results in the integer that is nearest to its argument, but that if the argument is exactly half-way between 2 integers, it is rounded upwards. This is the most commonly used method of rounding.
sys.Core.Rat.round_to_even of Int (Rat $topic): This rounding function results in the integer that is nearest to its argument, but that if the argument is exactly half-way between 2 integers, it is rounded to the one that is even. Note that this operation is also legitimately known as: unbiased rounding, convergent rounding, statistician's rounding, bankers' rounding.
sys.Core.Rat.round_to_floor of Int (Rat $topic): This rounding function results in the integer that is either equal to its argument or that is the nearest integer that is above the argument.
sys.Core.Rat.round_to_ceiling of Int (Rat $topic): This rounding function results in the integer that is either equal to its argument or that is the nearest integer that is below the argument.
sys.Core.Rat.round_to_zero of Int (Rat $topic): This rounding function results in the integer that is either equal to its argument or that is the nearest integer to the argument that is closer to zero. Note that this operation is also legitimately known as truncate.

These functions round and/or truncate rational values to make them easier to deal with in various contexts.

sys.Core.Rat.rebase of Rat (Rat $topic, PInt $radix, PInt $max_denom, Cat.RoundMeth $round): This function results in the rational that is as close as possible to $topic but that its denominator is a positive power of $radix and said denominator is not larger than $max_denom; if rounding is needed, then $round dictates the rounding method.

Functions for sys.Core.Blob.Blob

These functions implement commonly used binary string operations.

sys.Core.Blob.catenate of Blob (Seq{Blob} $topic): This function results in the catenation of the N element values of its argument; it is a reduction operator that recursively takes each consecutive pair of input values and catenates (which is associative) them together until just one is left, which is the result. If $topic has zero values, then catenate results in the empty string value, which is the identity value for catenate.
sys.Core.Blob.repeat of Blob (Blob $topic, UInt $count): This function results in the catenation of $count instances of $topic.
sys.Core.Blob.length_in_bits of UInt (Blob $topic): This function results in the length of its argument in bits.
sys.Core.Blob.contains of Bool (Blob $look_in, Blob $look_for, Bool $fixed_start, Bool $fixed_end): This function results in Bool:true iff its $look_for argument is a substring of its $look_in argument as per the optional $fixed_start and $fixed_end constraints, and Bool:false otherwise. If $fixed_start or $fixed_end are Bool:true, then $look_for must occur right at the start or end, respectively, of $look_in in order for contains to results in Bool:true; if either flag is Bool:false, its additional constraint doesn't apply.
sys.Core.Blob.not of Blob (Blob $topic): This function results in the bitwise not of its argument.
sys.Core.Blob.and of Blob (Set{Blob} $topic): This function is a reduction operator that recursively takes each pair of its N input element values and does a bitwise and (which is both commutative and associative) on them until just one is left, which is the function's result. This function's argument values must all be of the same length in bits, that length being part of the argument's declared type (that is, Blob subtype) definition, and that is also the length in bits of the function's result. If $topic has zero values, then and will result in an appropriate-length string of identity/1 valued bits.
sys.Core.Blob.or of Blob (Set{Blob} $topic): This function is the same as sys.Core.Blob.and but that it recursively does a bitwise inclusive-or rather than a bitwise and, and its identity value is composed of zero valued bits.
sys.Core.Blob.xor of Blob (Bag{Blob} $topic): This function is the same as sys.Core.Blob.or but that it recursively does a bitwise exclusive-or rather than a bitwise inclusive-or.

These functions convert between Blob values and canonically formatted representations of binary strings as character strings.

sys.Core.Blob.Blob_from_Text of Blob (Text $text, PInt1_4 $size): This selector function results in the Blob value that its $text argument maps to when each input character represents a sequence of 1-4 bits, the number of bits per character being determined by the $size argument; for example, if $size is 1, then each input character is a [0-1] and represents a bit; or, if $size is 4, then each input character is a [0-9A-F] and represents 4 bits. This function will fail if $text can't be mapped as specified.
sys.Core.Blob.Text_from_Blob of Text (Blob $blob, PInt1_4 $size): This selector function results in the Text value where its argument is encoded using a character for each sequence of 1-4 bits, the number of bits per character being determined by the $size argument. This function will fail if $blob doesn't have a length in bits which is a multiple of $size.

Functions for sys.Core.Text.Text

These functions implement commonly used character string operations.

sys.Core.Text.catenate of Text (Seq{Text} $topic): This function results in the catenation of the N element values of its argument; it is a reduction operator that recursively takes each consecutive pair of input values and catenates (which is associative) them together until just one is left, which is the result. If $topic has zero values, then catenate results in the empty string value, which is the identity value for catenate.
sys.Core.Text.repeat of Text (Text $topic, UInt $count): This function results in the catenation of $count instances of $topic.
sys.Core.Text.length_in_nfd_graphs of UInt (Text $topic): This function results in the length of its argument in Unicode Normalization Form D graphemes.
sys.Core.Text.length_in_nfc_graphs of UInt (Text $topic): This function results in the length of its argument in Unicode Normalization Form C graphemes.
sys.Core.Text.length_in_nfd_codes of UInt (Text $topic): This function results in the length of its argument in Unicode Normalization Form D code points.
sys.Core.Text.length_in_nfc_codes of UInt (Text $topic): This function results in the length of its argument in Unicode Normalization Form C code points.
sys.Core.Text.contains of Bool (Text $look_in, Text $look_for, Bool $fixed_start, Bool $fixed_end): This function results in Bool:true iff its $look_for argument is a substring of its $look_in argument as per the optional $fixed_start and $fixed_end constraints, and Bool:false otherwise. If $fixed_start or $fixed_end are Bool:true, then $look_for must occur right at the start or end, respectively, of $look_in in order for contains to result in Bool:true; if either flag is Bool:false, its additional constraint doesn't apply.
sys.Core.Text.fold_case_to_upper (Text $topic): This function results in the transformation of its argument where any Latin letters are folded to (capital) uppercase.
sys.Core.Text.fold_case_to_lower (Text $topic): This function results in the transformation of its argument where any Latin letters are folded to (small) lowercase.
sys.Core.Text.trim_whitespace (Text $topic): This function results in the value of its argument but that any leading or trailing whitespace characters are trimmed.

SYSTEM-DEFINED CORE GENERIC NONSCALAR FUNCTIONS

These functions are applicable to mainly nonscalar types, but are generic in that they typically work with any nonscalar types.

Functions for sys.Core.Tuple.Tuple Having Single Input Tuples

sys.Core.Tuple.degree of UInt (Tuple $topic): This function results in the degree of its argument (that is, the count of attributes it has).
sys.Core.Tuple.rename of Tuple (Tuple $topic, Cat.BiDiNameMap $map): This function results in a Tuple value that is the same as its $topic argument but that some of its attributes have different names. Each tuple of the argument $map specifies how to rename one $topic attribute, with the key and value attributes of a $map tuple representing the old and new names of a $topic attribute, respectively. As a trivial case, this function's result is $topic if $map has no tuples. This function supports renaming attributes to each others' names. This function will fail if $map specifies any old names that $topic doesn't have, or any new names that are the same as $topic attributes that aren't being renamed.
sys.Core.Tuple.project of Tuple (Tuple $topic, Cat.SetOfName $attrs): This function results in the projection of its $topic argument that has just the subset of attributes of $topic which are named in its $attrs argument. As a trivial case, this function's result is $topic if $attrs lists all attributes of $topic; or, it is the nullary tuple if $attrs is empty. This function will fail if $attrs specifies any attribute names that $topic doesn't have.
sys.Core.Tuple.remove of Tuple (Tuple $topic, Cat.SetOfName $attrs): This function is the same as project but that it results in the complementary subset of attributes of $topic when given the same arguments.
sys.Core.Tuple.wrap of Tuple (Tuple $topic, Cat.SetOfName $inner, Cat.Name $outer): This function results in a Tuple value that is the same as its $topic argument but that some of its attributes have been wrapped up into a new Tuple-typed attribute, which exists in place of the original attributes. The $inner argument specifies which $topic attributes are to be removed and wrapped up, and the $outer argument specifies the name of their replacement attribute. As a trivial case, if $inner is empty, then the result has all the same attributes as before plus a new nullary tuple attribute; or, if $inner lists all attributes of $topic, then the result has a single attribute whose value is the same as $topic. This function supports the new attribute having the same name as an old one being wrapped into it. This function will fail if $inner specifies any attribute names that $topic doesn't have, or if $outer is the same as $topic attributes that aren't being wrapped.
sys.Core.Tuple.unwrap of Tuple (Tuple $topic, Cat.Name $outer): This function is the inverse of sys.Core.Tuple.wrap, such that it will unwrap a Tuple-type attribute into its member attributes. This function will fail if $outer specifies any attribute name that $topic doesn't have, or if an attribute of $topic{$outer} is the same as a $topic attribute.

Functions for sys.Core.Tuple.Tuple Having Multiple Input Tuples

sys.Core.Tuple.product of Tuple (QuasiSet{Tuple} $topic): This function is similar to sys.Core.Relation.product but that it works with tuples rather than relations. This function is mainly intended for use in connecting tuples that have all disjoint headings, such as for extending one tuple with additional attributes.

Functions for sys.Core.Relation.Relation Having Single Input Relations

sys.Core.Relation.degree of UInt (Relation $topic): This function results in the degree of its argument (that is, the count of attributes it has).
sys.Core.Relation.cardinality of UInt (Relation $topic): This function results in the cardinality of its argument (that is, the count of tuples its body has).
sys.Core.Relation.is_empty of Bool (Relation $topic): This function results in Bool:true iff its argument has zero tuples, and Bool:false otherwise. Note that if you are using a Maybe to represent a sparse data item, analagously to a SQL nullable context, then testing the Maybe with is_empty is analagous to testing a SQL nullable with is null.
sys.Core.Relation.is_not_empty of Bool (Relation $topic): This function is exactly the same as sys.Core.Relation.empty except that it results in the opposite boolean value when given the same argument. And following the analogy with is_empty, is_not_empty is analagous to SQL's is not null.
sys.Core.Relation.exists of Bool (Relation $r, Tuple $t): This function results in Bool:true iff its $t argument matches a tuple of its $r argument, and Bool:false otherwise. This function is like sys.Core.Relation.contains except that the tuple being looked for doesn't have to be wrapped in a relation. This function will fail if the 2 arguments don't have the same heading.
sys.Core.Relation.Tuple_from_Relation of Tuple (Relation $topic): This function results in the Tuple that is the sole member tuple of its argument. This function will fail if its argument does not have exactly one tuple.
sys.Core.Relation.Relation_from_Tuple of Relation (Tuple $topic): This function results in the Relation value those body has just the one Tuple that is its argument.
sys.Core.Relation.insert of Relation (Relation $r, Tuple $t): This function results in a Relation that is the relational union of $r and a relation whose sole tuple is $t; that is, conceptually the result is $t inserted into $r. As a trivial case, if $t already exists in $r, then the result is just $r.
sys.Core.Relation.delete of Relation (Relation $r, Tuple $t): This function results in a Relation that is the relational difference of a relation whose sole tuple is $t and $r; that is, conceptually the result is $t deleted from $r. As a trivial case, if $t already doesn't exist in $r, then the result is just $r.
sys.Core.Relation.evacuate of Relation (Relation $topic): This function results in a Relation that has the same heading as its argument, but with an empty body.
sys.Core.Relation.negation of Relation (Relation $topic): This function results in the relational negation/not of its argument. The result relation has the same heading as its argument, and its body contains all of the tuples that are in the universal relation of the same heading (that is, the relation having all the tuples that could ever exist in a relation with that heading) and that are not in the argument; that is, the result is the relational difference when the argument is subtracted from said universal relation. This function will fail if said universal relation is impossible or impractically large to represent, such as when any attributes are of infinite types.
sys.Core.Relation.rename of Relation (Relation $topic, Cat.BiDiNameMap $map): This function is the same as sys.Core.Tuple.rename but that it operates on and results in a Relation rather than a Tuple.
sys.Core.Relation.project of Relation (Relation $topic, Cat.SetOfName $attrs): This function is the same as sys.Core.Tuple.project but that it operates on and results in a Relation rather than a Tuple. But note that the result relation will have fewer tuples than $topic if any $topic tuples were non-distinct for just the projected attributes.
sys.Core.Relation.remove of Relation (Relation $topic, Cat.SetOfName $attrs): This function is the same as sys.Core.Tuple.remove but that it operates on and results in a Relation rather than a Tuple.
sys.Core.Relation.wrap of Relation (Relation $topic, Cat.SetOfName $inner, Cat.Name $outer): This function is the same as sys.Core.Tuple.wrap but that it operates on and results in a Relation rather than a Tuple, where each of its member tuples was transformed as per sys.Core.Tuple.wrap.
sys.Core.Relation.unwrap of Relation (Relation $topic, Cat.Name $outer): This function is the inverse of sys.Core.Relation.wrap as sys.Core.Tuple.unwrap is to sys.Core.Tuple.wrap.
sys.Core.Relation.group of Relation (Relation $topic, Cat.SetOfName $inner, Cat.Name $outer): TODO.
sys.Core.Relation.ungroup of Relation (Relation $topic, Cat.Name $outer): TODO.
sys.Core.Relation.tclose of Relation (Relation $topic): This function results in the transitive closure of its argument. The argument is a binary relation whose attributes are both of the same declared type, and the result is a relation having the same heading and a body which is a superset of the argument's tuples. Assuming that the argument represents all of the node pairs in a directed graph that have an arc between them, and so each argument tuple represents an arc, tclose will determine all of the node pairs in that graph which have a path between them (a recursive operation), so each tuple of the result represents a path. The result is a superset since all arcs are also complete paths. The tclose function is intended to support recursive queries, such as in connection with the "part explosion problem" (the problem of finding all components, at all levels, of some specified part).
sys.Core.Relation.restrict of Relation (Relation $topic, Cat.NameChain $func, Tuple $assuming): This function results in the relational restriction of its $topic argument as determined by applying the Bool-returning function named in its $func argument when the latter function is curried by its $assuming argument. The result relation has the same heading as $topic, and its body contains the subset of $topic tuples where, for each tuple, the function named by $func results in Bool:true when passed the tuple as its $topic argument and $assuming as its $assuming argument. As a trivial case, if $func is defined to unconditionally result in Bool:true, then this function results simply in $topic; or, for an unconditional Bool:false, this function results in the empty relation with the same heading. Note that this operation is also legitimately known as where. Note that sys.Core.Relation.semijoin is recommended for use instead of sys.Core.Relation.restrict to implement some common kinds of relational restrictions (those composed simply of anded or ored tests for attribute value equality), due to the former's greater simplicity.
sys.Core.Relation.extend of Relation (Relation $topic, Cat.NameChain $func, Tuple $assuming): This function results in a relation that is the same as its $topic argument, but that it has zero or more additional attributes, as determined by applying the Tuple-returning function named in its $func argument when the latter function is curried by its $assuming argument. The result relation has a heading that is a superset of that of $topic, and its body contains the same number of tuples, with all attribute values of $topic retained, and possibly extra present, determined as follows; for each $topic tuple, the function named by $func results in a second tuple when passed the first tuple as its $topic argument and $assuming as its $assuming argument; the first and second tuples have no attribute names in common, and the result tuple is derived by joining the tuples together.
sys.Core.Relation.summarize: TODO.
sys.Core.Relation.substitute: TODO.

Functions for sys.Core.Relation.Relation Having Multiple Input Relations

sys.Core.Relation.contains of Bool (Relation $look_in, Relation $look_for): This function results in Bool:true iff the set of tuples comprising $look_for is a subset of the set of tuples comprising $look_in, and Bool:false otherwise. This function will fail if the 2 arguments don't have the same heading.
sys.Core.Relation.union of Relation (Set{Relation} $topic): This function results in the relational union/inclusive-or of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and relationally unions (which is both commutative and associative) them together until just one is left, which is the result. The result relation has the same heading as all of its inputs, and its body contains every tuple that is in any of the input relations. If $topic has zero values, then union results in the empty relation with the same heading, which is the per-distinct-heading identity value for relational union.
sys.Core.Relation.exclusion of Relation (Bag{Relation} $topic): This function results in the relational exclusion/exclusive-or of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and relationally excludes (which is both commutative and associative) them together until just one is left, which is the result. The result relation has the same heading as all of its inputs, and its body contains every tuple that is in just an odd number of the input relations. If $topic has zero values, then exclusion results in the empty relation with the same heading, which is the per-distinct-heading identity value for relational exclusion. Note that this operation is also legitimately known as symmetric difference, or disjoint union.
sys.Core.Relation.intersection of Relation (Set{Relation} $topic): This function results in the relational intersection/and of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and relationally intersects (which is both commutative and associative) them together until just one is left, which is the result. The result relation has the same heading as all of its inputs, and its body contains only the tuples that are in every one of the input relations. If $topic has zero values, then intersection results in the universal relation with the same heading (that is, the relation having all the tuples that could ever exist in a relation with that heading), which is the per-distinct-heading identity value for relational intersection. Note that this intersection operator is conceptually a special case of join, applicable when the headings of the inputs are the same, and the other will produce the same result as this when given the same inputs, but with the exception that intersection has a different identity value when given zero inputs. This function will fail on a $topic of zero values if the result type's universal relation is impossible or impractically large to represent, such as when any attributes are of infinite types.
sys.Core.Relation.difference of Relation (Relation $source, Relation $filter): This function results in the relational difference when its $filter argument is subtracted from its $source argument. The result relation has the same heading as both of its arguments, and its body contains only the tuples that are in $source and are not in $filter. This function will fail if its 2 arguments do not have the same heading. Note that this difference operator is conceptually a special case of semidifference, applicable when the headings of the inputs are the same.
sys.Core.Relation.semidifference of Relation (Relation $source, Relation $filter): This function is the same as semijoin but that it results in the complementary subset of tuples of $source when given the same arguments. Note that this operation is also legitimately known as antijoin.
sys.Core.Relation.semijoin of Relation (Relation $source, Relation $filter): This function results in the relational semijoin of its $source and $filter arguments. The result relation has the same heading as $source, and its body contains the subset of $source tuples that match those of $filter as per join. Note that relational semijoin is conceptually a short-hand for first doing an ordinary relational join between its 2 arguments, and then performing a relational projection on all of the attributes that just $source has. This function will fail any time that join would fail on the same 2 input relations.
sys.Core.Relation.join of Relation (QuasiSet{Relation} $topic): This function results in the relational join of the N element values of its argument; it is a reduction operator that recursively takes each pair of input values and relationally joins (which is both commutative and associative) them together until just one is left, which is the result. The result relation has a heading that is a union of all of the headings of its inputs, and its body is the result of first pairwise-matching every tuple of each input relation with every tuple of each other input relation, then where each member of a tuple pair has attribute names in common, eliminating pairs where the values of those attributes differ and unioning the remaining said tuple pairs, then eliminating any result tuples that duplicate others. If $topic has zero values, then join results in the nullary relation with one tuple, which is the identity value for relational join. As a trivial case, if any input relation has zero tuples, then the function's result will too; or, if any input is the nullary relation with one tuple, that input can be ignored (see identity value); or, if any 2 inputs have no attribute names in common, then the join of just those 2 is a cartesian product; or, if any 2 inputs have all attribute names in common, then the join of just those 2 is an intersection; or, if for 2 inputs, one's set of attribute names is a proper subset of another's, then the join of just those to is a semijoin with the former filtering the latter. This function will fail if any input relations have attributes with common names but different/incompatible declared types. Note that this operation is also legitimately known as natural inner join.
sys.Core.Relation.product of Relation (QuasiSet{Relation} $topic): This function results in the relational cartesian/cross product of the N element values of its argument; it is conceptually a special case of join where all input relations have mutually distinct attribute names; unlike join, product will fail if any inputs have attribute names in common. Note that this operation is also legitimately known as cartesian/cross join.
sys.Core.Relation.compose of Relation (Relation $r1, Relation $r2): This function results in the relational composition of its 2 arguments. It is conceptually a short-hand for first doing an ordinary relational join between its 2 arguments, and then performing a relational projection on all of the attributes that only one of the arguments has; that is, the result has all of and just the attributes that were not involved in matching the tuples of the 2 arguments. This function will fail any time that join would fail on the same 2 input relations.
sys.Core.Relation.quotient of Relation (Relation $dividend, Relation $divisor): This function results in the quotient when its $dividend argument is divided by its $divisor argument using relational division. Speaking informally, say the relations $dividend and $divisor are called A and B, and their attribute sets are respectively named {X,Y} and {Y}, then the result relation has a heading composed of attributes {X} (so the result and $divisor headings are both complementary subsets of the $dividend heading); the result has all tuples {X} such that a tuple {X,Y} appears in A for all tuples {Y} appearing in B; that is, A / B is shorthand for A{X} - ((A{X} join B) - A){X}.

SYSTEM-DEFINED CORE GENERIC QUASI-NONSCALAR FUNCTIONS

This documentation is pending.

SYSTEM-DEFINED CORE UPDATE OPERATORS

Generic Universal

These update operators are applicable to values of any data type at all.

sys.Core.Universal.assign (UPD: Some.Universal $target ; RO: Some.Universal $v): This update operator will update the variable supplied as its $target argument so that it holds the value supplied as its $v argument. This update operator's arguments must be of compatible declared types; in this case, $v must be a subtype of $target.

Scalar

These update operators are applicable to just one or more specific system-defined core scalar data type.

This documentation is pending.

Nonscalar

These update operators are applicable to mainly nonscalar types, but are generic in that they typically work with any nonscalar types.

Quasi-Nonscalar

This documentation is pending.

SYSTEM-DEFINED CORE SYSTEM SERVICES

These system services are applicable to just one or more specific system-defined core scalar data type.

This documentation is pending.

SYSTEM-DEFINED CORE PROCEDURES

Generic Control-Flow Procedures

These procedures are applicable to use in all kinds of procedures.

sys.Core.Control.fail (RO: Cat.Exception $topic): This procedure will throw the exception given as its argument; this results in the call stack unwinding, and transaction rollbacks, until it is caught.
sys.Core.Control.try_catch (UPD: Tuple $try_updating, Tuple $catch_updating ; RO: Cat.NameChain $try, Cat.NameChain $catch, Tuple $try_assuming, Tuple $catch_assuming): This procedure invokes the procedure named in its $try argument, giving it the arguments $try_updating and $try_assuming as its $updating and $assuming arguments, respectively. If the $try procedure throws an exception, then any state changes it made roll back (but changes made before that don't), and the call stack unwinds to the try_catch itself; then the procedure named by $catch is invoked similarly to $try was, with corresponding arguments, but with the extra read-only argument $topic whose value is a Cat.Exception; if the $catch procedure also throws an exception (such as to say its not handling the thrown one), then that one is not caught and the call stack unwinding plus applicable transaction rollback carries on to the caller of the try_catch. If the $try procedure succeeds (doesn't throw an exception), then the $catch procedure is not called.

AUTHOR

Darren Duncan (perl@DarrenDuncan.net)

LICENSE AND COPYRIGHT

This file is part of the formal specification of the Muldis D language.

See the LICENSE AND COPYRIGHT of Language::MuldisD for details.

ACKNOWLEDGEMENTS

The ACKNOWLEDGEMENTS in Language::MuldisD apply to this file too.

To install Language::MuldisD, copy and paste the appropriate command in to your terminal.

cpanm

cpanm Language::MuldisD

CPAN shell

perl -MCPAN -e shell
install Language::MuldisD

For more information on module installation, please visit the detailed CPAN module installation guide.

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