XQuery 1.0 and XPath 2.0 Formal Semantics

2.1.1 Notations from grammar productions

Grammar productions are used to describe "objects" (values, types, [XPath/XQuery] expressions, etc.) manipulated by the Formal Semantics. The Formal Semantics makes use of several kinds of grammar productions.

XQuery grammar productions describe the XQuery language and expressions. XQuery productions are identified by a number, which corresponds to their number in the [XQuery 1.0: A Query Language for XML] document, and are annotated with "(XQuery)". For instance, the following production describes FLWR expressions in XQuery.

[For/FLWR] Expressions

For the purpose of this document, the differences between the XQuery 1.0 and the XPath 2.0 grammars are mostly irrelevant. By default, this document uses XQuery 1.0 grammar productions. Whenever the grammar for XPath 2.0 differs from the one for XQuery 1.0, the corresponding XPath 2.0 productions are also given. XPath productions are identified by a number, which corresponds to their number in [XML Path Language (XPath) 2.0], and are annotated with "(XPath)". For instance, the following production describes for expressions in XPath.

[For/FLWR] Expressions

XQuery Core grammar productions describe the XQuery Core. The Core grammar is given in [A Normalized core grammar]. Core productions are identified by a number, which corresponds to their number in [A Normalized core grammar], and are annotated by "(Core)". For instance, the following production describes the simpler form of the "FLWR" expression in the XQuery Core.

Core FLWOR Expressions

The Formal Semantics manipulates "objects" (values, types, expressions, etc.) for which there is no existing grammar production in the [XQuery 1.0: A Query Language for XML] document. In these cases, specific grammar productions are introduced. Notably, additional productions are used to describe values in the [Data Model], and to describe the [XPath/XQuery] type system. Formal Semantics productions are identified by a number, and are annotated by "(Formal)". For instance, the following production describes global type definitions in the [XPath/XQuery] type system.

Type Definitions

Note that grammar productions that are specific to the Formal Semantics (i.e., with the "(Formal)" annotation) are not part of [XPath/XQuery]. They are not accessible to the user and are only used in the course of defining the language's semantics.

2.1.2 Notations for judgments

The basic building block of the formal specification is called a judgment. A judgment expresses whether a property holds or not.

For example:

Notation

The judgment

holds if the object Painting is beautiful.

Notation

Here are three judgments that are used extensively in this document.

The judgment

holds if the expression Expr yields (or evaluates to) the value Value.

The judgment

holds when the expression Expr has type Type.

The judgment

holds when the expression Expr raises the error Error.

A judgment can contain symbols and patterns.

Symbols are purely syntactic and are used to write the judgment itself. In general, symbols in a judgment are chosen to reflect its meaning. For example, 'is beautiful', '=>' and ':' are symbols, the second and third of which should be read "yields", and "has type" respectively.

Patterns are written with italicized words. The name of a pattern is significant: each pattern name corresponds to an "object" (a value, a type, an expression, etc.) that can be substituted legally for the pattern. By convention, all patterns in the Formal Semantics correspond to grammar non-terminals, and are used to represent entities that can be constructed through application of the corresponding grammar production. For example, Expr represents any [XPath/XQuery] expression, and Value represents any value in the [XPath/XQuery] data model.

When applying the judgment, each pattern must be instantiated to an appropriate sort of "object" (value, type, expression, etc). For example, '3 => 3' and '$x+0 => 3' are both instances of the judgment 'Expr => Value'. Note that in the first judgment, '3' corresponds to both the expression '3' (on the left-hand side of the => symbol) and to the the value '3' (on the right-hand side of the => symbol).

Patterns may appear with subscripts (e.g. Expr₁, Expr₂) to distinguish different instances of the same sort of pattern. Each distinct pattern must be instantiated to a single "object" (value, type, expression, etc.). If the same pattern occurs twice in a judgment description then it should be instantiated with the same "object". For example, '3 => 3' is an instance of the judgment 'Expr₁ => Expr₁' but '$x+0 => 3' is not since the two expressions '$x+0' and '3' cannot be both instance of the pattern Expr₁. The judgment'$x+0 => 3' is an instance of the judgment 'Expr₁ => Expr₂'.

In a few cases, patterns may have a name that is not exactly the name of a grammar production but is based on it. For instance, a BaseTypeName is a pattern that stands for a type name, as would TypeName, or TypeName₂. This usage is limited, and only occurs to improve the readability of some of the inference rules.

2.1.3 Notations for inference rules

Inference rules are used to specify whether a judgment holds or not. Inference rules express the logical relation between judgments and describe how complex judgments can be concluded from simpler premise judgments.

A logical inference rule is written as a collection of premises and a conclusion, written respectively above and below a dividing line:

All premises and the conclusion are judgments. The interpretation of an inference rule is: if all the premise judgments above the line hold, then the conclusion judgment below the line must also hold.

Here is a simple example of inference rule, which uses the example judgment 'Expr => Value' from above:

This inference rule expresses the following property of the judgment 'Expr => Value': if the variable expression '$x' yields the value '0', and the literal expression '3' yields the value '3', then the expression '$x + 3' yields the value '3'.

An inference rule may have no premises above the line, which means that the expression below the line always holds:

This inference rule expresses the following property of the judgment 'Expr => Value': evaluating the literal expression '3' always yields the value '3'.

The two above rules are expressed in terms of specific variables and values, but usually rules are more abstract. That is, the judgments they relate contain patterns. Here is a rule that says that for any variable Variable that yields the integer value Integer, adding '0' yields the same integer value:

As in a judgment, each pattern in a particular inference rule must be instantiated to the same "object" within the entire rule. This means that one can talk about "the value of Variable" instead of the more precise "what Variable is instantiated to in (this particular instantiation of) the inference rule".

Note

In effect, inference rules are just a notation that describes a bottom-up algorithm. In the examples above, the rules describe an evaluation algorithm where the result of an expression depends on the result for its sub-expressions.

2.1.4 Notations for environments

Logical inference rules use environments to record information computed during static type analysis or dynamic evaluation so that this information can be used by other logical inference rules. For example, the type signature of a user-defined function in a [expression/query] prolog can be recorded in an environment and used by subsequent rules. Similarly, the value assigned to a variable within a "let" expression can be captured in an environment and used for further evaluations.

An environment is a dictionary that maps a symbol (e.g., a function name or a variable name) to an "object" (e.g., a function body, a type, a value). One can access information in an environment or update the environment.

The two main environments used in the Formal Semantics are: a dynamic environment (dynEnv), which captures the [XPath/XQuery]'s dynamic context, and a static environment (statEnv), which captures the [XPath/XQuery]'s static context. Both are defined in [3.1 Expression Context].

If "env" is an environment, then "env(symbol)" denotes the "object" to which symbol is mapped. The notation is intentionally similar to function application, because an environment can be considered a function from the argument symbol to the "object" to which the symbol is mapped.

This document uses environment groups that group related environments. If "env" is an environment group with the member "mem", then that environment is denoted "env.mem" and the value that it maps symbol to is denoted "env.mem(symbol)". For example, dynEnv.varValue denotes the dynamic environment that maps variables to values and dynEnv.varValue(Variable) denotes the value of the variable Variable in the dynamic environment.

Updating is only defined on environment groups:

• "env + mem(symbol => object) " denotes the new environment group that is identical to env except that the mem environment has been updated to map symbol to object. The notation symbol => object indicates that symbol is mapped to object in the new environment.

• The following shorthand is also allowed: "env + mem( symbol₁ => object₁ ; ... ; symbol_n => object_n ) " in which each symbol is mapped to a corresponding object in the new environment.

  This notation is equivalent to nested updates, as in " (env + mem( symbol₁ => object₁) + ... ) + mem(symbol_n => object_n)".

• If the "object" is a type then the following notation relates a symbol to a type: "env + mem(symbol : object) ".

Updating the environment overrides any previous binding that might exist for the same name. Updating the environment captures the scope of a symbol (e.g., a variable, a namespace prefix, etc.) Also, note that there are no operations to remove entries from environments: this is never necessary because updating an the environment group effectively creates a new extended copy of the original environment group, and the original environment group remains accessible along with the updated copy.

Environments are used in a judgment to capture some of the context in which the judgment is computed, and most judgments are computed assuming that some environment is given. This assumption is denoted by prefixing the judgment with "env |-". The "|-" symbol is called a "turnstile" and is used in almost all inference rules.

For instance, the judgment

is read as: Assuming the dynamic environment dynEnv, the expression Expr yields the value Value.

2.1.5 Putting it together

Putting the above notations together, here is an example of an inference rule that occurs later in this document:

This rule is read as follows: if two expressions Expr₁ and Expr₂ are known to have the static types types Type₁ and Type₂ (the two premises above the line), then it is the case that the expression below the line "Expr₁ , Expr₂" must have the static type "Type₁, Type₂", which is the sequence of types Type₁ and Type₂. The above inference rule does not modify the (static) environment.

The following rule defines the static semantics of a "let" expression. The binding of the new variable is captured by an update to the varType component of the original static environment.

This rule is read as follows: First, because the variable is a QName, it is first expanded into an expanded QName. Second, the type Type₁ for the "let" input expression Expr₁ is computed. Ten the "let" variable with expanded name, expanded-QName with type Type₁ is added into the varType component of the static environment group statEnv. Finally, the type Type₂ of Expr₂ is computed in that new environment.

2.2.1 Formal values

For reference, a summary of the data model is given below, followed by the formal notation for data model values. Although not specified in this document, all the normative constraints specified in [Data Model] apply to the formal notation for data model values.

A value is a sequence of zero or more items. An item is either an atomic value or a node.

An atomic value is a value in the value space of an atomic type and is labeled with the name of that atomic type. An XML Schema atomic type [Schema Part 2] may be primitive or derived, or xdt:untypedAtomic.

A node is either an element, an attribute, a document, a text, a comment, or a processing-instruction node. Elements have a type annotation and contain a value. Attributes have a type annotation and contain a simple value, which is a sequence of atomic values. Text nodes always contain one string value of type xdt:untypedAtomic, therefore the corresponding type annotation is omitted.

A type annotation can be either the QName of a declared type or an anonymous type. An anonymous type corresponds to an XML Schema type for which the schema writer did not provide a name. Anonymous type names are not visible to the user, but are generated during schema validation and used to annotate nodes in the data model. By convention, anonymous type names are written in the Formal Semantics as: [Anon0], [Anon1], etc.

Untyped elements (e.g., from well-formed documents) are annotated with xdt:untypedAny, untyped attributes are annotated with xdt:untypedAtomic, and untyped atomic values (i.e., text content or attribute content in well-formed documents) are annotated with xdt:untypedAtomic.

An element has an optional "nilled" marker. This marker can only be present if the element has been validated against an element type in the schema which is "nillable", and the element has no content and an attribute xsi:nil set to "true".

An element also has a sequence of namespace annotations, which are the set of active namespace declarations that are in-scope for the element. Each namespace annotation is a prefix, URI pair. Namespace annotations are not values, i.e., they are never the result of evaluating an expression, and they only occur as annotations on elements. In examples, we omit the namespace annotations when they are empty. For example, the following two values are equivalent:

  element weight of type xs:integer { text { "42" } } {}
  element weight of type xs:integer { text { "42" } } 

Values

Notation

In the above grammar, "String" indicates the value space of xs:string, "Decimal" indicates the value space of xs:decimal, etc.

Note that the same rule about constructing sequences apply to the values described by that grammar. Notably sequences cannot be nested. For example, the sequence (10, (1, 2), (), (3, 4)) is equivalent to the sequence (10, 1, 2, 3, 4).

2.2.2 Examples of values

A well-formed document

  <fact>The cat weighs <weight units="lbs">12</weight> pounds.</fact>

In the absence of a Schema, this document is represented as

  element fact of type xdt:untyped {
    text { "The cat weighs " },
    element weight of type xdt:untyped {
      attribute units of type xdt:untypedAtomic {
        "lbs" of type xdt:untypedAtomic
      }
      text { "12" }
    },
    text { " pounds." }
  }

A document before and after validation.

  <weight xsi:type="xs:integer">42</weight>

The formal model for values can represent values before and after validation. Before validation, this element is represented as:

  element weight of type xdt:untyped {
    attribute xsi:type of type xdt:untypedAtomic {
      "xs:integer" of type xdt:untypedAtomic
    },
    text { "42" }
  }

After validation, this element is represented as:

  element weight of type xs:integer {
    attribute xsi:type of type xs:QName {
      "xs:integer" of type xs:QName
    },
    42 of type xs:integer
  }

An element with a list type

  <sizes>1 2 3</sizes>

Before validation, this element is represented as:

  element sizes of type xdt:untyped {
    text { "1 2 3" }
  }

Assume the following Schema.

  <xs:element name="sizes" type="sizesType"/>
  <xs:simpleType name="sizesType">
    <xs:list itemType="sizeType"/>
  </xs:simpleType>
  <xs:simpleType name="sizeType">
    <xs:restriction base="xs:integer"/>
  </xs:simpleType>

After validation against this Schema, the element is represented as:

  element sizes of type sizesType {
    1 of type sizeType,
    2 of type sizeType,
    3 of type sizeType
  }

An element with an anonymous type

  <sizes>1 2 3</sizes>

Before validation, this element is represented as:

  element sizes of type xdt:untyped {
    text { "1 2 3" }
  }

Assume the following Schema.

  <xs:element name="sizes">
    <xs:simpleType>
      <xs:list itemType="xs:integer"/>
    </xs:simpleType>
  </xs:element>

After validation, this element is represented as:

  element sizes of type [Anon1] {
    1 of type xs:integer,
    2 of type xs:integer,
    3 of type xs:integer
  }

where [Anon1] stands for the internal anonymous name generated by the system for the sizes element.

An nillable element with xsi:type set to true

  <sizes xsi:nil="true"/>

Before validation, this element is represented as:

  element sizes of type xdt:untyped {
    attribute xsi:nil of type xdt:untypedAtomic { "true" }
  }

Assume the following Schema.

  <xs:element name="sizes" type="sizesType" nillable="true"/>

After validation against this Schema, the element is represented as:

  element sizes nilled of type sizesType {
    attribute xsi:nil of type xs:boolean { "true" }
  }

An element with a union type

  <sizes>1 two 3 four</sizes>

Before validation, this element is represented as:

  element sizes of type xdt:untypedAtomic {
    text { "1 two 3 four" }
  }

Assume the following Schema:

  <xs:element name="sizes" type="sizesType"/>
  <xs:simpleType name="sizesType">
    <xs:list itemType="sizeType"/>
  </xs:simpleType>
  <xs:simpleType name="sizeType">
    <xs:union memberType="xs:integer xs:string"/>
  </xs:simpleType>

After validation against this Schema, the element is represented as:

  element sizes of type sizesType {
    1 of type xs:integer,
    "two" of type xs:string,
    3 of type xs:integer,
    "four" of type xs:string
  }

2.3.1 XML Schema and the [XPath/XQuery] Type System

The [XPath/XQuery] type system is based on [Schema Part 1] and [Schema Part 2]. [Schema Part 1] and [Schema Part 2] specify normatively the schema information available in [XPath/XQuery]. Formalizing the treatment of types in [XPath/XQuery], however, requires some adjustments.

Use of formal notations for types. The Formal Semantics uses formal notations for types instead of XML Schema syntax. These notations are used extensively to describe and manipulate types in the inference rules. The formal notations for types introduced here are not exposed to the [XPath/XQuery] user.

Representation of content models. For the purpose of static typing, the [XPath/XQuery] type system only describes minOccurs, maxOccurs combinations that correspond to the DTD operators +, *, and ?. Choices are represented using the DTD operator |. All groups are represented using the & notation.

Representation of anonymous types. To clarify the semantics, the [XPath/XQuery] type system makes all anonymous types explicit.

Representation of XML Schema simple type facets and identity constraints. For simplicity, XML Schema simple type facets and identity constraints are not formally represented in the [XPath/XQuery] type system. However, an [XPath/XQuery] implementation supporting XML Schema import and validation must must take simple type facets and identity constraints into account.

The mapping from XML Schema into the [XPath/XQuery] type system is given in [8 Importing Schemas]. The rest of this section is organized as follows. [2.3.2 Item types] describes types items, [2.3.3 Content models] describes content models, and [2.3.4 Top level definitions] describe top-level type declarations.

2.3.2 Item types

An item type is either an atomic type, an element type, an attribute type, a document node type, a text node type, a comment node type, or a processing instruction type.

Item Types

An element or attribute type has an optional name and an optional type reference. A name alone corresponds to a reference to a global element or attribute declaration. A name with a type reference corresponds to a local element or attribute declaration. The word "element" or "attribute" alone refers to the wildcard types for any element or any attribute. In addition, an element type has an optional nillable flag that indicates whether the element can be nilled or not.

A document type has an optional content type. If no content type is given, then it refers to the wildcard type describing any document.

Note

Generic node types (e.g., node()), are interpreted in the type system as union types (e.g., element | attribute | text | comment | processing-instruction) and therefore do not appear here. The semantics of sequence types is described in [3.4.4 SequenceType Matching].

Examples

The following is a text node type

  text

The following is a type for all elements

  element

The following is a type for all elements of type string

  element of type xs:integer 

The following is a type for a nillable element of type string

  element size nillable of type xs:integer 

The following is a reference to a global attribute declaration

  attribute sizes

The following is a type for elements with anonymous type [Anon1]

  element sizes of type [Anon1]

2.3.3 Content models

Following XML Schema, types in [XPath/XQuery] are composed from item types by optional, one or more, zero or more, all group, sequence, choice, empty sequence, or empty choice (written none).

The type empty matches the empty sequence. The type none matches no values. It is called the empty choice because it is the identity for choice, that is (Type | none) = Type)). The type none is the static type for [6.2.5 The fn:error function].

Types

The [XPath/XQuery] type system includes three binary operators on types: ",", "|" and "&", corresponding respectively to sequence, choice and all groups in Schema. The [XPath/XQuery] type system includes three unary operators on types: "*", "+", and "?", corresponding respectively to zero or more instances of the type, one or more instances of the type, or an optional instance of the type.

The "&" operator builds the "interleaved product" of two types. The type Type₁ & Type₂ matches any sequence that is an interleaving of a sequence that matches Type₁ and a sequence that matches Type₂. The interleaved product represents all groups in XML Schema. All groups in XML Schema are restricted to apply only on global or local element declarations with lower bound 0 or 1, and upper bound 1.

Examples

A sequence of elements

The "," operator builds the "sequence" of two types. For example,

  element title of type xs:string, element year of type xs:integer

is a sequence of an element title of type string followed by an element year of type integer.

The union of two element types

The "|" operator builds the "union" of two types. For example,

  element editor of type xs:string | element bib:author

means either an element editor of type string, or a reference to the global element bib:author.

An all group of two elements

The "&" operator builds the "interleaved product" of two types. For example,

  (element a & element b) =
    element a, element b
  | element b, element a

which specifies that the a and b elements can occur in any order.

An empty type

The following type matches the empty sequence.

  empty

A sequence of zero or more elements

The following type matches zero or more elements each of which can be a surgeon or a plumber.

  (element surgeon | element plumber)*

2.3.4 Top level definitions

Top level definitions correspond to global element declarations, global attribute declarations and type definitions in XML Schema.

Type Definitions

A type definition has a name (possibly anonymous) and a type derivation. In the case of a complex type, or a simple type derived by list or union, derivation indicates if the type is derived by extension or restriction from its base type, gives its content model, and an optional flag indicating if it has mixed content. In the case of an atomic type, it just indicates from which type it is derived. When the type derivation is omitted, the type derives by restriction from xs:anyType, as in:

  define type Bib { xs:integer } =
  define type Bib restricts xs:anyType { xs:integer }

Empty content can be indicated with the explicit empty sequence, or omitted, as in:

  define type Bib { } =
  define type Bib { empty }

Global element and attribute declarations always have a name and a reference to a (possibly anonymous) type. A global element declaration also may declare a substitution group for the element and whether the element is nillable.

Examples

A type declaration with complex content

  define type Address {
    element name of type xsd:string,
    element street of type xsd:string*
  }

A type declaration with complex content derived by extension

  define type USAddress extends Address {
    element zip name of type xsd:integer
  }

A type declaration with mixed content

  define type Section mixed {
    (element h1 of type xsd:string |
     element p of type xsd:string |
     element div of type Section)*
  }

A type declaration with simple content derived by restriction

  define type SKU restricts xsd:string { xsd:string }

An element declaration

  define element address of type Address

An element declaration with a substitution group

  define element usaddress substitutes for address of type USAddress

An element declaration which is nillable

  define element zip nillable of type xs:integer

2.3.5 Example of a complete Schema

Here is a schema describing purchase orders from [XML Schema Part 0].

  <xsd:schema xmlns:xsd="http://www.w3.org/2001/XMLSchema">
  
   <xsd:annotation>
    <xsd:documentation xml:lang="en">
     Purchase order schema for Example.com.
     Copyright 2000 Example.com. All rights reserved.
    </xsd:documentation>
   </xsd:annotation>
  
   <xsd:element name="purchaseOrder" type="PurchaseOrderType"/>
  
   <xsd:element name="comment" type="xsd:string"/>
  
   <xsd:complexType name="PurchaseOrderType">
    <xsd:sequence>
     <xsd:element name="shipTo" type="USAddress"/>
     <xsd:element name="billTo" type="USAddress"/>
     <xsd:element ref="comment" minOccurs="0"/>
     <xsd:element name="items"  type="Items"/>
    </xsd:sequence>
    <xsd:attribute name="orderDate" type="xsd:date"/>
   </xsd:complexType>
  
   <xsd:complexType name="USAddress">
    <xsd:sequence>
     <xsd:element name="name"   type="xsd:string"/>
     <xsd:element name="street" type="xsd:string"/>
     <xsd:element name="city"   type="xsd:string"/>
     <xsd:element name="state"  type="xsd:string"/>
     <xsd:element name="zip"    type="xsd:decimal"/>
    </xsd:sequence>
    <xsd:attribute name="country" type="xsd:NMTOKEN"
        fixed="US"/>
   </xsd:complexType>
  
   <xsd:complexType name="Items">
    <xsd:sequence>
     <xsd:element name="item" minOccurs="0" maxOccurs="unbounded">
      <xsd:complexType>
        <xsd:sequence>
         <xsd:element name="productName" type="xsd:string"/>
         <xsd:element name="quantity">
          <xsd:simpleType>
           <xsd:restriction base="xsd:positiveInteger">
            <xsd:maxExclusive value="100"/>
           </xsd:restriction>
          </xsd:simpleType>
         </xsd:element>
         <xsd:element name="USPrice"  type="xsd:decimal"/>
         <xsd:element ref="comment"   minOccurs="0"/>
         <xsd:element name="shipDate" type="xsd:date" minOccurs="0"/>
        </xsd:sequence>
        <xsd:attribute name="partNum" type="SKU" use="required"/>
      </xsd:complexType>
     </xsd:element>
    </xsd:sequence>
   </xsd:complexType>
  
   <!-- Stock Keeping Unit, a code for identifying products -->
   <xsd:simpleType name="SKU">
    <xsd:restriction base="xsd:string">
     <xsd:pattern value="\d{3}-[A-Z]{2}"/>
    </xsd:restriction>
   </xsd:simpleType>
  
  </xsd:schema>

Here is the mapping of the above schema into the [XPath/XQuery] type system.

  namespace xsd = "http://www.w3.org/2001/XMLSchema"

  define element purchaseOrder of type PurchaseOrderType
 
  define element comment of type xsd:string
  
  define type PurchaseOrderType {
    attribute orderDate of type xsd:date?,
    element shipTo of type USAddress,
    element billTo of type USAddress,
    element comment?,
    element items of type Items
  }

  define type USAddress {
    attribute country of type xsd:NMTOKEN?,
    element name of type xsd:string,
    element street of type xsd:string,
    element city of type xsd:string,
    element state of type xsd:string,
    element zip of type xsd:decimal
  }

  define type Items {
    attribute partNum of type SKU,
    element item of type [Anon1]*
  }

  define type [Anon1] {
    element productName of type xsd:string,
    element quantity of type [Anon2],
    element USPrice of type xsd:decimal,
    element comment?,
    element shipDate of type xsd:date?
  }

  define type [Anon2] restricts xsd:positiveInteger

  define type SKU restrict xsd:string

Note that the two anonymous types in the item element declarations are mapping to types with names [Anon1] and [Anon2].

The following additional definitions illustrate how more advanced XML Schema features (a complex type derived by extension, an anonymous simple type derived by restriction, and substitution groups) are represented in the [XPath/XQuery] type system.

  <complexType name="NYCAddress">
    <complexContent>
     <extension base="USAddress">
      <sequence>
       <element ref="apt"/>
      </sequence>
     </extension>
    </complexContent>
  </complexType>

  <element name="apt">
    <xsd:simpleType>
     <xsd:restriction base="xsd:positiveInteger">
      <xsd:maxExclusive value="10000"/>
     </xsd:restriction>
    </xsd:simpleType>
  </element>

  <element name="usaddress" substitutionGroup="address" type="USAddress"/>
  <element name="nycaddress" substitutionGroup="usaddress" type="NYCAddress"/>

The above definitions are mapped into the [XPath/XQuery] type system as follows:

  define type NYCAddress extends USAddress {
    element apt
  }

  define element apt of type [Anon3]

  define type [Anon3] restricts xsd:positiveInteger

  define element usaddress  substitutes for address of type USAddress
  define element nycaddress substitutes for usaddress of type NYCAddress

2.4.1 Processing model

The [XPath/XQuery] processing model is defined in Section 2.2 Processing Model^XQ, which contains the following figure depicting the processing model.

Figure 1: Processing Model Overview

This processing model is not intended to describe an actual implementation, although a naive implementation might be based upon it. It does not prescribe an implementation technique, but any implementation should produce the same results as obtained by following this processing model and applying the rest of the Formal Semantics specification.

Query processing consists of two phases: a static analysis phase and a dynamic evaluation phase. Static analysis is further divided into four sub-phases. Each phase consumes the result of the previous phase and generates output for the next phase. For each processing phase, we point to the relevant notations introduced later in the document.

The static analysis phase (sometimes also called "compilation"), can be performed on a [expression/query] before examining any input document. Indeed, static analysis can be performed before input documents even exist. Static analysis can detect many errors early, e.g., syntax errors or type errors. If no error occurs, the result of static analysis could be some compiled form of [expression/query], suitable for execution by a compiled-[expression/query] processor. Static analysis consists of the following sub-phases:

1. Parsing. (Step SQ1 in Figure 1). The grammar for the [XPath/XQuery] syntax is defined in [XQuery 1.0: A Query Language for XML]. Parsing may generate syntax errors. If no error occurs, an internal operation tree of the parsed query is created.

2. Static Context Processing. (Steps SQ2, SQ3, and SQ4 in Figure 1). The static semantics of [expression/query] depends on the static input context. The static input context needs to be generated before the [expression/query] can be analysed. In XQuery, static the input context may be defined by the processing environment and by declarations in the Query Prolog (See [5 Modules and Prologs]). In XPath, the static input context is defined by the processing environment. The static input context is denoted by statEnv.

3. Normalization. (Step SQ5 in Figure 1). To simplify the semantics specification, some normalization is performed on the [expression/query]. The [XPath/XQuery] language provides many powerful features that make [expression/query]s simpler to write and use, but are also redundant. For instance, a complex for expression might be rewritten as a composition of several simple for expressions. The language composed of these simpler [expression/query] is called the [XPath/XQuery] Core language and is described by a grammar which is a subset of the XQuery grammar. The grammar of the [XPath/XQuery] Core language is given in [A Normalized core grammar].

   During the normalization phase, each [XPath/XQuery] [expression/query] is mapped into its equivalent [expression/query] in the core. (Note that this has nothing to do with Unicode Normalization, which works on character strings.) Normalization works by bottom-up application of normalization rules over expressions, starting with normalization of literal expressions and variables.

   Specifically the normalization phase is defined in terms of the static part of the context (statEnv) and a [expression/query] (Expr) abstract syntax tree. Formal notations for the normalization phase are introduced in [2.4.2 Normalization judgment].

   After normalization, the full semantics is obtained by giving a semantics to the normalized Core [expression/query]. This is done during the last two phases.

4. Static type analysis. (Step SQ6 in Figure 1). Static type analysis is optional. If this phase is not supported, then normalization is followed directly by dynamic evaluation.

   Static type analysis checks whether each [expression/query] is type safe, and if so, determines its static type. Static type analysis is defined only for Core [expression/query]. Static type analysis works by bottom-up application of type inference rules over expressions, taking the type of literals and of input documents into account.

   If the [expression/query] is not type-safe, static type analysis yields a type error. For instance, a comparison between an integer value and a string value might be detected as an type error during the static type analysis. If static type analysis succeeds, it yields an abstract syntax tree where each sub-expression is "annotated" with its static type.

   More precisely, the static analysis phase is defined in terms of the static context (statEnv) and a core [expression/query] (CoreExpr). Formal notations for the static analysis phase are introduced in [2.4.3 Static typing judgment].

   Static typing does not imply that the content of XML documents must be rigidly fixed or even known in advance. The [XPath/XQuery] type system accommodates "flexible" types, such as elements that can contain any content. Schema-less documents are handled in [XPath/XQuery] by associating a standard type with the document, such that it may include any legal XML content.

The dynamic evaluation phase (sometimes also called "execution") evaluates a query on input document(s).

1. Dynamic Context Processing. (Steps DQ2 and DQ3 in Figure 1).The dynamic semantics of [expression/query] depends on the dynamic input context. The dynamic input context needs to be generated before the [expression/query] can be evaluated. The dynamic input context may be defined by the processing environment and by statements in the Query Prolog (See [5 Modules and Prologs]). In XPath, the dynamic input context is defined by the processing environment. The static input context is denoted by dynEnv.

2. Dynamic Evaluation. (Steps DQ4 and DQ5 in Figure 1). This phase computes the value of an [expression/query]. The semantics of evaluation is defined only for Core [expression/query] terms. Evaluation works by bottom-up application of evaluation rules over expressions, starting with evaluation of literals and variables. Evaluation may result in a value OR a dynamic error, which may be a non-type error or a type error. If static typing of an expression does not raise a type error, then dynamic evaluation of the same expression will not raise a type error, although dynamic evaluation may raise some non-type error.

   The dynamic evaluation phase is defined in terms of the static context (statEnv) and evaluation context (dynEnv), and a core [expression/query] (CoreExpr). Formal notations for the dynamic evaluation phase are introduced in [2.4.4 Dynamic evaluation judgment].

Static type analysis catches only certain classes of errors. For instance, it can detect a comparison operation applied between incompatible types (e.g., xs:int and xs:date). Some other classes of errors cannot be detected by the static analysis and are only detected at evaluation time. For instance, whether an arithmetic expression on 32 bits integers (xs:int) yields an out-of-bound value can only be detected at run-time by looking at the data.

While implementations are free to implement different processing models, the [XPath/XQuery] static semantics relies on the existence of a static type analysis phase that precedes any access to the input data. Statically typed implementations are required to find and report type errors during static analysis, as specified in this document. Dynamically typed implementations are required to find and report type errors during evaluation, but are permitted to report them during static analysis.

Notice that the separation of logical processing into phases is not meant to imply that implementations must separate the static analysis phase from the dynamic evaluation phase; processors may choose to perform all phases simultaneously at evaluation-time and may even mix the phases in their internal implementations. The processing model simply defines the final result.

The above processing phases are all internal to the [XPath/XQuery] processor. They do not deal with how the [XPath/XQuery] processor interacts with the outside world, notably how it accesses actual documents and types. A typical [expression/query] engine would support at least three other important processing phases:

1. Schema Import Processing. The [XPath/XQuery] type system is based on XML Schema. In order to perform dynamic or static typing, the [XPath/XQuery] processor needs to build type descriptions that correspond to the schema(s) of the input documents. This phase is achieved by mapping all schemas required by the [expression/query] into the [XPath/XQuery] type system. The XML Schema import phase is described in [8 Importing Schemas].

2. Data Model Generation. Expressions are evaluated on values in the [Data Model]. XML documents must be loaded into the [Data Model] before the evaluation phase. This is described in the [Data Model] and is not discussed further here.

3. Serialization. Once the [expression/query] is evaluated, processors might want to serialize the result of the [expression/query] as actual XML documents. Serialization of data model instances is described in [Data Model Serialization] and is not discussed further here.

The parsing phase is not specified formally; the formal semantics does not define a formal model for the syntax trees, but uses the [XPath/XQuery] concrete syntax directly. More details about parsing for XQuery 1.0 can be found in the [XQuery 1.0: A Query Language for XML] document and more details about parsing for XPath 2.0 can be found in the [XML Path Language (XPath) 2.0] document. No further discussion of parsing is included here.

2.4.2 Normalization judgment

Normalization is specified using mapping rules, which describe how a [XPath/XQuery] expression is rewritten into an expression in the [XPath/XQuery] Core. Mapping rules are also used in [8 Importing Schemas] to specify how XML Schemas are imported into the [XPath/XQuery] type system.

Notation

Mapping rules are written using a square bracket notation, as follows:

The original "object" is written above the == sign. The rewritten "object" is written beneath the == sign. The subscript is used to indicate what kind of "object" is mapped, and sometimes to pass some information between mapping rules.

Since normalization is always applied in the presence of a static context, the above rule is a shorthand for:

The static environment is used in certain normalization rules (e.g. for normalization of function calls). To keep the notation simpler, the static environment is not written in the normalization rules, but it is assumed to be available.

The normalization rule that is used to map "top-level" expressions in the [XPath/XQuery] syntax into expressions in the [XPath/XQuery] Core is:

which indicates that the expression Expr is normalized to the expression CoreExpr in the [XPath/XQuery] core (with the implied statEnv).

Example

For instance, the following [expression/query]

    for $i in (1, 2),
        $j in (3, 4)
    return
      element pair { ($i,$j) }

is normalized to the core expression

    for $i in (1, 2) return
      for $j in (3, 4) return
          element pair { ($i,$j) }

in which the "FWLR" expression is mapped into a composition of two simpler "for" expressions.

2.4.3 Static typing judgment

The static semantics is specified using type inference rules, which relate [XPath/XQuery] expressions to types and specify under what conditions an expression is well typed.

Notation

The judgment

holds when, in the static environment statEnv, the expression Expr has type Type.

Example

The result of static type inference is to associate a static type with every [expression/query], such that any evaluation of that [expression/query] is guaranteed to yield a value that belongs to that type.

For instance, the following expression.

   let $v := 3 return $v+5

has type xs:integer. This can be inferred as follows: the input literals '3' and '5' have type integer, so the variable $v also has type integer. Since the sum of two integers is an integer, the complete expression has type integer.

Note

The type of an expression is computed by inference. Static type inference rules define for each kind of expression how to compute the type of the expression given the types of its sub-expressions. Here is a simple example:

This rule states that if the conditional expression of an "if" expression has type boolean, then the type of the entire expression is one of the two types of its "then" and "else" clauses. Note that the resulting type is represented as a union: '(Type₂|Type₃)'.

The "left half" (the part before the :) of the expression below the line corresponds to some [expression/query], for which a type is computed. If the [expression/query] has been parsed into an internal abstract syntax tree, this usually corresponds to some node in that tree. The expression usually has patterns in it (here Expr₁, Expr₂, and Expr₃) that need to be matched against the children of the node in the abstract syntax tree. The expressions above the line indicate things that need to be computed to use this rule; in this case, the types of the condition expression and the two branches of the if-then-else expression. Once those types are computed (by further applying static inference rules recursively to the expressions on each side), then the type of the expression below the line can be computed. This example illustrates a general feature of the [XPath/XQuery] type system: the type of an expression depends only on the type of its sub-expressions. The overall static type inference algorithm is recursive, following the abstract syntax of the [expression/query]. At each point in the recursion, an appropriate matching inference rule is sought; if at any point there is no applicable rule, then static type inference has failed and the [expression/query] is not type correct.

2.4.4 Dynamic evaluation judgment

The dynamic, or operational, semantics is specified using value inference rules, which relate [XPath/XQuery] expressions to values, and in some cases specify the order in which an [XPath/XQuery] expression is evaluated.

Notation

The judgment

holds when, in the static environment statEnv and dynamic environment dynEnv, the expression Expr yields the value Value.

The judgment

holds when, in the static environment statEnv and dynamic environment dynEnv, the expression Expr raises the error Error.

The static environment is used in certain cases (e.g. for type matching) during evaluation. To keep the notation simpler, the static environment is not written in the dynamic inference rules, but it is assumed to be available.

Example

For instance, the following expression.

   let $v := 3 return $v+5

yields the integer value 8. This can be inferred as follows: the input literals '3' and '5' denote the values 3 and 5, respectively, so the variable $v has the value 3. Since the sum of 3 and 5 is 8, the complete expression has the value 8.

Note

As with static type inference, logical inference rules are used to determine the value of each expression, given the dynamic environment and the values of its sub-expressions.

The inference rules used for dynamic evaluation, like those for static type inference, follow a bottom-up recursive structure, computing the value of expressions from the values of their sub-expressions.

2.5.1 Namespaces

The Formal Semantics uses the following namespace prefixes.

• fn: for functions and operators from the [Functions and Operators] document.

• xs: for XML Schema components and built-in types.

• xdt: for XML Schema components and built-in types.

All these prefixes are assumed to be bound to the appropriate URIs.

In addition, the Formal Semantics uses the following special prefixes for the means of specification.

• dm: for accessors of the [Data Model].

• op: for operators in [Functions and Operators].

• fs: for functions and types defined in the formal semantics.

These prefixes are always italicized to emphasize that the corresponding functions, variables, and types are abstract: they are not and cannot be made accessible in the host language. None of these special prefixes are given a URI.

2.5.2 Functions and operators

The [Functions and Operators] document defines built-in functions available in [XPath/XQuery]. A number of these functio