Etapa Intermedia

7.3.1

Preliminary Notions

A first intuitive notion of substitutions has already been given in Section 4.4. This notion was rather straightforward and similar to the usual definition of substitutions, since Section 4.4 only considered query and data terms. However, variable restrictions occurring in query terms have to be taken into account. As a variable might be restricted, not every substitution is applicable to every query term.

Also, Xcerpt construct terms extend the usual terms by grouping constructs that group several substitu- tions within a single ground instance by using the constructsallandsome. For instance, given a construct term f{all var X}and three alternative substitutions{X7→a},{X7→b}and{X7→c}, the resulting data term is f{a,b,c}.

In order to define such groupings, it is therefore necessary to provide a construct that represents all possible alternatives and can be applied to a construct term. This is called a substitution set below. Substi- tution sets are used in Section 7.4 which defines satisfaction for Xcerpt term formulas and later in Section 8.2.1 to define the notion of a simulation unifier. In the following, substitutions are denoted by lowercase greek letters (likeσ orπ), while substitution sets are denoted by uppercase greek letters (likeΣorΠ).

7.3. SUBSTITUTIONS AND SUBSTITUTION SETS

Substitutions

A substitution is a mapping from the set of (all) variables to the set of (all) construct terms. In the following, lower case greek letters (likeσ orτ) are usually used to denote substitutions. As usual in mathematics, a substitution is a mapping of infinite sets. Of course, finite representations are usually used, as the number of variables occurring in a term is finite. Substitutions are often conveniently denoted as sets of variable assignments instead of as functions. For example, we write X 7→a,Y 7→b to denote a substitution that maps the variable X to a and the variable Y to b, and any other variable to itself. In general, a substitution provides assignments for all variables, but “irrelevant” variables are not given in the description of substitutions.

If a substitution is applied to a query term tq, all occurrences of variables for which the substitution provides assignments are replaced by the respective assignments (see Section 7.3.2 below). The resulting term is called an instance of tqand the substitution. Not every substitution can be applied to every query term: variable assignments in the substitution have to respect variable restrictions occurring in the pattern for a substitution to be applicable (see also 7.3.2). If a substitutionσ respects the variable restrictions in a query term tq_{, it is said to be a substitution for t}q_{. For example, the substitutionX 7→ f{a} is a}

substitution for var X → f{{}}, but not for var X → g{{}}. Note that a substitution cannot be applied to a construct term, because construct terms may contain grouping constructs that group several instances of subterms together. Instead, substitution sets are used for this purpose (see below).

A substitutionσ is called a grounding substitution for a term t, ifσ(t)is a ground query term. Con- sequently, a grounding substitution is always a mapping from the set of variable names to the set of data terms (i.e. ground construct terms). A substitutionσ is called an all-grounding substitution, if it maps every variable to a data term. Naturally, every all-grounding substitution is a grounding substitution for every query term to which it is applicable. Note that the reverse does not hold: a grounding substitution is grounding wrt. some term t and does not necessarily assign ground terms to variables not occurring in t.

A substitutionσ1is a subset of a substitutionσ2(i.e.σ1⊆σ2), ifσ1(X)∼=σ2(X)for every variable

name X withσ1(X)6=X (i.e.σ1does not map X to itself), where∼=denotes simulation equivalence (cf.

Section 4.4.4). Correspondingly, two substitutionsσ1 andσ2are considered to be equal (i.e. σ1=σ2),

ifσ1⊆σ2andσ2⊆σ1. For example,

X 7→ f{a,b} andX 7→ f{b,a} are equal. This definition is reasonable because the data terms resulting from applying two such substitutions are treated equally in the model theory described below.

The composition of two substitutionsσ1 and σ2, denoted by σ1◦σ2 is defined as (σ1◦σ2)(t) =

σ1(σ2(t))for every query term t. Note that the assignments inσ2take precedence, becauseσ2is applied

first. Consider for exampleσ1={X 7→ a,Y 7→ b}andσ2={X 7→ c}, and a term t=f{var X,var Y}.

Applying the compositionσ1◦σ2to t yields(σ1◦σ2)(t) =f{c,b}.

The restriction of a substitutionσ to a set of variable names V , denoted byσ_|V, is the mapping that agrees withσ on V and with the identical mapping on the other variables.

Substitution Sets

A substitution set is simply a set containing substitutions. In the following, upper case greek letters (likeΣ andΦ) are usually used to denote substitution sets.

Substitution sets can be applied to a query or construct term (cf. Sections 7.3.2 and 7.3.3). The result of this application is in general a set of terms called the instances of the substitution set and the term. A substitution setΣis only applicable to a query term tq, if all substitutions inΣare applicable to tq. In this case,Σis called a substitution set for tq. Since construct terms do not contain variable restrictions, every substitution set except for the empty set is a substitution set for a construct term. There exists no query or construct term t such that the empty substitution set{}is a substitution set for t.

A substitution setΣfor a term t is called a grounding substitution set, if all instances of t andΣare ground query terms or data terms. A substitution setΣis called an all-grounding substitution set, if all

σ∈Σare all-grounding substitutions.

The composition of two substitution setsΣ1andΣ2, denoted asΣ1◦Σ2, is defined as

Σ1◦Σ2=

CHAPTER 7. DECLARATIVE SEMANTICS

Consider for example the substitution setsΣ1=

{X 7→ a} andΣ2= {Y 7→ b},{Y 7→ c} . Then Σ1◦Σ2= {X 7→ a,Y 7→ b},{X 7→ a,Y 7→ c} .

The restriction of a substitution setΣto a set of variables V , denoted byΣ_|V, is the set of substitutions inΣrestricted to V .

Similarly, the extension of a substitution setΣrestricted to a set of variables V to a set of variables V0 with V⊆V0, extends every substitutionσ inΣto substitutionsσ0by adding all possible assignments of variables in V0\V to data terms. For example, the extension of the restricted substitution set{X7→a} to the set of variables{X,Y}is the (infinite) set{X7→a,Y7→a},{X7→a,Y 7→b}, . . .

Note that in practice, it would be desirable to define substitution sets as multi-sets that may contain duplicate elements: if an XML document contains two persons named “Donald Duck”, then it should be assumed that these are different persons with the same name. Providing a proper formalisation with multi- sets is, however, not in the scope of this thesis, as subsequent definitions and proofs would be much more complicated without adding an interesting aspect to the formalisation.

Maximal Substitution Sets

So as to properly convey the meaning ofall, it is not sufficient to consider arbitrary substitution sets. The interesting substitution sets are those that are maximal for the satisfaction of the query part Q of a rule. As satisfaction is not yet formally defined, this property shall for now simply be called P.

Intuitively, the definition of maximal substitution sets is straightforward: a substitution setΣsatisfying

P is a maximal substitution set, if there exists no substitution setΦsatisfying P such thatΣis a proper subset ofΦ. However, this informal definition does not take into account that there might be substitution sets that differ only in that some substitutions contain bindings that are irrelevant because they do not occur in the considered term formula Q. Maximal substitution sets are therefore formally defined as follows:

Definition 7.1 (Maximal Substitution Set)

Let Q be a quantifier free query term formula with set of variables V , let P be a property, and letΣbe a set of substitutions such that P holds forΣ.Σis called a maximal substitution set wrt. P and Q, if there exists no substitution setΦsuch that P holds forΦandΣ_|V is a proper subset ofΦ_|V(i.e.Σ_|V(Φ|V).

7.3.2

Application to Query Terms

Since query terms do not contain the grouping constructs all and some, applying substitutions and substi- tution sets is straightforward. Application of a single substitution yields a single term where some variable occurrences are substituted, while application of a substitution set yields a set of terms where some vari- ables are substituted.

Definition 7.2 (Substitutions: Application to Query Terms)

Let tq_{be a query term.}

1. The application of a substitutionσto tq, writtenσ(tq)is recursively defined as follows: • σ(var X) =t0if(X7→t0)∈σ • σ(var X → s) =t0if(X7→t0)∈σ andσ(s)t0 • σ(f{t1, . . . ,tn}) =σ(f){σ(t1), . . . ,σ(tn)} • σ(f[t1, . . . ,tn]) =σ(f)[σ(t1), . . . ,σ(tn)] • σ(f{{t1, . . . ,tn}}) =σ(f){{σ(t1), . . . ,σ(tn)}} • σ(f[[t1, . . . ,tn]]) =σ(f)[[σ(t1), . . . ,σ(tn)]] • σ(without t) =withoutσ(t) • σ(optional t) =optionalσ(t) for some n≥0. Sebastian Schaffert 147

7.3. SUBSTITUTIONS AND SUBSTITUTION SETS

2. The application of a substitution setΣto tqis defined as follows: Σ(tq) =σ(tq) | σ∈Σ

Note that not every substitution can be applied to a query term tq. If a variable in tqis restricted as in

var X → s, then a substitution can only be applied if it provides bindings for X that are compatible to

this restriction. Likewise, a substitution set is only applicable to a query term tq, if all its substitutions are applicable to tq.

Since query terms never contain grouping constructs, the cardinality ofΣ(t)always equals the cardi- nality ofΣ. In particular, ifΣ=/0, thenΣ(t) =/0, even if t is a ground query term. Since an interpretation with an empty substitution set would be a model for any formula, substitution sets in the following are con- sidered to be non-empty. In case no variables are bound, substitution sets are usually defined asΣ={/0}.

7.3.3

Application to Construct Terms

Applying a single substitution to a construct term is not reasonable as the meaning of the grouping con- structs all and some is unclear in such cases. In the following, the application is thus only defined for substitution sets. On substitution sets, the grouping constructs group such substitutions that have the same assignment on the free variables of a construct term. For each such group, the application of the substitu- tionΣyields a different construct term. A variable is considered free in a construct term if it is not in the scope of a grouping construct. The set of free variables of a construct term tcis denoted by FV(tc). Recall also that∼=denotes simulation equivalence between two ground terms.

Definition 7.3 (Grouping of a Substitution Set)

Given a substitution setΣand a set of variables V ={X1, . . . ,Xn}such that allσ∈Σhave bindings for all

Xi,1≤i≤n.

• The equivalence relation'V⊆Σ×Σis defined as:σ1'Vσ2iffσ1(X)∼=σ2(X)for all X∈V .

• The set of equivalence classesΣ/'V with respect to'Vis called the grouping ofΣon V .

• Each of the equivalence classesJσK∈Σ/'V is accordingly defined asJσK = τ∈Σ | τ 'V σ}.

Informally, each equivalence classJσK∈Σ/'V contains such substitutions that have the same assign- ment for each of the variables in V .

Example 7.4

Given the substitution setΣ=σ1,σ2,σ3 with

σ1={X17→a,X27→b},σ2={X17→a,X27→c}, andσ3={X17→c,X27→b} The grouping ofΣon V ={X1}is • Jσ1K=Jσ2K= {X17→a,X27→b},{X17→a,X27→c} • Jσ3K= {X17→c,X27→b}

The application of a substitution set to a construct term (possibly containing grouping constructs) is defined in terms of this grouping. Given a substitution set Σ, the applicationΣ(tc_{)to a construct term}

tc_{with free variables FV(t}c_{)yields exactly|}Σ_/

'_FV(tc)|results, one for each different binding of the free variables in tc.

Example 7.5

Given a term t=f{X1,g{all X2}}, i.e. FV(t) ={X1}. Consider again

Σ={X17→a,X27→b},{X17→a,X27→c},{X17→c,X27→b}

from Example 7.4. The result of applyingΣto t is Σ(t) =f{a,g{b,c}},f{c,g{b}}

CHAPTER 7. DECLARATIVE SEMANTICS

The following definition specifies how a substitution set is applied to a construct term tc. The defini- tion is divided into two parts: In the first part, it is assumed that all substitutions in the substitution setΣ contain the same assignments for the free variables of tc(variables occurring within the scope of grouping constructs are unrestricted). As the quotientΣ/'FV(tc) in this case obviously only contains a single equiva- lence class, the application of this restrictedΣto tcyields only a single term, which simplifies the recursive definition. In the second part of Definition 7.4, this restriction is lifted.

Since the construction of data terms requires to construct new lists of subterms, the following defini- tion(s) use the notion of term sequences introduced in Section 4.4.2. Recall that a sequence is a binary relation between a set of integers and a set of terms, and usually denoted by S=hx1, . . . ,xnifor some n and

terms xi. Recall furthermore the definitions of subsequences and concatenation (Definition 4.5 on page 81).

Defining the semantics oforder by furthermore requires a function sort_f(V)(·,·), where V is a se- quence of variables, that takes as arguments a grouping of a substitution set on V and returns a sequence of substitution sets ordered according to f(V)and the variables in V . f(V)is a total ordering on the set of substitution sets that assign ground terms to the variables in V comparing variable bindings for the variables in V .1

Definition 7.4 (Substitutions: Application to Construct Terms)

1. LetΣbe a substitution set and let tc be a construct term such that all free variables of tc have the same assignment in all substitutions ofΣ, i.e.Σ/'FV(tc) ={JσK}. The restricted application ofΣto

tc, writtenJσK(tc), is recursively defined as follows: • JσK(var V) =hσ(V)i2

• JσK(f{t1, . . . ,tn}) =hJσK(f){JσK(t1)◦ · · · ◦JσK(tn)}ifor some n≥0

• JσK(f[t1, . . . ,tn]) =hJσK(f)[JσK(t1)◦ · · · ◦JσK(tn)]ifor some n≥0

• JσK(all t) =Jτ1K(t)◦ · · · ◦JτkK(t)where{Jτ1K, . . . ,JτkK}=JσK/'FV(t)

• JσK(all t group by V) =Jτ1K(t)◦ · · · ◦JτkK(t)where{Jτ1K, . . . ,JτkK}=JσK/'FV(t)∪V • JσK(all t order by f V) =Jτ1K(t)◦ · · · ◦JτkK(t)

wherehJτ1K, . . . ,JτkKi=sort(f(V),JσK/'FV(t)∪V) • JσK(some k t) =Jτ1K(t)◦ · · · ◦JτkK(t)where{Jτ1K, . . . ,JτkK} ⊆JσK/'FV(t)

• JσK(some k t group by V) =Jτ1K(t)◦ · · · ◦JτkK(t)where{Jτ1K, . . . ,JτkK} ⊆JσK/'FV(t)∪V • JσK(some k t order by f V) =Jτ1K(t)◦ · · · ◦JτkK(t)

wherehJτ1K, . . . ,JτkKi vsort(f(V),JσK/'FV(t)∪V) • JσK(optional t) =

JσK(t) if the ground instanceJσK(t)exists

hi otherwise

• JσK(optional t with de f ault t0) =

JσK(t) if the ground instanceJσK(t)exists

JσK(t0) otherwise whereJτK1, . . . ,JτKk are pairwise different substitution sets.

2. Let tcbe a term, and let FV(tc)be the free variables in tc. The application of a substitution setΣto

tcis defined as follows:

Σ(t) =tc0|JσK∈Σ/'FV(tc) ∧ ht

c0_i=JσK(tc₎

Although not explicitly defined above, integrating aggregations and functions in this definition is straightforward.

1_{As the substitution set is grouped on V , all substitutions inJσK(respectivelyJτK) provide identical bindings for variables in V .} 2_{Note thatσis the representative of the equivalence classJσK}