2.1 Multisets

A multiset is a set where repetitions are allowed. We denote by $\{\!\!\{ x_1, \ldots, x_m \}\!\!\}$ a multiset, where $x_1, \ldots, x_m$ is a sequence with (possible) repetitions, and by $\{\!\!\{ \}\!\!\}$ the empty multiset. We will often use the polynomial notation $v_1^{i_1} \ldots v_n^{i_n}$ , where $v_1, \ldots, v_n$ is a sequence without repetitions, to denote a multiset $A$ whose element $v_j$ appears $i_j$ times. The set $\{v_j \mid i_j \gt 0\}$ is called the support of $A$ and is denoted by $\lfloor \!\lfloor A \rfloor \!\rfloor$ . We also use the functional notation $A: \{v_1, \ldots, v_n\} \rightarrow \mathbb{N}$ , where $A(v_j)=i_j$ .

In this paper, we only consider multisets whose support is finite. We denote with $\mathscr{P}_m(X)$ the set of all the multisets whose support is any finite subset of $X$ . For example, $a^3c^5$ and $a^3b^2c^1$ are elements of $\mathscr{P}_m(\{a,b,c\})$ . The fundamental operation for multisets is the sum, defined as

\begin{align*} A \uplus B = \lambda v \in \lfloor \!\lfloor A \rfloor \!\rfloor \cup \lfloor \!\lfloor B \rfloor \!\rfloor . A(v)+B(v) \enspace . \end{align*}

Note that we also use $\uplus$ to denote disjoint union for standard sets. The context will allow us to discern the proper meaning of $\uplus$ . Given a multiset $A$ and $X \subseteq \lfloor \!\lfloor A \rfloor \!\rfloor$ , the restriction of $A$ over $X$ , denoted by $A_{|X}$ , is the only multiset $B$ such that $\lfloor \!\lfloor B \rfloor \!\rfloor =X$ and $B(v)=A(v)$ for each $v \in X$ .

2.2 The domain of existential substitutions

We work in the framework of existential substitutions (Amato and Scozzari, Reference Amato and Scozzari2009), which allows us to simplify those semantic definitions which are heavily based on renaming apart objects and to avoid variable clashes. In this framework, all the details concerning renamings are moved to the inner level of the semantic domain, where they are more easily manageable. We briefly recall the basic definitions of the domain.

The set of substitutions, idempotent substitutions, and renamings are denoted by $\mathit{Subst}$ , $\mathit{ISubst}$ , and $\mathit{Ren}$ , respectively. Given $\theta _1, \theta _2 \in \mathit{Subst}$ and $U \in \mathscr{P}_f({\mathcal{V}})$ , the preorder $\preceq _U$ is defined as follows:

\begin{align*} \theta _1 \preceq _U \theta _2 \iff \exists \delta \in \mathit {Subst}. \forall x \in U.\ \theta _1(x)=\delta (\theta _2(x)) \enspace . \end{align*}

The notation $\theta _1 \preceq _U \theta _2$ states that $\theta _1$ is an instance of $\theta _2$ w.r.t. the variables in $U$ . The equivalence relation induced by the preorder $\preceq _U$ is given by

\begin{align*} \theta _1 \sim _U \theta _2 \iff \exists \rho \in \mathit {Ren}. \forall x \in U.\ \theta _1(x)=\rho (\theta _2(x)) \enspace . \end{align*}

Let $\mathit{ISubst}_{\sim _U}$ be the quotient set of $\mathit{ISubst}$ w.r.t. $\sim _U$ . The domain $\mathit{ISubst}_\sim$ of existential substitutions is defined as the disjoint union of all the $\mathit{ISubst}_{\sim _U}$ for $U\in \mathscr{P}_f({\mathcal{V}})$ , namely:

\begin{align*} \mathit {ISubst}_\sim = \biguplus _{U\in \mathscr {P}_f({\mathcal {V}})} \mathit {ISubst}_{\sim _U} \enspace . \end{align*}

In the following, we write $[\theta ]_{U}$ for the equivalence class of $\theta$ w.r.t. $\sim _U$ . The partial order $\preceq$ over $\mathit{ISubst}_\sim$ is given by

\begin{align*} [\theta ]_U \preceq [\theta ^{\prime}]_V \iff U \supseteq V \wedge \theta \preceq _V \theta ^{\prime} \enspace . \end{align*}

Intuitively, $[\theta ]_U \preceq [\theta ^{\prime}]_V$ means that $\theta$ is an instance of $\theta ^{\prime}$ w.r.t. the variables in $V$ , provided that they are all variables of interest of $\theta$ .

To ease notation, we often omit braces from the sets of variables of interest when they are given extensionally. So we write $[\theta ]_{x,y}$ or $[\theta ]_{xy}$ instead of $[\theta ]_{\{x,y\}}$ and $\sim _{x,y,z}$ instead of $\sim _{\{x,y,z\}}$ . When the set of variables of interest is clear from the context or when it is not relevant, it will be omitted. Finally, we omit the braces that enclose the bindings of a substitution when the latter occurs inside an equivalence class; that is, we write $[x/y]_U$ instead of $[\{x/y\}]_U$ .

2.2.1 Unification

Given $U,V \in \mathscr{P}_f({\mathcal{V}})$ , $[\theta _1]_U, [\theta _2]_V \in \mathit{ISubst}_{\sim }$ , the most general unifier between these two classes is defined as the mgu of suitably chosen representatives, where variables not of interest are renamed apart. In formulas:

(1) \begin{align} \mathsf{mgu}([\theta _1]_U,[\theta _2]_V)= [\mathsf{mgu}(\theta ^{\prime}_1, \theta ^{\prime}_2)]_{U \cup V} \enspace, \end{align}

where $\theta _1 \sim _U \theta ^{\prime}_1 \in \mathit{ISubst}$ , $\theta _2 \sim _V \theta ^{\prime}_2 \in \mathit{ISubst}$ , and $(U \cup \mathrm{vars}(\theta ^{\prime}_1)) \cap (V \cup \mathrm{vars}(\theta ^{\prime}_2)) \subseteq U \cap V$ . The last condition is needed to avoid variable clashes between the chosen representatives $\theta ^{\prime}_1$ and $\theta ^{\prime}_2$ . It turns out that $\mathsf{mgu}$ is the greatest lower bound of $\mathit{ISubst}_{\sim }$ ordered by $\preceq$ .

A different version of unification is obtained when one of the two arguments is an existential substitution and the other one is a standard substitution. In this case, the latter argument may be viewed as an existential substitution where all the variables are of interest:

(2) \begin{align} \mathsf{mgu}([\theta ]_U,\delta )= \mathsf{mgu}([\theta ]_U,[\delta ]_{\mathrm{vars}(\delta )}) \enspace . \end{align}

Note that deriving the general unification in (1) from the special case in (2) is not possible. This is because there are elements in $\mathit{ISubst}_\sim$ , which cannot be obtained as $[\delta ]_{\mathrm{vars}(\delta )}$ for any $\delta \in \mathit{ISubst}$ .

This is the form of unification that is better suited for analysis of logic programs, where existential substitutions are the denotations of programs while standard substitutions are the result of unification between goals and heads of clauses. Devising optimal abstract operators for (2) in different abstract domains is the topic of Amato and Scozzari (Reference Amato and Scozzari2010).

2.2.2 Matching

Given $U_1,U_2 \in \mathscr{P}_f({\mathcal{V}})$ , $[\theta _1]_{U_1} \in \mathit{ISubst}_{\sim }$ , and $[\theta _2]_{U_2} \in \mathit{ISubst}$ , the matching of $[\theta _1]_{U_1}$ with $[\theta _2]_{U_2}$ (Amato and Scozzari, Reference Amato and Scozzari2009) is defined in the same way as unification, as soon as none of the variables in $U_1$ get instantiated in the result. If this is not the case, the matching is undefined.

Definition 2.1 (Matching). Given $[\theta _1]_{U_1}, [\theta _2]_{U_2} \in \mathit{ISubst}_{\sim }$ , we have that

(3) \begin{align} \mathsf{match}([\theta _1]_{U_1},[\theta _2]_{U_2})=\begin{cases} \mathsf{mgu}([\theta _1]_{U_1},[\theta _2]_{U_2}) & \text{if } \theta _1 \preceq _{U_1 \cap U_2} \theta _2, \\[3pt] \text{undefined} & \text{otherwise.} \end{cases} \end{align}

Note that the condition $\theta _1 \preceq _{U_1 \cap U_2} \theta _2$ is equivalent to $[\theta _1]_{U_1}= \mathsf{mgu}([\theta _1]_{U_1},[\theta _2]_{U_2})_{|U_1}$ (Amato and Scozzari, Reference Amato and Scozzari2009).

Example 2.2. If we unify $[\theta _1]_{x,y}=[x/a, y/b]_{x,y}$ with $[\theta _2]_{y,z}=[z/r(y)]_{y,z}$ , we obtain $[\theta _3]_{x,y,z}=[x/a, y/b, z/r(b)]_{x,y,z}$ . Note that the variables $y$ and $z$ in $\theta _3$ are instantiated w.r.t. $\theta _2$ ; therefore, $\mathsf{match}([\theta _2]_{y,z}, [\theta _1]_{x,y})$ is undefined. However, $x$ and $y$ in $\theta _3$ are not instantiated w.r.t. $\theta _1$ ; therefore, $\mathsf{match}([\theta _1]_{x,y},[\theta _2]_{y,z})=[\theta _3]_{x,y,z}$ .

Most of the time, when matching is applied in goal-dependent analysis of logic programs, we have that $U_1 \subseteq U_2$ . This is because $U_1$ is the set of variables in a clause, while $U_2$ contains both the variables in the clause and in the call substitution. Nonetheless, we study here the general case so that it can be applied in any framework.

2.2.3 Other operations

Given $V \in \mathscr{P}_f({\mathcal{V}})$ and $[\theta ]_{U} \in \mathit{ISubst}_\sim$ , we denote by $([\theta ]_U)_{|V}$ the projection of $[\theta ]_U$ on the set of variables $V$ , defined as:

(4) \begin{align} ([\theta ]_{U})_{|V} = [\theta ]_{U \cap V} \enspace . \end{align}

2.3 Abstract interpretation

Given two sets $C$ and $A$ of concrete and abstract objects, respectively, an abstract interpretation (Cousot and Cousot, Reference Cousot and Cousot1992b) is given by an approximation relation $\triangleright \subseteq A \times C$ . When $a \triangleright c$ holds, this means that $a$ is a correct abstraction of $c$ . We are interested in the case when $(A,\leq _A)$ is a poset and $a \leq _A a^{\prime}$ means that $a$ is more precise than $a^{\prime}$ . In this case, we require that if $a \triangleright c$ and $a \leq _A a^{\prime}$ , then $a^{\prime} \triangleright c$ , too. In more detail, we require what Cousot and Cousot (Reference Cousot and Cousot1992b) call the existence of the best abstract approximation assumption; that is, the existence of a map $\alpha : C \rightarrow A$ such that for all $a \in A, c \in C$ , it holds that $a \triangleright c \iff \alpha (c) \leq _A a$ . The map $\alpha$ is called the abstraction function and maps each $c$ to its best approximation in $A$ .

Given a (possibly partial) function $f: C \rightarrow C$ , we say that $\tilde{f}: A \rightarrow A$ is a correct abstraction of $f$ , and write $\tilde{f} \triangleright f$ , whenever

\begin{align*} a \triangleright c \Rightarrow \tilde {f}(a) \triangleright f(c) \enspace, \end{align*}

assuming that $\tilde{f}(a) \triangleright f(c)$ is true whenever $f(c)$ is not defined. We say that $\tilde{f}: A \rightarrow A$ is the optimal abstraction of $f$ when it is the best correct approximation of $f$ , that is, when $\tilde{f} \triangleright f$ and

\begin{align*} \forall f^{\prime}:A \rightarrow A.\ f^{\prime} \triangleright f \Rightarrow \tilde {f} \leq _{A \rightarrow A} f^{\prime} \enspace . \end{align*}

In some cases, we prefer to deal with a stronger framework, in which the domain $C$ is also endowed with a partial order $\leq _C$ and $\alpha : C \rightarrow A$ is a left adjoint to $\gamma : A \rightarrow C$ , that is,

\begin{align*} \forall c \in C. \forall a \in A. \alpha (c) \leq _A a \iff c \leq _C \gamma (a) \enspace . \end{align*}

The pair $\langle \alpha, \gamma \rangle$ is called a Galois connection. In particular, we will only consider the case of Galois insertions, which are Galois connections such that $\alpha \circ \gamma$ is the identity map. If $\langle \alpha, \gamma \rangle$ is a Galois insertion and $f: C \rightarrow C$ is a monotone map, the optimal abstraction $\tilde{f}$ always exists, and it is definable as $\tilde{f}= \alpha \circ f \circ \gamma$ .

3.1 The domain $\mathtt{ShLin}^{\omega }$

We call $\omega$ -sharing group a multiset of variables, that is, an element of $\mathscr{P}_m({\mathcal{V}})$ . Given a substitution $\theta$ and a variable $v \in{\mathcal{V}}$ , we denote by $\theta ^{-1}(v)$ the $\omega$ -sharing group $\lambda w \in{\mathcal{V}}. \mathit{occ}(v,\theta (w))$ , which maps each variable $w$ to the number of occurrences of $v$ in $\theta (w)$ .

Given a set of variables $U$ and a set of $\omega$ -sharing groups $S \subseteq \mathscr{P}_m(U)$ , we say that $[S]_U$ correctly approximates a substitution $[\theta ]_W$ if $U=W$ and, for each $v \in{\mathcal{V}}$ , $\theta ^{-1}(v)_{|U} \in S$ . We write $[S]_U \triangleright [\theta ]_W$ to mean that $[S]_U$ correctly approximates $[\theta ]_W$ . Therefore, $[S]_U \triangleright [\theta ]_U$ when $S$ contains all the $\omega$ -sharing groups in $\theta$ , restricted to the variables in $U$ .

Definition 3.1 ( $\mathtt{ShLin}^{\omega }$ ). The domain $\mathtt{ShLin}^{\omega }$ is defined as

(5) \begin{align}{\mathtt{ShLin}^{\omega }} =\{ [S]_U \mid U \in \mathscr{P}_f({\mathcal{V}}), S \subseteq \mathscr{P}_m(U), S \neq \emptyset \Rightarrow \{\!\!\{ \}\!\!\} \in S\} \enspace, \end{align}

and ordered by $[S_1]_{U_1} \leq _\omega [S_2]_{U_2}$ iff $U_1=U_2$ and $S_1 \subseteq S_2$ .

The existence of the empty multiset, when $S$ is not empty, is required in order to have a surjective abstraction function.

In order to ease the notation, we write $[\{\{\!\!\{ \}\!\!\}, B_1,\ldots, B_n\}]_U$ as $[B_1,\ldots, B_n]_U$ by omitting the braces and the empty multiset, and variables in each $\omega$ -sharing group are sorted lexicographically. Moreover, if $X \in{\mathtt{ShLin}^{\omega }}$ , we write $B \in X$ in place of $X=[S]_U \wedge B \in S$ . Analogously, if $S^{\prime} \in{\mathtt{ShLin}^{\omega }}$ , we write $S^{\prime} \subseteq X$ in place of $X=[S]_U \wedge S^{\prime} \subseteq S$ . The best correct abstraction of a substitution $[\theta ]_U$ is

(6) \begin{align} \alpha _\omega ([\theta ]_U)=[\{ \theta ^{-1}(v)_{|U} \mid v \in{\mathcal{V}} \}]_U \enspace . \end{align}

3.2 Matching operator

In the matching operation, when $[\theta _1]_{U_1}$ is matched with $[\theta _2]_{U_2}$ , sharing groups in $\theta _1$ and $\theta _2$ are joined together in the resulting substitution. However, not all combinations are allowed. Assume $\alpha _{\mathrm{\omega }}([\theta _i]_{U_i})=[S_i]_{U_i}$ for $i \in \{1,2\}$ . If $\mathsf{match}([\theta _1]_{U_1},[\theta _2]_{U_2})$ is defined, $\theta _1$ will not be further instantiated and thus $\alpha _{\mathrm{\omega }}(\mathsf{match}([\theta _1]_{U_1}, [\theta _2]_{U_2})_{|U_1} ) \subseteq S_1$ . Moreover, the sharing groups in $S_2$ which do not contain any variable in $U_1$ are not affected by the unification, since the corresponding existential variable does not appear in $\theta _2(v)$ for any $v \in U_1$ .

Example 3.3. Let $\theta _1 = \{ x/r(w_1,w_2,w_2,w_3,w_3), y/a, z/r(w_1) \}$ with $U_1=\{x,y,z\}$ and $\theta _2 = \{ x/r(w_4, w_5, w_6,w_8,w_8), u/r(w_4,w_7), v/r(w_7,w_8)\}$ with $U_2=\{u,v,x\}$ . We have that

\begin{align*} [\theta ]_U = \mathsf{match}([\theta _1]_{U_1}, [\theta _2]_{U_2}) = [u/r(w_1,w_7),& v/r(w_7,w_3), \\ & x/r(w_1,w_2,w_2,w_3,w_3), y/a, z/r(w_1)]_U \enspace, \end{align*}

with $U=\{u,v,x,y,z\}$ . At the abstract level, we have $[S_1]_{U_1}= \alpha _{\mathrm{\omega }}([\theta _1]_{U_1})= [x^2, xz]_{U_1}$ , $[S_2]_{U_2}= \alpha _{\mathrm{\omega }}([\theta _2]_{U_2})= [uv,ux,vx^2,x]_{U_2}$ , and $[S]_U = \alpha _{\mathrm{\omega }}([\theta ]_U)= [uv, uxz, vx^2, x^2]_U$ .

Note that the new sharing group $uxz$ has the property that its restriction to $U_1$ is in $S_1$ . More generally, if we abstract $\theta _1$ w.r.t. the variables in $U_1$ , we get $\alpha _{\mathrm{\omega }}([\theta ]_{U_1}) = [x^2, xz]_{U_1}$ . This is equal to $[S_1]_{U_1}$ , showing that no new sharing group has been introduced by the matching operation relative to the variables in $U_1$ . Moreover, the sharing group $uv$ , which does not contain variables in $U_1$ , is brought unchanged from $S_2$ to $S$ .

Following the idea presented above, we may design an abstract matching operation for the domain $\mathtt{ShLin}^{\omega }$ .

Definition 3.4 (Matching over $\mathtt{ShLin}^{\omega }$ ). Given $[S_1]_{U_1}, [S_2]_{U_2} \in{\mathtt{ShLin}^{\omega }}$ , we define

\begin{align*} \mathsf {match}_{\mathrm {\omega }}([S_1]_{U_1},[S_2]_{U_2})=\big [S^{\prime}_2 \cup \big \{ X \in \mathscr {P}_m(U_1 \cup U_2) \mid \\ X_{|U_1} \in S_1 \wedge X_{|U_2} \in (S^{\prime\prime}_2)^* \big \} \big ]_{U_1 \cup U_2} \end{align*}

where

\begin{align*} S^{\prime}_2=\{ B \in S_2 \mid B_{|U_1} = \emptyset \} \qquad S^{\prime\prime}_2=S_2 \setminus S^{\prime}_2 \enspace \qquad S^*=\left \{ \uplus \mathcal {S} \mid \mathcal {S} \in \mathscr {P}_m(S) \right \} \enspace . \end{align*}

We now show an example of computing the matching over $\mathtt{ShLin}^{\omega }$ .

Example 3.5. Consider $[S_1]_{U_1}=[x^2, xz]_{xyz}$ and $[S_2]_{U_2}=[uv,ux,vx^2,x]_{uvx}$ as in Example 3.3. We show that from the definition of $\mathsf{match}_{\mathrm{\omega }}$ , it holds

\begin{align*}\mathsf {match}_{\mathrm {\omega }}([S_1]_{U_1},[S_2]_{U_2}) = [uv, uxz, xz, u^2x^2, ux^2, vx^2, x^2]_{uvxyz} \enspace .\end{align*}

First, we have that $S^{\prime}_2 = \{uv\}$ and $S^{\prime\prime}_2 = \{ux, vx^2, x\}$ . Apart from $uv$ , which directly comes from $S^{\prime}_2$ , all the other $\omega$ -sharing groups in the result may be obtained by choosing a multiset $\mathcal{M}$ of sharing groups in $S^{\prime\prime}_2$ and summing them together, obtaining $\uplus \mathcal{M}$ . Then, we consider if it is possible to add to $\uplus \mathcal{M}$ some occurrences of the variables $y$ and $z$ , for instance, $n$ occurrences of $y$ and $m$ of $z$ , in such a way that $(\uplus \mathcal{M}) \uplus \{\!\!\{ y^{n} z^{m} \}\!\!\}$ restricted to $U_1$ is a sharing group in $S_1$ .

We start by considering $\mathcal{M}$ with a single $\omega$ -sharing group.

• • If $\mathcal{M} = \{\!\!\{ ux \}\!\!\}$ , then $\uplus \mathcal{M} = ux$ but $(\uplus \mathcal{X})|_{U_1} = x \notin S_1$ . However, we can add the variable $z$ to get $uxz \in S_1$ ; hence $uxz$ is an $\omega$ -sharing group in the result of $\mathsf{match}_{\mathrm{\omega }}$ .

• • If $\mathcal{M} = \{\!\!\{ vx^2 \}\!\!\}$ , then $\uplus \mathcal{M} = vx^2$ and $(\uplus \mathcal{M})|_{U_1} = x^2$ , which is already an element of $S_1$ . Therefore, $vx^2$ is in the result of $\mathsf{match}_{\mathrm{\omega }}$ .

• • If $\mathcal{M} = \{\!\!\{ x \}\!\!\}$ , then $\uplus \mathcal{M} = x$ , but $(\uplus \mathcal{M})|_{U_1} = x \notin S_1$ . However, we can add the variable $z$ to get $xz \in S_1$ ; hence $xz$ is in the result of $\mathsf{match}_{\mathrm{\omega }}$ .

We now consider the cases when $\mathcal{M}$ has two (possibly equal) elements. Note that if $\mathcal{M}$ contains $vx^2$ , it cannot contain anything else; otherwise, $\uplus \mathcal{M}$ would contain at least three occurrences of $x$ , and no sharing group in $S_1$ could be matched. Therefore, the only choices are

• • if $\mathcal{M} = \{\!\!\{ ux, ux \}\!\!\}$ , then $\uplus \mathcal{M} = u^2x^2$ and $(\uplus \mathcal{M})|_{U_1} = x^2$ , which is already in $S_1$ ; hence $u^2x^2$ is in the result of $\mathsf{match}_{\mathrm{\omega }}$ ;

• • if $\mathcal{M} = \{\!\!\{ x, x \}\!\!\}$ , then $\uplus \mathcal{M} = x^2$ and $(\uplus \mathcal{M})|_{U_1} = x^2$ , which is already in $S_1$ ; hence $x^2$ is in the result of $\mathsf{match}_{\mathrm{\omega }}$ ;

• • if $\mathcal{M} = \{\!\!\{ ux, x \}\!\!\}$ , then $\uplus \mathcal{M} = ux^2$ and $(\uplus \mathcal{M})|_{U_1} = x^2$ , which is already in $S_1$ ; hence $ux^2$ is in the result of $\mathsf{match}_{\mathrm{\omega }}$ .

In theory, we should also consider the case when $\mathcal{M}$ has more than two elements, but in the example, this would not lead to new results, for the same reason why $vx^2$ may only be used alone.

We will prove that $\mathsf{match}_{\mathrm{\omega }}$ is correct and therefore comes as no surprise that

\begin{align*} \alpha _{\mathrm {\omega }}(\mathsf {match}([\theta _1]_{U_1},[\theta _2]_{U_2}])))= [uv, uxz, vx^2, x^2]_U \leq [uv, uxz, xz, u^2x^2, ux^2, vx^2, x^2]_{U} \enspace . \end{align*}

The lack of equality means that $\mathsf{match}_{\mathrm{\omega }}$ is not ( $\alpha$ -)complete (Giacobazzi et al., Reference Giacobazzi, Ranzato and Scozzari2000; Amato and Scozzari, Reference Amato and Scozzari2011). We will show later that it is optimal.

We try to give the intuition behind the definition of $\mathsf{match}$ , especially in the context of the backward unification. First note that the operator is additive on the first argument, namely:

\begin{align*}\mathsf {match}_{\mathrm {\omega }}([S_1]_{U_1},[S_2]_{U_2}) = \left [\bigcup _{B\in S_1} \mathsf {match}_{\mathrm {\omega }}([\{B\}]_{U_1},[S_2]_{U_2}) \right ]_{U_1\cup U_2}\end{align*}

This immediately implies that we can reason on matching considering one sharing group at a time. Given a sharing group $B\in S_1$ , which represents the exit substitution from a sub-computation, we try to guess which of the sharing groups in $S_2$ are part of the entry substitution that has generated the sub-computation ending with $B$ . In the simple case, when $U_1 \supseteq U_2$ , we simply check that $B$ can be generated by the sharing groups in $S_2$ ; that is, there exists $\mathcal{S} \in \mathscr{P}_m(S_2)$ such that $B_{|U_2} = \uplus \mathcal{S}$ .

The difficult case is when $U_2$ contains some variables that are not in $U_1$ . These variables come from the call substitution; they are removed from the abstraction before entering the sub-computation and now should be re-introduced as precisely as possible. In this case, we build a new sharing group $X$ such that $X_{|U_1}$ coincides with $B$ and $X_{|U_2}$ is generated by $S_2$ ; namely, there exists $\mathcal{S} \in \mathscr{P}_m(S_2)$ such that $X_{|U_2} = \uplus \mathcal{S}$ . This condition ensures that we pair each exit substitution $\theta _1$ (in the concretization of $B$ ) with some entry substitution $\theta _2$ (in the concretization of $\mathcal{S}$ ), which is less instantiated than $\theta _1$ .

Note that, although the abstract unification operator $\mathsf{mgu}_{\mathrm{\omega }}$ defined in Amato and Scozzari (Reference Amato and Scozzari2010) takes an abstract object and a substitution as inputs, the operator $\mathsf{match}_{\mathrm{\omega }}$ is designed in such a way that both the arguments are abstract objects. The reason for this choice is that these are the variants needed for static analysis of logic programs. However, it would be possible to devise a variant of $\mathsf{mgu}_{\mathrm{\omega }}$ with two abstract arguments and variants of $\mathsf{match}_{\mathrm{\omega }}$ with one abstract argument and a concrete one.

In order to prove the correctness of $\mathsf{match}_{\mathrm{\omega }}$ , we first extend the definition of $\theta ^{-1}$ to the case when it is applied to a sharing group $B$ , elementwise as

(7) \begin{align} \theta ^{-1}(v_1^{i_1} \cdots v_n^{i_n} ) = \biguplus \{\!\!\{ (\theta ^{-1}(v_1))^{i_1}, \dots, (\theta ^{-1}(v_n))^{i_n} \}\!\!\} \enspace . \end{align}

It turns out that

(8) \begin{align} (\eta \circ \theta )^{-1}(B)=\theta ^{-1}(\eta ^{-1}(B)) \enspace, \end{align}

a result that has been proved in Amato and Scozzari (Reference Amato and Scozzari2010).

Example 3.6. Given $\eta =\{x/s(y,u,y), z/s(u,u), v/u\}$ and $\theta = \{ v/a, w/s(x, x) \}$ , we have $\eta \circ \theta = \{ v/a, w/s(s(y,u,y), s(y,u,y)), x/s(y,u,y), z/s(u,u))\}$ and $(\eta \circ \theta )^{-1}(u) = uw^2xz^2$ . The same result may be obtained as $\theta ^{-1}(\eta ^{-1}(u))$ since $\eta ^{-1}(u) = uvxz^2$ and

\begin{align*} \theta ^{-1}(uvxz^2) & = \uplus \{\!\!\{ \theta ^{-1}(u), \theta ^{-1}(v), \theta ^{-1}(x), (\theta ^{-1}(z))^2 \}\!\!\} \\ &= \uplus \{\!\!\{ u, \{\!\!\{ \}\!\!\}, xw^2, (z)^2 \}\!\!\} = uw^2xz^2 \enspace . \end{align*}

Proof Given $[S_1]_{U_1} \triangleright [\theta _1]_{U_1}$ and $[S_2]_{U_2} \triangleright [\theta _2]_{U_2}$ such that $\mathsf{match}([\theta _1]_{U_1},[\theta _2]_{U_2})$ is defined, we need to prove that

\begin{align*} \mathsf {match}_{\mathrm {\omega }}([S_1]_{U_1},[S_2]_{U_2}) \triangleright \mathsf {match}([\theta _1]_{U_1},[\theta _2]_{U_2}) \enspace . \end{align*}

Assume without loss of generality that $\mathrm{dom}(\theta _1)=U_1$ , $\mathrm{dom}(\theta _2)=U_2$ , and $\mathrm{vars}(\theta _1) \cap \mathrm{vars}(\theta _2) \subseteq U_1 \cap U_2$ . In particular, this implies $\mathrm{rng}(\theta _1) \cap \mathrm{rng}(\theta _2)=\emptyset$ . By hypothesis $\theta _1 \preceq _{U_1 \cap U_2} \theta _2$ , that is, there exists $\delta \in \mathit{ISubst}$ such that $\theta _1(x)= \delta (\theta _2(x))$ for each $x \in U_1 \cap U_2$ , $\mathrm{dom}(\delta )=\mathrm{vars}(\theta _2(U_1 \cap U_2))$ and $\mathrm{rng}(\delta )=\mathrm{vars}(\theta _1(U_1 \cap U_2))$ . We have

\begin{eqnarray*} &&\mathsf{mgu}(\theta _1,\theta _2)\\ &=&\mathsf{mgu}(\{ \theta _2(x)=\theta _2(\theta _1(x)) \mid x \in U_1 \}) \circ \theta _2\\ &&\quad \text{[by assumptions on the $\theta _i$'s]}\\ &=&\mathsf{mgu}(\{ \theta _2(x)=\theta _1(x) \mid x \in U_1 \}) \circ \theta _2\\ &=&\mathsf{mgu}(\{ \theta _2(x)=\theta _1(x) \mid x \in U_1 \cap U_2 \}) \circ (\theta _1)_{|U_1 \setminus U_2} \circ \theta _2\\ &&\quad \text{[since $\mathrm{vars}(\theta _2) \cap \mathrm{vars}((\theta _1)_{U_1 \setminus U_2})=\emptyset ]$}\\ &=&\mathsf{mgu}(\{ \theta _2(x)=\delta (\theta _2(x)) \mid x \in U_1 \cap U_2\}) \circ ((\theta _1)_{|U_1 \setminus U_2} \uplus \theta _2)\\ &=&\delta \circ ((\theta _1)_{|U_1 \setminus U_2} \uplus \theta _2)\\ &&\quad \text{[since $\mathrm{dom}(\delta ) \cap \mathrm{vars}((\theta _1)_{U_1 \setminus U_2})=\emptyset ]$}\\ &=&(\theta _1)_{|U_1 \setminus U_2} \uplus (\delta \circ \theta _2) \enspace . \end{eqnarray*}

With an analogous derivation, we obtain

\begin{align*} \mathsf {mgu}(\theta _1,\theta _2)= \theta _1 \uplus (\delta \circ (\theta _2)_{|U_2 \setminus U_1}) \enspace . \end{align*}

Now, if $\eta =\mathsf{mgu}(\theta _1,\theta _2)$ , we need to prove that for each $v \in{\mathcal{V}}$ , $\eta ^{-1}(v)_{|U_1 \cup U_2} \in \mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1},[S_2]_{U_2})$ . We distinguish several cases.

• • $v \notin \mathrm{rng}(\theta _1)$ and $v \notin \mathrm{rng}(\theta _2)$ . In this case $v \notin \mathrm{vars}(\delta )$ and $\eta ^{-1}(v)_{|U_1 \cup U_2}=\{\!\!\{ \}\!\!\} \in \mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1},[S_2]_{U_2})$ .

• • $v \in \mathrm{rng}(\theta _1)$ and $v \notin \mathrm{vars}(\theta _1(U_2))$ . In this case $v \notin \mathrm{rng}(\theta _2)$ and $v \notin \mathrm{vars}(\delta )$ . We have $\eta ^{-1}(v)_{|U_1 \cup U_2} = \theta _1^{-1}(v)_{|U_1} \in S_1 \subseteq \mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1},[S_2]_{U_2})$ .

• • $v \in \mathrm{rng}(\theta _2)$ and $v \notin \mathrm{vars}(\theta _2(U_1))$ . In this case $v \notin \mathrm{rng}(\theta _1)$ and $v \notin \mathrm{vars}(\delta )$ . We have $\eta ^{-1}(v)_{|U_1 \cup U_2} = \theta _2^{-1}(v)_{|U_2} \in S_2$ . Since $v \notin \mathrm{vars}(\theta _2(U_1))$ , then $\theta _2^{-1}(v)_{|U_2} \in S^{\prime}_2 \subseteq \mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1},[S_2]_{U_2})$ .

• • $v \in \mathrm{vars}(\theta _2(U_1 \cap U_2))$ . In this case $v \notin \mathrm{rng}(\theta _1)$ and $v \in \mathrm{dom}(\delta )$ ; therefore, $\eta ^{-1}(v)_{|U_1 \cup U_2}=\{\!\!\{ \}\!\!\}$ .

• • $v \in \mathrm{vars}(\theta _1(U_1 \cap U_2))$ . Now, $v \in \mathrm{rng}(\delta )$ and $\eta ^{-1}(v)_{|U_2}=\theta _2^{-1}( \delta ^{-1}(v))_{|U_2}=\biguplus X$ , where $X \in \mathscr{P}_m(S_2)$ and each sharing group $B \in X$ is of the form $\theta _2^{-1}(w)_{|U_2}$ for some $w \in \lfloor \!\lfloor \delta ^{-1}(v) \rfloor \!\rfloor$ . Note that every such $w$ is an element of $\mathrm{vars}(\theta _2(U_1 \cap U_2))$ ; therefore, $\theta _2^{-1}(w)_{|U_2} \in S^{\prime\prime}_2$ , $X \in \mathscr{P}_m(S^{\prime\prime}_2)$ , and $\biguplus X \in (S_2^{\prime\prime})^*$ . On the other side, since $\eta =\theta _1 \uplus (\delta \circ (\theta _2)_{|U_2 \setminus U_1})$ , we have $\eta ^{-1}(v)_{|U_1}=\theta _1^{-1}(v)_{|U_1} \in S_1$ . Hence, $\eta ^{-1}(v)_{|U_1 \cup U_2} \in \mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1},[S_2]_{U_2})$ .

This concludes the proof.

Now, we may prove the optimality of $\mathsf{match}_{\mathrm{\omega }}$ . Actually, we prove a stronger property, a sort of weak completeness of $\mathsf{match}_{\mathrm{\omega }}$ , which will be used later to derive optimality.

We distill the idea presented above in the following result.

Lemma 3.9 (Completeness on the second argument). Given $[S_1]_{U_1} \in{\mathtt{ShLin}^{\omega }}$ and $[\theta _2]_{U_2} \in \mathit{ISubst}_\sim$ , there exist $\delta _1,\dots, \delta _n \in \mathit{ISubst}_\sim$ such that for all $i\in \{1,\dots, n\}$ , $[S_1]_{U_1} \triangleright [\delta _i]_{U_1}$ and

\begin{align*}\mathsf {match}_{\mathrm {\omega }}([S_1]_{U_1}, \alpha _{\mathrm {\omega }}([\theta _2]_{U_2}))= \left [\bigcup _ {i\in \{1,\ldots, n\}} \alpha _\omega (\mathsf {match}([\delta _i]_{U_1}, [\theta _2]_{U_2}))\right ]_{U_1\cup U_2} \end{align*}

Proof It is enough to prove that for each $[S_1]_{U_1}, [S_2]_{U_2} \in{\mathtt{ShLin}^{\omega }}$ and $B \in \mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1},[S_2]_{U_2})$ , there are substitutions $\theta _1$ , $\theta _2$ such that $[S_1]_{U_1} \triangleright [\theta _1]_{U_1}$ , $[S_2]_{U_21} \triangleright [\theta _2]_{U_2}$ and $B \in \alpha _{\mathrm{\omega }}(\mathsf{match}([\theta _1]_{U_1}, [\theta _2]_{U_2})$ . Assume $B_{|U_2}=\biguplus \mathcal{X}$ where $\mathcal{X} \in \mathscr{P}_m(S^{\prime\prime}_2)$ . Consider a substitution $[\theta _2]_{U_2}$ such that $[\lfloor \!\lfloor \mathcal{X} \rfloor \!\rfloor ]_{U_2} \leq \alpha _{\mathrm{\omega }}([\theta _2]_{U_2}) \leq [S_2]_{U_2}$ . It means that, for each $H \in \lfloor \!\lfloor \mathcal{X} \rfloor \!\rfloor$ , there is $v_H \in{\mathcal{V}}$ such that $\theta _2^{-1}(v_H)_{|U_2}=H$ . If $\lfloor \!\lfloor \mathcal{X} \rfloor \!\rfloor =\{ H_1, \ldots, H_n \}$ , we may define $\theta _2$ as the substitution with $\mathrm{dom}(\theta _2)=U_2$ and, for each $u \in U_2$ ,

\begin{align*} \theta _2(u) = t( \underbrace {v_{H_1}, \ldots, v_{H_1}}_{H_1(u) \text { times}}, \underbrace {v_{H_2}, \ldots, v_{H_2}}_{H_2(u) \text { times}}, \ldots, \underbrace {v_{H_n}, \ldots, v_{H_n}}_{H_n(u) \text { times}} ) \enspace . \end{align*}

By Lemma 3.9, there exists $[\theta _1]_{U_1}$ , such that $[S_1]_{U_1} \triangleright [\theta _1]_{U_1}$ and $B \in \alpha _{\mathrm{\omega }}(\mathsf{match}([\theta _1]_{U_1}, [\theta _2]_{U_2}))$ . Therefore, $\mathsf{match}_{\mathrm{\omega }}$ is the optimal approximation of $\mathsf{match}$ .

4.1 The domain $\mathtt{ShLin}^2$

Here we recall the main definitions for the domain $\mathtt{ShLin}^2$ , viewed as an abstraction of $\mathtt{ShLin}^{\omega }$ , following the presentation given in Amato and Scozzari (Reference Amato and Scozzari2010).

The idea is to simplify the domain $\mathtt{ShLin}^{\omega }$ by only recording whether a variable in a sharing group is linear or not, but forgetting its actual multiplicity. Intuitively, we abstract an $\omega$ -sharing group by replacing any exponent equal to or greater than $2$ with a new symbol $\infty$ .

A 2-sharing group is a map $o:{\mathcal{V}} \rightarrow \{0,1,\infty \}$ such that its support $\lfloor \!\lfloor o \rfloor \!\rfloor =\{ v \in{\mathcal{V}} \mid o(v) \neq 0\}$ is finite. We use a polynomial notation for 2-sharing groups as for $\omega$ -sharing groups. For instance, $o=xy^\infty z$ denotes the 2-sharing group whose support is $\lfloor \!\lfloor o \rfloor \!\rfloor =\{x,y,z\}$ , such that $o(x)=o(z)=1$ and $o(y)=\infty$ . We denote with $\emptyset$ the 2-sharing group with empty support and by ${\mathit{Sg}^2}(V)$ the set of 2-sharing groups whose support is a subset of $V$ . Note that in King (Reference King1994) the number $2$ is used as an exponent instead of $\infty$ , but we prefer our notation to be coherent with $\omega$ -sharing groups.

An $\omega$ -sharing group $B$ may be abstracted into the 2-sharing group $\alpha _{\mathrm{2}}(B)$ given by

(9) \begin{align} \alpha _{\mathrm{2}}(B)=\lambda v\in{\mathcal{V}} . \begin{cases} B(v) & \text{if $B(v)\leq 1$,} \\[3pt] \infty & \text{otherwise.} \end{cases} \end{align}

For instance, the $\omega$ -sharing groups $xy^{2} z,xy^{3} z,xy^{4} z,xy^{5} z,\ldots$ are all abstracted into $xy^\infty z$ .

There are operations on 2-sharing groups that correspond to variable projection and multiset union. For projection

(10) \begin{align} o_{|V} = \lambda v \in{\mathcal{V}}. \begin{cases} o(V) & \text{if $v \in V$,} \\[3pt] 0 & \text{otherwise,} \end{cases} \end{align}

while for multiset union

(11) \begin{align} o \uplus o^{\prime} = \lambda v \in{\mathcal{V}}. o(v) \oplus o^{\prime}(v) \enspace, \end{align}

where $0 \oplus x= x \oplus 0=x$ and $\infty \oplus x=x \oplus \infty =1 \oplus 1= \infty$ . We will use $\biguplus \{\!\!\{ o_1, \ldots, o_n \}\!\!\}$ for $o_1 \uplus \cdots \uplus o_n$ . Given a sharing group $o$ , we also define the delinearization operator:

(12) \begin{align} o^2=o \uplus o \enspace . \end{align}

This operator is extended pointwise to sets and multisets. The next proposition shows some properties of these operators.

Since we do not want to represent definite nonlinearity, we introduce an order relation over sharing groups as follows:

\begin{align*} o \leq o^{\prime} \iff \lfloor \!\lfloor o \rfloor \!\rfloor =\lfloor \!\lfloor o^{\prime} \rfloor \!\rfloor \wedge \forall x \in \lfloor \!\lfloor o \rfloor \!\rfloor .\ o(x) \leq o^{\prime}(x) \enspace, \end{align*}

and we restrict our attention to downward closed sets of sharing groups. The domain we are interested in is the following:

\begin{align*} {\mathtt {ShLin}^2}=\bigl \{ [T]_U \mid T \in \mathscr {P}_{{\mathop {\downarrow }}}({\mathit {Sg}^2}(U)), U \in \mathscr {P}_f({\mathcal {V}}), T \neq \emptyset \Rightarrow \emptyset \in T \bigr \} \enspace, \end{align*}

where $\mathscr{P}_{{\mathop{\downarrow }}}({\mathit{Sg}^2}(U))$ is the powerset of downward closed subsets of ${\mathit{Sg}^2}(U)$ according to $\leq$ and $[T_1]_{U_1} \leq _{\mathrm{2}} [T_2]_{U_2}$ iff $U_1=U_2$ and $T_1 \subseteq T_2$ . For instance, the set $\{xy^\infty z\}$ is not downward closed, while $\{xyz,xy^\infty z\}$ is downward closed. There is a Galois insertion of $\mathtt{ShLin}^2$ into $\mathtt{ShLin}^{\omega }$ given by the pair of adjoint maps $\gamma _{\mathrm{2}}:{\mathtt{ShLin}^2} \rightarrow{\mathtt{ShLin}^{\omega }}$ and $\alpha _{\mathrm{2}}:{\mathtt{ShLin}^{\omega }} \rightarrow{\mathtt{ShLin}^2}$ :

\begin{align*} \gamma _{\mathrm{2}}([T]_U)=\left [ \alpha _{\mathrm{2}}^{-1}(T) \right ]_U \qquad \alpha _{\mathrm{2}}([S]_U)=\left [{\mathop{\downarrow }} \alpha _{\mathrm{2}}(S) \right ]_U \enspace, \end{align*}

where ${\mathop{\downarrow }} T = \{o \mid o^{\prime} \in T, o \leq o^{\prime}\}$ is the downward closure of $T$ . With an abuse of notation, we also apply $\gamma _{\mathrm{2}}$ and $\alpha _{\mathrm{2}}$ to subsets of $\omega$ -sharing groups and 2-sharing groups, respectively, by ignoring the set of variables of interest. For instance, $\gamma _{\mathrm{2}}([\emptyset, xyz,xy^\infty z]_{x,y,z})= [ \{\!\!\{\}\!\!\}, xyz, xy^{2} z,xy^{3} z,xy^{4} z,xy^{5} z,\ldots ]_{x,y,z}$ . Moreover, we write ${\mathop{\downarrow }} [S]_U$ as an alternative form for $[{\mathop{\downarrow }} S]_U$ .

Example 4.2. Consider the substitution $[\theta ]_U = [\{x/s(y,u,y), z/s(u,u), v/u \}]_{w,x,y,z}$ in Example 3.2. Its abstraction in $\mathtt{ShLin}^2$ is given by

\begin{align*} \alpha _{\mathrm {2}}(\alpha _{\mathrm {\omega }}([\theta ]_U)) = [\{xy, x^\infty y, xz, xz^\infty, w\}]_U = [ {\mathop {\downarrow }} \{x^\infty y, xz^\infty, w \} ]_U \enspace . \end{align*}

Analogously, substitutions $[\theta _1]_{U_1} = [\{ x/r(w_1,w_2,w_2,w_3,w_3), y/a, z/r(w_1) \}]_{x,y,z}$ and $[\theta _2]_{U_2} = [\{ x/r(w_4, w_5, w_6, w_8, w_8), u/r(w_4,w_7), v/r(w_7,w_8)\}]_{u,v,x}$ from Example 3.3 are abstracted into $[T_1]_{U_1}={\mathop{\downarrow }}[x^\infty, xz]_{U_1}$ and $[T_2]_{U_2} ={\mathop{\downarrow }}[uv, ux, vx^\infty, x]_{U_2}$ , respectively.

4.2 Matching operator

By composition, $\alpha _{\mathrm{2}} \circ \alpha _{\mathrm{\omega }}$ is an abstraction from $\mathit{ISubst}_\sim$ to $\mathtt{ShLin}^2$ . Properties of the Galois connection $\langle \alpha _{\mathrm{2}}, \gamma _{\mathrm{2}} \rangle$ may be lifted to properties of $\alpha _{\mathrm{2}} \circ \alpha _{\mathrm{\omega }}$ . In our case, we need to define an abstract matching $\mathsf{match}_{\mathrm{2}}$ over $\mathtt{ShLin}^2$ , which is an optimal w.r.t. $\mathsf{match}$ . However, optimality of $\mathsf{match}_{\mathrm{2}}$ w.r.t. $\mathsf{match}$ is immediately derived by optimality w.r.t. $\mathsf{match}_{\mathrm{\omega }}$ . Since the correspondence between $\mathtt{ShLin}^{\omega }$ and $\mathtt{ShLin}^2$ is straightforward, the same happens for $\mathsf{match}_{\mathrm{\omega }}$ and $\mathsf{match}_{\mathrm{2}}$ .

Definition 4.3 (Matching over $\mathtt{ShLin}^2$ ). Given $[T_1]_{U_1}, [T_2]_{U_2} \in{\mathtt{ShLin}^2}$ , we define

\begin{align*} \mathsf {match}_{\mathrm {2}}([T_1]_{U_1},[T_2]_{U_2})=\bigl [T^{\prime}_2 \cup {\mathop {\downarrow }} \bigl \{ o \in {\mathit {Sg}^2}({\mathcal {V}}) \mid o_{|U_1} \in T_1 \wedge o_{|U_2} \in (T^{\prime\prime}_2)^* \bigr \} \bigr ]_{U_1 \cup U_2} \end{align*}

where $T^{\prime}_2=\{ B \in T_2 \mid B_{|U_1} = \emptyset \}$ , $T^{\prime\prime}_2 = T_2 \setminus T^{\prime}_2$ and $T^* = \{ \biguplus X \mid X \subseteq T \cup T^2 \}$ .

Example 4.4. Under the hypothesis of Examples 3.5 and 4.2, and according to the definition of $\mathsf{match}_{\mathrm{2}}$ , we have that $T^{\prime}_2 = \{ uv \}$ , $T^{\prime\prime}_2 = \{ ux, vx, vx^\infty, x \}$ and

\begin{align*} \mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2})={\mathop{\downarrow }} [ uv, u^\infty v^\infty x^\infty, uxz, u^\infty x^\infty, v^\infty x^\infty, vxz, x^\infty, xz ]_{u,v,x,y,z} \enspace . \end{align*}

Note that

\begin{align*} \alpha _{\mathrm{2}}(\mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1},[S_2]_{U_2})) ={\mathop{\downarrow }} [ uv, u^\infty x^\infty, uxz, vx^\infty, x^\infty, & xz]_{u,v,x,y,z} \\ &\leq \mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2}) \enspace . \end{align*}

This is consistent with the fact that $\mathsf{match}_{\mathrm{2}}$ is correct w.r.t. $\mathsf{match}_{\mathrm{\omega }}$ . The 2-sharing groups $u^\infty v^\infty x^\infty$ , $v^\infty x^\infty$ and $vxz$ do not appear in $\alpha _{\mathrm{2}}(\mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1},[S_2]_{U_2}))$ since $\mathsf{match}_{\mathrm{2}}$ is not complete w.r.t. $\mathsf{match}_{\mathrm{\omega }}$

As anticipated before, we prove the optimality of $\mathsf{match}_{\mathrm{2}}$ w.r.t. $\mathsf{match}_{\mathrm{\omega }}$ , which automatically entails optimality w.r.t. $\mathsf{match}$ .

Proof We need to prove that, for each $[T_1]_{U_1}, [T_2]_{U_2} \in{\mathtt{ShLin}^2}$ ,

(13) \begin{align} \mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2})=\alpha _{\mathrm{2}}(\mathsf{match}_{\mathrm{\omega }}(\gamma _{\mathrm{2}}([T_1]_{U_1}), \gamma _{\mathrm{2}}([T_2]_{U_2}))) \enspace . \end{align}

To ease notation, we denote $\gamma _{\mathrm{2}}([T_1]_{U_1})$ and $\gamma _{\mathrm{2}}([T_2]_{U_2})$ by $[S_1]_{U_1}$ and $[S_2]_{U_2}$ , respectively. Moreover, we denote with $S^{\prime}_2$ , $S^{\prime\prime}_2$ , $T^{\prime}_2$ , and $T^{\prime\prime}_2$ the subsets of $S_2$ and $T_2$ given accordingly to Definitions 3.4 and 4.3. Since $\lfloor \!\lfloor B \rfloor \!\rfloor =\lfloor \!\lfloor \alpha _{\mathrm{2}}(B) \rfloor \!\rfloor$ , given $B \in S_2$ , we have that $B \in S_2^{\prime}$ iff $\alpha _{\mathrm{2}}(B) \in T_2^{\prime}$ .

Let $o \in \mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2})$ . If $o \in T^{\prime}_2$ , consider any $B \in \alpha ^{-1}_{\mathrm{2}}(o) \subseteq S_2$ . Then, $B \in S^{\prime}_2 \subseteq \mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1},[S_2]_{U_2})$ . Therefore, $o = \alpha _{\mathrm{2}}(B)$ is in the right-hand side of (13). If $o \notin T^{\prime}_2$ , then $o \leq o^{\prime}$ where $o^{\prime}_{|U_1} \in T_1$ and $o^{\prime}_{|U_2} \in (T^{\prime\prime}_2)^*$ ; that is, there is $X \subseteq T^{\prime\prime}_2 \cup (T^{\prime\prime}_2)^2$ such that $o^{\prime}_{|U_2}=\biguplus X$ . For each $o^{\prime\prime} \in X \cap T_2$ , let $B_{o^{\prime\prime}} \in \alpha ^{-1}_{\mathrm{2}}(o^{\prime\prime})$ . For each $o^{\prime\prime} \in X \setminus T_2$ , we have $o^{\prime\prime} = (o^{\prime\prime\prime})^2$ for some $o^{\prime\prime\prime} \in T_2$ , and let $B_{o^{\prime\prime}} \in \alpha ^{-1}_{\mathrm{2}}(o^{\prime\prime\prime})$ . Let $\mathcal{X}$ be a multiset containing a single copy of each $B^{\prime\prime}_o$ for $o \in X \cap T_2$ and two copies of $B^{\prime\prime}_o$ for each $o \in X \setminus T_2$ , for example, $\mathcal{X} = \{\!\!\{ B_{o^{\prime\prime}} \mid o^{\prime\prime} \in X \cap T_2 \}\!\!\} \uplus \biguplus \{\!\!\{ \{\!\!\{ B_{o^{\prime\prime}}, B_{o^{\prime\prime}} \}\!\!\} \mid o^{\prime\prime} \in X \setminus T_2 \}\!\!\})$ . Note that $\alpha _{\mathrm{2}}(\biguplus \mathcal{X})=\biguplus \alpha _2(X)=o^{\prime}_{|U_2}$ by (3) of Proposition 4.1. Then, consider the $\omega$ -sharing group $C$ such that

\begin{align*} C(v)=\begin {cases} (\biguplus \mathcal {X})(v) & \text {for $v \in U_2$,} \\[3pt] o^{\prime}(v) & \text {if $v \in U_1 \setminus U_2$ and $o(v)\leq 1$,} \\[3pt] 2 & \text {otherwise}. \end {cases} \end{align*}

It is clear that $\alpha _{\mathrm{2}}(C_{|U_1})= o^{\prime}_{|U_1}$ , hence $C_{|U_1} \in S_1$ . Moreover, $C_{|U_2} = \biguplus X$ with $\mathcal{X} \in \mathscr{P}_m(S_2^{\prime\prime})$ . Therefore, we have $C \in \mathsf{match}_{\mathrm{\omega }}([T_1]_{U_1}, [T_2]_{U_2})$ and $\alpha _{\mathrm{2}}(C)=o^{\prime}$ is in the right-hand side of (13). The same holds for $o$ by downward closure of $\alpha _{\mathrm{2}}$ .

Conversely, let $o \in \alpha _{\mathrm{2}}(\mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1}, [S_2]_{U_2}))$ . As a consequence, there exists $B \in \mathsf{match}_{\mathrm{\omega }}([S_1]_{U_1}, [S_2]_{U_2})$ such that $o \leq o^{\prime}$ and $o^{\prime} = \alpha _{\mathrm{2}}(B)$ . It is enough to prove that $o^{\prime} \in \mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2})$ . If $B \in S^{\prime}_2$ then $o^{\prime} \in T^{\prime}_2$ , hence $o^{\prime} \in \mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2})$ . If $B \notin S^{\prime}_2$ , then $B_{|U_1} \in S_1$ and $B_{|U_2} = \biguplus \mathcal{X}$ , where $\mathcal{X} \in \mathscr{P}_m(S^{\prime\prime}_2)$ . By $B_{|U_1} \in S_1$ , we have $o^{\prime}_{|U_1} = \alpha _{\mathrm{2}}(B_{|U_1}) \in T_1$ . For each $C \in \mathcal{X}$ with $\mathcal{X}(C)=1$ , we define $o_{C}=\alpha _{\mathrm{2}}(C) \in T^{\prime\prime}_2$ , while for each $C$ with $\mathcal{X}(C)\gt 1$ , we define $o_{C}=\alpha _{\mathrm{2}}(C)^2 \in (T^{\prime\prime}_2)^2$ . We have that $o^{\prime\prime}_{|U_2} = \alpha _{\mathrm{2}}(B)_{|U_2} = \alpha _{\mathrm{2}}(B_{|U_2})=\alpha _{\mathrm{2}}(\biguplus \mathcal{X})=\biguplus X$ where $X = \{ o_C \mid C \in \mathcal{X}\}$ is an element of $\mathscr{P}(T^{\prime\prime}_2 \cup (T^{\prime\prime}_2)^2)$ . This means that $o^{\prime} \in \mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2})$ .

Although $\mathsf{match}_{\mathrm{2}}$ is not complete w.r.t. either $\mathsf{match}_{\mathrm{\omega }}$ or $\mathsf{match}$ , we claim it enjoys a property analogous to the one in Lemma 3.9.

4.3 Optimization

In a real implementation, we would like to encode an element of $\mathtt{ShLin}^2$ with the set of its maximal elements. This works well only if we may compute $\mathsf{match}_{\mathrm{2}}$ starting from its maximal elements, without implicitly computing the downward closure. We would also like $\mathsf{match}_{\mathrm{2}}$ to compute as few non-maximal elements as possible. We provide a new algorithm for $\mathsf{match}_{\mathrm{2}}$ following this approach.

Definition 4.6. Given $T_1, T_2$ sets of $2$ -sharing groups and $U_1, U_2 \subseteq{\mathcal{V}}$ , we define

\begin{align*} \mathsf {match}^{\prime}_{\mathrm {2}}(T_1,U_1,T_2,U_2) = T^{\prime}_2 \cup \bigcup _{o \in T_1} \mathsf {match}^{\prime}_{\mathrm {2}}(o) \enspace, \end{align*}

where

\begin{align*} \mathsf {match}^{\prime}_{\mathrm {2}}(o) = \left \{ \left (o \wedge \biguplus X\right ) \uplus \biguplus (X \cap \mkern 1.5mu\overline {\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu) \Bigm | X \subseteq T^{\prime\prime}_2, \left (\biguplus \lfloor \!\lfloor X \rfloor \!\rfloor \right )_{|U_1} \leq o_{|U_2} \right \} \enspace, \end{align*}

with $T^{\prime}_2$ and $T^{\prime\prime}_2$ as in Definition 4.3, $\mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu = \{ o^{\prime} \in T^{\prime\prime}_2 \mid \forall v \in o^{\prime} \cap U_1, o(v)=\infty \}$ and

\begin{align*} o \wedge o^{\prime}= \lambda v.\begin {cases} o(v) & \text {if $v \in U_1 \setminus U_2$}, \\ min(o(v), o^{\prime}(v)) & \text {if $v \in U_1 \cap U_2$,} \\ o^{\prime}(v) & \text {otherwise.} \end {cases} \end{align*}

The operator $\mathsf{match}^{\prime}_{\mathrm{2}}$ aims at computing the set of maximal $2$ -sharing groups in $\mathsf{match}_{\mathrm{2}}([{\mathop{\downarrow }} T_1]_{U_1},[{\mathop{\downarrow }} T_2]_{U_2})$ . It works by considering one sharing group $o \in T_1$ at a time and calling an auxiliary operator that computes the maximal $2$ -sharing groups compatible with $o$ . A $2$ -sharing group $o^{\prime}$ is compatible with $o$ if $\lfloor \!\lfloor o^{\prime} \rfloor \!\rfloor \cap U_1 = \lfloor \!\lfloor o \rfloor \!\rfloor \cap U_1$ .

When computing the auxiliary operator $\mathsf{match}^{\prime}_{\mathrm{2}}(o)$ we choose a subset $X$ of $T^{\prime\prime}_2$ . Given $o^{\prime} \in X$ , note that $\lfloor \!\lfloor o^{\prime} \rfloor \!\rfloor$ may be viewed as a $2$ -sharing group which is the linearized version of $o^{\prime}$ : it has the same support as $o^{\prime}$ , but all variables are linear. If $o^{\prime} \in T^{\prime\prime}_2$ , its linearization is in ${\mathop{\downarrow }} T_2$ . The choice of $X$ is valid if the multiset sum of all its linearizations is smaller than $o$ for all the variables in $U_1 \cap U_2$ . Once established that $X$ is a valid choice, we do not take directly $\biguplus \lfloor \!\lfloor X \rfloor \!\rfloor$ as the resulting $2$ -sharing group, but we try to find an $o^{\prime\prime} \geq \biguplus X$ .

To this purpose, we observe that, given $o^{\prime} \in X$ , if $o(v)=\infty$ for each $v \in \lfloor \!\lfloor o^{\prime} \rfloor \!\rfloor \cap U_1$ , then we may take $o^{\prime}$ twice. We denote with $\mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu$ the set of all the sharing groups in $T^{\prime\prime}_2$ , which is such a property, and we unconditionally add to the result the sharing groups in $X \cap \mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu$ so that all these sharing groups are taken twice. Therefore, the biggest element compatible with $o$ is $(o \wedge \biguplus X) \uplus \biguplus (X \cap \mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu)$ .

Example 4.7. Under the hypothesis of Examples 3.5 and 4.2, let us define $T_1=\{x^\infty, xz\}$ and $T_2 = \{ uv, ux, vx^\infty, x \}$ . We have $T^{\prime}_2 = \{uv\}$ and $T^{\prime\prime}_2 = \{ux, vx^\infty, x \}$ .

Let us compute $\mathsf{match}^{\prime}_{\mathrm{2}}(x^\infty )$ . We have $\mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu = T^{\prime\prime}_2$ . If we take $X=\{ux, xv^\infty \}$ , we have $\left (\biguplus \lfloor \!\lfloor X \rfloor \!\rfloor \right )_{|U_1} = x^\infty$ , hence the choice is valid. The corresponding result is $(x^\infty \wedge \biguplus X) \uplus \biguplus X = uv^\infty x^\infty \uplus uv^\infty x^\infty = u^\infty v^\infty x^\infty$ . Overall, we have $\mathsf{match}^{\prime}_{\mathrm{2}}(x^\infty )= \{ u^\infty x^\infty, v^\infty x^\infty, x^\infty, u^\infty x^\infty v^\infty \}$ . Note that some results are computed by different choices of $X$ . For example, both $\{ux, x\}$ and $\{ux\}$ generates $u^\infty x^\infty$ .

Let us compute $\mathsf{match}^{\prime}_{\mathrm{2}}(xz)$ and take $X=\{ v x^\infty \}$ . We have that $\left (\biguplus \lfloor \!\lfloor X \rfloor \!\rfloor \right )_{|U_1} = x = xz_{|U_2}$ . In this case, $\mkern 1.5mu\overline{\mkern -1.5muT^{\prime\prime}_2\mkern -1.5mu}\mkern 1.5mu = \emptyset$ . Therefore, the result is $x \wedge vx^\infty = vx$ . Note that if we take $X= \{xz, x\}$ , then $\left (\biguplus \lfloor \!\lfloor X \rfloor \!\rfloor \right )_{|U_1} = x \uplus x= x^\infty$ , and the choice is not valid. This shows that, due to the downward closure, choosing in $X$ a sharing group with a nonlinear variable is very different from choosing the same variable twice. At the end, we have $\mathsf{match}^{\prime}_{\mathrm{2}}(xz) = \{ uxz, vxz, xz \}$ . Finally,

\begin{align*} \mathsf {match}_{\mathrm {2}}(T_1,U_1,T_2,U_2) = \{ uv, u^\infty x^\infty, v^\infty x^\infty, x^\infty, u^\infty x^\infty v^\infty, uxz, vxz, xz \} \enspace . \end{align*}

This is exactly the set of maximal elements in $\mathsf{match}_{\mathrm{2}}([{\mathop{\downarrow }} T_1]_{U_1}, [{\mathop{\downarrow }} T_2]_{U_2})$ .

We can show that the correspondence between $\mathsf{match}_{\mathrm{2}}$ and $\mathsf{match}^{\prime}_{\mathrm{2}}$ in the previous example was not by chance, proving the following:

Theorem 4.8. Given $T_1, T_2$ sets of $2$ -sharing groups such that ${\mathop{\downarrow }} [T_1]_{U_1},{\mathop{\downarrow }} [T_2]_{U_2} \in{\mathtt{ShLin}^2}$ , we have

\begin{align*} \mathsf {match}_{\mathrm {2}}({\mathop {\downarrow }}[T_1]_{U_1},{\mathop {\downarrow }} [T_2]_{U_2}) = {\mathop {\downarrow }} [\mathsf {match}^{\prime}_{\mathrm {2}}(T_1,U_1,T_2,U_2) ]_{U_1 \cup U_2} \enspace . \end{align*}

5.1 The domain $\mathtt{ShLin}$

The domain $\mathtt{ShLin}$ couples an object of $\mathtt{Sharing}$ with the set of variables known to be linear. Each element of $\mathtt{ShLin}$ is therefore a triple: the first component is an object of $\mathtt{Sharing}$ ; the second component is an object of $\mathtt{Lin}$ , that is, the set of variables that are linear in all the sharing groups of the first component; and the third component is the set of variables of interest. It is immediate that $\mathtt{ShLin}$ is an abstraction of $\mathtt{ShLin}^2$ (and thus of $\mathtt{ShLin}^{\omega }$ ).

\begin{align*} \mathtt {ShLin} = \{ [S,L,U]\mid S \subseteq \mathscr {P}(U), (S \neq \emptyset \Rightarrow \emptyset \in S), L \supseteq U \setminus \mathrm {vars}(S), U \in \mathscr {P}_f({\mathcal {V}}) \} \enspace, \end{align*}

with the approximation relation $\leq _{\mathit{sl}}$ defined as $[S,L,U] \leq _{\mathit{sl}} [S^{\prime},L^{\prime},U^{\prime}]$ iff $U=U^{\prime}$ , $S \subseteq S^{\prime}$ , $L \supseteq L^{\prime}$ . There is a Galois insertion of $\mathtt{ShLin}$ into $\mathtt{ShLin}^2$ given by the pair of maps:

\begin{align*} \alpha _{\mathit{sl}}([T]_U) & =[\{ \lfloor \!\lfloor o \rfloor \!\rfloor \mid o \in T \},\{ x \in U \mid \forall o \in T.\ o(x) \leq 1\},U] \enspace, \\ \gamma _{\mathit{sl}}([S,L,U]) & =[{\mathop{\downarrow }} \{ B_L \mid B \in S \}]_U \enspace, \end{align*}

where $B_L$ is the 2-sharing group that has the same support of $B$ , with linear variables dictated by the set $L$ :

\begin{align*} B_L & =\lambda v \in{\mathcal{V}}. \begin{cases} \infty & \text{if $v \in B \setminus L$,} \\ 1 & \text{if $v \in B \cap L$,} \\ 0 & \text{otherwise.} \end{cases} \end{align*}

The functional composition of $\alpha _{\mathrm{\omega }}$ , $\alpha _{\mathrm{2}}$ , and $\alpha _{\mathit{sl}}$ gives the standard abstraction map from substitutions to $\mathtt{ShLin}$ . We still use the polynomial notation to represent sharing groups, but now all the exponents are fixed to one. Note that the last component $U$ in $[S,L,U]$ is redundant since it can be retrieved as $L \cup \mathrm{vars}(S)$ . This is because the set $L$ contains all the ground variables.

Example 5.1. Consider the substitution $[\theta ]_U = [\{x/s(y,u,y), z/s(u,u), v/u \}]_{w,x,y,z}$ in Example 3.2. Its abstraction in $\mathtt{ShLin}$ is given by

\begin{align*} \alpha _{\mathit {sl}}(\alpha _{\mathrm {2}}(\alpha _{\mathrm {\omega }}([\theta ]_U))) = [ \{ xy, xz, w\}, \{ y, w \}, U ] \enspace . \end{align*}

Analogously, substitutions $[\theta _1]_{U_1} = [\{ x/r(w_1,w_2,w_2,w_3,w_3), y/a, z/r(w_1) \}]_{x,y,z}$ and $[\theta _2]_{U_2} = [\{ x/r(w_4, w_5, w_6, w_8, w_8), u/r(w_4,w_7), v/r(w_7,w_8)\}]_{u,v,x}$ from Example 3.3 are abstracted into $[S_1,L_1,U_1]=[\{x, xz\},\{y, z\}, U_1]$ and $[S_2, L_2, U_2] = [\{ uv, ux, vx, x\}, \{u,v\}, U_2]$ , respectively.

5.2 Matching operator

We want to provide an optimal abstract matching operator for $\mathtt{ShLin}$ . We may effectively compute $\mathsf{match}_{\mathit{sl}}$ by composing $\gamma _{\mathit{sl}}$ , $\mathsf{match}_{\mathrm{2}}$ , and $\alpha _{\mathit{sl}}$ . However, we provide a more direct characterization of $\mathsf{match}_{\mathit{sl}}$ , which may potentially improve performance.

First, we define the auxiliary function $nl: \mathscr{P}(\mathscr{P}({\mathcal{V}})) \rightarrow \mathscr{P}({\mathcal{V}})$ which takes a set $X$ of sharing groups and returns the set of variables that appear in $X$ more than once. In formulas:

(14) \begin{align} nl(X)= \{ v \in{\mathcal{V}} \mid \exists B_1, B_2 \in X, B_1 \neq B_2, v \in B_1 \cap B_2 \} \end{align}

The name $nl$ stands for nonlinear since it is used to recover those variables that, after joining sharing groups in $X$ , are definitively not linear.

Definition 5.2 (Matching over $\mathtt{ShLin}$ ). Given $[S_1,L_1,U_1]$ and $[S_2,L_2,U_2] \in \mathtt{ShLin}$ , we define

\begin{align*} \mathsf {match}_{\mathit {sl}}([S_1,L_1, U_1],[S_2,L_2,U_2])= \left [s(S^{\prime}_0 \cup S^{\prime\prime}_0), l(S^{\prime}_0 \cup S^{\prime\prime}_0), U \right ], \end{align*}

where $U=U_1 \cup U_2$ , $S^{\prime}_2=\{ B \in S_2 \mid B \cap U_1 = \emptyset \}$ , $S^{\prime\prime}_2=S_2 \setminus S^{\prime}_2$ , $\mkern 1.5mu\overline{\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu = \{B \in S^{\prime\prime}_2 \mid B \cap L_1 = \emptyset \}$ and

\begin{align*} S^{\prime}_0 & = \bigl \{ \langle B, L_2 \rangle \mid B \in S^{\prime}_2 \bigr \}, \\ S^{\prime\prime}_0 & = \Big \{ \left \langle B \cup \bigcup X, L_2 \setminus nl(X) \setminus \bigcup (X \cap \mkern 1.5mu\overline{\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu) \right \rangle \Bigm | B \in S_1, \\ & \qquad X \subseteq S^{\prime\prime}_2, B \cap U_2 = \left (\bigcup X\right ) \cap U_1, L_1 \cap nl(X)=\emptyset \Big \}, \end{align*}

\begin{align*} s(H) & = \left \{ B \mid \langle B, L \rangle \in H \right \}, \\ l(H) & = \bigcap \left \{ L_1 \cup L \cup (U \setminus B) \mid \langle B,L \rangle \in H \right \}. \end{align*}

This operator is more complex than previous ones because linearity information is not connected to sharing groups and needs to be handled separately. In view of this and a similar situation that happens for unification (Amato and Scozzari, Reference Amato and Scozzari2010), it seems that the idea of embedding linearity within sharing groups from King (Reference King1994) was particularly insightful.

We give an intuitive explanation of $\mathsf{match}_{\mathit{sl}}$ . It essentially works by simulating the optimized version of $\mathsf{match}_{\mathrm{2}}$ starting from $[T_1]_{U_1}=\gamma _{\mathrm{2}}([S_1,L_1,U_1])$ and $[T_2]_{U_2}=\gamma _{\mathrm{2}}([S_2, L_2, U_2])$ . The sets $S^{\prime}_2$ and $S^{\prime\prime}_2$ are defined as for the other matching operators. Each element of $H$ , $S^{\prime}_0$ , or $S^{\prime\prime}_0$ is a pair $\langle B, L \rangle$ that corresponds to the $2$ -sharing group $B_{L \cup L_1}$ in $\mathsf{match}_{\mathrm{2}}([T_1]_{U_1}, [T_2]_{U_2})$ . The component $B$ is the support of the $2$ -sharing group, while $L$ is the set of linear variables. We only record a subset of the linear variables, since those in $L_1$ are always linear.

The set $S^{\prime}_0$ encodes all the maximal $2$ -sharing groups derived from $S^{\prime}_2$ . The set $S^{\prime\prime}_0$ encodes $2$ -sharing groups which may be generated by gluing the sharing groups in $S^{\prime\prime}_2$ in a way which is compatible with $S_1$ . A given choice of $X \subseteq T^{\prime\prime}_2$ is compatible with a sharing group $B \in S_1$ only if $\bigcup X$ and $B$ have the same support on the common variables $U_1 \cap U_2$ and if $X$ does not conflict with the linearity of variables given by $L_1$ . This means that we cannot use a variable in $L_1$ more than once; therefore, $L_1 \cap nl(X)=\emptyset$ . Note that we may use a variable $v \in L_1 \setminus L_2$ since $v \notin L_2$ means that $v$ is possibly, but not definitively, nonlinear. On the contrary, if $v \in nl(X)$ , then $v$ is definitively nonlinear in the resultant sharing group. Once established that $X$ is compatible with $B$ , the set $\mkern 1.5mu\overline{\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu$ plays the same role of $\mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu$ in Definition 4.6: we may join another copy of the sharing groups in $X \cap \mkern 1.5mu\overline{\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu$ , making all their variables nonlinear.

Finally, the maps $s$ and $l$ extract from $S^{\prime}_0$ and $S^{\prime\prime}_0$ , the set of all the sharing groups and the set of variables that are linear in all the sharing groups.

Example 5.3. Consider the substitutions in Example 5.1. According to the definition of $\mathsf{match}_{\mathit{sl}}$ , we have $S^{\prime}_2 = \{uv\}$ , $S^{\prime\prime}_2 = \mkern 1.5mu\overline{\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu = \{ux, vx, x \}$ , $S^{\prime}_0 = \{ \langle uv, \{ u,v \} \rangle \}$ , and $S^{\prime\prime}_0 = \{ \langle ux, \emptyset \rangle, \langle vx, \emptyset \rangle, \langle x, \emptyset \rangle, \langle uvx, \emptyset \rangle, \langle uxz, \emptyset \rangle, \langle vxz, \emptyset \rangle, \langle xz, \emptyset \rangle, \langle uvxz, \emptyset \rangle \}$ . The final result is

(15) \begin{align} \mathsf{match}_{\mathit{sl}}([S_1,L_1,U_1],[S_2,L_2,U_2])&=\left [\left \{uv, uvx, ux, vx, x,\right .\right . \nonumber \\ &\left .\left .\quad uvxz, uxz, xz, vxz \right \}, \{ y,z\}, U_1 \cup U_2\right ] \enspace . \end{align}

Considering the results for $\mathsf{match}_{\mathrm{2}}$ in Example 4.4, we have

\begin{align*} &\alpha _{\mathit{sl}}(\mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2}))\\ &= \alpha _{\mathit{sl}}({\mathop{\downarrow }} [ uv, u^\infty v^\infty x^\infty, uxz, u^\infty x^\infty, v^\infty x^\infty, vxz, x^\infty, xz ]_{u,v,x,y,z} )\\ &= [\{ uv, uvx, uxz, ux, vx, vxz, x, xz\}, \{ y,z\},\{u,v,x,y,z\}] \end{align*}

and $\mathsf{match}_{\mathit{sl}}([S_1,L_1,U_1],[S_2,L_2,U_2]) \gt \alpha _{\mathit{sl}}(\mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2}))$ . In particular, the sharing group $uvxz$ does not appear in $\alpha _{\mathit{sl}}(\mathsf{match}_{\mathrm{2}}([T_1]_{U_1},[T_2]_{U_2}))$ which proves that $\mathsf{match}_{\mathit{sl}}$ is not a complete abstraction of $\mathsf{match}_{\mathrm{2}}$ . However, the next theorem shows that $\mathsf{match}_{\mathit{sl}}$ is optimal w.r.t. $\mathsf{match}_{\mathrm{2}}$ and by composition also w.r.t. $\mathsf{match}_{\mathrm{\omega }}$ and $\mathsf{match}$ .

Proof We prove that, given $[S_1, L_1, U_1]$ and $[S_2, L_2, U_2] \in \mathtt{ShLin}$ , we have

\begin{align*} \mathsf {match}_{\mathit {sl}}([S_1, L_1, U_1],[S_2, L_2, U_2])= \alpha _{\mathit {sl}}({\mathop {\downarrow }} [\mathsf {match}^{\prime}_{\mathrm {2}}(T_1, U_1, T_2, U_2)]_{U}) \end{align*}

where $T_i = \{B_{L_i} \mid B \in S_i\}$ is the set of maximal elements of $\gamma _{\mathit{sl}}([S_i, L_i, U_i])$ and $U=U_1 \cup U_2$ .

Let us define the function $\gamma _0$ which maps a pair $\langle B, L \rangle$ with $B, L \in \mathscr{P}({\mathcal{V}})$ to the $2$ -sharing group $B_{L \cup L_1}$ . We show that, given $H \subseteq \mathscr{P}({\mathcal{V}}) \times \mathscr{P}({\mathcal{V}})$ , we have $\alpha _{\mathit{sl}}([{\mathop{\downarrow }} \gamma _0(H)]_U) = [s(H), l(H), U]$ . Assume $\alpha _{\mathit{sl}}([{\mathop{\downarrow }} \gamma _0(S)]_U])=[R,V,U]$ . We have $R= \{\lfloor \!\lfloor o \rfloor \!\rfloor \mid o \in{\mathop{\downarrow }} \gamma _0(S)\}= \{\lfloor \!\lfloor o \rfloor \!\rfloor \mid o \in \gamma _0(S)\}=\{ \lfloor \!\lfloor \gamma _o(\langle B,L \rangle ) \rfloor \!\rfloor \mid \langle B,L \rangle \in S \}=\{ B \mid \langle B,L \rangle \in S \} = s(S)$ and $V=\{ x\in U \mid \forall o \in{\mathop{\downarrow }} \gamma _0(S), o(x) \leq 1\} = \{ x \in U \mid \forall o \in \gamma _0(S), o(x) \leq 1\} = \{ x \in U \mid \forall \langle B, L \rangle \in S, B_{L \cup L_1}(x) \leq 1\} \} = \{ x \in U \mid \forall \langle B, L \rangle \in S, x \in L_1 \cup L \cup (U \setminus B) \} = \bigcap \{ L_1 \cup L \cup (U \setminus B) \mid \langle B, L \rangle \in S \} = l(S)$ .

Therefore, if we prove that $\gamma _0(S^{\prime}_0 \cup S^{\prime\prime}_0) = \mathsf{match}^{\prime}_{\mathrm{2}}(T_1, U_1, T_2, U_2)$ , then we have $\mathsf{match}_{\mathit{sl}}([S_1,T_1,U_1],[S_2,T_2,U_2])= [s(S^{\prime}_0 \cup S^{\prime\prime}_0), l(S^{\prime}_0 \cup S^{\prime\prime}_0), U] = \alpha _{\mathit{sl}}([{\mathop{\downarrow }} \gamma _0(S^{\prime}_0 \cup S^{\prime\prime}_0)]_U]) = \alpha _{\mathit{sl}}({\mathop{\downarrow }} [\mathsf{match}^{\prime}_{\mathrm{2}}(T_1, U_1, T_2, U_2)]_{U}))$ .

Let us take $o \in \gamma _0(S^{\prime}_0 \cup S^{\prime\prime}_0)$ and prove $o \in \mathsf{match}^{\prime}_{\mathrm{2}}(T_1, U_1, T_2, U_2)$ . If $o=B_{L \cup L_1}$ for some $\langle B,L \rangle \in S^{\prime}_0$ , then $B \in S^{\prime}_2$ and $L=L_2$ . Therefore, $B_{L \cup L_1} = B_{L_2 \cup L_1} = B_{L_2} \in T_2$ since $B \cap U_1 = \emptyset$ . In particular, $B_{L_2} \in T^{\prime}_2$ , hence $B_{L_2} \in \mathsf{match}^{\prime}_{\mathrm{2}}(T_1, U_1, T_2, U_2)$ . Otherwise, $o= \gamma _0(\langle C, L \rangle )$ for $\langle C, L \rangle \in S^{\prime\prime}_0$ , where $C = B \cup \bigcup X$ and $L=L_2 \setminus nl(X) \setminus \bigcup (X \cap \mkern 1.5mu\overline{\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu)$ according to Definition 5.2. For each sharing group $B^{\prime} \in X$ , consider the $2$ -sharing group $B^{\prime}_{L_2} \in T^{\prime\prime}_2$ , and let $Y$ be the set of all those $2$ -sharing groups. We want to prove that $o$ is generated by $B_{L_1} \in T_1$ and $Y \subseteq T^{\prime\prime}_2$ , according to Definition 4.6. First, note that $\lfloor \!\lfloor (\biguplus \lfloor \!\lfloor Y \rfloor \!\rfloor )_{|U_1} \rfloor \!\rfloor =\lfloor \!\lfloor (\biguplus X)_{|U_1} \rfloor \!\rfloor = (\bigcup X) \cap U_1 = B \cap U_2 = \lfloor \!\lfloor (B_{L_1})_{|U_2} \rfloor \!\rfloor$ . Now, assume $v \in \lfloor \!\lfloor (B_{L_1})_{|U_2} \rfloor \!\rfloor$ . If $v$ is linear in $(B_{L_1})_{|U_2}$ , then $v \in L_1$ , hence $v \notin nl(X)$ and $v$ is linear in $(\biguplus X)_{|U_1}=(\biguplus \lfloor \!\lfloor Y \rfloor \!\rfloor )_{|U_1}$ . Therefore, $Y$ is a valid choice for $\mathsf{match}_{\mathrm{2}}^{\prime}$ and $\mathsf{match}^{\prime}_{\mathrm{2}}(B)$ generates $o^{\prime}=(B_{L_1} \wedge \biguplus Y) \uplus \biguplus (Y \cap \mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu)$ . We prove $o^{\prime}=o$ .

First, we prove that $B^{\prime} \in \mkern 1.5mu\overline{\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu$ iff $B^{\prime}_{L_2} \in \mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu$ . We have $B^{\prime}_{L_2} \in \mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu$ iff $B^{\prime}_{L_2} \in T^{\prime\prime}_2$ and $\forall v \in \lfloor \!\lfloor B^{\prime}_{L_2} \rfloor \!\rfloor \cap U_1, B^{\prime}_{L_1}(v)=\infty$ , iff $B^{\prime} \in S^{\prime\prime}_2$ and $\forall v \in B^{\prime} \cap U_1$ , $v \notin L_1$ iff $B^{\prime} \cap L_1 = \emptyset$ iff $B^{\prime} \in S^{\prime\prime}_2$ and $B^{\prime} \cap L_1=\emptyset$ iff $B \in S$ .

Now, it is immediate to check that, $\lfloor \!\lfloor o \rfloor \!\rfloor = B \cup \bigcup X =\lfloor \!\lfloor o^{\prime} \rfloor \!\rfloor$ . Then, consider $v \in \lfloor \!\lfloor o^{\prime} \rfloor \!\rfloor$ . We have that

\begin{align*} & o^{\prime}(v)=1 \Leftrightarrow \\ & (B_{L_1}(v)=1 \vee \left (\biguplus Y\right )(v)=1) \wedge \left (\biguplus (Y \cap \mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu)\right )=0 \Leftrightarrow \\ & (v \in L_1 \vee (v \in L_2 \setminus nl(X))) \wedge v \notin \bigcup (X \cap \mkern 1.5mu\overline{\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu) \Leftrightarrow \\ & v \in L_1 \vee (v \in L_2 \setminus nl(X) \setminus \bigcup (X \cap \mkern 1.5mu\overline{\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu) \Leftrightarrow \\ & v \in L_1 \vee v \in L \Leftrightarrow \\ & B_{L \cup L_1}(v) = o(v)=1 \end{align*}

It remains to prove that given $o \in \mathsf{match}^{\prime}_{\mathrm{2}}(T_1, U_1, T_2, U_2)$ , we have $o \in \gamma _0(S^{\prime}_0 \cup S^{\prime\prime}_0)$ . If $o \in T^{\prime}_2$ , then $o=B_{L_2}$ with $B \in S^{\prime}_2$ . Therefore, $\langle B, L_2 \rangle \in S^{\prime}_0$ and $o= \gamma _0(\langle B,L_2 \rangle )$ . Otherwise, $o=(o^{\prime} \wedge \biguplus Y) \uplus \biguplus (Y \cap \mkern 1.5mu\overline{\mkern -1.5muT\mkern -1.5mu}\mkern 1.5mu)$ . We know $o^{\prime}=B^{\prime}_{L_1}$ with $B^{\prime} \in S_1$ and let $X=\{ \lfloor \!\lfloor o^{\prime\prime} \rfloor \!\rfloor \mid o^{\prime\prime} \in Y \}$ . By Definition 4.6, $\bigcup X \cap U_1 = \lfloor \!\lfloor \uplus Y \rfloor \!\rfloor \cap U_1 = \lfloor \!\lfloor o^{\prime} \rfloor \!\rfloor \cap U_2 = B^{\prime} \cap U_2$ . Moreover, if $v \in L_1$ , then $o^{\prime}(v) \leq 1$ , and therefore, $v$ cannot appear twice in $Y$ , which means $v \notin nl(X)$ ; hence $L_1 \cap nl(X) = \emptyset$ . Therefore, $B^{\prime}$ and $X$ make a valid choice for $\mathsf{match}_{\mathit{sl}}$ and generate the pair

\begin{align*} \langle B, L \rangle = \left \langle B^{\prime} \cup \bigcup X, L_2 \setminus nl(X) \setminus \bigcup (X \cap \mkern 1.5mu\overline {\mkern -1.5muS\mkern -1.5mu}\mkern 1.5mu) \right \rangle . \end{align*}

Using the first half of the proof, it is easy to check that $B_{L \cup L^{\prime}} = o$ , which terminates the proof.

6.1 Matching and backward unification

First, we show with a simple example how the matching operator strictly improves the result of a standard goal-dependent analysis using forward and backward unification. Consider the goal $p(x,f(x,z),z)$ with the (abstract) call substitution $[x,z]_{xz}$ and the trivial program with just one clause:

$p(u,v,w).$

In order to analyze the goal, we first need to perform the forward unification between the call substitution $[x,z]_{xz}$ , the goal $p(x,f(x,z),z)$ , and the head of the clause $p(u,v,w)$ , and then project the result on the variables of the clause. In order to keep the notation simple, we do not perform renaming, unless necessary. This amounts to computing:

\begin{align*} \mathsf {mgu}_{\mathrm {\omega }}([x,z]_{xz}, \{u/x, v/f(x,z), w/z\}) = [uvx, vwz]_{uvwxz} \enspace . \end{align*}

By projecting the result on the variables of the clause, we obtain the entry substitution $[uv, vw]_{uvw}$ . Since the clause has no body, it is immediate to see that the exit substitution coincides with the entry substitution.

We now need to compute the backward unification of the exit substitution $[uv, vw]_{uvw}$ , the call substitution $[x, z]_{xz}$ and $\theta =\{u/x, v/f(x,z), w/z\}$ . If we implement this operator with the aid of matching, we may first unify $[x, z]_{xz}$ with $\theta$ , obtaining $[uvx, vwz]_{uvwxz}$ as above, and then apply the matching with the exit substitution $[uv, vw]_{uvw}$ obtaining

\begin{align*} \mathsf {match}_{\mathrm {\omega }}([uv, vw]_{uvw}, [uvx, vwz]_{uvwxz} ) = [uvx, vwz]_{uvwxz} \enspace . \end{align*}

By projecting the result on the variables of the goal, we obtain the result $[x, z]_{xz}$ , which proves that $x$ and $z$ do not share.

On the contrary, if we avoid matching, the backward-unification operator must unify $[x, z]_{xz}$ with $[uv, vw]_{uvw}$ and $\theta$ in any order it deems fit. However, this means that the operator must correctly approximate the mgu of any substitution $\theta _1$ in the concretization of the call substitution $[x,z]_{xz}$ , with any substitution $\theta _2$ in the concretization of the exit substitution $[uv, vw]_{uvw}$ and $\theta =\{u/x, v/f(x,z), w/z\}$ . If we choose $\theta _1 = \epsilon$ and $\theta _2 = \{v/f(w,u)\}$ , we obtain that the unification of $\theta _1$ , $\theta _2$ , and $\theta$ is

\begin{align*} \{u/x, v/f(x,x), w/x, z/x \} \end{align*}

which is not in the concretization of $[uvx, vwz]_{uvwxz}$ since $u$ , $v$ , and $w$ share a common variable. Moreover, in this substitution, $x$ and $z$ share. This means that, after the projection on the variables of the goals $x$ and $z$ , the result will always include the $\omega$ -sharing group $xz$ . Note that this consideration holds for any correct abstract unification operator that avoids matching.

It is worth noting that the above example, with minimal changes, also works for $\mathtt{ShLin}^2$ and $\mathtt{ShLin}$ : also in these cases, matching improves the precision of the analysis.

6.2 Example on a nontrivial program

Consider a program implementing the $\mathit{member}$ predicate. Using the Prolog notation for lists, we have

$\mathit{member}(u, [u|v]).$

$\mathit{member}(u, [v|w]) \leftarrow \mathit{member}(u, w).$

We want to analyze the goal $\mathit{member}(x,[y])$ in the domain $\mathtt{ShLin}^2$ using the call substitution $[xy, xz]_{xyz}$ .

We start by considering the first clause of $\mathit{member}$ . The concrete unification of the goal $\mathit{member}(x,[y])$ and the head of the clause $\mathit{member}(u, [u|v])$ yields the most general unifier $\theta = \{ x/u, y/u, v/[] \}$ . Forward unification computes the entry substitution as the abstract mgu between the call substitution $[xy, xz]_{xyz}$ and $\theta$ . Proceeding one binding at a time, we have

• • $\mathsf{mgu}_{\mathrm{2}}([xy, xz]_{xyz}, \{x/u\}) = [uxy, uxz]_{uxyz}$ ;

• • $\mathsf{mgu}_{\mathrm{2}}([uxy, uxz]_{uxyz}, \{y/u\}) ={\mathop{\downarrow }} [u^\infty x^\infty y^\infty ]_{uxyz}$ ;

• • $\mathsf{mgu}_{\mathrm{2}}({\mathop{\downarrow }} [u^\infty x^\infty y^\infty ]_{uxyz}, \{v/[]\} ) ={\mathop{\downarrow }} [u^\infty x^\infty y^\infty ]_{uvxyz}$ .

Projecting over the variables of the clause, we get the entry substitution ${\mathop{\downarrow }} [u^\infty ]_{uv}$ . Since this clause has no body, the entry substitution is equal to the exit substitution, and we may proceed to compute the answer substitution through backward unification.

First, we consider the case when the backward unification is performed using the standard $\mathsf{mgu}_{\mathrm{2}}$ operator. We need to unify the call substitution $[xy, xz]_{xyz}$ , the exit substitution ${\mathop{\downarrow }} [u^\infty ]_{uv}$ , and the concrete substitution $\theta$ (the same as before). Unifying call and exit substitution is immediate since they are relative to disjoint variables of interest: the result is ${\mathop{\downarrow }} [u^\infty, xy, xz]_{uvxyz}$ , obtained by collecting all sharing groups together. This should be unified with $\theta$ . Proceeding one binding at a time, and omitting the set of variables of interest since it does not change, we have

• • $\mathsf{mgu}_{\mathrm{2}}({\mathop{\downarrow }} [u^\infty, xy, xz], \{x/u\}) ={\mathop{\downarrow }} [u^\infty x^\infty y^\infty, u^\infty x^\infty y^\infty z^\infty, u^\infty x^\infty z^\infty ]$ ;

• • $\mathsf{mgu}_{\mathrm{2}}({\mathop{\downarrow }} [u^\infty x^\infty y^\infty, u^\infty x^\infty y^\infty z^\infty, u^\infty x^\infty z^\infty ], \{y/u\}) ={\mathop{\downarrow }} [u^\infty x^\infty y^\infty, u^\infty x^\infty y^\infty z^\infty ]$ ;

• • $\mathsf{mgu}_{\mathrm{2}}({\mathop{\downarrow }} [u^\infty x^\infty y^\infty, u^\infty x^\infty y^\infty z^\infty ],\{v/[]\})={\mathop{\downarrow }} [u^\infty x^\infty y^\infty, u^\infty x^\infty y^\infty z^\infty ]$ .

Projecting over the set of variables in the goal, we get ${\mathop{\downarrow }} [x^\infty y^\infty, x^\infty y^\infty z^\infty ]_{xyz}$ .

On the contrary, if we perform backward unification using the matching operation, we need to compute the matching of the exit substitution $[u^\infty ]_{uv}$ with the entry substitution before variable projection $[u^\infty x^\infty y^\infty ]_{uvxyz}$ , namely:

\begin{align*} \mathsf {match}_{\mathrm {2}}({\mathop {\downarrow }} [u^\infty ]_{uv}, {\mathop {\downarrow }} [u^\infty x^\infty y^\infty ]_{uvxyz}) = {\mathop {\downarrow }} [u^\infty x^\infty y^\infty ]_{uvxyz} \enspace . \end{align*}

Projecting over the variables of the goal, we get ${\mathop{\downarrow }} [x^\infty y^\infty ]_{xyz}$ : using matching we can prove that $z$ is ground in the answer substitution.

However, we still need to check what happens when we analyze the second clause of the $\mathit{member}$ predicate. In this case, the concrete unification between $\mathit{member}(x, [y])$ and $\mathit{member}(u, [v|w])$ gives the substitution $\theta = \{x/u, y/v, w/[]\}$ . Then, forward unification between the call substitution and $\theta$ gives

• • $\mathsf{mgu}_{\mathrm{2}}([xy, xz]_{xyz}, \{x/u\}) = [uxy, uxz]_{uxyz}$ ;

• • $\mathsf{mgu}_{\mathrm{2}}([uxy, uxz]_{uxyz}, \{y/v\}) = [uvxy, uxz]_{uvxyz}$ ;

• • $\mathsf{mgu}_{\mathrm{2}}([uvxy, uxz]_{uvxyz}, \{w/[]\}) = [uvxy, uxz]_{uvwxyz}$ .

Projecting over the variables of the clause, we get the entry substitution $[uv, v]_{uvw}$ .

Now we should compute the answer substitution of the body $\mathit{member}(u, w)$ under the call substitution $[uv, u]_{uvw}$ . We could proceed by showing all the details, but we try to be more concise. In the abstract substitution $[uv, w]_{uvw}$ , the variable $w$ is known to be ground. When $\mathit{member}$ is called with its second argument ground, the first argument becomes ground too. This property is easily captured by $\mathtt{Sharing}$ and more precise domains, independently of the fact that we use matching or not for the backward unification. Therefore, we can conclude that the answer substitution for the goal $\mathit{member}(u, w)$ under the entry substitution $[uv, u]_{uvw}$ is $[\emptyset ]_{uvw}$ . Although we do not generally write the empty $\mathrm{2}$ -sharing group $\emptyset$ in an element of $\mathtt{ShLin}^2$ , in this case, it is important to write it in order to distinguish $[\emptyset ]_{uvw}$ , denoting those substitutions in which $u, v, w$ are ground, from $[]_{uvw}$ , denoting a non-succeeding derivation.

Performing the backward unification of $[\emptyset ]_{uvw}$ , we get the answer substitution $[\emptyset ]_{xyz}$ , independently of the use of matching.

Putting together the results we got for analyzing the goal $\mathit{member}(x, [y])$ according to the two clauses of the program, we have shown that using matching, we get $[x^\infty y^\infty ]_{xyz}$ , while using standard unification, we get $[x^\infty y^\infty, x^\infty y^\infty z^\infty ]_{xyz}$ , and we are not able to prove that $z$ is ground after the goal returns.