Corrigendum to "Counting Database Repairs that Satisfy Conjunctive Queries with Self-Joins"

TL;DR

This paper corrects a false lemma in a previous work on counting database repairs satisfying conjunctive queries with self-joins, providing a new proof for the main theorem.

Contribution

It offers a corrected proof for a key theorem in database repair counting, addressing an error in prior literature.

Findings

The original lemma was false, affecting the proof of the main theorem.

A new, valid proof for Theorem 3 is provided.

Ensures correctness of counting repairs in databases with self-joins.

Abstract

The helping Lemma 7 in [Maslowski and Wijsen, ICDT, 2014] is false. The lemma is used in (and only in) the proof of Theorem 3 of that same paper. In this corrigendum, we provide a new proof for the latter theorem.

Equations

$${\mathsf{enc}}_{\mathbf{S}}(q_0)$$

$${\mathsf{enc}}_{\mathbf{S}}^{\ast }(q_0)$$

$$\mathbf{r}\models q^{\prime }⟺f(\mathbf{r})\models q.$$

$$R\left({{\theta }_R}^{-1}(x_1),\dots ,{{\theta }_R}^{-1}(x_n)\right)\in q.$$

$$R\left(\theta \left({{\theta }_R}^{-1}(x_1)\right),\dots ,\theta \left({{\theta }_R}^{-1}(x_n)\right)\right)\in f(\mathbf{r}).$$

$$R\left(⟨{\theta }_L({{\theta }_R}^{-1}(x_1)),x_1⟩,\dots ,⟨{\theta }_L({{\theta }_R}^{-1}(x_n)),x_n⟩\right)\in f(\mathbf{r}).$$

$$R\left(⟨\mu (x_1),x_1⟩,\dots ,⟨\mu (x_n),x_n⟩\right)\in f(\mathbf{r}).$$

$$\mathbf{r}\models q⟺f(\mathbf{r})\models {\mathsf{enc}}_{\mathbf{S}}^{\ast }(q).$$

Full text

1 The Flaw

The helping Lemma 7 in [MW14] is false. A counterexample is given next.

Example 1**.**

For $\mathbf{S}=\{R,S\}$ and $q=\{R(\underline{x},y)$, $S(\underline{y})\}$, we have ${\mathsf{enc}}_{\mathbf{S}}({q})=\{N(\underline{R,x},y)$, $N(\underline{S,y},0)\}$. From [MW14, Lemma 8], it follows that $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}_{\mathbf{S}}({q})})$ is $\sharp\P$-hard. From [MW13, Theorem 4], it follows that $\mathsf{\sharp CERTAINTY}({q})$ is in $\FP$. Consequently, assuming $\sharp\P\neq\FP$, there exists no polynomial-time many-one reduction from $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}_{\mathbf{S}}({q})})$ to $\mathsf{\sharp CERTAINTY}({q})$. Lemma 7 in [MW14] is thus false. ∎

The first part in the proof of Lemma 7 in [MW14] is correct; it shows a polynomial-time many-one reduction from $\mathsf{\sharp CERTAINTY}({q})$ to $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}_{\mathbf{S}}({q})})$. However, the second part in that proof is flawed when it claims “We can compute in polynomial time the (unique) database ${\mathbf{db}}_{0}^{\prime}$ with schema $\mathbf{S}$ such that ${\mathsf{enc}}_{\mathbf{S}}({{\mathbf{db}}_{0}^{\prime}})={\mathbf{db}}_{0}$.” The flaw is that the database ${\mathbf{db}}_{0}^{\prime}$ does not generally exist, as shown next. Let $\mathbf{S}=\{R,S\}$ and $q=\{R(\underline{x},y)$, $S(\underline{y})\}$, as in Example 1. Then, ${\mathsf{enc}}_{\mathbf{S}}({q})=\{N(\underline{R,x},y)$, $N(\underline{S,y},0)\}$. A legal input to $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}_{\mathbf{S}}({q})})$ is ${\mathbf{db}}_{0}=\{N(\underline{R,b},c)$, $N(\underline{S,c},0)$, $N(\underline{S,c},1)\}$. However, there exists no database ${\mathbf{db}}_{0}^{\prime}$ such that ${\mathsf{enc}}_{\mathbf{S}}({{\mathbf{db}}_{0}^{\prime}})={\mathbf{db}}_{0}$. Indeed, for every database ${\mathbf{db}}_{0}^{\prime}$ with schema $\mathbf{S}$, if $N(\underline{S,c},s)\in{\mathsf{enc}}_{\mathbf{S}}({{\mathbf{db}}_{0}^{\prime}})$, then $s=0$.

2 The Solution

The following treatment is relative to a database schema $\mathbf{S}$. Let $k,m$ be non-negative integers such that every relation name in $\mathbf{S}$ has at most $k$ primary-key positions, and at most $m$ non-primary-key positions. We define a new function ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})$ which encodes Boolean conjunctive queries $q$ into unirelational Boolean conjunctive queries. For ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})$, we use a fresh relation name $N$ with $k+1$ primary-key positions, and $m$ non-primary-key positions. For every atom $R(\underline{\vec{x}},\vec{y})$ in $q$, the query ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})$ will contain some atom $N(\underline{R,\vec{x},\vec{0}},\vec{y},\vec{z})$, where $\vec{0}$ is a sequence of padding zeros, and $\vec{z}$ is a sequence of padding fresh variables, all distinct and not occurring elsewhere. This encoding is different from [MW14, Definition 3] where a sequence of padding zeros was used instead of $\vec{z}$.

Example 2**.**

We illustrate the difference between the old encoding ${\mathsf{enc}}_{\mathbf{S}}({\cdot})$ of [MW14, Definition 3] and the newly proposed encoding ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({\cdot})$. For $q_{0}=\{R(\underline{x},y)$, $S(\underline{y})\}$, we have

[TABLE]

We recall from [MW14, p. 156] that the complex part of a Boolean conjunctive query contains every atom $F\in q$ such that some non-primary-key position in $F$ contains either a variable with two or more occurrences in $q$ or a constant. Note that $N(\underline{S,y},0)$ belongs to the complex part of ${\mathsf{enc}}_{\mathbf{S}}({q_{0}})$, while $N(\underline{S,y},z)$ is not in the complex part of ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q_{0}})$. ∎

Definition 1**.**

We define skBCQ as the class of Boolean conjunctive queries in which all relation names are simple-key. We say that a query $q\in\mbox{\sf skBCQ}$ is minimal if both

• •

$q$ contains no two distinct atoms $R_{1}(\underline{x_{1}},\vec{y}_{1})$, $R_{2}(\underline{{x}_{2}},\vec{y}_{2})$ such that $R_{1}=R_{2}$ and $x_{1}=x_{2}$; and

• •

there exists no substitution $\theta$ over $\mathsf{vars}({q})$ such that $\theta(q)\subsetneq q$.

We define cxBCQ as the class of unirelational Boolean conjunctive queries $q$ whose relation name has signature $[{n},{2}]$ (for some $n\geq 2$) such that for every $F\in q$, the first position of $F$ is a constant.

Definition 2**.**

The intersection graph of a Boolean conjunctive query is an undirected graph whose vertices are the atoms of $q$. There is an undirected edge between any two atoms that have a variable in common.

Lemma 1**.**

Assume $\sharp\P\neq\FP$. For every minimal query $q$ in skBCQ, if $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})})$ is $\sharp\P$-hard, then so is $\mathsf{\sharp CERTAINTY}({q})$.

Proof.

Let $q$ be a minimal query in skBCQ such that $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})})$ is $\sharp\P$-hard. Note that $q$ does not need to be unirelational or self-join-free. The query ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})$, which is unirelational, is a legal input to the function IsEasy of [MW14, p. 163].222For uniformity of notation, we will assume that the unirelational query uses relation name $N$. Since $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})})$ is $\sharp\P$-hard, the function IsEasy will return ${\mathbf{false}}$ on input ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})$. This function will repeat, as long as possible, the following step: pick some atom $N(\underline{R,c},\vec{y})$ and some variable $y\in\mathsf{vars}({\vec{y}})$, with $R$ some relation name (treated as a constant) and $c$ some constant, and replace all occurrences of $y$ with an arbitrary constant. Let $\bar{q}$ be the query that results from these steps. Clearly, for every atom $N(\underline{R,s},\vec{t})$ in $\bar{q}$, either $s$ is a constant or $\vec{t}$ is variable-free. Since IsEasy returns ${\mathbf{false}}$ on input $\bar{q}$, it follows that $\bar{q}$ does not satisfy the premise of [MW14, Lemma 5]. Therefore, it must be the case that $\bar{q}$ contains two distinct atoms $N(\underline{R,x},\vec{u})$ and $N(\underline{S,y},\vec{w})$ that are connected in the intersection graph of $\bar{q}$ such that

• •

$R$ and $S$ are relation names (serving as constants), not necessarily distinct;

• •

$x$ and $y$ are distinct variables; and

• •

neither $\vec{u}$ nor $\vec{w}$ is exclusively composed of variables occurring only once in the query. That is, $N(\underline{R,x},\vec{u})$ and $N(\underline{S,y},\vec{w})$ belong to the complex part of $\bar{q}$.

For every relation name $R$ that appears in $q$, we assume fresh relation names $R_{1},R_{2},R_{3},\ldots$ with the same signature as $R$. Using these relation names, we can construct a self-join-free Boolean conjunctive query $q^{\prime}$ such that $|{q^{\prime}}|=|{q}|$ and for every atom $R(\underline{x},\vec{y})$ in $q$, the query $q$ contains some atom $R_{i}(\underline{x},\vec{y})$. For example, if $q=\{R(\underline{x},y)$, $R(\underline{y},z)$, $S(\underline{z},x)\}$, then we can let $q^{\prime}=\{R_{1}(\underline{x},y)$, $R_{2}(\underline{y},z)$, $S_{1}(\underline{z},x)\}$. It can now be shown that the function IsSafe in [MW14, p. 158] will return ${\mathbf{false}}$ on input $q^{\prime}$, and thus $\mathsf{\sharp CERTAINTY}({q^{\prime}})$ is $\sharp\P$-hard. Indeed, whenever IsEasy picked $N(\underline{R,c},\vec{y})$ and some variable $y\in\mathsf{vars}({\vec{y}})\cap\mathsf{vars}({q})$, the function IsSafe can execute SE3 on the corresponding $R_{i}$-atom of $q^{\prime}$. This eventually leads to a query whose complex part contains two atoms $R_{i}(\underline{x},\vec{u}^{\prime})$ and $S_{j}(\underline{y},\vec{w}^{\prime})$, $x\neq y$, that are connected in the intersection graph, at which point IsSafe will return ${\mathbf{false}}$. In this reasoning, one needs that non-primary-key positions are padded with fresh variables occurring only once, as can be seen from Example 2.

In the remainder of this proof, we show the existence of a polynomial-time many-one reduction from $\mathsf{\sharp CERTAINTY}({q^{\prime}})$ to $\mathsf{\sharp CERTAINTY}({q})$. We incidentally note that the remaining reasoning, which generalizes the proof of [MW14, Lemma 2], does not require that relation names are simple-key.

Let $f$ be a mapping from facts to facts such that for every atom $R_{i}(x_{1},\dots,x_{n})\in q^{\prime}$, for every $R_{i}$-fact $A\mathrel{\mathop{:}}=R_{i}(a_{1},\dots,a_{n})$, $f(A)\mathrel{\mathop{:}}=R(\langle{a_{1}},{x_{1}}\rangle,\dots,\langle{a_{n}},{x_{n}}\rangle)$. Notice that $f$ maps $R_{i}$-facts to $R$-facts. Here, every couple $\langle{a_{i}},{x_{i}}\rangle$ denotes a constant such that $\langle{a_{i}},{x_{i}}\rangle=\langle{a_{j}},{x_{j}}\rangle$ if and only if both $a_{i}=a_{j}$ and $x_{i}=x_{j}$. Moreover, if $c$ is a constant, then $\langle{c},{c}\rangle\mathrel{\mathop{:}}=c$. Since no two distinct atoms of $q$ agree on both their relation name and primary key, it will be the case that for all facts $A$ and $B$, $A\sim B$ if and only if $f(A)\sim f(B)$, where $\sim$ denotes “is key-equal-to.”

We extend the function $f$ in the natural way to databases ${\mathbf{db}}$ that use only relation names from $q^{\prime}$: $f({\mathbf{db}})\mathrel{\mathop{:}}=\{f(A)\mid A\in{\mathbf{db}}\}$. Clearly, $f({\mathbf{db}})$ can be computed in polynomial time in the size of ${\mathbf{db}}$. Let ${\mathbf{db}}$ be a set of facts with relation names in $q^{\prime}$. It can be easily seen that $|{\mathsf{rset}({{\mathbf{db}}})}|=|{\mathsf{rset}({f({\mathbf{db}})})}|$ and $\mathsf{rset}({f({\mathbf{db}})})=\{f(\mathbf{r})\mid\mathbf{r}\in\mathsf{rset}({{\mathbf{db}}})\}$. Let $\mathbf{r}$ be an arbitrary repair of ${\mathbf{db}}$. It suffices to show that

[TABLE]

For the implication $\implies$, assume that $\mathbf{r}\models q^{\prime}$. We can assume a valuation $\theta$ over $\mathsf{vars}({q^{\prime}})$ such that $\theta(q^{\prime})\subseteq\mathbf{r}$. Let $\mu$ be the valuation such that for every variable $x\in\mathsf{vars}({q^{\prime}})$, $\mu(x)=\langle{\theta(x)},{x}\rangle$. By our construction of $q^{\prime}$ and $f$, it will be the case that $\mu(q)\subseteq f(\mathbf{r})$, thus $f(\mathbf{r})\models q$.

For the implication $\impliedby$, assume that $f(\mathbf{r})\models q$. We can assume a valuation $\theta$ over $\mathsf{vars}({q})$ such that $\theta(q)\subseteq f(\mathbf{r})$. Notice that if $c$ is a constant in $q$, then it must be the case that $\theta(c)=\langle{c},{c}\rangle\mathrel{\mathop{:}}=c$. We define $\theta_{L}$ as the substitution that maps every variable $x$ in $\mathsf{vars}({q})$ to the first coordinate of $\theta(x)$; and $\theta_{R}$ maps every $x$ to the second coordinate of $\theta(x)$. It is convenient to think of $L$ and $R$ as references to the Left and the Right coordinates, respectively. Thus, by definition, $\theta(x)=\langle{\theta_{L}(x)},{\theta_{R}(x)}\rangle$.

By inspecting the right-hand coordinates of couples $\langle{a_{i}},{x_{i}}\rangle$ in $f(\mathbf{r})$, it can be easily seen that $\theta(q)\subseteq f(\mathbf{r})$ implies $\theta_{R}(q)\subseteq q$. Since the query $q$ is minimal, it follows that $\theta_{R}(q)=q$, i.e., $\theta_{R}$ is an automorphism. Since the inverse of an automorphism is an automorphism, ${\theta_{R}}^{-1}$ is an automorphism as well. Note that $\theta_{R}$ will be the identity on constants that appear in $q$. We now define $\mu\mathrel{\mathop{:}}=\theta_{L}\circ{\theta_{R}}^{-1}$ (i.e., $\mu$ is the composed function $\theta_{L}$ after the inverse of $\theta_{R}$), and show that $\mu(q^{\prime})\subseteq\mathbf{r}$, which implies the desired result that $\mathbf{r}\models q^{\prime}$. To this extent, let $R_{i}(x_{1},\dots,x_{n})$ be an arbitrary atom of $q^{\prime}$. It suffices to show $R_{i}(\mu(x_{1}),\dots,\mu(x_{n}))\in\mathbf{r}$, which can be proved as follows. From $R_{i}(x_{1},\dots,x_{n})\in q^{\prime}$, it follows $R(x_{1},\dots,x_{n})\in q$. Thus, since ${\theta_{R}}^{-1}$ is an automorphism,

[TABLE]

Since $\theta(q)\subseteq f(\mathbf{r})$,

[TABLE]

Since, for every symbol $s$, $\theta(s)=\langle{\theta_{L}(s)},{\theta_{R}(s)}\rangle$ and $\theta_{R}\left({{\theta_{R}}^{-1}(s)}\right)=s$, we obtain

[TABLE]

That is, by our definition of $\mu$,

[TABLE]

From this, it is correct to conclude that $R_{i}(\mu(x_{1}),\dots,\mu(x_{n}))\in\mathbf{r}$. This concludes the proof. ∎

Lemma 2**.**

For every Boolean conjunctive query $q$, there exists a polynomial-time many-one reduction from $\mathsf{\sharp CERTAINTY}({q})$ to $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})})$.

Proof.

Let $q$ be a Boolean conjunctive query. Let $R$ be a relation name that occurs in $q$. Let $\{R(\underline{\vec{x}_{i}},\vec{y}_{i})\}_{i=1}^{m}$ be the set of $R$-atoms of $q$. Then, ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})$ will contain, for every $i\in\{1,\dots,m\}$, some atom $N(\underline{R,\vec{x}_{i},\vec{0}},\vec{y}_{i},\vec{z}_{i})$, where $\vec{z}_{i}$ is a (possibly empty) sequence of distinct fresh variables not occurring elsewhere. For every $R$-fact $A\mathrel{\mathop{:}}=R(\underline{\vec{a}},\vec{b})$, we define $f(A)\mathrel{\mathop{:}}=N(\underline{R,\vec{a},\vec{0}},\vec{b},\vec{0})$. Note here that $f(A)$ depends on the signatures of $R$ and $N$, but not on the $R$-atoms of $q$. The mapping $f$ is defined similarly for all relation names that appear in $q$. It can be easily seen that for all facts $A$ and $B$ whose relation names appear in $q$, $A\sim B$ if and only if $f(A)\sim f(B)$.

If ${\mathbf{db}}$ is an instance of $\mathsf{\sharp CERTAINTY}({q})$, we can assume without loss of generality that every relation name in ${\mathbf{db}}$ also appears in $q$. We extend the function $f$ to such instances ${\mathbf{db}}$ of $\mathsf{\sharp CERTAINTY}({q})$: $f({\mathbf{db}})\mathrel{\mathop{:}}=\{f(A)\mid A\in{\mathbf{db}}\}$. Obviously, $f({\mathbf{db}})$ can be computed in polynomial time in the size of ${\mathbf{db}}$. It is also obvious that $|{\mathsf{rset}({{\mathbf{db}}})}|=|{\mathsf{rset}({f({\mathbf{db}}})}|$ and $\mathsf{rset}({f({\mathbf{db}})})=\{f(\mathbf{r})\mid\mathbf{r}\in\mathsf{rset}({{\mathbf{db}}})\}$. It suffices to show that for every repair $\mathbf{r}$ of ${\mathbf{db}}$,

[TABLE]

For the implication $\implies$, assume $\mathbf{r}\models q$. We can assume a valuation $\theta$ over $\mathsf{vars}({q})$ such that $\theta(q)\subseteq\mathbf{r}$. Let $\theta^{\prime}$ be the valuation that extends $\theta$ from $\mathsf{vars}({q})$ to $\mathsf{vars}({{\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})})$ such that $\theta^{\prime}(z)=0$ for every variable $z$ that appears in ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})$ but not in $q$. By the construction of $f$, it will be the case that $\theta^{\prime}({\mathsf{enc}}^{\ast}_{\mathbf{S}}({q}))\subseteq f(\mathbf{r})$. Indeed, if ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})$ contains $N(\underline{R,\vec{x}_{i},\vec{0}},\vec{y}_{i},\vec{z}_{i})$, then $\mathbf{r}$ will contain $R(\underline{\theta(\vec{x}_{i})},\theta(\vec{y}_{i}))$, hence $f(\mathbf{r})$ will contain $N(\underline{R,\theta^{\prime}(\vec{x}_{i}),\vec{0}},\theta^{\prime}(\vec{y}_{i}),\vec{0})$ and $\theta^{\prime}(\vec{z}_{i})=\vec{0}$.

For the implication $\impliedby$, assume $f(\mathbf{r})\models{\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})$. We can assume a valuation $\theta$ over $\mathsf{vars}({{\mathsf{enc}}^{\ast}_{\mathbf{S}}({q})})$ such that $\theta({\mathsf{enc}}^{\ast}_{\mathbf{S}}({q}))\subseteq f(\mathbf{r})$. It is straightforward to see that $\theta(q)\subseteq\mathbf{r}$. ∎

We now give the new proof for Theorem 3 in [MW14].

Theorem 1** ([MW14, Theorem 3]).**

The set $\{\mathsf{\sharp CERTAINTY}({q})\mid q\in\mbox{\sf skBCQ}\}$ exhibits an effective $\FP$-$\sharp\P$-dichotomy.

New proof.

Let $q\in\mbox{\sf skBCQ}$. It can be decided whether $q$ can be satisfied by a consistent database. If $q$ cannot be satisfied by a consistent database, then for every database ${\mathbf{db}}$, the number of repairs of ${\mathbf{db}}$ satisfying $q$ is [math]. An example is $q=\{R(\underline{x},0)$, $R(\underline{x},1)\}$. Assume next that $q$ can be satisfied by a consistent database. Then, we can compute a minimal query $q_{m}$ such that for every database, the number of repairs satisfying $q_{m}$ is equal to the number of repairs satisfying $q$. That is, the problems $\mathsf{\sharp CERTAINTY}({q_{m}})$ and $\mathsf{\sharp CERTAINTY}({q})$ are identical.

Then, ${\mathsf{enc}}^{\ast}_{\mathbf{S}}({q_{m}})$ belongs to cxBCQ. By [MW14, Lemma8], the set $\{\mathsf{\sharp CERTAINTY}({q})\mid q\in\mbox{\sf cxBCQ}\}$ exhibits an effective $\FP$-$\sharp\P$-hard dichotomy. If the problem $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}^{\ast}_{\mathbf{S}}({q_{m}})})$ is in $\FP$, then $\mathsf{\sharp CERTAINTY}({q})$ is in $\FP$ by Lemma 2; and if $\mathsf{\sharp CERTAINTY}({{\mathsf{enc}}^{\ast}_{\mathbf{S}}({q_{m}})})$ is $\sharp\P$-hard, then $\mathsf{\sharp CERTAINTY}({q})$ is $\sharp\P$-hard by Lemma 1. Consequently, $\mathsf{\sharp CERTAINTY}({q})$ is in $\FP$ or $\sharp\P$-hard, and it is is decidable which of the two cases applies. ∎