ODIn
/
paper-BagRelationalPDBsAreHard
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

Finished changes @atri021122.

master
Aaron Huber 2022-02-16 09:41:00 -05:00
parent ff7d0c3630
commit dd701e3478
6 changed files with 32 additions and 27 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
Draft Comments/Bag_PDBs-AtriCommentsFeb11.pdf
View File

Binary file not shown.

24
circuits-model-runtime.tex
View File

@ -24,7 +24,7 @@
% In practice there is often a limited number of alternatives for each block (e.g., which of five conflicting data sources to trust). Note that all \tis trivially fulfill this condition (i.e., $c = 1$).}
%That is for \bis that fulfill this restriction approximating the expectation of results of SPJU queries is only has a constant factor overhead over deterministic query processing (using one of the algorithms for which we prove the claim).
% with the same complexity as it would take to evaluate the query on a deterministic \emph{bag} database of the same size as the input PDB.
In~\Cref{sec:intro}, we introduced the structure $T_{det}\inparen{\cdot}$ to analyze the runtime complexity of~\Cref{prob:expect-mult}.
To decouple our results from specific join algorithms, we first abstract the cost of a join.
\begin{Definition}[Join Cost]
@ -41,24 +41,24 @@ For these algorithms, $\jointime{R_1, \ldots, R_n}$ is linear in the {\em AGM bo
\noindent\resizebox{1\linewidth}{!}{
\begin{minipage}{1.0\linewidth}
\begin{align*}
\qruntimenoopt{R,\db,\bound} & = |\db.R| &
\qruntimenoopt{\sigma \query, \db,\bound} & = \qruntimenoopt{\query,\db} &
\qruntimenoopt{\pi \query, \db,\bound} & = \qruntimenoopt{\query,\db,\bound} + \abs{\query(\db)}
\qruntimenoopt{R,\gentupset,\bound} & = |\gentupset.R| &
\qruntimenoopt{\sigma \query, \gentupset,\bound} & = \qruntimenoopt{\query,\gentupset} &
\qruntimenoopt{\pi \query, \gentupset,\bound} & = \qruntimenoopt{\query,\gentupset,\bound} + \abs{\query(\gentupset)}
\end{align*}\\[-15mm]
\begin{align*}
\qruntimenoopt{\query \cup \query', \db,\bound} & = \qruntimenoopt{\query, \db,\bound} +
\qruntimenoopt{\query', \db,\bound} +
\abs{\query(D)}+\abs{\query'(D)} \\
\qruntimenoopt{\query_1 \bowtie \ldots \bowtie \query_m, \db,\bound}
& = \qruntimenoopt{\query_1, \db,\bound} + \ldots +
\qruntimenoopt{\query_m,\db,\bound} +
\jointime{\query_1(\db), \ldots, \query_m(\db)}
\qruntimenoopt{\query \cup \query', \gentupset,\bound} & = \qruntimenoopt{\query, \gentupset,\bound} +
\qruntimenoopt{\query', \gentupset,\bound} +
\abs{\query\inparen{\gentupset}}+\abs{\query'\inparen{\gentupset}} \\
\qruntimenoopt{\query_1 \bowtie \ldots \bowtie \query_m, \gentupset,\bound}
& = \qruntimenoopt{\query_1, \gentupset,\bound} + \ldots +
\qruntimenoopt{\query_m,\gentupset,\bound} +
\jointime{\query_1(\gentupset), \ldots, \query_m(\gentupset)}
\end{align*}
\end{minipage}
}\\
Under this model, an $\raPlus$ query $\query$ evaluated over database $\db$ has runtime $O(\qruntimenoopt{Q,\db})$.
Under this model, an $\raPlus$ query $\query$ evaluated over database $\gentupset$ has runtime $O(\qruntimenoopt{Q,\gentupset})$.
We assume that full table scans are used for every base relation access. We can model index scans by treating an index scan query $\sigma_\theta(R)$ as a base relation.
%Observe that
% () .\footnote{This claim can be verified by e.g. simply looking at the {\em Generic-Join} algorithm in~\cite{skew} and {\em factorize} algorithm in~\cite{factorized-db}.} It can be verified that the above cost model on the corresponding $\raPlus$ join queries correctly captures the runtime of current best known .

8
intro-rewrite-070921.tex
View File

@ -3,8 +3,8 @@
\section{Introduction}\label{sec:intro}
\secrev{
This work explores the problem of computing the expectation of a tuple's multiplicity in an important special case of bag \abbrTIDB, which we call a \abbrCTIDB. A \abbrCTIDB,
$\pdb = \inparen{\worlds, \bpd}$ encodes a bag of uncertain tuples such that each tuple in $\pdb$ has a multiplicity of at most $\bound$. $\tupset$ is the set of tuples appearing across all possible worlds, and the set of all worlds is encoded in $\worlds$, which is the set of all vectors of length $\numvar=\abs{\tupset}$ such that each index corresponds to a distinct $\tup \in \tupset$ storing its multiplicity. $\bpd$ is a product distribution over the set of all worlds. A given world $\worldvec \in\worlds$ can be interpreted such that, for each $\tup \in \tupset$, $\worldvec\pbox{\tup}$ is the multiplicity of $\tup$ in $\worldvec$. The resulting product distribution can then be encoded as $\prob_{\tup} = \probOf\pbox{W\pbox{\tup} = j}$ (for $j \in\pbox{\bound}$), where each tuple multiplicity combination $\inparen{\inparen{\tup, \bound} \in \tupset\times\pbox{\bound}}$ %distribution
This work explores the problem of computing the expectation of a tuple's multiplicity in a specific construction of bag \abbrTIDB, which we call a \abbrCTIDB. A \abbrCTIDB,
$\pdb = \inparen{\worlds, \bpd}$ encodes a bag of uncertain tuples such that each possible tuple encoded in $\pdb$ has a multiplicity of at most $\bound$. $\tupset$ is the set of tuples appearing across all possible worlds, and the set of all worlds is encoded in $\worlds$, which is the set of all vectors of length $\numvar=\abs{\tupset}$ such that each index corresponds to a distinct $\tup \in \tupset$ storing its multiplicity and $\bpd$ is the probability distribution over $\worlds$. A given world $\worldvec \in\worlds$ can be interpreted such that, for each $\tup \in \tupset$, $\worldvec_{\tup}$ is the multiplicity of $\tup$ in $\worldvec$. The probability distribution $\bpd$ for any tuple $\tup$ can then be encoded as $\prob_{\tup, j} = \probOf\pbox{\worldvec_{\tup} = j}$ (for $j \in\pbox{\bound}$), where each tuple multiplicity combination $\inparen{\inparen{\tup, \bound} \in \tupset\times\pbox{\bound}}$ %distribution
is an independent random event. %for $\tup \in \tupset$.
}
%\mypar{For a later section}
@ -126,7 +126,7 @@ Specifically, in this work we ask if~\Cref{prob:expect-mult} can be solved in ti
%Define $\gentupset$ to be the set of tuples appearing across all the possible worlds of a $\abbrCTIDB$, formally $\gentupset = \inset{\tup_i ~|~ \forall \worldvec \in \worlds,~\forall i \in \abs{\tupset}:~\worldvec\pbox{i} > 0}$. When a specific $\pdb = \inparen{\worlds, \bpd}$ is being referred to, we will use $\tupset$ to denote the set of tuples.
%\end{Definition}
Let $\qruntime{\optquery{\query},\gentupset,\bound}$ denote the runtime for query $\optquery{\query}$, deterministic database $\gentupset$, and multiplicity bound $\bound$. Being we consider $\raPlus$ queries in which order of operators can impact runtime, we denote the optimal query as $\optquery{\query} = \min_{\query'\in\raPlus, \query'\equiv\query}\qruntime{\query', \gentupset, \bound}$.
Let $\qruntime{\optquery{\query},\gentupset,\bound}$ (see~\Cref{sec:gen} for further details) denote the runtime for query $\optquery{\query}$, deterministic database $\gentupset$, and multiplicity bound $\bound$. Being we consider $\raPlus$ queries in which order of operators can impact runtime, we denote the optimal query as $\optquery{\query} = \min_{\query'\in\raPlus, \query'\equiv\query}\qruntime{\query', \gentupset, \bound}$.
%let $\qruntim{\optquery{\query}, \gentupset, \bound} = \min_{\query'\in\raPlus,~\query'\equiv\query}T_{det}\inparen{\query, \gentupset, \bound}$ be the runtime for the optimally structured equivalent $\raPlus$ query $\query'$ (with some caveats; discussed in~\Cref{sec:gen}). % of query $\query$ on deterministic database $\tupset$.
%{\newline\noindent\centerline{\Huge \textcolor{black}{Or instead$\ldots$}}}
%\newline\noindent Let $T_{det}\inparen{\query, \gentupset, \bound}$ denote the runtime for $\raPlus$ query $\query$, deterministic database $\gentupset$, and multiplicity bound $\bound$. Since this paper does not consider optimization schemes, we leave optimization to the reader and show that our results hold across all inputs.
@ -323,7 +323,7 @@ Our negative results (\Cref{tab:lbs}) indicate that \abbrCTIDB{}s (even for $\bo
We adopt the two-step intensional model of query evaluation used in set-\abbrPDB\xplural, as illustrated in \Cref{fig:two-step}:
(i) \termStepOne (\abbrStepOne): Given input $\tupset$ and $\query$, output every tuple $\tup$ that possibly satisfies $\query$, annotated with its lineage polynomial ($\poly(\vct{X})=\apolyqdt\inparen{\vct{X}}$);
(ii) \termStepTwo (\abbrStepTwo): Given $\poly(\vct{X})$ for each tuple, compute $\expct_{\randWorld\sim\bpd}\pbox{\poly(\vct{\randWorld})}$.
Let $\timeOf{\abbrStepOne}(Q,\tupset,\circuit)$ denote the runtime of \abbrStepOne when it outputs $\circuit$ (which is a representation of $\poly$ as an arithmetic circuit --- more on this representation shortly).
Let $\timeOf{\abbrStepOne}(Q,\tupset,\circuit)$ denote the runtime of \abbrStepOne when it outputs $\circuit$ (which is a representation of $\poly$ as an arithmetic circuit --- more on this representation in~\Cref{sec:expression-trees}).
Denote by $\timeOf{\abbrStepTwo}(\circuit, \epsilon)$ (recall $\circuit$ is the output of \abbrStepOne) the runtime of \abbrStepTwo, which we can leverage~\Cref{def:reduced-poly} and~\Cref{lem:tidb-reduce-poly} to address the next formal objective: % to formally define our objective:
\begin{Problem}[\abbrCTIDB linear time approximation]\label{prob:big-o-joint-steps}

11
poly-form.tex
View File

@ -43,11 +43,14 @@ Let $\abs{\poly}$ be the number of operators in $\poly$.
If $\poly$ is a $1$-\abbrBIDB lineage polynomial already in \abbrSMB, then the expectation of $\poly$, i.e., $\expct\pbox{\poly} = \rpoly\left(\prob_1,\ldots, \prob_\numvar\right)$ can be computed in $\bigO{\abs{\poly}}$ time.
\end{Corollary}
\secrev{
Queries over probabilistic databases are traditionally viewed as being evaluated using the so-called possible world semantics. A general bag-\abbrPDB can be defined as the pair $\bpd = \inparen{\Omega, \bpd}$ where $\Omega$ is the set of possible worlds represented by $\pdb$. Under the possible world semantics, the result of a query $\query$ over an incomplete database $\Omega$ is the set of query answers produced by evaluating $\query$ over each possible world $\omega\in\Omega$: $\inset{\query\inparen{\omega}: \omega\in\Omega}$.
\subsubsection{Possible World Semantics}\label{subsub:possible-world-sem}
Queries over probabilistic databases are traditionally viewed as being evaluated using the so-called possible world semantics. A general bag-\abbrPDB can be defined as the pair $\pdb = \inparen{\Omega, \bpd}$ where $\Omega$ is the set of possible worlds represented by $\pdb$. Under the possible world semantics, the result of a query $\query$ over an incomplete database $\Omega$ is the set of query answers produced by evaluating $\query$ over each possible world $\omega\in\Omega$: $\inset{\query\inparen{\omega}: \omega\in\Omega}$.
The result of a query is the pair $\inparen{\query\inparen{\omega}, \bpd'}$ where $\bpd'$ is a probability distribution that assigns to each possible query result the sum of the probabilites of the worlds that produce this answer: $\probOf\pbox{\omega\in\Omega} = \sum_{\omega'\in\Omega,\\\query\inparen{\omega'}=\query\inparen{\omega}}\probOf\pbox{\omega'}$.
}
Suppose that $\bpd$ is a $1$-\abbrBIDB. Instead of looking only at the possible worlds of $\pdb$, one can consider all worlds, including those that cannot exist due to, e.g., disjointness. The all worlds set can be modeled by $\worldvec\in \{0, 1\}^{\bound\numvar}$, such that $\worldvec_{\tup, j} \in \worldvec$ represents whether or not the multiplicity of $\tup$ is $j$ (\emph{here and later, especially in \Cref{sec:algo}, we will rename the variables as $X_1,\dots,X_n$, where $n=\sum_{i=1}^\ell \abs{b_i}$}).%(where $k = \sum_{\ell = 1}^{i - 1} \abs{b_\ell} + j$).
\AH{
I am not sure the following paragraph is needed, since the reduction definition says pretty much the same thing. Unless that definition changes, we can get rid of this paragraph.
}
Suppose that $\pdb$ is a $1$-\abbrBIDB. Instead of looking only at the possible worlds of $\pdb$, one can consider all worlds, including those that cannot exist due to, e.g., disjointness. The all worlds set can be modeled by $\worldvec\in \{0, 1\}^{\bound\numvar}$, such that $\worldvec_{\tup, j} \in \worldvec$ represents whether or not the multiplicity of $\tup$ is $j$ (\emph{here and later, especially in \Cref{sec:algo}, we will rename the variables as $X_1,\dots,X_n$, where $n=\sum_{\tup\in\tupset}\abs{b_\tup}$}).%(where $k = \sum_{\ell = 1}^{i - 1} \abs{b_\ell} + j$).
We can denote a probability distribution over all $\worldvec \in \{0, 1\}^{\bound\numvar}$ as $\bpd'$. When $\bpd'$ is the one induced from each $\prob_{\tup, j}$ while assigning $\probOf\pbox{\worldvec} = 0$ for any $\worldvec$ with $\worldvec_{\tup, j}, \worldvec_{\tup, j'} \geq 1$ for $j\neq j'$, we end up with a bijective mapping from $\bpd$ to $\bpd'$, such that each mapping is equivalent, implying the distributions are equivalent.
\Cref{subsec:supp-mat-ti-bi-def} has more details.
@ -60,7 +63,7 @@ we have (denoting $\randDB$ as the random variable over $\Omega$):
$ \expct_{\randDB \sim \bpd}[\query(\randDB)(t)] = \expct_{\vct{\randWorld}\sim \pdassign}\pbox{\apolyqdt\inparen{\vct{\randWorld}}}. $
\end{Proposition}
\noindent A formal proof of \Cref{prop:expection-of-polynom} is given in \Cref{subsec:expectation-of-polynom-proof}.\footnote{Although \Cref{prop:expection-of-polynom} follows, e.g., as an obvious consequence of~\cite{IL84a}'s Theorem 7.1, we are unaware of any formal proof for bag-probabilistic databases.}
We focus on the problem of computing $\expct_\pdassign\pbox{\apolyqdt\inparen{\vct{\randWorld}}}$ from now on, assume implicit $\query, \tupset, \tup$, and drop them from $\apolyqdt$ (i.e., $\poly\inparen{\vct{X}}$ will denote a polynomial).
We focus on the problem of computing $\expct_{\worldvec\sim\pdassign}\pbox{\apolyqdt\inparen{\vct{\randWorld}}}$ from now on, assume implicit $\query, \tupset, \tup$, and drop them from $\apolyqdt$ (i.e., $\poly\inparen{\vct{X}}$ will denote a polynomial).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

8
prob-def.tex
View File

@ -6,12 +6,14 @@
%We first formally define circuits, an encoding of polynomials that we use throughout the paper.
%
%For illustrative purposes consider the polynomial $\poly(\vct{X}) = 2X^2 + 3XY - 2Y^2$ over $\vct{X} = [X, Y]$.
We represent lineage polynomials via {\em arithmetic circuits}~\cite{arith-complexity}, a standard way to represent polynomials over fields (particularly in the field of algebraic complexity) that we use for polynomials over $\mathbb N$ in the obvious way. Since we are particularly using circuits to model lineage polynomials, we can refer to these circuits as lineage circuits. However, when the meaning is clear, we will drop the term lineage and only refer to them as circuits.
\secrev{
~\Cref{prob:intro-stmt} asks if there exists a linear time approximation algorithm in the size of a given circuit \circuit which encodes $\poly\inparen{\vct{X}}$. In this work we
}
represent lineage polynomials via {\em arithmetic circuits}~\cite{arith-complexity}, a standard way to represent polynomials over fields (particularly in the field of algebraic complexity) that we use for polynomials over $\mathbb N$ in the obvious way. Since we are particularly using circuits to model lineage polynomials, we can refer to these circuits as lineage circuits. However, when the meaning is clear, we will drop the term lineage and only refer to them as circuits.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Definition}[Circuit]\label{def:circuit}
A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source gates (in degree of $0$) consist of elements in either $\domN$ or $\vct{X}$. For each result tuple there exists one sink gate. The internal gates have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source gates (in degree of $0$) consist of elements in either $\domN$ or $\vct{X} = \inparen{X_1,\ldots,X_\numvar}$. For each result tuple there exists one sink gate. The internal gates have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
%
Each gate has the following members: \type, \vpartial, \vari{input}, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the value type $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} the list of inputs. Source gates have an extra member \val storing the value. $\circuit_\linput$ ($\circuit_\rinput$) denotes the left (right) input of \circuit.
\end{Definition}

8
ra-to-poly.tex
View File

@ -49,18 +49,18 @@ or simply lineage polynomial), if there exists a $\raPlus$ query $\query$, \abbr
\noindent\secrev{
A block independent database \abbrBIDB $\pdb'$ can viewed as a $1$-\abbrTIDB $\pdb$ with the added flexibility that each $\tup\in\tupset$ has multiple disjoint alternatives, i.e., all $\tup \in \tupset'$ are partitioned into $m$ independent blocks with the condition that tuples $\tup \in \block_i$ for $i \in \pbox{m}$ are disjoint events. We define next a specific construction of \abbrBIDB that is useful for our work.
\begin{Definition}[$1$-\abbrBIDB]\label{def:one-bidb}
Define a $1$-\abbrBIDB to be the pair $\pdb' = \inparen{\prod_{\tup\in\tupset'}\inset{0, \bound_\tup}, \bpd'},$ where $\tupset'$ is the set of possible tuples such that each $\tup \in \tupset'$ has a multiplicity domain of $\inset{0, \bound_\tup}$, with $\bound_\tup \in \mathbb{N}$. The operation $\prod_{\tup\in\tupset'}$ is the direct product of all such multiplicity domain pairs. The tuples $\tup\in\tupset'$ are partitioned into $m$ independent blocks $\block_i,~i\in\pbox{m}$, of disjoint tuples. $\bpd$ is the probability distribution across all worlds such that, given $\worldvec\in\prod_{\tup\in\tupset'}\inset{0,\bound_\tup},\tup,~\tup'\in\block_i~:~\probOf\pbox{\worldvec_\tup, \worldvec_\tup'>0} = 0$.
Define a $1$-\abbrBIDB to be the pair $\pdb' = \inparen{\prod_{\tup\in\tupset'}\inset{0, \bound_\tup}, \bpd'},$ where $\tupset'$ is the set of possible tuples such that each $\tup \in \tupset'$ has a multiplicity domain of $\inset{0, \bound_\tup}$, with $\bound_\tup\in\mathbb{N}$. The operation $\prod_{\tup\in\tupset'}$ is the direct product of all such multiplicity domain pairs. The tuples $\tup\in\tupset'$ are partitioned into $m$ independent blocks $\block_i,~i\in\pbox{m}$, of disjoint tuples. $\bpd'$ is the probability distribution across all worlds such that, given $\worldvec\in\prod_{\tup\in\tupset'}\inset{0,\bound_\tup},\tup,~\tup'\in\block_i~:~\probOf\pbox{\worldvec_\tup, \worldvec_\tup'>0} = 0$.
\end{Definition}
%A \abbrCTIDB $\pdb$ is a pair $\inparen{\worlds, \bpd}$ such that $\worlds$ is an incomplete database whose set of possible worlds is the $c+1^\numvar$ tuple/multiplicity combinations across all $\tup\in\tupset$, where $\abs{\tupset} = \numvar$, $\tupset = \bigcup_{\worldvec\in\worlds,~\worldvec_{\tup}\geq 1}\tup$ is the set of possible tuples across possible worlds, and $\bpd$ is a probability distribution over $\worlds$.
%\begin{Definition}[$\bound$-Block Independent Disjoint Database ($\bound$-\abbrBIDB)]\label{def:bidb}
%A $\bound$-block independent database ($\bound$-\abbrBIDB) $\pdb' = \inparen{\inset{0,\ldots,\bound}^{\tupset'}, \bpd'}$ is a probabilistic database such that the all worlds set is encoded as the set of vectors $\worldvec\in\inset{0,\ldots,\bound}^{\abs{\tupset'}}$ where $\worldvec_\tup\leq\bound$ is the multiplicity for tuple $\tup$. $\pdb'$ requires the set of all possible tuples $\tupset = \bigcup_{\worldvec\in\inset{0,\ldots, \bound}^{\tupset'},~\worldvec_\tup \geq 1}\tup$ to be partitioned into $m$ independent blocks $\block_i$ ($i\in\pbox{m}$) where all tuples $\tup_{i, j}\in \block_i$ are disjoint. $\bpd'$ is the probability distribution where, for all $\worldvec\in\inset{0,\ldots,\bound}^{\tupset'}$ such that $\worldvec_{\tup_{i, j}},\worldvec_{\tup_{i, j'}}\neq 0, j\neq j'$ for any $\block_i$, $\probOf\pbox{\worldvec} = 0$, where all other $\worldvec$ has $0<\probOf\pbox{\worldvec}\leq 1$.%bpd'$ set with the all worlds set $\worlds$ and probability distribution $\bpd'$ such that $\tupset' = \bigcup_{\worldvec\in\worlds, \worldvec_\tup \geq 1}\tup$ is the set of all possible tuples for which all $\tup\in\tupset'$ can be partitioned into $\numedge$ blocks $\block_i$ where the set of tuples $\tup_j \in \block_i$ are all disjoint, while blocks $\block_i$ are independent of one another. Each $\tup\in\tupset'$ has a multiplicity of at most $\bound$. $\bpd'$ is the distribution such that for any $\worldvec\in\worlds$ with $\worldvec_{\tup_{i, j}}\geq 1$ and $\worldvec_{\tup_{i, j'}}\geq 1$, $j\neq j'$ in any $\block_i$ more than one tuple present from the same block $\block_i$ has probability $\probOf\pbox{\worldvec} = 0$.
%\end{Definition}
%A block independent database (\abbrBIDB) is a related probabilistic data model $\pdb=\inparen{\Omega, \bpd}$ such that the base set of tuples $\tupset = \bigcup_{\omega\in\Omega,~\tup\in\omega}\tup$ is partitioned into a set of $\numvar$ independent blocks $\inset{\inparen{\block_\tup}_{\tup\in\pbox{\numvar}}}$ such that the set of tuples $\inset{\inparen{\tup_j}_{j\in\pbox{\abs{\block}}}}$ in block $\block_\tup$ are disjoint from one another. This construction produces the set of possible worlds $\Omega$ that consists of all unique combinations of tuples in $\tupset$ with the constraint that for any $\omega\in\Omega$, no two tuples $\tup_j, \tup_{j'}, j\neq j'$ from the same block $\block_\tup$ exist together. A $\bound$-\abbrBIDB has the further requirement that each block has a multiplicity of at most $c$.
We now present a reduction that is useful in derivign our results:
We now present a reduction that is useful in deriving our results:
\begin{Definition}[\abbrCTIDB reduction]\label{def:ctidb-reduct}
Given \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\Omega, \bpd'}$ be the \abbrOneBIDB obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup = \inset{\intup{\tup, jX_{\tup, j}}_{j\in\pbox{\bound}}}$, for all $j\in\pbox{\bound}$ such that $X_{\tup, j}\in\inset{0,1}$.
The probability distribution $\bpd'$ is the one induced by $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$ and the \abbrBIDB disjoint requirement that for any $X_{\tup, j} = 1, j'\in\pbox{\bound} - \inset{j}, X_{\tup, j'} = 0$.
Given \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\prod_{\tup\in\tupset'}\inset{0, \bound_\tup}^{\tupset'}, \bpd'}$ be the \abbrOneBIDB obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup = \inset{\intup{\tup, j}_{j\in\pbox{\bound}}}$ of disjoint tuples, for all $j\in\pbox{\bound}$.% such that $X_{\tup, j}\in\inset{0,1}$.
The probability distribution $\bpd'$ is the one induced by $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$ and the \abbrBIDB disjoint requirement, where given any $\worldvec\in\prod_{\tup\in\tupset'}\inset{0, \bound_\tup}^{\tupset'}$, $\probOf\pbox{\worldvec_{\tup, j}, \worldvec_{\tup, j'} > 0} = 0$ for any $j \neq j' \in \pbox{\bound}$.% that for any $X_{\tup, j} = 1, j'\in\pbox{\bound} - \inset{j}, X_{\tup, j'} = 0$.
% $\block_\tup,~j\in\pbox{\bound}~|~X_{\tup, j} = 1,\not\exists j'\neq j~|~X_{\tup, j'} = 1$.
%$\tup_j\geq1\implies \tup_{j'} = 0$.$\forall j, j' \in \pbox{\bound},\forall \tup\in\tupset, \tup_j\geq 1\implies \tup_{j'} = 0$ for any block $\block_\tup$.
\end{Definition}