Contents

Chapter 1
Introduction

1.1 What is MayBMS?

The MayBMS system (note: MayBMS is read as “maybe-MS”, like DBMS) is a complete probabilistic database management system that leverages robust relational database technology: MayBMS is an extension of the Postgres server backend. MayBMS is open source and the source code is available under the BSD license at

The MayBMS system has been under development since 2005. While the development has been carried out in an academic environment, care has been taken to build a robust, scalable system that can be reliably used in real applications. The academic homepage of the MayBMS project is at

MayBMS stands alone as a complete probabilistic database management system that supports a powerful, compositional query language for which nevertheless worst-case efficiency and result quality guarantees can be made. We are aware of several research prototype probabilistic database management systems that are built as front-end applications of Postgres, but of no other fully integrated and available system. The MayBMS backend is accessible through several APIs, with efficient internal operators for computing and managing probabilistic data.

In summary, MayBMS has the following features:

• Full support of all features of PostgreSQL 8.3.3, including unrestricted query functionality, query optimization, APIs, updates, concurrency control and recovery, etc.
• Essentially no performance loss on PostgreSQL 8.3.3 functionality: After parsing a query or DML statement, a fast syntactic check is made to decide whether the statement uses the extended functionality of MayBMS. If it does not, the subsequently executed code is exactly that of PostgreSQL 8.3.3.
• Support for efficiently creating and updating probabilistic databases, i.e., uncertain databases in which degrees of belief can be associated with uncertain data.
• A powerful query and update language for processing uncertain data that gracefully extends SQL with a small number of well-designed language constructs.
• State-of-the-art efficient techniques for exact and approximate probabilistic inference.

1.2 Applications

Database systems for uncertain and probabilistic data promise to have many applications. Query processing on uncertain data occurs in the contexts of data warehousing, data integration, and of processing data extracted from the Web. Data cleaning can be fruitfully approached as a problem of reducing uncertainty in data and requires the management and processing of large amounts of uncertain data. Decision support and diagnosis systems employ hypothetical (what-if) queries. Scientific databases, which store outcomes of scientific experiments, frequently contain uncertain data such as incomplete observations or imprecise measurements. Sensor and RFID data is inherently uncertain. Applications in the contexts of fighting crime or terrorism, tracking moving objects, surveillance, and plagiarism detection essentially rely on techniques for processing and managing large uncertain datasets. Beyond that, many further potential applications of probabilistic databases exist and will manifest themselves once such systems become available.

The MayBMS distribution comes with a number of examples that illustrate its use in these application domains. Some of these examples are described in the tutorial chapter of this manual.

The experiments section at the end of this manual reports on some performance experiments with MayBMS. Unfortunately, at the time of writing this, no benchmark for probabilistic database systems exists, so these experiments are necessarily somewhat ad-hoc.

1.3 Acknowledgments

Michaela Goetz, Thomas Jansen and Ali Baran Sari are alumni of the MayBMS team. The MayBMS project was previously supported by German Science Foundation (DFG) grant KO 3491/1-1 and by funding provided by the Center for Bioinformatics (ZBI) at Saarland University, Saarbruecken, Germany. It is currently supported by grant IIS-0812272 of the US National Science Foundation.

Chapter 2
First Steps

2.1 Installing MayBMS

Using the installers

Installers for MayBMS are available for both Windows and Linux operating systems and can be downloaded at https://sourceforge.net/projects/maybms/

After you have obtained a copy of the installer, start it and follow the instructions.

Compiling from scratch

Alternatively, you can obtain the latest snapshot from the repository by issuing the following command:

This creates a directory maybms/ with a subdirectory postgresql-8.3.3/ that contains the source code of the system.

To compile and install MayBMS, just follow the instructions for installing PostgreSQL 8.3.3. The latter is documented at http://www.postgresql.org/docs/8.3/interactive/installation.html.

2.2 Running MayBMS

After you have installed MayBMS (in either of the described ways), you can set up a database and start using it. Creating and accessing databases is the same as in PostgreSQL 8.3.3. Follow the links http://www.postgresql.org/docs/8.3/interactive/tutorial-createdb.html

and http://www.postgresql.org/docs/8.3/interactive/tutorial-accessdb.html.

See next section for short instructions on how to run MayBMS.

2.3 Short Instructions

Alternatively, you can follow the following set of instructions.¹

On most UNIX machines, Postgres is by default installed in the directory /usr/local/pgsql/ and run under user “postgres”. MayBMS uses the same defaults. If you prefer to install MayBMS in your home directory and run it with your user privileges, you do not need root privileges to install it. You just need to use the --prefix option when calling the configure script (e.g., ./configure --prefix /home/myname/pgsql/ if your home directory is /home/myname/).

To compile, install, and start the Postgres server, execute the following statements:

When compiling with a recent version of gcc, one needs to disable some optimizations that are not compatible with PostgreSQL 8.3. To do this, replace “./configure” with “./configure CFLAGS=-fno-aggressive-loop-optimizations”.

Note: In these minimal instructions, we did not create a special database using createdb (so the default, template1, has to be used), and error messages are written to the console.

Now MayBMS is available for connections from applications.

For example, the Postgres command line interface psql in which you can issue MayBMS queries and data manipulation language statements is started with

(depending on your installation, you may need to add the -h localhost option to force the use of TCP connections).

Now you can enter the examples from, e.g., the following tutorial. The psql program is terminated using the command “\q”. The database server is stopped with

Remark

Since Postgres and MayBMS use the same process identifiers, MayBMS and Postgres cannot run concurrently on the same machine. If you start Postgres when MayBMS is already running (or vice versa), there will be an error message stating that Postgres is already running. Since MayBMS always identifies itself as Postgres, standard Postgres applications and middleware can run on MayBMS.

Chapter 3
Probabilistic Databases

We first give an informal definition of probabilistic databases, followed by a formal definition.

3.1 Informal Definition

Given a relational database schema (i.e., the structural information usually specified by SQL CREATE TABLE statements). A probabilistic database is a finite set of possible worlds, where each possible world has a weight greater than 0 but no greater than 1 such that the sum of the weights of all the worlds is one. Each possible world is a relational database over the given schema. That is, the schema is common to all possible worlds.

Possible worlds are a means of expressing uncertainty.

• In a frequentist interpretation, the probabilistic database represents the possible outcomes of a random experiment, the outcomes of which are relational databases (or can be conveniently represented as relational databases). The probability weight of a possible world is (the limit of) the relative frequency of that possible world occurring as outcome of the random experiment over a large number of trials.
• In a Bayesian interpretation, one of the possible worlds is “true”, but we do not know which one, and the probabilities represent degrees of belief in the various possible worlds.

Note that these interpretations of probabilistic databases are completely standard in probability theory (and formalized via the notion of probability spaces). The only aspect particular to probabilistic databases is the fact that possible worlds are relational databases.

Note that the idea of a probabilistic database as a set of possible worlds is only the conceptual model. The physical representation of the set of possible worlds in the MayBMS system is quite different (see Section 5.4) and allows for the efficient and space-saving (compressed) representation of very large sets of possible worlds.

3.2 Formal Definition

The following is a standard definition from probability theory and shall only be recalled to demonstrate the close connection of probabilistic databases to classical concepts in mathematics.

Formally, a probabilistic database over a relational database schema sch is a finite probability space (Ω, = 2^Ω, Pr) with an associated function I (for instance) that maps each ω Ω to a relational database over schema sch.

We call the elements ω of Ω the possible worlds of the probabilistic database.

We can identify events with Boolean queries Q that are true on a subset of Ω. Of course, the probability of such an event is given by

One particular type of event is membership of a given tuple in the result of a (nonboolean) query, i.e., an event

The probability of this event is called the tuple confidence for tuple .

A random variable X is a function from Ω to a set D (the “values” of the random variable). We can associate each expression X = x, where x D, with an event

Again, this is the usual notion from probability theory.

3.3 An Example

Consider a finite probability space with

and Pr[ω_rain,wet] = 0.35, Pr[ω_rain,¬wet] = 0.05, Pr[ω_¬rain,wet] = 0.1, and Pr[ω_¬rain,¬wet] = 0.5.

Let Wet be the event {ω_rain,wet,ω_¬rain,wet}. Then Pr[Wet] = 0.35 + 0.1 = 0.45. We define Boolean random variables Wet and Rain as follows:

Then, Pr[Wet = true] is again 0.45.

The first example of the following tutorial chapter captures this example in the framework of the MayBMS query and update language.

Chapter 4
Tutorial

This tutorial introduces the main features of MayBMS in an informal way. The full examples can be run using the psql command line interface.

4.1 A Really Simple Example

We start by creating a simple table using SQL commands. The table encodes that we see rain and wet ground with probability 0.4, no rain but wet ground with probability 0.1, and no rain and dry ground with probability 0.5.

Table R is a completely standard relational database table, created using standard SQL statement. One of the columns, P, stores probabilities, but to the system these are only numbers without any particular meaning so far.

The following statement creates a probabilistic database table S:

The repair-key statement is one of the extensions of the MayBMS query language over standard SQL, and it associates a special meaning to the values taken from the “weight by” column.

The statement creates a probability space with a sample space consisting of three possible databases – each one consisting just of one tuple from R – with an associated probability measure given by the P column.

There are at least two natural interpretations of this example, one using random variables and one using a possible worlds semantics.

• We can think of S as a table specifying the joint probability distribution of two discrete random variables Weather (with values “rain” and “no rain”) and Ground (with values “wet” and “dry”).
• Alternatively, there are three possible worlds. Each of these worlds is a relation S with a single tuple from R. The probability of such a world is the value given for the tuple in column P of R.

We can compute the probabilities Pr[Ground=’wet’] and Pr[Weather=’rain’ and Ground=’wet’] as follows using the MayBMS conf() aggregate (which stands for “confidence”).

The conditional probability Pr[Weather=’rain’ | Ground=’wet’] can be computed as the ratio

Since conf() is an aggregate, we can compute the marginal probability table for random variable Ground as

4.2 Example: Triangles in Random Graphs

In this tutorial, we compute the probability that a triangle occurs in a random graph with k named (and thus distinguishable) nodes. That is, we ask for the probability that an undirected graph, chosen uniformly at random among the graphs of k nodes, contains at least one triangle. This is equivalent to computing the count n of graphs that contain a triangle among the 2^{k⋅(k-1)∕2} undirected graphs of k distinguished nodes. Indeed, an undirected graph of k nodes has at most k ⋅ (k - 1)∕2 edges, and we obtain all the graphs over the given k nodes by considering all subsets of this maximal set of edges.

We start by creating a unary “node” relation, say with five nodes. We do this with the standard SQL “create table” and “insert” commands, which behave as usual in a relational database system.

Next we create the total order over the nodes, i.e., a binary relation with exactly one edge between any two nodes. This is again a standard SQL “create table” statement where we compute the tuples to be inserted with a standard SQL query over the “node” relation.

We create a table to represent that each edge is either in the graph (bit=1) or missing (bit=0).

The following operation introduces uncertainty into the database and creates a probabilistic database with 2^5⋅4∕2 = 1024 possible worlds, one for each possible edge relation over the five nodes (=subset of the total order). We do this by a query operation “repair key” that for each edge of the total order nondeterministically chooses whether the edge is in the graph (bit=1) or not. (That is, since we do not indicate at what probability either of the two alternatives for bit is to be chosen, the system makes the decision uniformly at random, choosing bit=1 with probability 0.5.) The resulting probabilistic database represents all the alternative edge relations as possible worlds.

The “repair key” operation is the most difficult to understand and at the same time the most interesting addition to SQL that MayBMS provides. Conceptually, “repair key” takes a set of attributes and a relation R (in this case the relational product of total_order and inout) as arguments and nondeterministically chooses a maximal repair of key in R, that is, it removes a minimal set of tuples from R such that ceases to violate a key constraint on columns u,v. In this case, there are exactly two tuples for each pair (u,v), namely (u,v, 1) and (u,v, 0), and repair key chooses exactly one of them. The consequence is that, overall, the operation nondeterministically chooses a subset of the set of all edges. It chooses from these subsets uniformly. The “repair key” operation accepts an additional argument that allows us to assign nonuniform probabilities to the possible choices, but in this case we do want uniform probabilities.

We have now created a probabilistic database. Conceptually, queries and updates are evaluated in all possible worlds in parallel. Viewed differently, there is only one to_subset relation (but we do not know which one), and we continue to run queries and updates on this uncertain relation.

To actually create the edge relation, we select those tuples that have bit=1 and compute their symmetric closure (to really represent an undirected graph).

Now we can compute the probability that the chosen graph has a triangle as

where the conf aggregate computes the probability (“confidence”) that the query given by the from-where statement returns a nonempty result. This results in

This is the correct probability: out of the 1024 possible graphs of five nodes, 636 have a triangle, and 636∕1024 ≈ .623355. Indeed, the query

computes at least one tuple in exactly those possible worlds (=on those graphs) that have a triangle. The conf() aggregate applied to this query conceptually computes the sum of the probability weights of the worlds in which the query has a nonempty result. (The actual implementation does not naively iterate over possible worlds, because this would be very inefficient.)

A more efficient implementation of the same query starts from the “edge0” relation:

Finally, an even more efficient implementation uses the aconf(ϵ,δ) aggregate to compute an (ϵ,δ)-approximation of the probability, i.e., the probability that the computed value returned by aconf deviates from the correct probability p by more than ϵ ⋅ p is less than δ.

This result may be somewhat off, but the probability that the error is greater than 5% is less than 5%.

Note that in the example we have seen only two extensions of SQL, “repair key” and “[a]conf”. The good news is that this is essentially all there is. SQL extended by just these two features allows for very powerful queries, including the computation of conditional probability tables, maximum likelihood estimates, maximum-a-posteriori, Bayesian learning, and much more.

4.3 Example: Skills Management

The following example demonstrates that probabilistic databases can be useful even if the input data is not uncertain and the desired result is a classical relational table. We define a hypothetical query in the context of skills management. Assume we are given a classical relational database with two tables, one, CE, stores possible takeover targets – companies that we might decide to buy with the employees that work in these companies. The second table, ES, stores each employee’s skills.

Here is an example database. We can build this database in MayBMS with the standard SQL “create table” and “insert” statements.

Now suppose that we want to buy exactly one of those companies, and we expect exactly one employee to leave as a result of the takeover. Which skills can we gain for certain?

We express this query in two steps. First we randomly choose a company to buy and an employee who leaves, and compute the remaining employees in the chosen company. We obtain this uncertain table using the following query:

Note that the probabilistic database thus created contains five possible worlds (since there are five tuples in CE), with a uniform probability distribution. Not all these worlds have the same number of tuples: If we chose Google and Bob, the world contains one tuple, Google and Joe. If we choose Yahoo and Dan, the world contains two tuples, (Yahoo, Bill) and (Yahoo, Fred).

Now we compute which skills we gain for certain:

The result is the table

indicating that if we buy Google, we gain the skill “Web” for certain, and if we buy Yahoo, we gain the skill “Java” for certain.

It is worth looking at the auxiliary table SkillGained:

This table consists of the tuples (x,y,p1,p2,p) such that

• x is a company,
• y is a skill,
• p1 is the probability that the chosen company is x and the skill y is gained (e.g., for x=Yahoo and y=Web, this is true in two of the five possible worlds),
• p2 is the probability that x is the chosen company (e.g., for x=Yahoo, this is true in three of the five possible worlds), and
• p = p1∕p2 is the probability that skill y is gained if company x is bought (e.g., for x=Yahoo and y=Web, the probability is 2/3: of the three possible worlds in which Yahoo was bought, only two worlds guarantee that the skill Web is gained).

Thus, indeed, if we select those tuples of SkillGained for which p = 1, we obtain the desired pairs of companies and skills – those skills that we obtain for certain if we buy a company.

4.4 Data Cleaning

The following example is in the domain of data cleaning. Consider a census in which a number of individuals complete forms, that are subsequently digitized using an OCR system that will in some cases indicate a number of alternative readings, together with probabilities. For simplicity, let us assume that the forms only ask for a social security number (SSN).

For example, if two individuals complete their forms and the OCR system recognizes the SSN of the first to be either 185 (with probability .4) or 785 and the SSN of the second to be either 185 (with probability .7) or 186, we store this information in a probabilistic database constructed as follows:

We can view the alternatives and their probability weights by the following query:

We can determine the probability that at least one individual has any particular SSN (assuming that the OCR system did not miss the correct SSN as an alternative) using the following query:

Indeed, the probability that at least one individual has SSN 185 is 1 - .6 ⋅ .3 = .82.

We now perform data cleaning using a single integrity constraint, namely that no two individuals can have the same ssn. Conceptually, we want to exclude worlds that violate the functional dependency

i.e., the constraint that ssn must be a key for the relation.

We start by computing an auxiliary relation that computes, in each possible worlds, the ssn values that violate the integrity constraint.

Note that two tuples violate the constraint if they have the same ssn but different tid. We express this in the above query using a slightly changed condition: (S1.tid < S2.tid and S1.ssn = S2.ssn) instead of (S1.tid <> S2.tid and S1.ssn = S2.ssn). However, both conditions select the same set of distinct ssn values that violate the integrity constraint.

This query computes the uncertain table that holds 185 in the world in which both forms have ssn value 185. In all other worlds it is empty.

Next we compute the conditional probability table

This table stores, for each pair of form tid and ssn, the posterior probability that the individual who completed the form tid has the social security number ssn given that no two individuals can have the same ssn.

We can compute, for each form, the maximum-a-posteriori ssn (the most likely ssn given the evidence specified by the integrity constraint) as

In a sense, these map values are the locally best values that we could decide upon for each uncertain answer in our census database. Note, however, that, if we always choose the map value, we may sometimes create a database that again violates the integrity constraints used for data cleaning. This would have been the case if we had indicated probability .9 for both 185 alternatives in the input database.

A further example that computes conditional probabilities and MAP values in a different context can be found in Chapter 6.1 (Example 6.1.1).

Chapter 5
Formal Foundations

This chapter describes the formal foundations of MayBMS, including the principles used for representing and storing probabilistic data, the design of the query language, and efficient algorithms for query processing.

It is safe for a reader who has gained sufficient intuitive understanding of the workings of MayBMS from the tutorial to skip this chapter on first reading and to directly proceed to the query language reference chapter that follows.

5.1 Probabilistic Databases: Notation

Given a schema with relation names R₁,…,R_k. We use sch(R_l) to denote the attributes of relation schema R_l. Formally, a probabilistic database is a finite set of structures

of relations R₁ⁱ,…,R _kⁱ and numbers 0 < p^[i] ≤ 1 such that

We call an element ⟨R₁ⁱ,…,R _kⁱ,p^[i]⟩ W a possible world, and p^[i] its probability. We use superscripts for indexing possible worlds. To avoid confusion with exponentiation, we sometimes use bracketed superscripts ⋅^[i]. We call a relation R complete or certain if its instantiations are the same in all possible worlds of W, i.e., if R¹ = = Rⁿ.

Tuple confidence refers to the probability of the event R, where R is one of the relation names of the schema, with

5.2 Query Language Desiderata

At the time of writing this, there is no accepted standard query language for probabilistic databases. In fact, we do not even agree today what use cases and functionality such systems should support. It seems to be proper to start the query language discussion with the definition of design desiderata. The following are those used in the design of MayBMS.

1. Efficient query evaluation.
2. The right degree of expressive power. The language should be powerful enough to support important queries. On the other hand, it should not be too strong, because expressiveness generally comes at a price: high evaluation complexity and infeasibility of query optimization.
3. Genericity. The semantics of a query language should be independent from details of how the data is represented. Queries should behave in the same way no matter how the probabilistic data is stored. This is a basic requirement that is even part of the traditional definition of what constitutes a query (cf. e.g. [1]), but it is nontrivial to achieve for probabilistic databases [4].
4. The ability to transform data. Queries on probabilistic databases are often interpreted quite narrowly in the literature. It is the authors’ view that queries in general should be compositional mappings between databases, in this case probabilistic databases. This is a property taken for granted in relational databases. It allows for the definition of clean database update languages.
5. The ability to introduce additional uncertainty. This may appear to be a controversial goal, since uncertainty is commonly considered undesirable, and probabilistic databases are there to deal with it by providing useful functionality despite uncertainty. However, it can be argued that an uncertainty-introduction operation is important for at least three reasons: (1) for compositionality, and to allow construction of an uncertain database from scratch (as part of the update language); (2) to support what-if queries; and (3) to extend the hypothesis space modeled by the probabilistic database. The latter is needed to accommodate the results of experiments or new evidence, and to define queries that map from prior to posterior probabilistic databases. This is a nontrivial issue, and will be discussed in more detail later.

The next section introduces a query algebra and argues that it satisfies each of these desiderata.

5.3 The Algebra

This section covers the core query algebra of MayBMS: probabilistic world-set algebra (probabilistic WSA) [4, 13, 12]. Informally, probabilistic world-set algebra consists of the operations of relational algebra, an operation for computing tuple confidence conf, and the repair-key operation for introducing uncertainty. The operations of relational algebra are evaluated individually, in “parallel”, in each possible world. The operation conf(R) computes, for each tuple that occurs in relation R in at least one world, the sum of the probabilities of the worlds in which the tuple occurs. The result is a certain relation, or viewed differently, a relation that is the same in all possible worlds. Finally, repair-key_@P (R), where ,P are attributes of R, conceptually nondeterministically chooses a maximal repair of key . This operation turns a possible world Rⁱ into the set of worlds consisting of all possible maximal repairs of key . A repair of key in relation Rⁱ is a subset of Rⁱ for which is a key. It uses the numerically-valued column P for weighting the newly created alternative repairs.

Formally, probabilistic world-set algebra consists of the following operations:

• The operations of relational algebra (selection σ, projection π, product ×, union ∪, difference -, and attribute renaming ρ), which are applied in each possible world independently.

  The semantics of operations Θ on probabilistic database W is

  

  for unary operations (1 ≤ l ≤ k). For binary operations, the semantics is

  

  Selection conditions are Boolean combinations of atomic conditions (i.e., negation is permitted even in the positive fragment of the algebra). Arithmetic expressions may occur in atomic conditions and in the arguments of π and ρ. For example, ρ_A+B→C(R) in each world adds up the A and B values of each tuple of R and keeps them in a new C attribute.

• An operation for computing tuple confidence,

  where, w.l.o.g., P ⁄ sch(R_l), and

  

  with schema sch(S) = sch(R_l) ∪{P}. The result of conf(R_l), the relation S, is the same in all possible worlds, i.e., it is a certain relation.

  By our definition of probabilistic databases, each possible world has nonzero probability. As a consequence, conf does not return tuples with probability 0.

  For example, on probabilistic database

  R1   A   B       a   b       b   c       p[1] = .3

  R2   A   B       a   b       c   d       p[2] = .2

  R3   A   B       a   c       c   d       p[3] = .5

  conf(R) computes, for each possible tuple, the sum of the weights of the possible worlds in which it occurs, here

  conf(R)	A	B	P
  	a	b	.5
  	a	c	.5
  	b	c	.3
  	c	d	.7

• An uncertainty-introducing operation, repair-key, which can be thought of as sampling a maximum repair of a key for a relation. Repairing a key of a complete relation R means to compute, as possible worlds, all subset-maximal relations obtainable from R by removing tuples such that a key constraint is satisfied. We will use this as a method for constructing probabilistic databases, with probabilities derived from relative weights attached to the tuples of R.

  We say that relation R′ is a maximal repair of a functional dependency (fd, cf. [1]) for relation R if R′ is a maximal subset of R which satisfies that functional dependency, i.e., a subset R′⊆ R that satisfies the fd such that there is no relation R′′ with R′⊂ R′′⊆ R that satisfies the fd.

  Let ,B sch(R_l). For each possible world ⟨R₁,…,R_k,p⟩ W, let column B of R contain only numerical values greater than 0 and let R_l satisfy the fd (sch(R_l) - B) → sch(R_l). Then,

  Such a repair operation, apart from its usefulness for the purpose implicit in its name, is a powerful way of constructing probabilistic databases from complete relations.

  Example 5.3.1 Consider the example of tossing a biased coin twice. We start with a certain database

  R	Toss	Face	FProb
  	1	H	.4
  	1	T	.6
  	2	H	.4
  	2	T	.6

  p = 1

  that represents the possible outcomes of tossing the coin twice. We turn this into a probabilistic database that represents this information using alternative possible worlds for the four outcomes using the query S := repair-key_Toss@FProb(R). The resulting possible worlds are

  S1	Toss	Face
  	1	H
  	2	H

  S2	Toss	Face
  	1	H
  	2	T

  
  

  S3	Toss	Face
  	1	T
  	2	H

  S4	Toss	Face
  	1	T
  	2	T

  with probabilities p^[1] = p ⋅ ⋅ = .16, p^[2] = p^[3] = .24, and p^[4] = .36. □

The fragment of probabilistic WSA which excludes the difference operation is called positive probabilistic WSA.

Computing possible and certain tuples is redundant with conf:

Example 5.3.2 A bag of coins contains two fair coins and one double-headed coin. We take one coin out of the bag but do not look at its two faces to determine its type (fair or double-headed) for certain. Instead we toss the coin twice to collect evidence about its type.

---

Coins Type   Count    fair   2     2headed  1        Faces Type Face FProb    fair H .5     fair T .5     2headed H 1      Tosses Toss    1     2       Rf Type fair Rdh Type 2headed  Sf.HH   Type  Toss   Face       fair   1    H        fair   2    H         Sf.HT    Type  Toss   Face       fair   1    H        fair   2    T         Sdh   Type   Toss   Face       2headed  1    H        2headed  2    H        pf.HH = 1∕6 pf.HT = 1∕6 pdh = 1∕3        Sf.TH   Type  Toss   Face       fair   1    T        fair   2    H         Sf.TT    Type  Toss   Face       fair   1    T        fair   2    T        pf.TH = 1∕6 pf.TT = 1∕6  Ev   Toss   Face       1    H        2    H        Q   Type    P    fair    (1∕6)∕(1∕2) = 1∕3    2headed   (1∕3)∕(1∕2) = 2∕3    Figure 5.1: Tables of Example 5.3.2.

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We start out with a complete database (i.e., a relational database, or a probabilistic database with one possible world of probability 1) consisting of three relations, Coins, Faces, and Tosses (see Figure 5.1 for all tables used in this example). We first pick a coin from the bag and model that the coin be either fair or double-headed. In probabilistic WSA this is expressed as

This results in a probabilistic database of two possible worlds,

The possible outcomes of tossing the coin twice can be modeled as

This turns the two possible worlds into five, since there are four possible outcomes of tossing the fair coin twice, and only one for the double-headed coin.

Let T := π_Toss,Face(S). The posterior probability that a coin of type x was picked, given the evidence Ev (see Figure 5.1) that both tosses result in H, is

Let A be a relational algebra expression for the Boolean query T = Ev. Then we can compute a table of pairs ⟨x, Pr[x R∣T = Ev]⟩ as

The prior probability that the chosen coin was fair was 2/3; after taking the evidence from two coin tosses into account, the posterior probability Pr[the coin is fair | both tosses result in H] is only 1/3. Given the evidence from the coin tosses, the coin is now more likely to be double-headed. □

Discussion The repair-key operation admits an interesting class of queries: Like in Example 5.3.2, we can start with a probabilistic database of prior probabilities, add further evidence (in Example 5.3.2, the result of the coin tosses) and then compute interesting posterior probabilities. The adding of further evidence may require extending the hypothesis space first. For this, the repair-key operation is essential. Even though our goal is not to update the database, we have to be able to introduce uncertainty just to be able to model new evidence – say, experimental data. Many natural and important probabilistic database queries cannot be expressed without the repair-key operation. The coin tossing example was admittedly a toy example (though hopefully easy to understand). Real applications such as diagnosis or processing scientific data involve technically similar questions.

Regarding our desiderata, it is quite straightforward to see that probabilistic WSA is generic (3): see also the proof for the non-probabilistic language in [4]. It is clearly a data transformation query language (4) that supports powerful queries for defining databases. The repair-key operation is our construct for uncertainty introduction (5). The evaluation efficiency (1) of probabilistic WSA is studied in Section 5.5. Expressiveness (2) is best demonstrated by the ability of a language to satisfy many relevant use cases. While there are no agreed upon expressiveness benchmarks for probabilistic databases yet, this manual provides numerous examples that are closely related to natural use cases.

5.4 Representing Probabilistic Data

This section discusses the method used for representing and storing probabilistic data and correlations in MayBMS. We start by motivating the problem of finding a practical representation system.

This leads to three natural desiderata for a representation system: (*) Expressiveness, that is, the power to represent all (relevant) probabilistic databases, (*) succinctness, that is, space-efficient storage of the uncertain data, and (*) efficient real-world query processing.

Often there are many rather (but not quite) independent local alternatives in probabilistic data, which multiply up to a very large number of possible worlds. For example, the US census consists of many dozens of questions for about 300 million individuals. Suppose forms are digitized using OCR and the resulting data contains just two possible readings for 0.1% of the answers before cleaning. Then, there are on the order of 2^10,000,000 possible worlds, and each one will take close to one Terabyte of data to store. Clearly, we need a way of representing this data that is much better than a naive enumeration of possible worlds.

Also, the repair-key operator of probabilistic world-set algebra in general causes an exponential increase in the number of possible worlds.

There is a trade-off between succinctness on one hand and efficient processing on the other. Computing confidence conf(Q) of conjunctive queries Q on tuple-independent databases is #P-hard – one such hard query [6] (in datalog notation [1]) is

At the same time, much more expressive queries can be evaluated efficiently on nonsuccinct representations (enumerations of possible worlds) [4]. Query evaluation in probabilistic databases is not hard because of the presence of probabilities, but because of the succinct storage of alternative possible worlds! We can still have the goal of doing well in practice.

Conditional tables MayBMS uses a purely relational representation system for probabilistic databases called U-relational databases, which is based on probabilistic versions of the classical conditional tables (c-tables) of the database literature [9]. Conditional tables are a relational representation system based on the notion of labeled null values or variables, that is, null values that have a name. The name makes it possible to use the same variable x in several fields of a database, indicating that the value of x is unknown but must be the same in all those fields in which x occurs. Tables with variables are also known as v-tables.

Formally, c-tables are v-tables extended by a column for holding a local condition. That is, each tuple of a c-table has a Boolean condition constructed using “and”, “or”, and “not” from atomic conditions of the form x = c or x = y, where c are constants and x and y are variables. Possible worlds are determined by functions θ that map each variable that occurs in at least one of the tuples or local conditions in the c-tables of the database to a constant. The database in that possible world is obtained by (1) selecting those tuples whose local condition ϕ satisfies the variable assignment θ, i.e., that becomes true if each variable x in ϕ is replaced by θ(x), (2) replacing all variables y in the value fields of these tuples by θ(y), and (3) projecting away the local condition column. For example, the following c-table represents the possible worlds for the census forms:

The variables y and z have domains {1, 2} and {1, 2, 3, 4}, respectively and encode the marital statuses of the two persons, and variable x with domain {1, 2, 3} is used to encode the uniqueness of the social security constraint. Indeed, under any valuation θ the tuples having social security status of 185 do not have their local conditions satisfied at the same time.

Conditional tables are sometimes defined to include a notion of global condition, which we do not use: We want each probabilistic database to have at least one possible world. We can encode the same information as above using the following c-table with global condition Φ = (u≠v), where u : dom(u) = {185, 785},v : dom(v) = {185, 186} are the variables holding the social security numbers:

Conditional tables are a so-called strong representation system: They are closed under the application of relational algebra queries. The set of worlds obtained by evaluating a relational algebra query in each possible world represented by a conditional table can again be straightforwardly represented by a conditional table. Moreover, the local conditions are in a sense the most natural and simple formalism possible to represent the result of queries on data with labeled nulls.

U-Relational Databases In our model, probabilistic databases are finite sets of possible worlds with probability weights. It follows that each variable naturally has a finite domain, the set of values it can take across all possible worlds. This has several consequences. First, variables can be considered finite random variables. Second, only allowing for variables to occur in local conditions, but not in attribute fields of the tuples, means no restriction of expressiveness. Moreover, we may assume without loss of generality that each atomic condition is of the form x = c (i.e., we never have to compare variables).

If we start with a c-table in which each local condition is a conjunction of no more than k atomic conditions, then a positive relational algebra query on this uncertain database will result in a c-table in which each local condition is a conjunction of no more than k′ atoms, where k′ only depends on k and the query, but not on the data. If k is small, it is reasonable to actually hard-wire it in the schema, and represent local conditions by k pairs of columns to store atoms of the form x = c.

These are the main ideas of our representation system, U-relations. Random variables are assumed independent in the current MayBMS system, but as we will see, this means no restriction of generality. Nevertheless, it is one goal of future work to support graphical models for representing more correlated joint probability distributions below our U-relations. This would allow us to represent learned distributions in the form of e.g. Bayesian networks directly in the system (without the need to map them to local conditions) and run queries on top, representing the inferred correlations using local conditions. The latter seem to be better suited for representing the incremental correlations constructed by queries.

One further idea employed in U-relational databases is to use vertical partitioning for representing attribute-level uncertainty, i.e., to allow to decompose tuples in case several fields of a tuple are independently uncertain.

Formally, a U-relational database consists of a set of independent random variables with finite domains (here, x,y,v,w), a set of U-relations, and a ternary table W (the world-table) for representing distributions. The W table stores, for each variable, which values it can take and with what probability. The schema of each U-relation consists of a set of pairs (V _i,D_i) of condition columns representing variable assignments and a set of value columns for representing the data values of tuples.

The semantics of U-relational databases is as follows. Each possible world is identified by a valuation θ that assigns one of the possible values to each variable. The probability of the possible world is the product of weights of the values of the variables. A tuple of a U-relation, stripped of its condition columns, is in a given possible world if its variable assignments are consistent with θ. Attribute-level uncertainty is achieved through vertical decompositioning, so one of the value columns is used for storing tuple ids and undoing the vertical decomposition on demand.

Example 5.4.3 Consider the U-relational database of Example 5.4.2 and the possible world

The probability weight of this world is .4 ⋅ .3 ⋅ .8 ⋅ .25 = .024. By removing all the tuples whose condition columns are inconsistent with θ and projecting away the condition columns, we obtain the relations

R[SSN]	TID	SSN
	t1	185
	t2	186

R[M]	TID	M
	t1	1
	t2	1

R[N]	TID	N
	t1	Smith
	t2	Brown

which are just a vertically decomposed version of R in the chosen possible world. That is, R is R[SSN]R[M]R[N] in that possible world. □

Properties of U-relations U-relational databases are a complete representation system for (finite) probabilistic databases [3]. This means that any probabilistic database can be represented in this formalism. In particular, it follows that U-relations are closed under query evaluation using any generic query language, i.e., starting from a represented database, the query result can again be represented as a U-relational database. Completeness also implies that any (finite) correlation structure among tuples can be represented, despite the fact that we currently assume that the random variables that our correlations are constructed from (using tuple conditions) are independent: The intuition that some form of graphical model for finite distributions may be more powerful (i.e., able to represent distributions that cannot be represented by U-relations) is false.

5.5 Conceptual Evaluation and Rewritings

This section gives a complete solution for efficiently evaluating a large fragment of probabilistic world-set algebra using relational database technology. Then we discuss the evaluation of the remaining operations of probabilistic WSA, namely difference and tuple confidence. Finally, an overview of known worst-case computational complexity results is given.

Translating queries down to the representation relations Let rep be the representation function, which maps a U-relational database to the set of possible worlds it represents. Our goal is to give a reduction that maps any positive relational algebra query Q over probabilistic databases represented as U-relational databases T to an equivalent positive relational algebra query Q of polynomial size such that

where the ⁱ are relational database instances (possible worlds) or, as a commutative diagram,

The following is such a reduction, which maps the operations of positive relational algebra, poss, and repair-key to relational algebra over U-relational representations:

For simplicity, let us assume that the elements of π_⟨⟩(U_R) are not yet used as variable names. Moreover, let us assume that the B value column of U_R, which is to provide weights for the alternative values of the columns sch(R) - ( ∪ B) for each tuple in π_⟨⟩(U_R), are probabilities, i.e., sum up to one for each and do not first have to be normalized as described in the definition of the semantics of repair-key in Section 5.3. The operation S := repair-key_@B(R) for complete relation R is translated as

with

Here, ⟨⋅⟩ turns tuples of values into atomic values that can be stored in single fields.

That is, repair-key starting from a complete relation is just a projection/copying of columns, even though we may create an exponential number of possible worlds.

The projection technique only works if the relation that repair-key is applied to is certain. However, this means no loss of generality (cf. [12], and see also Example 5.3.3).

The next example demonstrates the application of the rewrite rules to compile a query down to relational algebra on the U-relations.

Example 5.5.2 We revisit our census example with U-relations U_R[SSN] and U_R[N]. We ask for possible names of persons who have SSN 185,

To undo the vertical partitioning, the query is evaluated as

We rewrite the query using our rewrite rules into

where ψ ensures that we only generate tuples that occur in some worlds,

and ϕ ensures that the vertical partitioning is correctly undone,

□

Properties of the relational-algebra reduction The relational algebra rewriting down to positive relational algebra on U-relations has a number of nice properties. First, since relational algebra has PTIME (even AC₀) data complexity, the query language of positive relational algebra, repair-key, and poss on probabilistic databases represented by U-relations has the same. The rewriting is in fact a parsimonious translation: The number of algebra operations does not increase and each of the operations selection, projection, join, and union remains of the same kind. Query plans are hardly more complicated than the input queries. As a consequence, we were able to observe that off-the-shelf relational database query optimizers do well in practice [3].

5.6 Asymptotic Efficiency

We have seen in the previous section that for all but two operations of probabilistic world-set algebra, there is a very efficient solution that builds on relational database technology. These remaining operations are confidence computation and relational algebra difference.

Approximate confidence computation To compute the confidence in a tuple of data values occurring possibly in several tuples of a U-relation, we have to compute the probability of the disjunction of the local conditions of all these tuples. We have to eliminate duplicate tuples because we are interested in the probability of the data tuples rather than some abstract notion of tuple identity that is really an artifact of our representation. That is, we have to compute the probability of a DNF, i.e., the sum of the weights of the worlds identified with valuations θ of the random variables such that the DNF becomes true under θ. This problem is #P-complete [8, 6]. The result is not the sum of the probabilities of the individual conjunctive local conditions, because they may, intuitively, “overlap”.

Confidence computation can be efficiently approximated by Monte Carlo simulation [8, 6, 13]. The technique is based on the Karp-Luby fully polynomial-time randomized approximation scheme (FPRAS) for counting the number of solutions to a DNF formula [10, 11, 5]. There is an efficiently computable unbiased estimator that in expectation returns the probability p of a DNF of n clauses (i.e., the local condition tuples of a Boolean U-relation) such that computing the average of a polynomial number of such Monte Carlo steps (= calls to the Karp-Luby unbiased estimator) is an (ϵ,δ)-approximation for the probability: If the average is taken over at least ⌈3 ⋅n⋅ log(2∕δ)∕ϵ²⌉ Monte Carlo steps, then Pr|p - |≥ ϵ ⋅ p ≤ δ. The paper [5] improves upon this by determining smaller numbers (within a constant factor from optimal) of necessary iterations to achieve an (ϵ,δ)-approximation.

Avoiding the difference operation Difference R -S is conceptually simple on c-tables. Without loss of generality, assume that S does not contain tuples ⟨,ψ₁⟩,…,⟨,ψ_n⟩ that are duplicates if the local conditions are disregarded. (Otherwise, we replace them by ⟨,ψ₁ ∨∨ ψ_n⟩.) For each tuple ⟨,ϕ⟩ of R, if ⟨,ψ⟩ is in S then output ⟨,ϕ ∧¬ψ⟩; otherwise, output ⟨,ϕ⟩. Testing whether a tuple is possible in the result of a query involving difference is already NP-hard [2]. For U-relations, we in addition have to turn ϕ ∧¬ψ into a DNF to represent the result as a U-relation. This may lead to an exponentially large output and a very large number of columns may be required to represent the conditions. For these reasons, MayBMS currently does not implement the difference operation.

In many practical applications, the difference operation can be avoided. Difference is only hard on uncertain relations. On such relations, it can only lead to displayable query results in queries that close the possible worlds semantics using conf, computing a single certain relation. Probably the most important application of the difference operation is for encoding universal constraints, for example in data cleaning. But if the confidence operation is applied on top of a universal query, there is a trick that will often allow to rewrite the query into an existential one (which can be expressed in positive relational algebra plus conf, without difference) [13].

Example 5.6.2 The example uses the census scenario and the uncertain relation R[SSN] with columns TID and SSS discussed earlier; below we will call this relation just simply R. Consider the query of finding, for each TID t_i and SSN s, the confidence in the statement that s is the correct SSN for the individual associated with the tuple identified by t_i, assuming that social security numbers uniquely identify individuals, that is, assuming that the functional dependency SSN → TID (subsequently called ψ) holds. In other words, the query asks, for each TID t_i and SSN s, to find the probability Pr[ϕ∣ψ], where ϕ(t_i,s) = ∃t R t.TID = t_i ∧ t.SSN = s. Constraint ψ can be thought of as a data cleaning constraint that ensures that the SSN fields in no two distinct census forms (belonging to two different individuals) are interpreted as the same number.

We compute the desired conditional probabilities, for each possible pair of a TID and an SSN, as Pr[ϕ∣ψ] = Pr[ϕ ∧ ψ]∕Pr[ψ]. Here ϕ is existential (expressible in positive relational algebra) and ψ is an equality-generating dependency (i.e., a special universal query) [1]. The trick is to turn relational difference into the subtraction of probabilities, Pr[ϕ∧ψ] = Pr[ϕ]-Pr[ϕ∧¬ψ] and Pr[ψ] = 1 - Pr[¬ψ], where ¬ψ = ∃t,t′ R t.SSN = t′.SSN ∧ t.TID≠t′.TID is existential (with inequalities). Thus ¬ψ and ϕ ∧¬ψ are expressible in positive relational algebra. This works for a considerable superset of the equality-generating dependencies [13], which in turn subsume useful data cleaning constraints.

Let R_¬ψ be the relational algebra expression for ¬ψ,

and let S be

The overall example query can be expressed as

For the example table R given above, S and T are

S	TID	SSN	Pϕ	Pϕ∧¬ψ	P¬ψ
	t1	185	.4	.28	.28
	t1	785	.6	0	.28
	t2	185	.7	.28	.28
	t2	186	.3	0	.28

 

T	TID	SSN	P
	t1	185	1/6
	t1	785	5/6
	t2	185	7/12
	t2	186	5/12

Complexity Overview Figure 5.2 gives an overview over the known complexity results for the various fragments of probabilistic WSA.

Difference [2] and confidence computation [6] independently make queries NP-hard. Full probabilistic world-set algebra is essentially not harder than the language of [6], even though it is substantially more expressive.

It is worth noting that repair-key by itself, despite the blowup of possible worlds, does not make queries hard. For the language consisting of positive relational algebra, repair-key, and poss, we have shown by construction that it has PTIME complexity: We have given a positive relational algebra rewriting to queries on the representations earlier in this section. Thus queries are even in the highly parallelizable complexity class AC₀.

The final result in Figure 5.2 concerns the language consisting of the positive relational algebra operations, repair-key, (ϵ,δ)-approximation of confidence computation, and the generalized equality generating dependencies of [13] for which we can rewrite difference of uncertain relations to difference of confidence values (see Example 5.6.2). The result is that queries of that language that close the possible worlds semantics – i.e., that use conf to compute a certain relation – are in PTIME overall.

Chapter 6
The MayBMS Query and Update Language

6.1 Language Overview

This section describes the query and update language of MayBMS, which is based on SQL. In fact, our language is a generalization of SQL on classical relational databases. To simplify the presentation, a fragment of the full language supported in MayBMS is presented here.

The representation system used in MayBMS, U-relations, has as a special case classical relational tables, that is, tables with no condition columns. We will call these tables typed-certain (t-certain) tables in this section. Tables that are not t-certain are called uncertain. Note that this notion of certainty is purely syntactic, and

may well be equal to the projection of a U-relation U_R to its attribute (non-condition) columns despite R not being t-certain according to this definition.

Aggregates In MayBMS, full SQL is supported on t-certain tables. Beyond t-certain tables, some restrictions are in place to assure that query evaluation is feasible. In particular, we do not support the standard SQL aggregates such as sum or count on uncertain relations. This can be easily justified: In general, these aggregates will produce exponentially many different numerical results in the various possible worlds, and there is no way of representing these results efficiently. However, MayBMS supports a different set of aggregate operations on uncertain relations. These include the computations of expected sums and counts (using aggregates esum and ecount).

Moreover, the confidence computation operation is an aggregate in the MayBMS query language. This is a deviation from the language flavor of our algebra, but there is a justification for this. The algebra presented earlier assumed a set-based semantics for relations, where operations such as projections automatically remove duplicates. In the MayBMS query language, just like in SQL, duplicates have to be eliminated explicitly, and confidence is naturally an aggregate that computes a single confidence value for each group of tuples that agree on (a subset of) the non-condition columns. By using aggregation syntax for conf and not supporting select distinct on uncertain relations, we avoid a need for conditions beyond the special conjunctions that can be stored with each tuple in U-relations.

All supported aggregates on uncertain tables produce t-certain tables.

Duplicate tuples SQL databases in general support multiset tables, i.e., tables in which there may be duplicate tuples. There is no conceptual difficulty at all in supporting multiset U-relations. In fact, since U-relations are just relations in which some columns are interpreted to have a special meaning (conditions), just storing them in a standard relational database management system which supports duplicates in tables yields support for multiset U-relations.

Syntax The MayBMS query language is compositional and built from uncertain and t-certain queries. The uncertain queries are those that produce a possibly uncertain relation (represented by a U-relation with more than zero V and D columns). Uncertain queries can be constructed, inductively, from t-certain queries, select-from-where queries over uncertain tables, the multiset union of uncertain queries (using the SQL union construct), and the repair-key and pick-tuples statements that can be specified as follows

Note that repair-key is a query, rather than an update statement. Details on these constructs can be found in Section 6.2, Language reference.

The select-from-where queries may use any t-certain subqueries in the conditions, plus uncertain subqueries in atomic conditions of the form

that occur positively in the condition. (That is, if the condition is turned into DNF, these literals are not negated.)

The t-certain queries (i.e., queries that produce a t-certain table) are given by

• all constructs of SQL on t-certain tables and t-certain subqueries, extended by a new aggregate
     argmax(<argument-attribute>, <value-attribute>)

  which outputs one of the argument-attribute values in the current group (determined by the group-by clause) whose tuples have a maximum value-attribute value within the group. Thus, this is the typical argmax construct from mathematics added as an SQL extension.

• select-from-where-group-by on uncertain queries using the possible construct for computing possible tuples, or the aggregates conf, esum, and ecount, but none of the standard SQL aggregates. There is an exact and an approximate version of the conf aggregate. The latter takes two parameters ϵ and δ (see the earlier discussion of the Karp-Luby FPRAS).

The aggregates esum and ecount compute expected sums and counts across groups of tuples. While it may seem that these aggregates are at least as hard as confidence computation (which is #P-hard), this is in fact not so. These aggregates can be efficiently computed exploiting linearity of expectation. A query

is equivalent to a query

where R’ is obtained from the U-relation of R by replacing each local condition V ₁,D₁, … , V _k, D_k by the probability Pr[V ₁ = D₁ ∧∧ V _k = D_k], not eliminating duplicates. That is, expected sums can be computed efficiently tuple by tuple, and only require to determine the probability of a conjunction, which is easy, rather than a DNF of variable assignments as in the case of the conf aggregate. The ecount aggregate is a special case of esum applied to a column of ones.

Example 6.1.1 The query of Example 5.3.2 can be expressed in the query language of MayBMS as follows. Let R be repair key in Coins weight by Count and let S be

select R.Type, Toss, Face  
from (repair key Type, Toss in (select * from Faces, Tosses)  
      weight by FProb) S0, R  
where R.Type = S0.Type;

It is not hard to verify that π_Toss,Face(S)≠Ev exactly if there exist tuples S, Ev such that .Toss = .Toss and .Face≠ .Face. Let C be

select S.Type from S, Ev  
where S.Toss = Ev.Toss and S.Face <> Ev.Face;

Then we can compute Q using the trick of Example 5.6.2 as

select Type, (P1-P2)/(1-P3) as P  
from (select Type, conf() as P1 from S group by Type) Q1,  
     ((select Type, conf() as P2 from C group by Type)  
      union  
      (  
         (select Type, 0 as P2 from Coins)  
         except  
         (select Type, 0 as P2 from  
             (select Type, conf() from C group by Type) Dummy)  
      )) Q2,  
     (select conf() as P3 from C) Q3  
where Q1.Type = Q2.Type;

The argmax aggregate can be used to compute maximum-a-posteriori (MAP) and maximum-likelihood estimates. For example, the MAP coin type

can be computed as select argmax(Type, P) from Q because the normalizing factor (1-P3) has no impact on argmax. Thus, the answer in this example is the double-headed coin. (See table Q of Figure 5.1: The fair coin has P = 1∕3, while the double-headed coin has P = 2∕3.)

The maximum likelihood estimate

can be computed as

select argmax(Q.Type, Q.P/R’.P)  
from Q, (select Type, conf() as P from R) R’  
where Q.Type = R’.Type;

Here, again, the result is 2headed, but this time with likelihood 1. (The fair coin has likelihood 1/4). □

Supported Queries MayBMS supports full SQL on t-certain tables. In addition it supports a large subset of SQL on t-uncertain tables, with even more features supported when fragments of the uncertain query involve t-certain subqueries. The following restrictions apply:

• Exact aggregates and duplicate elimination using distinct in a select statement are supported as long as the from clause subqueries and the subqueries in the where condition are t-certain.
• If a t-certain subquery Q in the where condition of a select statement contains references to t-uncertain tables, then the containing query is supported if Q is not correlated with it.
• The set operations except and union with duplicate elimination are supported when both the left and the right argument are t-certain queries.
• repair-key and pick-tuples are supported on t-certain queries.

Restrictions on the update statements are discussed below.

Updates MayBMS supports the usual schema modification and update statements of SQL. In fact, our use of U-relations makes this quite easy. An insertion of the form

is just the standard SQL insertion for tables we interpret as U-relations. Thus, the table inserted into must have the right number (that is, a sufficient number) of condition columns. Schema-modifying operations such as

are similarly straightforward. A deletion

admits conditions that refer to the attributes of the current tuple and may use t-certain subqueries. One can also update an uncertain table with an update statement

where the set list does not modify the condition columns and the where condition satisfies the same conditions as that of the delete statement. MayBMS allows users to insert a constant tuple by specifying values for the data columns in an insert statement:

6.2 Language Reference

We next discuss the extensions to SQL by MayBMS. For a description of the standard SQL constructs please see the Postgres SQL language reference available at http://www.postgresql.org/docs/8.3/interactive/sql-commands.html

6.2.1 repair-key

Syntax:

Description: The repair-key operation turns a t-certain-query (or, as a special case, a t-certain-relation) into the set of worlds consisting of all possible maximal repairs of key attributes. A repair of key in relation R is a subset of R for which is a key. We say that relation R′ is a maximal repair of a functional dependency for relation R if R′ is a maximal subset of R which satisfies that functional dependency. The numerically-valued expression is used for weighting the newly created alternative repairs. If the weight by clause is omitted, a uniform probability distribution is assumed among all tuples with the same key. Suppose there are n tuples sharing the same key, each of them is associated with a probability of 1∕n. If the weight is specified by expression, the value of expression will be the probability of the tuple before normalization. Suppose there are n tuples sharing the same key, tuple t_i is associated with probability expression_i / ∑ _k=1ⁿexpression _k. In either case, the sum of the probabilities among all tuples with the same key is 1. There will be an error message if the value of expression in any tuple is negative. The tuples for which probability is 0 are ignored and not included in any resulting possible world.

repair-key can be placed wherever a select statement is allowed in SQL. See Section 5.3 for more details on repair-key.

Example: Suppose Customer is a certain relation with columns ID and name, the following query performs a repair-key operation on column ID in Customer:

Suppose Accounts is a certain relation with columns ID and account, the following is an example of repair-key operation on column ID in the output of a certain query:

6.2.2 pick-tuples

Syntax:

Description: The pick-tuples operation generates the set of worlds which can be obtained from a t-certain-query or a t-certain-relation by selecting a subset of the tuples of that query or relation. In the current version of MayBMS, the presence of independently does not affect query evaluation. It is the default; in the future, MayBMS may be extended by other options.

By default, every tuple in a possible world is associated with probability 0.5. If with probability expression is specified, the numerical value of expression is the probability of the tuple. Note that only values in (0,1] are valid. There will be an error message if the value of expression is negative or larger than 1. Tuples for which expression are 0 are ignored.

pick-tuples can be placed wherever a select statement is allowed in SQL.

6.2.3 possible

Syntax:

Description: The operation possible selects the set of tuples appearing in at least one possible world. This construct is a shortcut for the query which selects all distinct tuples with confidence greater than zero:

Example: Suppose R and S are uncertain relations, the following query displays distinct pairs (A,B) with positive probabilities.

6.2.4 Confidence computation and approximate aggregates

argmax, conf, aconf, tconf, esum and ecount are functions introduced by MayBMS. Following is the summary of the functions.

6.2.4.1 argmax(argument-attribute, value-attribute)

Outputs an argument-attribute value in the current group (determined by the group-by clause) whose tuples have a maximum value-attribute value within the group. If there are several tuples sharing the same maximum value-attribute value with different argument-attribute values, an arbitrary value among them is returned. For example,

retrieves one of the dates with the highest temperature for each location.

argmax can be used on all relations and queries.

6.2.4.2 conf()

Syntax:

Description: Computes for each possible distinct tuple of attribute values of the target list that occurs in an uncertain relation in at least one possible world, the sum of the probabilities of the worlds in which it occurs. conf can only be used on a t-uncertain query or a t-uncertain relation and the output of the query is a t-certain relation.

Example: Suppose weather_forecast is an uncertain relation storing information regarding weather prediction, the following query computes the probability of each weather condition for each location:

6.2.4.3 tconf()

Syntax:

Description: Computes for each possible tuple the sum of the probabilities of the worlds where it appears. tconf() is different from conf() in that it does not eliminate duplicates. tconf can only be used on a t-uncertain query or a t-uncertain relation and the output of the query is a t-certain relation.

6.2.4.4 aconf(ϵ, δ)

Syntax:

Description: Computes for each possible distinct tuples of the target list that occurs in at least one possible world, the approximate sum of the probabilities of the worlds in which it occurs. Let p be the exact sum (computed by conf) and be the approximate sum (computed by aconf), the approximation has the following property:

Pr|p - |≥ ϵ ⋅ p ≤ δ.

See the earlier discussion of the Karp-Luby FPRAS for more details. aconf can only be used on a t-uncertain query or a t-uncertain relation and the output of the query is a t-certain relation.

6.2.4.5 esum and ecount

Syntax:

Description: esum and ecount compute expected sum and count, respectively. ecount can take zero or one argument, and the number of arguments does not affect the results. esum and ecount can only be used on a t-uncertain query or a t-uncertain relation and the output of the query is a t-certain relation.

Example: The following query computes the expected total rainfall of seven days for each location:

Chapter 7
MayBMS Internals

Representations, relational encoding, and query optimization Our representation system, U-relations, is basically implemented as described earlier, with one small exception. With each pair of columns V _i, D_i in the condition, we also store a column P_i for the probability weight of alternative D_i for variable V _i, straight from the W relation. While the operations of relational algebra, as observed earlier, do not use probability values, confidence computation does. This denormalization (the extension by P_i columns) removes the need to look up any probabilities in the W table in our exact confidence computation algorithms.

Our experiments show that the relational encoding of positive relational algebra which is possible for U-relations is so simple – it is a parsimonious transformation, i.e., the number of relational algebra operations is not increased – that the standard Postgres query optimizer actually does well at finding good query plans (see [3]).

Approximate confidence computation MayBMS implements both an approximation algorithm and several exact algorithms for confidence computation. The approximation algorithm is a combination of the Karp-Luby unbiased estimator for DNF counting [10, 11] in a modified version adapted for confidence computation in probabilistic databases (cf. e.g. [13]) and the Dagum-Karp-Luby-Ross optimal algorithm for Monte Carlo estimation [5]. The latter is based on sequential analysis and determines the number of invocations of the Karp-Luby estimator needed to achieve the required bound by running the estimator a small number of times to estimate its mean and variance. We actually use the probabilistic variant of a version of the Karp-Luby estimator described in the book [17] which computes fractional estimates that have smaller variance than the zero-one estimates of the classical Karp-Luby estimator.

Exact confidence computation Our exact algorithm for confidence computation is described in [15]. It is based on an extended version of the Davis-Putnam procedure [7] that is the basis of the best exact Satisfiability solvers in AI. Given a DNF (of which each clause is a conjunctive local condition), the algorithm employs a combination of variable elimination (as in Davis-Putnam) and decomposition of the DNF into independent subsets of clauses (i.e., subsets that do not share variables), with cost-estimation heuristics for choosing whether to use the former (and for which variable) or the latter.

Figure 7.1: Exact confidence computation.

Our exact algorithm solves a #P-hard problem and exhibits exponential running time in the worst case. However, like some other algorithms for combinatorial problems, this algorithm shows a clear easy-hard-easy pattern. Outside a narrow range of variable-to-clause count ratios, it very pronouncedly outperforms the (polynomial-time) approximation techniques [15]. It is straightforward to extend this algorithm to condition a probabilistic database (i.e., to compute “assert”) [15].

Hierarchical queries The tuple-independent databases are those probabilistic databases in which, for each tuple, a probability can be given such that the tuple occurs in the database with that probability and the tuples are uncorrelated. It is known since the work of Dalvi and Suciu [6] that there is a class of conjunctive queries, the hierarchical queries Q, for which computing conf(Q) exactly on tuple-independent probabilistic databases is feasible in polynomial time.

In fact, these queries can essentially be computed using SQL queries that involve several nested aggregate-group-by queries. On the other hand, it was also shown in [6] that for any conjunctive query Q that is not hierarchical, computing conf(Q) is #P-hard with respect to data complexity. Dalvi and Suciu introduce the notion of safe plans that are at once certificates that a query is hierarchical and query plans with aggregation operators that can be used for evaluating the queries.

To deal with hierarchical queries, MayBMS runs SPROUT as part of its query engine [16]. SPROUT extends the early work by Suciu in three ways. First, the observation is used that in the case that a query has a safe plan [6], it is not necessary to use that safe plan for query evaluation. Instead, one can choose any unrestricted query plan, not only restricted safe plans, for the computation of the answer tuples; confidence computation is performed as an aggregation which can be pushed down or pull up past joins in relational query plans. Second, the aggregation function for confidence computation is implemented as a special low-level operator in the query engine. Finally, the fact is exploited that the #P-hardness result for any single nonhierarchical query of [6] only applies as long as the problem is that of evaluating the query on an arbitrary probabilistic database of suitable schema. If further information about permissible databases is available in the form of functional dependencies that the databases must satisfy, then a larger class of queries can be processed by our approach [16].

Updates, concurrency control and recovery As a consequence of our choice of a purely relational representation system, these issues cause surprisingly little difficulty. U-relations are just relational tables and updates are just modifications of these tables that can be expressed using the standard SQL update operations. While the structure of the rewritings could allow for optimizations in the concurrency and recovery managers, those are currently left to the underlying DBMS.

Chapter 8
The MayBMS Codebase

MayBMS is currently implemented in PostgreSQL 8.3.3. Integration into an existing full-fledged DBMS brings two major advantages. First, integration makes it possible to reuse the internal functions and structures of the DBMS. Secondly, it often increases the efficiency of query processing.

Figures 8.1 and 8.2 give a list of source files modified or added to the original PostgreSQL 8.3.3. All modifications are explicitly marked in the source files by

All files in directory maybms are newly created and the others are existing files in PostgreSQL8.3.3. Header files (*.h) refer to src/include/directory/filename. Source files (*.c and *.y) refer to src/backend/directory/filename.

Chapter 9
Experiments

This section reports on experiments performed with the first MayBMS release (beta) and a benchmark consisting of two parts, which are described in more detail in the remainder of this chapter:

1. Computing the probability of triangles in random graphs.
2. A modified subset of the TPC-H queries on uncertain TPC-H datasets.

By this benchmark, we do not attempt to simulate a representative set of use cases: the jury is still out on what such a set of use cases might be. Instead, we focus on a benchmark that allows us to see how the performance of MayBMS develops across releases on the two core technical problems solved by MayBMS: polynomial-time query evaluation for the polynomial-time fragment of our query language and the efficient approximation of query results for queries that do not belong to the polynomial-time fragment. (Finding triangles in random graphs is a near-canonical example of such queries.)

We will keep monitoring the development of the state of the art and will continue to survey applications and collect use cases; we will extend or replace this benchmark as consensus develops regarding the most important applications of probabilistic databases.

Experimental setup. All the experiments reported on in this chapter were conducted on an Athlon-X2(4600+)64bit / 1.8GB / Linux2.6.20 / gcc4.1.2 machine.

9.1 Random Graphs

9.1.1 Experiments with Varying Levels of Precision

In this experiment, we create undirected random graphs in which the presence of each edge is independent of that of the other edges. The probability that an edge is in the graph is 0.5 and this applies to each edge. Then we compute the probability that there exists a triangle in the graphs using approximation. The queries can be found in Appendix A.

We report wall-clock execution times of queries run in the PostgreSQL8.3.3 psql shell with a warm cache obtained by running a query once and then reporting the average execution time over three subsequent, identical executions. Figure 9.1 shows the execution time of approximation with different precision parameters for random graphs composed of 5 to 33 nodes. An (ϵ,δ) approximation has the following property: let p be the exact probability and be the approximate probability, then Pr|p - |≥ ϵ ⋅ p ≤ δ.

9.1.2 Experiments with Different Edge Probabilities

In the previous experiments, each edge had probability 0.5. We use other values as the edge probability(all edges still have the same probability) and run the experiment again with (0.05,0.05) approximation. The SQL statements in Appendix A should be modified accordingly. Let p be the probability, change the following statements

to

Figure 9.2 shows the execution time for queries of random graphs composed of 25 to 101 nodes with different fixed edge probabilities.

9.1.3 Experiments with General Random Graphs

The previous experiments were conducted on undirected graphs in which every pair of nodes had a possibly present edge. However, this may not be the case in general. In many scenarios, each pair of nodes may have a certainly present, certainly absent or possibly present edge. In our following experiments, we construct such general probabilistic random graphs from data representing directed links between webpage within nd.edu domain¹ . If a link between two pages is absent from the data, then it is also absent from our graphs. If a link is present in the data, then it is a certainly or possibly present edge in our graphs. We run again the queries computing the probabilities of existence of triangles in such graphs with (0.05,0.05) approximation. The probabilities that possibly present edges are in the graphs are randomly distributed in (0,0.1). The queries of the graph constructions and confidence computation can be found in Appendix B. Figure 9.3 shows the execution time for queries of such random graphs composed of 1000 to 30000 nodes.

9.2 Probabilistic TPC-H

SPROUT² is a part of the query engine of MayBMS and provides state-of-the-art techniques for efficient exact confidence computation. In this section, we show how TPC-H queries can benefit from these techniques. For each TPC-H query, we consider its largest subquery without aggregations and inequality joins but with conf() for specifying exact probability computation for distinct tuples in query answers. We consider two flavours of each of these queries: A version with original selection attributes (again, without aggregations), and a version where we drop keys from the selection attributes. Queries are included in the experiments if SPROUT’s techniques can be applied to them. Our data set consists of tuple-independent probabilistic databases obtained from deterministic databases produced by TPC-H 2.7.0 by associating each tuple with a Boolean random variable and by choosing at random a probability distribution over these variables. We perform experiments with TPC-H scale factor 1 (1GB database size) and evaluate the TPC-H-like queries mentioned above. The queries can be found in Appendix C. In addition, we compare our results with the reported time from [16] in which SPROUT was only partially integrated into PostgreSQL and storing temporary relations to the disk was sometimes necessary. The average time shown below is obtained from ten subsequent, identical executions with a warm cache by running the query once.

Appendix A
Queries in Random Graph Experiments

Appendix B
Queries in General Random Graph Experiments

Appendix C
Probabilistic TPC-H Queries

Query 1:

Query 4:

Query 12:

Query 15:

Boolean Version of Query 1:

Boolean Version of Query 4:

Boolean Version of Query 6:

Boolean Version of Query 12:

Boolean Version of Query 14:

Boolean Version of Query 15:

Boolean Version of Query 16:

Boolean Version of Query 17:

Bibliography