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November 20th, 2024

SQL, Homomorphisms and Constraint Satisfaction Problems

The article highlights SQL's ability to solve complex problems like Sudoku and CSPs, demonstrating efficiency in puzzles compared to Python and C, and its relationship with graph theory and homomorphisms.

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SQL, Homomorphisms and Constraint Satisfaction Problems

The article discusses the capabilities of SQL in solving complex problems, particularly through the lens of homomorphisms and constraint satisfaction problems (CSPs). It highlights how SQL can be utilized to tackle challenges like Sudoku and the "send more money" puzzle, demonstrating its power beyond traditional database queries. The author explains that SQL's core structure, represented by "SELECT FROM WHERE," can be viewed as a form of loops, allowing for the enumeration of possibilities in search problems. The article also compares the performance of SQL with Python and C implementations for solving the "send more money" puzzle, noting that SQL can be more efficient in certain contexts. Additionally, it touches on the concept of graph homomorphisms, illustrating how SQL queries can be mapped to graph structures, enabling the extraction of subgraphs from larger datasets. The author provides examples of converting graphs into SQL tables and vice versa, emphasizing the relationship between SQL queries and graph theory. Overall, the piece showcases SQL's versatility as a constraint solver and its potential applications in computational problems.

- SQL can solve complex problems like Sudoku and CSPs.

- The "send more money" puzzle illustrates SQL's efficiency compared to Python and C.

- SQL queries can be mapped to graph structures, enabling subgraph extraction.

- The article emphasizes the relationship between SQL and graph homomorphisms.

- SQL's core structure allows for powerful enumeration of possibilities in search problems.

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Link Icon 4 comments
By @mbid - 5 months
The post mentions the idea that querying a database D can be understood algebraically as enumerating all morphisms Q -> D, where Q is the "classifying" database of the query, i.e. a minimal database instance that admits a single "generic" result of the query. You can use this to give a neat formulation of Datalog evaluation. A Datalog rule then corresponds a morphism P -> H, where P is the classifying database instance of the rule body and H is the classifying database instance for matches of both body and head. For example, for the the transitivity rule

  edge(x, z) :- edge(x, y), edge(y, z).
you'd take for P the database instance containing two rows (a_1, a_2) and (a_2, a_3), and the database instance H contains additionally (a_1, a_3). Now saying that a Database D satisfies this rule means that every morphism P -> D (i.e., every match of the premise of the rule) can be completed to a commuting diagram

  P --> D
  |    ^
  |   /
  ⌄  /
  Q
where the additional map is the arrow Q -> D, which corresponds to a match of both body and head.

This kind of phenomenon is known in category theory as a "lifting property", and there's rich theory around it. For example, you can show in great generality that there's always a "free" way to add data to a database D so that it satisfies the lifting property (the orthogonal reflection construction/the small object argument). Those are the theoretical underpinnings of the Datalog engine I'm sometimes working on [1], and there they allow you to prove that Datalog evaluation is also well-defined if you allow adjoining new elements during evaluation in a controlled way. I believe the author of this post is involved in the egglog project [2], which might have similar features as well.

[1] https://github.com/eqlog/eqlog [2] https://github.com/egraphs-good/egglog

By @babel_ - 5 months
For anyone curious: the performance difference between Clang and GCC on the example C solution for verbal arithmetic comes down to Clang's auto-vectorisation (deducing SIMD) whilst GCC here sticks with scalar, which is why the counter brings Clang closer in line to GCC (https://godbolt.org/z/xfdxGvMYP), and it's actually a pretty nice example of auto-vectorisation (and its limitations) in action, which is a fun tangent from this article (given its relevance to high-performance SMT/SAT solving for CSP)
By @pcblues - 5 months
When SQL can't internally optimise a query into a more efficient constraint problem, unrolling joins is the key. This once MSSQL hacker got to the point of optimising queries with large amounts of joins or CTEs to just populating a single table's columns with one query per one to a few columns at a time (two minute locking queries down to about two seconds.) After that, I started using SQL to generate SQL and run that for really curly requirements. That gives you the ability to write queries that can search for a particular value in any column in any table, or find changes in the past 5 minutes in any column in any table within a fairly quick timeframe. And that's great for debugging applications that interface with the database or identifying rogue table changes. Without needing a transaction log. Programmer's paradise :)
By @lovasoa - 5 months
The topic of huge queries on tiny databases makes me think of this recent discussion on the SQLite forum: https://sqlite.org/forum/forumpost/0d18320369

Someone had an issue because SQLite failed to optimize the following query

    select * from t where x = 'x' or '' = 'x'
Someone said that SQLite could not optimize out the "or '' = 'x'" because it would be too expensive to compute. Which is obviously true only for huge queries on tiny datasets.