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Logical reduction of relations: from relational databases to Peirce's reduction thesis

Sergiy Koshkin

TL;DR

The paper addresses the problem of reducing relations to lower arity using primitive positive formulas, unifying Peirce's bond algebra with relational database operations and CSP clone theory. It introduces projoins and hypostatic abstraction, bonds and teridentities, and bonding diagrams to analyze reductions, establishing a framework for complete reductions and a refined notion of ternarity. On infinite domains, non-degenerate $n$-ary relations reduce to unaries, binaries, and teridentities with $\mathrm{ter}=n-2$, while on finite domains the situation is more intricate and there are counterexamples (e.g., Herzberger's quaternary) showing larger ternarity and non-uniform behavior. The paper raises open questions about irreducible higher-arity relations on finite domains, sharp ternarity bounds, and the possible information-theoretic interpretation of ternarity, suggesting further connections to clone theory.

Abstract

We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them, and introduce a new characteristic of relations, ternarity, that measures their `complexity of relating' and allows to refine reduction results. In particular, we refine Peirce's controversial reduction thesis, and show that reducibility behavior is dramatically different on finite and infinite domains.

Logical reduction of relations: from relational databases to Peirce's reduction thesis

TL;DR

The paper addresses the problem of reducing relations to lower arity using primitive positive formulas, unifying Peirce's bond algebra with relational database operations and CSP clone theory. It introduces projoins and hypostatic abstraction, bonds and teridentities, and bonding diagrams to analyze reductions, establishing a framework for complete reductions and a refined notion of ternarity. On infinite domains, non-degenerate -ary relations reduce to unaries, binaries, and teridentities with , while on finite domains the situation is more intricate and there are counterexamples (e.g., Herzberger's quaternary) showing larger ternarity and non-uniform behavior. The paper raises open questions about irreducible higher-arity relations on finite domains, sharp ternarity bounds, and the possible information-theoretic interpretation of ternarity, suggesting further connections to clone theory.

Abstract

We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them, and introduce a new characteristic of relations, ternarity, that measures their `complexity of relating' and allows to refine reduction results. In particular, we refine Peirce's controversial reduction thesis, and show that reducibility behavior is dramatically different on finite and infinite domains.
Paper Structure (10 sections, 25 theorems, 41 equations, 4 figures)

This paper contains 10 sections, 25 theorems, 41 equations, 4 figures.

Table of Contents

  1. Preliminaries
  2. Cartesian Products
  3. Joins and dependencies
  4. Projective joins and hypostatic abstraction
  5. Projections as unions
  6. Bonds and teridentity
  7. Bonding diagrams
  8. Complete reductions and ternarity
  9. Ternarity and Peirce's reduction thesis
  10. Conclusions and open problems

Key Result

Theorem 1

A relation $R\subseteq\mathcal{D}^\Sigma$ is a Cartesian product over a partition $\Sigma=\Lambda_1\cup\dots\cup\Lambda_m$ if and only if the values of its tuples on $\Lambda_i$ can be chosen independently, i.e. for any collection of $\alpha^i\in\pi_{\Lambda_i}R$ there exists a common $a\in R$ such

Figures (4)

  • Figure 1: Projoin graphs for a) $\exists\,t[P(x_1)\land Q(x_1,t)\land R(t,x_2)]$; b) $\exists\,s\exists\,t[P(x_1,s,t)\land Q(s,t,x_2)]$.
  • Figure 2: Bonding diagrams of projoins from Figure \ref{['ProGraphs']}.
  • Figure 3: a) Bond explication of $\exists\,t\left[P(t,x_1,x_2,t)\land\exists\,s\,Q(x_2,x_3,s,t)\right]$ as \ref{['ExpliExm']}, here $\widetilde{Q}(t_3,y_2,x_3):=\exists s\,Q(t_3,y_2,x_3,s)$; b) bonding diagram of the reduction \ref{['InviaI3']} of $n$-identity to teridentities.
  • Figure 4: Schematic bond diagrams, the dots stand for predicate vertices: a) subcubic multiedges; b) merging unaries; c) merging binaries.

Theorems & Definitions (67)

  • Definition 1
  • Theorem 1: Independence criterion
  • proof
  • Example 1
  • Theorem 2
  • proof
  • Example 2
  • Theorem 3
  • proof
  • Definition 2
  • ...and 57 more