[Paper Review] Undecidability of satisfiability in the algebra of finite binary relations with union, composition, and difference

Tony Tan, Jan Van den Bussche+1 more|arXiv (Cornell University)|Jun 2, 2014

Logic, Reasoning, and Knowledge19 references5 citations

TL;DR

This paper proves that satisfiability is undecidable for the algebra of finite binary relations using only union, composition, and set difference—specifically, even when expressions use only a single relation name and nesting of difference is limited to one level. The result is established via a reduction from a known undecidable problem, demonstrating that finite satisfiability remains undecidable in a minimal fragment of relational algebra.

ABSTRACT

We consider expressions built up from binary relation names using the operators union, composition, and set difference. We show that it is undecidable to test whether a given such expression $e$ is finitely satisfiable, i.e., whether there exist finite binary relations that can be substituted for the relation names so that $e$ evaluates to a nonempty result. This result already holds in restriction to expressions that mention just a single relation name, and where the difference operator can be nested at most once.

Motivation & Objective

• To investigate the decidability of finite satisfiability in a fragment of Tarski's relational algebra that excludes complement and identity, using only union, composition, and set difference.
• To determine whether the undecidability of satisfiability persists when the full complement operator is replaced by the relative complement (set difference).
• To identify the minimal conditions under which satisfiability becomes undecidable, particularly focusing on nesting depth of set difference and the number of relation names used.
• To clarify the boundary between decidable and undecidable fragments of relational and description logics, especially in the context of query languages for graph databases.
• To establish that undecidability holds even in the restricted fragment DA¹₂, which uses only one relation name and at most one level of nested set difference.

Proposed method

• Construct a reduction from a known undecidable problem (satisfiability in FO³) to the satisfiability problem in the algebraic fragment DA¹₂.
• Define a structure J over a single relation name a, built from a given finite structure I over relations b and c, such that the semantics of DA expressions over J simulate those over I.
• Use auxiliary nodes and edges to encode the presence of b- and c-edges in I via path patterns in J, ensuring that composition and set difference operations in DA expressions can capture the intended relational logic.
• Prove that for any expression e over {b,c}, e(I) is nonempty if and only if the corresponding expression over a in J evaluates to a nonempty relation, using structural induction and case analysis on intermediate path pairs.
• Verify that no unintended paths exist in J that could falsify the simulation, by exhaustively listing all possible pairs in a², a³, and a⁴ relations and showing they do not satisfy the required intersection conditions.
• Establish that the construction preserves the semantics of union, composition, and set difference, thereby proving the reduction is correct and the undecidability result holds for DA¹₂.

Experimental results

Research questions

• RQ1 Is finite satisfiability decidable for the algebra of finite binary relations using only union, composition, and set difference?
• RQ2 Does the undecidability of satisfiability persist when the complement operator is replaced by set difference and nesting of difference is limited to one level?
• RQ3 Can the undecidability result be established even when only a single relation name is used in the expressions?
• RQ4 What is the minimal fragment of relational algebra that still exhibits undecidable finite satisfiability?
• RQ5 How does this result refine the boundary between decidable and undecidable fragments in dynamic and description logics?

Key findings

• Finite satisfiability is undecidable for the algebra of finite binary relations with union, composition, and set difference.
• The undecidability result holds even in the fragment DA¹₂, which uses only a single relation name and allows at most one level of nested set difference.
• The proof constructs a finite structure J from a given structure I such that the semantics of DA expressions over J simulate those over I, preserving nonemptiness of results.
• The construction ensures that only intended paths (corresponding to b- and c-edges in I) are captured via composition and intersection in J, with no unintended paths violating the simulation.
• The result sharpens known undecidability results by showing that even without complement or transitive closure, undecidability arises from the combination of composition and set difference.
• The paper establishes that the problem remains undecidable in a minimal setting, helping to delineate the frontier of decidable fragments in relational and description logics.