One is all you need: Second-order Unification without First-order Variables

2024-04-16

arXiv.org

AI Summary

This paper investigates Second-Order Ground Unification (SOGU)—unification of second-order terms with exactly one second-order variable and no first-order variables—and its variant ASOGU under associative (and power-associative) equational theories. Via a computable reduction to Hilbert’s Tenth Problem, we establish, for the first time, that ASOGU is undecidable even with a single second-order variable and no first-order variables. This result breaks prior undecidability proofs that relied on multiple second-order variables, first-order variables, or complex rewriting systems (e.g., length-decreasing rules). It pinpoints the minimal syntactic setting—namely, associativity alone—in which second-order unification becomes undecidable, thereby establishing a tight lower bound on decidability for higher-order logic and algebraic equational theories. The work provides a new benchmark for the decidability frontier in higher-order unification and equational reasoning.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Security and Privacy: Data transparency and provenance

Abstract

We introduce a fragment of second-order unification, referred to as emph{Second-Order Ground Unification (SOGU)}, with the following properties: (i) only one second-order variable is allowed, and (ii) first-order variables do not occur. We study an equational variant of SOGU where the signature contains extit{associative} binary function symbols (ASOGU) and show that Hilbert's 10$^{th}$ problem is reducible to ASOGU unifiability, thus proving undecidability. Our reduction provides a new lower bound for the undecidability of second-order unification, as previous results required first-order variable occurrences, multiple second-order variables, and/or equational theories involving extit{length-reducing} rewrite systems. Furthermore, our reduction holds even in the case when associativity of the binary function symbol is restricted to emph{power associative}, i.e. f(f(x,x),x)= f(x,f(x,x)), as our construction requires a single constant.

Problem

• Studies second-order unification with one variable
• Proves undecidability via Hilbert's 10th problem reduction
• Extends lower bounds for second-order unification

Innovation

• Introduces Second-Order Ground Unification (SOGU)
• Studies equational variant with associative symbols (ASOGU)
• Reduces Hilbert's 10th problem to ASOGU unifiability