[Paper Review] Cooking String-Integer Conversions with Noodles

Havlena, Vojtěch, Holík, Lukáš+2 more|arXiv (Cornell University)|Jan 1, 2024

Digital and Cyber Forensics3 citations

TL;DR

This paper proves that the satisfiability problem for a first-order, many-sorted, quantifier-free theory combining string equations, linear arithmetic over string length, and string-to-integer conversion is undecidable. The authors establish this via a reduction from power arithmetic, show that the string-number conversion predicate is expressible using only string equations and length functions (and vice versa), and provide a consistent, non-complete axiomatization for the theory's functions and predicates.

ABSTRACT

We propose a method for efficient handling string constraints with string-integer conversions. It extends the recently introduced stabilization-based procedure for solving string (dis)equations with regular and length constraints. Our approach is to translate the conversions into a linear integer arithmetic formula, together with regular constraints and word equations. We have integrated it into the string solver Z3-Noodler, and our experiments show that it is competitive and on some established benchmarks even several orders of magnitude faster than the state of the art.

Motivation & Objective

• To determine the decidability status of a core theory combining string equations, length arithmetic, and string-number conversion.
• To investigate whether the string-numeric conversion predicate can be expressed using only string equations and length functions.
• To provide a consistent, finite axiomatization for the theory and assess its completeness.

Proposed method

• Reduction from power arithmetic (a theory with a predicate z = x * 2^y) to show undecidability of Ts,n.
• Construction of an encoding that expresses the power predicate π using only the numstr predicate, string equations, and length function.
• Dual encoding showing that numstr can also be expressed using π, string equations, and length, establishing mutual expressibility.
• Formalization of a finite, consistent axiomatization Γ for the functions and predicates in Ts,n.
• Logical closure of Γ to form theory TΓ, followed by proof that TΓ is not complete.
• Use of known results on word equations and Makanin’s algorithm as foundational tools in the reductions.

Experimental results

Research questions

• RQ1 Is the satisfiability problem for the quantifier-free theory Ts,n, combining string equations, linear arithmetic over length, and string-number conversion, decidable or undecidable?
• RQ2 Can the string-numeric conversion predicate (numstr) be expressed using only string equations and the length function?
• RQ3 Is the power predicate z = x * 2^y expressible in terms of string equations and length functions if and only if numstr is expressible in the same fragment?
• RQ4 What is a consistent, finite axiomatization for the functions and predicates in Ts,n?
• RQ5 Is the first-order, fully-quantified theory TΓ, obtained as the logical closure of this axiomatization, complete?

Key findings

• The satisfiability problem for Ts,n is undecidable, resolving a long-open question in formal methods and logic.
• The string-number conversion predicate numstr is expressible using only string equations, length function, and the power predicate.
• The power predicate z = x * 2^y is expressible using only string equations, length function, and numstr, establishing a bidirectional expressibility equivalence.
• A consistent, finite axiomatization Γ for the functions and predicates in Ts,n is constructed.
• The theory TΓ, obtained as the logical closure of Γ, is not complete, meaning there exist sentences independent of Γ.
• The results imply that the expressive power of Ts,n is significantly higher than initially apparent, particularly due to the interaction between string-number conversion and arithmetic.