[Paper Review] A Note on Semi-Algebraic Proofs and Gaussian Elimination over Prime Fields

Albert Atserias|arXiv (Cornell University)|Feb 13, 2015

Complexity and Algorithms in Graphs8 references3 citations

TL;DR

This paper demonstrates that unsatisfiable systems of linear equations over prime finite fields, with a constant number of variables per equation, admit polynomial-size, constant-degree semi-algebraic proofs of unsatisfiability. The key contribution is showing that general (dag-like) semi-algebraic proofs can efficiently simulate Gaussian elimination, contrasting with exponential lower bounds for static or tree-like variants of these proof systems.

ABSTRACT

In this note we show that unsatisfiable systems of linear equations with a constant number of variables per equation over prime finite fields have polynomial-size constant-degree semi-algebraic proofs of unsatisfiability. These are proofs that manipulate polynomial inequalities over the reals with variables ranging in $\{0,1\}$. This upper bound is to be put in contrast with the known fact that, for certain explicit systems of linear equations over the two-element field, such refutations require linear degree and exponential size if they are restricted to so-called static semi-algebraic proofs, and even tree-like semi-algebraic and sums-of-squares proofs. Our upper bound is a more or less direct translation of an argument due to Grigoriev, Hirsch and Pasechnik (Moscow Mathematical Journal, 2002) who did it for a family of linear systems of interest in propositional proof complexity. We point out that their method is more general and can be thought of as simulating Gaussian elimination.

Motivation & Objective

• To investigate the proof complexity of unsatisfiable linear systems over prime finite fields using semi-algebraic proof systems.
• To compare the power of general (dag-like) semi-algebraic proofs with static or tree-like variants, particularly in the context of Gaussian elimination.
• To resolve the gap between known exponential lower bounds for static semi-algebraic and sums-of-squares proofs and the feasibility of general semi-algebraic proofs.
• To establish that constant-degree, polynomial-size semi-algebraic refutations exist for systems with bounded equation size over prime fields.

Proposed method

• The proof system uses positive linear combinations and lifting rules to derive polynomial inequalities over {0,1}-valued variables.
• It leverages the identity $ D_c(L) = (L - c)(L - c + 1) \geq 0 $ to enforce that a linear form $ L $ avoids integer values in $ (c-1, c) $.
• The method recursively refines approximations of linear forms modulo $ p $, using auxiliary variables $ y $ to represent fractional parts in $ \mathbb{Q} $.
• It constructs proofs by induction on the number of variables, using lifting rules and substitution lemmas to maintain polynomial degree and size.
• The construction simulates Gaussian elimination over $ \mathbb{F}_p $ by encoding field arithmetic in real polynomial inequalities with $ \{0,1\} $-valued variables.
• It combines proofs of $ M_{\mathbf{z}}(\mathbf{x}) = 0 $ for $ \mathbf{z} \notin Z $ with a sum over $ \mathbf{z} \in Z $ to reconstruct the full $ D_c(L_k) \geq 0 $ inequality.

Experimental results

Research questions

• RQ1 Can general semi-algebraic proofs efficiently refute unsatisfiable linear systems over prime fields, even when static versions require exponential size?
• RQ2 To what extent can semi-algebraic proofs simulate Gaussian elimination over finite fields?
• RQ3 Why do known exponential lower bounds for static semi-algebraic and sums-of-squares proofs not extend to general (dag-like) semi-algebraic proofs?
• RQ4 What is the relationship between the structure of linear equations over $ \mathbb{F}_p $ and the degree and size of their semi-algebraic refutations?

Key findings

• Unsatisfiable linear systems over prime fields with a constant number of variables per equation admit semi-algebraic refutations of polynomial size and constant degree.
• The proof system can simulate Gaussian elimination over $ \mathbb{F}_p $ by encoding field arithmetic through real polynomial inequalities with $ \{0,1\} $-valued variables.
• The construction achieves a refutation of $ -1 \geq 0 $ by proving $ D_{q+1}(L_0) \geq 0 $ for a rational linear form $ L_0 $ with fractional value $ \frac{1}{p} $, then scaling to derive contradiction.
• The method avoids exponential lower bounds that apply to static or tree-like semi-algebraic and sums-of-squares proofs, showing that dag-like proofs are significantly more powerful.
• The size of the refutation is bounded by a polynomial in $ n $ and the maximum coefficient size, with degree at most 3 throughout the proof construction.