Lower Bounds against the Ideal Proof System in Finite Fields

TL;DR

This paper establishes new lower bounds for algebraic proof systems over finite fields, particularly for multilinear IPS and read-once algebraic branching programs, advancing understanding of proof complexity in finite field settings.

Contribution

It proves lower bounds for IPS fragments over finite fields, extending previous results to finite fields and connecting algebraic proof complexity with Boolean proof systems.

Findings

No polynomial-size multilinear IPS refutations for specific finite field instances.

Extended lower bounds for read-once algebraic branching program refutations in finite fields.

Lower bounds imply hardness results for related Boolean proof systems like $AC^0[p]$-Frege.

Abstract

Lower bounds against strong algebraic proof systems and specifically fragments of the Ideal Proof System (IPS), have been obtained in an ongoing line of work. All of these bounds, however, are proved only over large (or characteristic $0$) fields, yet finite fields are the more natural setting for propositional proof complexity, especially for progress toward lower bounds for Frege systems such as $AC^0[p]$-Frege. This work establishes lower bounds against fragments of IPS over fixed finite fields. Specifically, we show that a variant of the knapsack instance studied by Govindasamy, Hakoniemi, and Tzameret (FOCS'22) has no polynomial-size IPS refutation over finite fields when the refutation is multilinear and written as a constant-depth circuit. The key ingredient of our argument is the recent set-multilinearization result of Forbes (CCC'24), which extends the earlier result of Limaye, Srinivasan, and Tavenas (FOCS'21) to all fields, and an extension of the techniques of Govindasamy, Hakoniemi, and Tzameret to finite fields. We also separate this proof system from the one studied by Govindasamy, Hakoniemi, and Tzameret. In addition, we present new lower bounds for read-once algebraic branching program refutations, roABP-IPS, in finite fields, extending results of Forbes, Shpilka, Tzameret, and Wigderson (Theor. of Comput.'21) and Hakoniemi, Limaye, and Tzameret (STOC'24). Finally, we show that any lower bound against any proof system at least as strong as (non-multilinear) constant-depth IPS over finite fields for any instance, even a purely algebraic instance (i.e., not a translation of a Boolean formula or CNF), implies a hard CNF formula for the respective IPS fragment, and hence an $AC^0[p]$-Frege lower bound by known simulations over finite fields (Grochow and Pitassi (J. ACM'18)).