Superpolynomial Length Lower Bounds for Tree-Like Semantic Proof Systems with Bounded Line Size

Abstract: We prove superpolynomial length lower bounds for the semantic tree-like Frege refutation system with bounded line size. Concretely, for any function $n^{2-\varepsilon} \leq s(n) \leq 2^{n^{1-\varepsilon}}$ we exhibit an explicit family $\mathcal{A}$ of $n$-variate CNF formulas $A$, each of size $|A| \le s(n)^{1+\varepsilon}$, such that if $A$ is chosen uniformly from $\mathcal{A}$, then asymptotically almost surely any tree-like Frege refutation of $A$ in line-size $s(n)$ is of length super-polynomial in $|A|$. Our lower bounds apply also to tree-like degree-$d$ threshold systems, for $d \approx \log\bigl(s(n)\bigr)$, that is, for $d$ up to $n^{1-\varepsilon}$. More generally, our lower bounds apply to the semantic version of these systems and to any semantic tree-like proof system where the number of distinct lines is bounded by $\exp\bigl(s(n)\bigr)$.

• The paper establishes superpolynomial lower bounds by developing a linear pseudo-measure that forces many leaves in tree-like semantic refutations.
• It applies to systems including tree-like Frege, degree-d threshold, cutting planes, and resolution, all using explicit CNF encodings of the clique problem.
• The results delineate clear hierarchies in proof complexity and overcome limitations of traditional lower bound methods in syntactically constrained settings.

Superpolynomial Lower Bounds for Tree-Like Semantic Proof Systems with Bounded Line Size

Introduction and Motivation

This paper addresses a central topic in proof complexity: the development of lower bounds on the length of refutations within propositional proof systems under various constraints. In particular, the work focuses on tree-like semantic proof systems—specifically, systems that are not restricted by explicit syntactic derivation rules but by the set of Boolean functions available as proof lines, subject to an upper bound on the number of distinct lines. The principal goal is to establish superpolynomial refutation length lower bounds in these settings, targeting powerful systems where traditional lower bound techniques stall.

Semantic versions of proof systems allow each line to be an arbitrary Boolean function, making them strictly more powerful than most syntactic systems. However, bounding the number or size of allowed proof lines makes these systems tractable for complexity analysis. Of note, the results here are not limited to weak or artificial systems: the lower bounds apply to tree-like Frege with bounded line-size, tree-like degree-$d$ threshold systems with $d$ up to nearly linear in $n$, and related highly expressive systems, provided their "line" complexity is sufficiently bounded.

A key implication is that the lower bound technique—based on the construction of a pseudo-measure—does not depend on the hardness of simple combinatorial principles (like pigeonhole principle or Tseitin formulas), which are efficiently refuted in the considered semantic systems. This crucially distinguishes this work from prior approaches, which often fail in the face of such systems' power.

Technical Framework

Semantic Proof Systems and Line Size Restrictions

Tree-like semantic proof systems are defined by a family $F$ of $n$-variate Boolean functions with a cardinality constraint, and inference is based on semantic implication: a new line can be written whenever it is semantically implied by prior lines (under all assignments). A refutation derives the constant-1 function from the initial set of axioms (typically the clauses of a CNF formula).

The line-size parameter $s(n)$ determines the maximal number of possible distinct proof lines in $F$ (or, in some instances, the maximal syntactic size of a formula in each line). If line size or line complexity is permitted to grow too large (e.g., $\Omega(|A|)$ where $|A|$ is the CNF formula size), the systems become trivial: the whole formula itself can be included as a single proof line. Thus, nontrivial lower bounds are only possible for sub-maximal line size.

Encodings and Target Formulas

The hard formulas considered are explicit families of CNF encodings of the $k$-clique problem (with parameters tuned according to the regime): unary, binary, and interpolated block encodings are used, allowing control over width and variable count. These formulas are unsatisfiable for random Erdös–Rényi graphs whose maximum clique is strictly less than $d$0, and a key observation is that the number of such formulas is quasi-maximal for a given variable count, making them effective for average-case analysis.

Proof systems considered include (but are not limited to):

• Tree-like Frege systems with bounded line size—semantic inferences with lines restricted in formula size.
• Tree-like degree-$d$1 threshold systems—lines are polynomial inequalities of degree at most $d$2.
• Tree-like cutting planes and resolution over linear equations—special cases handled via the general framework.

Lower Bound Method: Pseudo-Measure Construction

A pivotal technique is the introduction of a linear pseudo-measure $d$3 over sets of satisfying assignments (tuples), adapting the methodology of [dRPR23UnarySA]'s lower bounds for Sherali–Adams. Here, $d$4 is constructed such that:

1. $d$5 (the measure of all tuples) is large (close to 1).
2. For any set of tuples corresponding to a single missing edge (i.e., an axiom that is falsified only due to the absence of a particular edge in $d$6), $d$7 is very small (inverse-superpolynomial in $d$8).
3. The measure is linear: the measure of an inferred line is the sum of the measures of the antecedents.

The measure's properties force any semantic tree-like refutation to contain many leaves (since the total measure must be accounted for by numerous tiny contributions), yielding superpolynomial lower bounds on refutation size.

An important technical challenge overcome in this paper is that, since semantic systems permit arbitrary Boolean functions as proof lines (subject to the cardinality bound), the structure of possible leaves in the proof tree is unstructured—necessitating delicate union bounds and careful parameter management to ensure that the measure-based argument remains sound.

Main Results

General Lower Bound Theorem

For any function $d$9, there exist explicit CNF formula families, each $n$0 in size, such that for $n$1 drawn uniformly at random from this family, and for any semantic tree-like proof system using lines from a set $n$2 with $n$3, any tree-like semantic refutation requires superpolynomial length in $n$4 (2604.28172).

This holds for all explicit families considered, including:

• $n$5-CNF and higher width ($n$6-CNF, $n$7 even) formulas, with tight bounds on line family size for a given width.
• Unary, binary, and interpolated block encodings of the clique problem.
• Formulas with parameter regimes adjusted for higher line family sizes, giving a tradeoff between formula complexity and allowable proof system strength.

Corollaries and System Instantiations

• Tree-like Cutting Planes and Tree-like Resolution over Linear Equations: Refutation length lower bounds of $n$8 are shown for these systems on random clique formulas. The VC-dimension techniques quantify the number of possible proof lines, enabling the main theorem's application.
• Tree-like Degree-$n$9 Threshold Systems: The first-size lower bounds for polynomial degree $F$0 (almost linear in $F$1) are established, far surpassing previous lower bounds which were confined to logarithmic $F$2.
• Tree-like Frege with Bounded Line Size: For $F$3, any line-size $F$4 tree-like Frege system refuting the constructed CNF formulas requires superpolynomial length.
• Hierarchy of Semantic Proof Systems: For increasing formula width or representation power (e.g., $F$5), strict hierarchies in refutation length are established, up to tight parameter boundaries.

Structural Consequences

• Typical Density of Minimally Unsatisfiable CNFs: It is shown that, in high-density regimes, almost all minimally unsatisfiable constant-width CNF formulas are dense, with minimally unsatisfiable $F$6-CNF formulas typically having at least $F$7 clauses. This aligns with and extends combinatorial expectations.

Implications and Future Directions

These results demonstrate that the pseudo-measure method yields substantive lower bounds for extremely strong semantic proof systems, including those that easily refute classical hard principles. This indicates that the difficulty arises not from these principles, but rather from the inherent limitations placed by bounding semantic line complexity.

Open questions and future directions include:

• Can the pseudo-measure method be adapted to syntactic rather than semantic systems, i.e., systems that restrict the form of inference rather than the available Boolean functions?
• Is it possible to obtain lower bounds for dag-like (as opposed to tree-like) semantic $F$8 proof systems, which would have significant implications for Lovász–Schrijver and related lift-and-project systems?
• What are the prospects for leveraging pseudo-measure-based arguments to establish lower bounds against randomized or non-deterministic communication models—specifically, can this method bridge to the proof complexity versus communication complexity paradigm?
• Further exploration of the structure and density properties of random and minimally unsatisfiable CNF formulas, as well as their refutation complexity.

Conclusion

This work establishes the first superpolynomial lower bounds on tree-like semantic refutations in highly expressive proof systems under plausible line-size or cardinality constraints, using a generalized pseudo-measure approach. The results are tight up to small parameter factors and apply to systems far stronger than those previously amenable to lower bound proofs. The separation from the hardness of combinatorial principles, and the ability to avoid reliance on their complexity, marks a substantial methodological advance in proof complexity. The paper opens new, nontrivial avenues for further study of both semantic and syntactic proof systems, as well as the complexity of propositional unsatisfiability (2604.28172).