Provability & Complexity Constraints

• Provability and Complexity Constraints define the limits of formal proof systems by linking unprovability in bounded arithmetic with hardness in propositional proof systems.
• They employ algebraic, modal, and combinatorial methods to demonstrate exponential proof length lower bounds and underscore separations in complexity classes.
• The topic also examines decision complexity in logical fragments, clarifying NP-hardness, PSPACE-completeness, and other resource-bound phenomena in proof search.

Provability and Complexity Constraints

Provability and complexity constraints delineate the interplay between the logical provability of mathematical statements (e.g., tautologies, consistency statements, complexity lower bounds) and the resource-bounded feasibility of their proofs in various formal systems. These constraints manifest through the limits of proof length and structure, the expressiveness of proof systems, the formal strength of arithmetic theories, and the complexity-theoretic characterization of the underlying logical or combinatorial problems. A central theme is how noncomputability, unprovability, and proof length lower bounds in logic translate into hard instances for propositional proof systems and vice versa, and how model-theoretic or algebraic properties of problems encode thresholds for efficient proof search or proof existence.

1. Bounded Arithmetic and Unprovability of Complexity Statements

Bounded arithmetic theories, notably Cook’s PV, Buss’s S^i_2, and Jeřábek’s APC_1, formalize resource-bounded reasoning within the syntactic confines of first-order logic with length-bounded or sharply bounded quantification. The foundational unprovability results for these systems show that they cannot feasibly establish major circuit upper bounds (e.g., P ⊆ SIZE[n^k]), strong average-case or worst-case circuit lower bounds, or the efficient existence of proof systems for all tautologies. For every k ≥ 1, PV does not prove “P ⊆ SIZE[n^k],” and no witness function in PV can establish “P = NP” even under feasible proof schemes (Oliveira, 6 Apr 2025). These results often extend—via game-theoretic Skolemization or witnessing—to higher fragments such as T^i_2 and APC_1, showing that even sentences of arbitrary quantifier complexity (alternations of ∃ and ∀) cannot be proven to establish strong separations within the polynomial hierarchy:

• For all i ≥ 1, T^i_PV cannot prove the existence, for all sufficiently large n, of a Π_i-language f_n of poly-size circuits that is 1/n-far from all Σ_i-circuits of size 2^{n^{Ω(1)}} (average-case separation) (Li et al., 2023).

The proof-theoretic analysis proceeds by extracting uniform winning strategies in “tree-exploration games” compatible with the quantifier structure, demonstrating that a hypothetical proof yields an efficiently computable approximation (oracle circuit) that contradicts the presumed hardness.

2. Proof-Complexity and Circuit Lower Bounds

Proof complexity studies the length and structure of formal proofs in systems such as Resolution, bounded-depth Frege, Extended Frege, Polynomial Calculus (PC), Sums-of-Squares (SOS), and Lovász–Schrijver (LS). A sharp “gap theorem” applies: for CSPs of bounded width (i.e., possessing weak near-unanimity polymorphisms), all major proof systems admit polynomial-size refutations, with constant width/degree/depth or sublinear degree proofs; however, for templates of unbounded width, there exist explicit instances (e.g., systems of linear equations over finite Abelian groups, or graph k-coloring for k ≥ 3) where any proof in these systems must have exponential size or linear degree (Atserias et al., 2017). The key algebraic characterization reduces the existence of efficient proofs in strong propositional systems to purely algebraic properties of the constraint language or template (Gaysin, 2022). The preservation of proof complexity under standard CSP reductions (pp-interpretability, homomorphic equivalence, core expansions) ensures robustness of these dichotomies.

A further interplay is found in promise refutation systems, where a permissible “promise” on the number of satisfying assignments leads to a threshold: under a large promise (e.g., any satisfying assignment set of size ≥ ε * 2^n), all unsatisfiable instances have polynomial-size proofs; under smaller promises (e.g., 2^{δn}), random 3-CNF formulas remain exponentially hard for Resolution (0707.4255).

3. Provability in Modal and Provability Logics

Polymodal provability logics (notably GLP_Λ and positive fragments) provide a modal framework for formalizing hierarchies of provability (e.g., with respect to Πn reflection in arithmetic) and for capturing the complexity of reasoning about consistency, reflection, and iterated provability (Joosten, 2019). The transition from tractable fragments (e.g., positive, conjunction-and-diamond only) to intractable full fragments is marked by a computational complexity barrier: the positive fragment with infinitely many modalities remains in polynomial time, while the closed fragment with the full modal signature is PSPACE-complete (Pakhomov, 2013). The completeness and arithmetical soundness of these logics is established for various levels of the arithmetical hierarchy, with each reflection principle corresponding to a modal operator. Soundness and completeness results show that GLPΛ is the optimal modal logic for ramified provability up to any “complexity” level Λ.

4. Interrelations Between Model Theory, Compactness, and Proof Complexity

Logical compactness principles for finite relational structures, when relativized to the existence of homomorphisms to a fixed structure, exhibit a correspondence with the complexity hierarchy of the associated CSPs. For “width one” structures (solvable by local consistency), compactness is provable in ZF, aligning with polynomial-time solvability. For templates corresponding to NP-complete problems (e.g., graph 3-colorability), compactness is equivalent to the Ultrafilter Axiom, an example of a powerful set-theoretic principle outside the scope of ZF, reflecting the increased proof complexity of these problems (Rorabaugh et al., 2016). This parallelism relates infinitary combinatorial properties to classical proof-theoretic intractability.

5. Unprovability of Short Proofs and the Isomorphism Between Unprovability and Hard Propositional Families

A central result (“Ruling Out Short Proofs of Unprovable Sentences is Hard”) is an equivalence, conditional upon the nonexistence of optimal propositional proof systems, between unprovable sentences in arithmetic and hard tautology families in propositional proof complexity (Monroe, 2023). For instance, statements such as “x∈R” with R the set of Kolmogorov-random strings are generally unprovable, and the family of tautologies expressing “no short proof exists for x∈R” is hard for any proof system. The mapping from arithmetic unprovability to propositional hardness is essentially isomorphic: if there is no optimal proof system, then for every unprovable arithmetic sentence, one can uniformly generate a hard family of tautologies. This yields natural NP-intermediate languages and strengthens the interpretation of complexity barriers in proof-theoretic terms.

A plausible implication, suggested by these results, is that any proof system admitting polynomial-size refutations for all tautologies would collapse the polynomial hierarchy (PH) or even enforce unlikely complexity-class containments (e.g., NP ⊆ coNP/poly, PSPACE ⊆ NP∩coNP) (Bokov, 2016). Thus, the existence of succinct proofs is generically precluded by fundamental complexity-theoretic separations.

6. Decision Complexity in Logical Systems and Fragments

The complexity of provability, or decision procedures for provability, is tightly constrained by the logical fragment under scrutiny:

• Full propositional linear logic is Σ^0_1-complete (undecidable but recursively enumerable), but its multiplicative fragment is NP-complete, while the additive and exponential-free fragments are P-complete or in P (Chudigiewitsch, 2021).
• Intuitionistic propositional logic (IPC) is PSPACE-complete, but the implicational fragment (IIPC) is NP via normal-form compression and diagrammatic proof analysis (Bokov, 2016, Schubert et al., 9 May 2024).
• Fragments of polymodal provability logic (GLP) restricted to finitely many modalities are in PTIME, but the full fragment is PSPACE-complete due to reductions from QBF (Pakhomov, 2013).
• Deep inference calculi (e.g., BV) have NP-complete provability, but non-commutative extensions (e.g., Pomset Logic) reach Σ^p_2-completeness due to the need to check global graph-theoretic correctness conditions (Nguyên et al., 2022).

The complexity-theoretic barriers within each logic mirror the collapses or separations in the computational complexity of their underlying proof search and refutation systems.

7. Open Problems and Future Directions

Key open questions include:

• Formalizing and separating hierarchy levels (e.g., T^i_2 ≠ T^{j}_2 or S^i_2 ≠ S^{j}_2) unconditionally within bounded arithmetic (Oliveira, 6 Apr 2025).
• Constructing and certifying explicit hard instances for increasingly broad proof systems (Atserias et al., 2017).
• Extending unprovability results to strong theories such as APC_1 for superpolynomial lower bounds against circuit classes at higher levels of the polynomial hierarchy (Li et al., 2023).
• Characterizing the reverse mathematics of circuit lower-bound principles and exploring combinatorial correspondences for polynomial-time tractable algorithms (Gaysin, 2022).
• Translating separation and unprovability phenomena for time-bounded Kolmogorov complexity (e.g., MCSP) and exploring implications for one-way functions and average-case cryptographic hardness (Monroe, 2023).

The ongoing refinement of these constraints continues to illuminate the intricate boundaries between provability, computational complexity, and the structure of mathematical logic.