Bounded Arithmetic VL

• Bounded arithmetic VL is a two-sorted formal theory that captures deterministic log-space reasoning using number- and string-sort variables.
• The system underpins the propositional proof system GL*, translating Σ1^B formulas into ΣCNF(2) tautologies that reflect log-space computability.
• VL formalizes key combinatorial and graph-theoretic arguments, establishing its role in proving Reingold’s L=SL theorem and connecting with symmetric log-space (VSL).

Bounded arithmetic VL is a two-sorted formal theory designed to capture reasoning precisely within deterministic log-space (L) computation. VL is formulated in a language with both number- and string-sort variables, axiomatically mirroring the structure and limitations of log-space algorithms. The theory provides a crucial bridge between complexity theory, proof complexity, and logic, and serves as the canonical bounded arithmetic for L. VL also underpins propositional proof systems such as GL*, providing uniform and non-uniform characterizations of L, and has recently been shown to coincide, at the level of provability, with the symmetric log-space theory VSL, giving a constructive bounded arithmetic formalization of Reingold's L=SL theorem (0801.4105, Buss et al., 15 Nov 2025).

1. Language, Vocabulary, and Axiomatization

VL is constructed in the two-sorted language ℒ₂, with variables for numbers ($x, y, z, \ldots$) and for finite strings or sets ($X, Y, Z, \ldots$). The vocabulary includes:

• Number constants and functions: $0, 1, +, \cdot$
• Number order and equality: $<, =$
• String bit-predicates: $X(i)$ meaning $i \in X$ (bit $i$ of string $X$ is set)
• String-length function: $|X|$ (one plus the largest bit set in $X$)

VL is axiomatized by:

• 2BASIC axioms: Standard arithmetic for numbers and axioms for strings, including extensionality and bit-based characterization of length.
• Bounded comprehension schema ($\Sigma_0^B$-COMP): For any $\Sigma_0^B$ formula $\varphi$ not containing $Y$,

$$\forall \vec x, \vec X\, \exists Y \le t(\vec x, \vec X)\, \forall i < t\ (Y(i) \leftrightarrow \varphi(i, \vec x, \vec X))$$

• Bounded recursion/recurrence schema ($\Sigma_0^B$-REC): Captures log-space reachability in graphs of bounded out-degree via path-encoding.
• Induction for $\Sigma_0^B$ formulas (derivable in VL): Reflects the necessity of reasoning about numbers in bounded ways (0801.4105).

This axiomatization ensures that every function or predicate that is provably total or defined in VL corresponds exactly to a log-space computable function or relation (Buss et al., 15 Nov 2025).

2. Proof-Theoretic Characterizations and Propositional Translations

VL's strength is precisely characterized via its connection to propositional proof systems. Every theorem of VL yields propositional tautologies with polynomial-size proofs in the quantified propositional system GL*. The translation uses:

• Cut-variable normal form (CVNF): Ensures each non-parameter variable appears as an eigenvariable exactly once, with additional constraints on cut formulas to control variable dependencies and maintain propositional correspondence.
• ΣCNF(2) formulas: Quantified-propositional formulas (quantifier blocks over CNF formulas) adhering to a "2-occurrence" restriction on quantified variables, ensuring evaluation in log-space and L-completeness under AC$^0$ reductions.

The translation of $\Sigma_1^B$ VL-formulas to families of ΣCNF(2) tautologies proceeds by propositionalization of the two-sorted input, yielding for each instance a quantified-propositional formula whose validity exactly mirrors the truth of the original bounded arithmetic formula in the standard model.

3. The GL* Proof System and Reflection Principle

GL* is a tree-like sequent calculus based on Gentzen's G$_1^*$ with the following restrictions:

• Cuts only on ΣCNF(2) formulas.
• Free variables in cut formulas must be parameter variables (appear free in the end-sequent if not quantifier-free).

Sequents in GL* manipulate multisets of quantified-propositional formulas, and every tree-like GL* proof may be put in free-variable normal form. VL proves the Σ$_1^q$-reflection schema for GL*: for any GL* proof of a Σ$_1^q$ sequent, the end-sequent is true for every assignment. This reflection relies critically on the ability to evaluate ΣCNF(2) formulas and to extract witnesses for existential quantifiers using log-space algorithms definable within VL (0801.4105).

4. Computational Correspondence: Log-space, Symmetric Log-space, and Reingold’s Theorem

VL captures exactly the class L (deterministic log-space):

• The Δ$_1^B$-definable predicates of VL are precisely those in L.
• Σ$_1^B$-definable functions in VL are exactly the log-space computable functions (Buss et al., 15 Nov 2025).

The theory VSL, extending VL by an axiom for undirected reachability (symmetric graphs), characterizes symmetric log-space (SL). However, VL alone suffices to formalize Reingold's theorem (SL=L), as shown by a detailed formalization using Rozenman–Vadhan's combinatorial approach to expansion and derandomized squaring. Every Π$_1$-theorem of VL is a theorem of VSL and vice versa. This demonstrates the full expressive and deductive equivalence of VL and VSL for log-space reasoning—including undirected s-t connectivity—resolving an open question in the field (Buss et al., 15 Nov 2025).

5. Graph-Theoretic Formalizations and Combinatorics in VL

VL leverages a structured methodology for formalizing graph-theoretic diffusion and expansion arguments:

• Representations of $d$-regular graphs, their rotation maps, neighbor functions, and path codes are all definable via Σ$_0^B$ formulas and comprehension.
• Edge expansion and mixing ratio are encoded using string-encoded rational vectors, fully formalizable in VL.
• Key combinatorial lemmas—such as Mihail's direction of Cheeger’s inequality for regular graphs with sufficient self-loops—are formalizable in VL.
• Rozenman–Vadhan’s derandomized squaring is realized via Σ$_0^B$-definable routines, with all matrix and vector-norm calculations reducible to bounded arithmetic.

This structure allows, within VL, the iterative construction of expander families that support the translation of undirected reachability into log-space-definable routines, confirming L=SL in the bounded arithmetic setting (Buss et al., 15 Nov 2025).

6. Theoretical Significance and Connections to Propositional Proof Complexity

VL's alignment with the propositional proof system GL* establishes a deep connection between logical theories, proof complexity, and complexity classes:

• Every Σ$_1^B$-theorem of VL translates to a family of quantified-propositional tautologies with polynomial-size tree-like GL* proofs.
• The restriction of cuts in GL* to ΣCNF(2) ensures that the reasoning remains within log-space, preventing overstrengthening.
• Soundness is formalisable entirely within VL: all inferences carried out in GL* can be witnessed by log-space algorithms and formalized by the Δ$_1^B$ predicates of VL.
• The correspondence VL ⇄ GL* provides a non-uniform analog to the uniform (arithmetic) interpretation of L and cement the role of bounded arithmetic proof theory in complexity theory (0801.4105).

VL thus serves as a canonical formal system for log-space reasoning, providing uniform and non-uniform characterizations through deep connections to both logical definability and propositional proof complexity. Its recent characterization of SL=L formalizes and consolidates its foundational role in the bounded arithmetic landscape (Buss et al., 15 Nov 2025).