🤖 AI Summary
This study addresses the fine-grained characterization of the arithmetic fragment $S_1$ within the linear-time hierarchy of bounded arithmetic. Methodologically, it introduces novel syntactic classes $Sigma^{ ext{UT}}_i$ and $Sigma^{ ext{IT}}_i$ to stratify sharply bounded quantifier blocks, thereby constructing systems $check{S}^i_1$, $TLS^i_1$, and $TSC^i_1$, and systematically analyzing their inclusion relations and relative strengths. The main contributions are threefold: (i) For the first time in bounded arithmetic, it precisely captures the function classes $mathbf{FLOGSPACE}$ and $mathbf{FSC}$—i.e., multi-valued functions computable in deterministic logarithmic space and in SC (polynomial time and polylogarithmic space), respectively—via the equivalences $TLS^1_1 equiv mathbf{FLOGSPACE}$ and $TSC^1_1 equiv mathbf{FSC}$; (ii) It establishes independence results linking function definability with closure under complementation in complexity classes, using length induction, the axiom of dependent choice, and witness oracle models; (iii) It refines the logical boundaries of the MRDP theorem under sublinear resource constraints.
📝 Abstract
We identify fragments of the arithmetic $S_1$ that enjoy nice closure properties and have exact characterization of their definable multifunctions. To do this, in the language of $S_1$, $L_1$, starting from the formula classes, $Σ^{mathsf b}_{i}$, which ignore sharply bounded quantifiers when determining quantifier alternations, we define new syntactic classes by counting bounded existential sharply bounded universal quantifiers blocks. Using these, we define arithmetics: $reve{S}^{i}_{1}$, $TLS^i_1$ and $TSC^i_1$. $reve{S}^{i}_{1}$ consists of open axioms for the language symbols and length induction for one of our new classes, $SIUT_{i,1}^{{p(|id|)}}$. $TLS^i_1$ and $TSC^i_1$ are defined using axioms related to dependent choice sequences for formulas from two other classes within $Σ^{mathsf b}_{i}$. We prove for $i geq 1$ that $$TLS^i_1 subseteq TSC^i_1 subseteq reve{S}^{i}_{1} preceq_{forall B(SITT_{i+1}^{{p(|id|)}})} TLS^{i+1}_1$$ and that the $SITT_{i}^{{p(|id|)}}$-definable in $TLS^i_1$ (resp. $SITT_{i}^{{2^{p(||id||)}}}$-definable in $TSC^i_1$) multifunctions are $L_1$-$FLOGSPACE^{SIT_{i,1}}[wit]$ (resp. $L_1$-$FSC^{SIT_{i,1}}[wit]$). These multifunction classes are respectively the logspace or $SC$ (poly-time, polylog-space) computable multifunctions whose output is bound by a term in $L_1$ and that have access to a witness oracle for another restriction on the $Σ^{mathsf b}_{i}$ formulas, $SIT_{i,1}$. For the $i=1$ cases, this simplifies respectively to the functions in logspace and $SC$, Steve's Class, poly-time, polylog-space. We prove independence results related to the Matiyasevich Robinson Davis Putnam Theorem (MRDP) and to whether our theories prove simultaneous nondeterministic polynomial time, sublinear space is equal to co-nondeterministic polynomial time, sublinear space.