Arithmetics within the Linear Time Hierarchy

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, $\Sigma^{\mathsf b}_{i}$, 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: $\breve{S}^{i}_{1}$, $TLS^i_1$ and $TSC^i_1$. $\breve{S}^{i}_{1}$ 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 $\Sigma^{\mathsf b}_{i}$. We prove for $i \geq 1$ that $$TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\}})} TLS^{i+1}_1$$ 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 $\Sigma^{\mathsf b}_{i}$ 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.