TLS^i_1: Bounded-Arithmetic & Logspace

• TLS^i_1 is a bounded-arithmetic theory in L_1 that focuses on logspace computation using dependent choice sequences and iteration principles.
• It introduces strict syntactic hierarchies and axiom schemes to exactly characterize definable multifunctions in complexity classes like FLOGSPACE and FSC.
• The theory isolates logspace and SC-type computations, providing insights into bounded arithmetic’s structure and related independence phenomena.

$TLS^i_1$ is a family of bounded-arithmetic theories in the language $L_1$ introduced in the study of fragments of $S_1$ with exact characterizations of definable multifunctions and strong closure properties within the linear time hierarchy (Pollett, 15 Aug 2025). For $i \ge 1$, the theory is defined over $LIOpen_1$ by adding axioms related to dependent choice sequences for formulas from syntactic classes constructed inside $\Sigma^{\mathsf b}_i$, with the resulting hierarchy satisfying

$TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$

The theory is designed so that its $SITT_{i}^{\{p(|id|)\}}$-definable multifunctions coincide with $L_1$-$FLOGSPACE^{SIT_{i,1}[wit]}$, and in the case $L_1$0 this simplifies to the functions in logspace (Pollett, 15 Aug 2025).

1. Language and bounded-arithmetic setting

The underlying language is $L_1$1, a conservative extension of $L_1$2, with non-logical symbols

$L_1$3

Their intended meanings are the successor operation, truncated subtraction, binary length, padding by powers of two, and most-significant-part extraction. The base open theory $L_1$4 consists of quantifier-free axioms specifying these symbols, and $L_1$5 is defined as $L_1$6 together with length induction for open formulas.

The framework starts from the usual bounded formula classes $L_1$7, which ignore sharply bounded quantifiers when determining quantifier alternations. A $L_1$8-formula is an $L_1$9-formula whose innermost quantifier is sharply bounded, and the corresponding closure classes are obtained under subformulas, boolean connectives, and sharply bounded quantification. This is the standard bounded-arithmetic environment in which the new theories are introduced.

A central issue is that the language $S_1$0 lacks exponentiation and $S_1$1, so the customary decomposition familiar from $S_1$2 does not directly yield fragments with clean complexity-theoretic interpretations. The theories $S_1$3, $S_1$4, and $S_1$5 are introduced precisely to isolate fragments whose definable multifunctions admit exact descriptions.

2. Syntactic classes behind $S_1$6

The paper defines new syntactic classes by counting bounded existential sharply bounded universal quantifier blocks. Their construction proceeds from special formulas that encode bounded iterative computation.

A formula $S_1$7 is $S_1$8-steppable if it has the disjunctive form

$S_1$9

with the $i \ge 1$0 drawn from an earlier stage of the hierarchy. This expresses a unique next-configuration relation.

Iteration is then encoded by formulas of the form

$i \ge 1$1

where a bounded witness $i \ge 1$2 codes a sequence of configurations and a sharply bounded universal quantifier verifies that each transition satisfies the relevant step predicate. From these building blocks the paper defines hierarchies such as $i \ge 1$3, $i \ge 1$4, $i \ge 1$5, and $i \ge 1$6, together with their query-iterable analogues.

Using these constructions, the families $i \ge 1$7, $i \ge 1$8, and $i \ge 1$9 are defined recursively from boolean combinations of $LIOpen_1$0-formulas. The growth-bounded variants $LIOpen_1$1 and $LIOpen_1$2 restrict the functions $LIOpen_1$3 appearing in iterations to a specified set $LIOpen_1$4. For $LIOpen_1$5, the relevant bound set is $LIOpen_1$6; for $LIOpen_1$7, it is $LIOpen_1$8 (Pollett, 15 Aug 2025).

3. Definition of $LIOpen_1$9

The theory $\Sigma^{\mathsf b}_i$0 is defined by adjoining an iteration axiom scheme to $\Sigma^{\mathsf b}_i$1. In the general $\Sigma^{\mathsf b}_i$2 setting,

$\Sigma^{\mathsf b}_i$3

and therefore

$\Sigma^{\mathsf b}_i$4

The $\Sigma^{\mathsf b}_i$5 axioms are the $\Sigma^{\mathsf b}_i$6 formulas with trivial accept state. Concretely, they assert the existence of bounded dependent-choice sequences of configurations generated by the iteration formulas described above. The restriction to $\Sigma^{\mathsf b}_i$7 is what makes $\Sigma^{\mathsf b}_i$8 the logspace-oriented member of the hierarchy.

This can be contrasted with the companion theories.

Theory	Definition
$\Sigma^{\mathsf b}_i$9	open axioms for the language symbols and length induction for $TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$0
$TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$1	$TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$2
$TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$3	$TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$4

A plausible implication is 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.$5 replaces a full induction principle by an iteration principle tailored to bounded configuration sequences, thereby isolating the logspace-type content of the corresponding fragment.

4. Structural position in the hierarchy

For $TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$6, the theories satisfy the inclusion and conservativity chain

$TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$7

The first inclusion reflects the fact that the $TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$8 bounds used in $TLS^i_1 \subseteq TSC^i_1 \subseteq \breve{S}^{i}_{1} \preceq_{\forall B(SITT_{i+1}^{\{p(|id|)\})} TLS^{i+1}_1.$9 are stricter than the $SITT_{i}^{\{p(|id|)\}}$0 bounds used in $SITT_{i}^{\{p(|id|)\}}$1. The second inclusion places both iteration-based theories below the induction-based fragment $SITT_{i}^{\{p(|id|)\}}$2.

The final relation is a conservativity statement over universal formulas whose matrices are boolean combinations of $SITT_{i}^{\{p(|id|)\}}$3-formulas. This shows that the next level $SITT_{i}^{\{p(|id|)\}}$4 already captures the universal content of $SITT_{i}^{\{p(|id|)\}}$5 over that fragment.

This hierarchy is intended to mirror a complexity-theoretic stratification. $SITT_{i}^{\{p(|id|)\}}$6 is the logspace-oriented theory, $SITT_{i}^{\{p(|id|)\}}$7 is the $SITT_{i}^{\{p(|id|)\}}$8-oriented theory, and $SITT_{i}^{\{p(|id|)\}}$9 is an induction-based intermediary. The construction is motivated by the search for fragments of $L_1$0 that enjoy exact closure properties and exact function-class characterizations.

5. Definable multifunctions and complexity characterizations

The principal technical result is that $L_1$1 has an exact multifunction characterization. The $L_1$2-definable multifunctions in $L_1$3 are precisely the $L_1$4-$L_1$5 multifunctions, while the $L_1$6-definable multifunctions in $L_1$7 are the $L_1$8-$L_1$9 multifunctions (Pollett, 15 Aug 2025).

These multifunction classes are respectively the logspace or $FLOGSPACE^{SIT_{i,1}[wit]}$0 computable multifunctions whose output is bound by a term in $FLOGSPACE^{SIT_{i,1}[wit]}$1 and that have access to a witness oracle for $FLOGSPACE^{SIT_{i,1}[wit]}$2. For the $FLOGSPACE^{SIT_{i,1}[wit]}$3 cases, this simplifies respectively to the functions in logspace and $FLOGSPACE^{SIT_{i,1}[wit]}$4, Steve's Class, poly-time, polylog-space.

At the predicate level, the theory identifies the corresponding bounded-arithmetic fragment with the relevant complexity class. This suggests that $FLOGSPACE^{SIT_{i,1}[wit]}$5 plays, inside the $FLOGSPACE^{SIT_{i,1}[wit]}$6 setting, the role of a logspace theory with exact provably total multifunctions, while higher levels $FLOGSPACE^{SIT_{i,1}[wit]}$7 capture oracle-refined versions of that behavior.

A plausible implication is that the hierarchy $FLOGSPACE^{SIT_{i,1}[wit]}$8 supplies a proof-theoretic organization of logspace-style computation analogous to the role played by the more familiar bounded-arithmetic hierarchies for polynomial time.

6. Independence phenomena and proof-theoretic significance

The paper also derives independence results linked to $FLOGSPACE^{SIT_{i,1}[wit]}$9, $L_1$00, and their surrounding hierarchy. One result concerns the Matiyasevich Robinson Davis Putnam Theorem. Another concerns whether the theories prove that simultaneous nondeterministic polynomial time, sublinear space is equal to co-nondeterministic polynomial time, sublinear space.

These results are tied to the exact function-class characterizations. Because $L_1$01 and $L_1$02 are calibrated to logspace and $L_1$03-type multifunctions, a proof of certain collapse statements inside these theories would force corresponding collapses in the associated computational hierarchies. The paper proves that such consequences do not follow in general.

The broader significance of $L_1$04 is therefore twofold. First, it gives a precisely axiomatized fragment of $L_1$05 whose definable multifunctions are exactly characterized. Second, it provides a setting in which independence questions about bounded arithmetic can be related directly to statements about logspace, $L_1$06, and the linear time hierarchy. This suggests that $L_1$07 is best understood not merely as a formal subtheory of $L_1$08, but as a proof-theoretic model of logspace computation within bounded arithmetic (Pollett, 15 Aug 2025).