Bounded arithmetic, propositional logic, and complexity theory
Bounded arithmetic, propositional logic, and complexity theory
On Sharply Bounded Length Induction
CSL '95 Selected Papers from the9th International Workshop on Computer Science Logic
On Proofs about Threshold Circuits and Counting Hierarchies
LICS '98 Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science
Logical Foundations of Proof Complexity
Logical Foundations of Proof Complexity
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Conservative subtheories of $${{R}^{1}_{2}}$$ and $${{S}^{1}_{2}}$$ are presented. For $${{S}^{1}_{2}}$$ , a slight tightening of Jeřábek's result (Math Logic Q 52(6):613---624, 2006) that $${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$$ is presented: It is shown that $${T^{0}_{2}}$$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this $${\forall\Sigma^{b}_{1}}$$ -theory, we define a $${\forall\Sigma^{b}_{0}}$$ -theory, $${T^{-1}_{2}}$$ , for the $${\forall\Sigma^{b}_{0}}$$ -consequences of $${S^{1}_{2}}$$ . We show $${T^{-1}_{2}}$$ is weak by showing it cannot $${\Sigma^{b}_{0}}$$ -define division by 3. We then consider what would be the analogous $${\forall\hat\Sigma^{b}_{1}}$$ -conservative subtheory of $${R^{1}_{2}}$$ based on Pollett (Ann Pure Appl Logic 100:189---245, 1999. It is shown that this theory, $${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$$ , also cannot $${\Sigma^{b}_{0}}$$ -define division by 3. On the other hand, we show that $${{S}^{0}_{2}+open_{\{||id||\}}}$$ -COMP is a $${\forall\hat\Sigma^{b}_{1}}$$ -conservative subtheory of $${R^{1}_{2}}$$ . Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium' 98, 262---279, 2000) and show that $${\hat{C}^{0}_{2}}$$ is $${\forall\hat\Sigma^{b}_{1}}$$ -conservative over a theory based on open cl-comprehension.