The first theme of this thesis investigates the complexity class CC and its associated bounded-arithmetic theory. Subramanian defined CC as the class of problems log-space reducible to the comparator circuit value problem (Ccv). Using the Cook-Nguyen method we define the two-sorted theory VCC whose provably-total functions are exactly the CC functions. To apply this method, we show CC is the same as the class of problems computed by uniform AC 0 circuits with unbounded Ccv oracle gates. We prove that VNL ⊆ VCC ⊆ VP, where VNL and VP are theories for the classes NL and P respectively. We strengthen Subramanian’s work by showing that the problems in his paper are indeed complete for CC under many-one AC 0 reductions. We then prove the correctness of these reductions in VCC. The second theme of this thesis is formalizing probabilistic proofs in bounded arithmetic. In a series of papers, Jerábek argued that the universal polynomial-time theory VPV augmented with the surjective weak pigeonhole principle sWPHP(LFP) for all VPV functions is the ‘right’ theory for randomized polynomial-time reasoning in bounded arithmetic. Motivated from the fact that no one had used Jeˇrábek’s framework for feasible reasoning about specific interesting randomized algorithms in classes such as RP and RNC 2, we formalize in VPV the correctness of two fundamental RNC 2 algorithms for testing if a bipartite graph has a perfect matching

We show that the well-known König’s Min-Max Theorem (KMM), a fundamental result in combinatorial matrix theory, can be proven in the first order theory LA with induction formulas. This is an improvement over the restricted to Σ B 1 standard textbook proof of KMM which requires Π B 2 induction, and hence does not yield feasible proofs — while our new approach does. LA is a weak theory that essentially captures the ring properties of matrices; however, equipped with Σ B 1 induction LA is capable of proving KMM, and a host of other combinatorial properties such as Menger’s, Hall’s and Dilworth’s Theorems. Therefore, our result formalizes Min-Max type of reasoning within a feasible framework.

Proof complexity is a research area that studies the concept of complexity from the point of view of logic. In proof complexity, an important question is: “how difficult is ittoproveatheorem?”Therearevariousways that one can measure the complexity of a theorem. We can ask what is the length of the shortest proof of a theorem in a given formal system (size of the proofs) or how strong a theory is needed to prove the theorem (that is, how complex are the concepts involved in the proof). Theformerisstudiedinthecontextofproof systems (in particular, propositional proof systems), the latter in bounded arithmetic. Naturally, the length of a shortest proof of a theorem very much depends on the type of proof system in which it is being proved. For a proof system, we also would liketoknowifthereisanefficientalgorithmthat would produce a proof of any tautology, and whether it would produce a shortest such proof. These questions, besides their mathematical and philosophical significance, havepracticalapplicationsinautomatedtheorem proving. From the computational point of view, the question of proving tautologiesisaco-NPquestion: thatis, acounterexampletoaformulawhichisnotatautologywouldbeshortandeasilyverifiable. Moreover, it is known that the existence of a propositional proof system inwhichalltautologieshaveshortproofsis equivalent to proving that NP is closed under complementation. This establishes an important link between

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