Clarithmetics are number theories based on computability logic (see http: www.csc.villanova.edu japaridz CL ). Formulas of these theories represent interactive computational problems, and their "truth" is understood as existence of an algorithmic solution. Various complexity constraints on such solutions induce various versions of clarithmetic. The present paper introduces a parameterized schematic version CLA11(P1,P2,P3,P4). By tuning the three parameters P1,P2,P3 in an essentially mechanical manner, one automatically obtains sound and complete theories with respect to a wide range of target tricomplexity classes, i.e. combinations of time (set by P3), space (set by P2) and so called amplitude (set by P1) complexities. Sound in the sense that every theorem T of the system represents an interactive number-theoretic computational problem with a solution from the given tricomplexity class and, furthermore, such a solution can be automatically extracted from a proof of T. And complete in the sense that every interactive number-theoretic problem with a solution from the given tricomplexity class is represented by some theorem of the system. Furthermore, through tuning the 4th parameter P4, at the cost of sacrificing recursive axiomatizability but not simplicity or elegance, the above extensional completeness can be strengthened to intensional completeness, according to which every formula representing a problem with a solution from the given tricomplexity class is a theorem of the system. This article is published in two parts. The present Part I introduces the system and proves its completeness, while Part II is devoted to proving soundness.
Preliminaries.- A.- I: Arithmetic as Number Theory, Set Theory and Logic.- II: Fragments and Combinatorics.- B.- III: Self-Reference.- IV: Models of Fragments of Arithmetic.- C.- V: Bounded Arithmetic.- Bibliographical Remarks and Further Reading.- Index of Terms.- Index of Symbols.  Abstract We define theories of bounded arithmetic, whose definable functions and relations are exactly those in certain complexity classes. Based on a recursion-theoretic characterization of NC in Clote (1988, 1990), the first-order theory TNC, whose principal axiom scheme is a form of short induction on notation for nondeterministic polynomial-time computable relations, has the property that those functions having nondeterministic polynomial-time graph Θ(x, y) such that TNC ⊢∀x ∃y Θ(x, y) are exactly the functions in NC, computable on a parallel random-access machine in polylogarithmic parallel time with a polynomial number of processors.0 We then define three theories of weak second-order arithmetic which respectively characterize relations in the classes of alternating logarithmic time, logspace and nondeterministic logspace.  Intuitionistic theories IS 2 i of Bounded Arithmetic are introduced and it is shown that the definable functions of IS 2 i are precisely the □ i p functions of the polynomial hierarchy. This is an extension of earlier work on the classical Bounded Arithmetic and was first conjectured by S. Cook. In contrast to the classical theories of Bounded Arithmetic where Σ i b -definable functions are of interest, our results for intuitionistic theories concern all the definable functions.  1. Introduction 2. Preliminaries 3. Basic complexity theory 4. Basic propositional logic 5. Basic bounded arithmetic 6. Definability of computations 7. Witnessing theorems 8. Definability and witnessing in second order theories 9. Translations of arithmetic formulas 10. Finite axiomatizability problem 11. Direct independence proofs 12. Bounds for constant-depth Frege systems 13. Bounds for Frege and extended Frege systems 14. Hard tautologies and optimal proof systems 15. Strength of bounded arithmetic References Index. “From two integers k, l one passes immediately to k l ; this process leads in a few steps to numbers which are far larger than any occurring in experience, e.g., 67 (257729) . Intuitionism, like ordinary mathematics, claims that this number can be represented by an arabic numeral. Could not one press further the criticism which intuitionism makes of existential assertions and raise the question: What does it mean to claim the existence of an arabic numeral for the foregoing number, since in practice we are not in a position to obtain it? This book treats bounded arithmetic and propositional proof complexity from the point of view of computational complexity. The first seven chapters include the necessary logical background for the material and are suitable for a graduate course. Associated with each of many complexity classes are both a two-sorted predicate calculus theory, with induction restricted to concepts in the class, and a propositional proof system. The result is a uniform treatment of many systems in the literature, including Buss's theories for the polynomial hierarchy and many disparate systems for complexity classes such as AC0, AC0(m), TC0, NC1, L, NL, NC, and P.