Build your own clarithmetic I: Setup and completeness

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

Build your own clarithmetic II: Soundness

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 previous Part I has introduced the system and proved its completeness, while the present Part II is devoted to proving soundness

On the system CL12 of computability logic

Computability logic (see http://www.csc.villanova.edu/~japaridz/CL/) is along-term project for redeveloping logic on the basis of a constructive gamesemantics, with games seen as abstract models of interactive computationalproblems. Among the fragments of this logic successfully axiomatized so far isCL12 --- a conservative extension of classical first-order logic, whoselanguage augments that of classical logic with the so called choice sorts ofquantifiers and connectives. This system has already found fruitfulapplications as a logical basis for constructive and complexity-orientedversions of Peano arithmetic, such as arithmetics for polynomial timecomputability, polynomial space computability, and beyond. The present paperintroduces a third, indispensable complexity measure for interactivecomputations termed amplitude complexity, and establishes the adequacy of CL12with respect to A-amplitude, S-space and T-time computability under certainminimal conditions on the triples (A,S,T) of function classes. This result verysubstantially broadens the potential application areas of CL12. The paper isself-contained, and targets readers with no prior familiarity with the subject.Comment: arXiv admin note: substantial text overlap with arXiv:1003.0425 and arXiv:1003.471