The theory of computability was launched in the 1930s, by a group of logicians who proposed new characterizations of the ancient idea of an algorithmic process. The most prominent of these iconoclasts were Kurt Gödel, Alonzo Church, and Alan Turing. The theoretical and philosophical work that they carried out in the 1930s laid the foundations for the computer revolution, and this revolution in turn fueled the fantastic expansion of scientific knowledge in the late 20th and early 21st centuries. Thanks in large part to these groundbreaking logico-mathematical investigations, unimagined number-crunching power was soon boosting all fields of scientific enquiry. The motivation of these three revolutionary thinkers was not to pioneer the disciplines now known as theoretical and applied computer science, although with hindsight this is indeed what they did.

The present article introduces ptarithmetic (short for "polynomial time arithmetic") -- a formal number theory similar to the well known Peano arithmetic, but based on the recently born computability logic (see http://www.cis.upenn.edu/~giorgi/cl.html) instead of classical logic. The formulas of ptarithmetic represent interactive computational problems rather than just true/false statements, and their "truth" is understood as existence of a polynomial time solution. The system of ptarithmetic elaborated in this article is shown to be sound and complete. Sound in the sense that every theorem T of the system represents an interactive number-theoretic computational problem with a polynomial time solution and, furthermore, such a solution can be effectively extracted from a proof of T. And complete in the sense that every interactive number-theoretic problem with a polynomial time solution is represented by some theorem T of the system. The paper is self-contained, and can be read without any previous familiarity with computability logic.

Have you ever wondered: What exactly is the device that you are reading this article on? Computational science dates back to a time long before these modern computing devices were even imagined. In an industry where the more frequently asked questions revolve around programming languages, frameworks, and libraries, we often taken for granted the fundamental concepts that make a computer tick. But these computers, which seem to possess endless potential--do they have any limitations? Are there problems that computers cannot be used to solve? In this article, we will address these questions by stepping away from the particulars of programming languages and computer architectures. By understanding the power and limitations of computers and algorithms, we can improve the way we think and better reason about different strategies. The abstract view of computing produces results that have stood the test of time, being as valuable to us today as they were when initially developed in the 1970s.

Computability logic (CoL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a recently introduced semantical platform and ambitious program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Its expressions represent interactive computational tasks seen as games played by a machine against the environment, and "truth" is understood as existence of an algorithmic winning strategy. With logical operators standing for operations on games, the formalism of CoL is open-ended, and has already undergone series of extensions. This article extends the expressive power of CoL in a qualitatively new way, generalizing formulas (to which the earlier languages of CoL were limited) to circuit-style structures termed cirquents. The latter, unlike formulas, are able to account for subgame/subtask sharing between different parts of the overall game/task. Among the many advantages offered by this ability is that it allows us to capture, refine and generalize the well known independence-friendly logic which, after the present leap forward, naturally becomes a conservative fragment of CoL, just as classical logic had been known to be a conservative fragment of the formula-based version of CoL. Technically, this paper is self-contained, and can be read without any prior familiarity with CoL.