It is not dfficult to point out the weaknesses of neural nets and deep learning. Simply put, neural nets are too weak to support general AI. They receive inputs (numbers), perform simple arithmetic operations and produce outputs (numbers). Consequently, they provide only primitive services such as object classifications. Although object classification has some interesting applications, the power of classification is in fact not much compared to all the complex services a human can provide. Complex services - making a coffee, withdrawing money from ATM, etc - are not well supported by neural nets.

The design and implementation of multi-agent systems is recognized as a key component of general AI. Implementing the Starbucks in AI is such an example. Yet it remains the case that researchers experience difficulties in this regard. Computability logic (CoL) [2]-[6], is an elegant theory of (multi-)agent computability. In CoL, computational problems are seen as games between a machine and its environment and logical operators stand for operations on games. It understands interaction among agents in its most general -- game-based -- sense. In this paper, we discuss a web-based implementation of multi-agent programming based on CL2[4]. We assume the following in our model: - Each agent correspondsto aweb site with a URL. An agent's resourcebase(RB) is described in its homepage.

In CoL, computational problems are seen as games between a machine and its environment and logical operators stand for operations on games. It understands interaction among agents in its most general -- game-based -- sense. On the other hand, other formalisms such as situation calculus appear to be too rudimentary to represent complex interactions among agents. In particular, CoL supports query/knowledge duality (or we call it'querying knowledge'): what is a query for one agent becomes new knowledge for another agent. This duality leads to dynamic knowledge migration from one agent to another agent.

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.