Solution of the Problem P = L

Abstract

The problems associated with the construction of polynomial complexity computer programs require new techniques and approaches from mathematicians. One of such approaches is representing some class of polynomial algorithms as a certain class of special logical programs. Goncharov and Sviridenko described a logical programming language L0, where programs inductively are obtained from the set of Δ0-formulas using special terms. In their work, a new idea has been proposed to look at the term as a program. The computational complexity of such programs is polynomial. In the same years, a number of other logical languages with similar properties were created. However, the following question remained: can all polynomial algorithms be described in these languages? It is a long-standing problem, and the method of describing some polynomial algorithm in a not Turing complete logical programming language was not previously clear. In this paper, special types of terms and formulas have been found and added to solve this problem. One of the main contributions is the construction of p-iterative terms that simulate the work of the Turing machine. Using p-iterative terms, the work showed that class P is equal to class L, which extends the programming language L0 with p-iterative terms. Thus, it is shown that L is quite expressive and has no halting problem, which occurs in high-level programming languages. For these reasons, the logical language L can be used to create fast and reliable programs. The main limitation of the language L is that the implementation of algorithms of complexity is not higher than polynomial.

Задачный Подход: На Пути К Доверительному Искусственному Интеллекту

“…С другой стороны, наличие процедурного слоя позволяет реализовать идею развития онтологии, управления её жизненным циклом. С этой точки зрения применение каждого метода или группы методов можно рассматривать как транзакцию, переводящую онтологию из одного состояния в другое [57] [62], что несомненно является плюсом, так как не требует никаких внешних программных средств для их реализации.…”

“…Due to the fact that smart contracts are programs of a special type in the p-complete language L*, this significantly reduces the level of labor costs when auditing them. In the language L* itself, it is possible to describe the operation of both neural networks [61] and blockchains [62], which is undoubtedly an advantage, since it does not require any external software for their implementation.…”

Всемирный Конгресс

“…В силу того, что смарт-контракты являются программами специального вида в p-полном языке L*, то это заметно снижает уровень трудозатрат при их аудите. В самом же языке L* можно описывать работу как нейронных сетей [61], так и блокчейнов [62], что несомненно является плюсом, так как не требует никаких внешних программных средств для их реализации.…”

Axiomatization of Blockchain Theory

“…The need for transparent and humanized AI algorithms is paramount to gaining trust in artificial intelligence. The focus now is on explainable artificial intelligence (XAI) [13] using hybrid methods with neural networks and high-level logic programming languages (such as L [14] and L * [15]) that provide clear execution logic, to ensure accurate results and logical explanations. To protect these programs from code alterations and breaches, blockchain structures [16] secured through consensus algorithms are required.…”