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**1 - 3**of**3**### Time Cut-off and the Halting Problem

, 2010

"... Abstract. This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts suppor ..."

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Abstract. This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view. In this second part, I address some of the issues raised in [Ma3] and provide their development in three contexts: a categorification of the algorithmic computations; time cut–off and Anytime Algorithms; and finally, a Hopf algebra renormalization of the Halting Problem.

### DYSON–SCHWINGER EQUATIONS IN THE THEORY OF COMPUTATION

"... Abstract. Following Manin’s approach to renormalization in the theory of computation, we investigate Dyson–Schwinger equations on Hopf algebras, operads and properads of flow charts, as a way of encoding self-similarity structures in the theory of algorithms computing primitive and partial recursive ..."

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Abstract. Following Manin’s approach to renormalization in the theory of computation, we investigate Dyson–Schwinger equations on Hopf algebras, operads and properads of flow charts, as a way of encoding self-similarity structures in the theory of algorithms computing primitive and partial recursive functions and in the halting problem. 1.

### MOTIVATION AND BACKGROUND

, 904

"... Abstract. The main observable quantities in Quantum Field Theory, correlation functions, are expressed by the celebrated Feynman path integrals which are not well defined mathematical objects. Perturbation formalism interprets such an integral as a formal series of finite– dimensional but divergent ..."

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Abstract. The main observable quantities in Quantum Field Theory, correlation functions, are expressed by the celebrated Feynman path integrals which are not well defined mathematical objects. Perturbation formalism interprets such an integral as a formal series of finite– dimensional but divergent integrals, indexed by Feynman graphs, the list of which is determined by the Lagrangian of the theory. Renormalization is a prescription that allows one to systematically “subtract infinities ” from these divergent terms producing an asymptotic series for quantum correlation functions. On the other hand, graphs treated as “flowcharts”, also form a combinatorial skeleton of the abstract computation theory and various operadic formalisms in abstract algebra. In this role of descriptions of various (classes of) computable functions, such as recursive functions, functions computable by a Turing machine with oracles etc., graphs can be used to replace standard formalisms having linguistic flavor, such as Church’s λ–calculus and various programming languages. The functions in question are generally not everywhere defined due to potentially infinite loops and/or necessity to search in an infinite haystack for a needle which is not there. In this paper I argue that such infinities in classical computation theory can be addressed in the same way as Feynman divergences, and that meaningful versions of renormalization in this context can be devised. Connections with quantum computation are also touched upon.