Computational Completeness of Tissue P Systems with Conditional Uniport

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The Proof Is in the Process: A Preamble for a Philosophy of Computer-Assisted Mathematics

According to some well-known mathematicians well-versed in computer-assisted mathematics (CaM), “Computers are changing the way we are doing mathematics”. To what extent this is really true is still an open question. Indeed, even though some philosophers of math have taken up the challenge to think about CaM, it is unclear in what sense exactly a machine (can) affect(s) the so-called “queen of the sciences”. In fact, some have concluded that issues raised by the use of the computer in mathematics are not specific to the use of the computer per se. However, such findings seem precarious since a systematic study of computer-assisted mathematics is still lacking. In this paper I argue that in order to understand the impact of CaM, it is necessary to take more seriously the computer itself and how it is actually used in the process of doing mathematics. Within such an approach, one searches for characteristics that are specific to the use of the computer in mathematics. I will focus on a feature that is beyond any doubt inherently connected to the use of computing machinery, viz. mathematician-computer interactions. I will show how such interactions are fundamentally different from the usual interactions between mathematicians and non-human aids (a piece of paper, a blackboard etc) and how such interactions determine at least two more characteristics of CaM, viz. the significance of time and processes and the steady process of internalization of mathematical tools and knowledge into the machine. I will restrict myself to the use of the computer within so-called experimental mathematics since this is the main object of CaM within the philosophical literature

From Turing machines to computer viruses

International audienceSelf-replication is one of the fundamental aspects of computing where a program or a system may duplicate, evolve and mutate. Our point of view is that Kleene's (second) recursion theorem is essential to understand self-replication mechanisms. An interesting example of self-replication codes is given by computer viruses. This was initially explained in the seminal works of Cohen and of Adleman in the 1980s. In fact, the different variants of recursion theorems provide and explain constructions of self-replicating codes and, as a result, of various classes of malware. None of the results are new from the point of view of computability theory. We now propose a self-modifying register machine as a model of computation in which we can effectively deal with the self-reproduction and in which new offsprings can be activated as independent organisms

Theoretical Computer Science: Computability, Decidability and Logic

International audienceThis chapter deals with a question in the very core of IA: what can be computed by a machine? An agreement has been reached on the answer brought by Alan Turing in 1936. Indeed, all other proposed approaches have led to exactly the same answer. Thus, there is a mathematical model of what can be done by a machine. And this has allowed to prove surprising results which feed the reflection on intelligence and machines