Computing Over Generalisations of the Reals Transfinite - PowerPoint PPT Presentation

Computing Over Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Spaces Bristol, 4 February 2020 Computing Over Generalisations of the Reals Transfinite Computability Surreal Numbers Lorenzo Galeotti Generalised Real 5th Workshop on Generalised Baire Spaces Line Bristol, 4 February 2020 Generalised Type 2 Computability Surreal Blum-Shub-Smale machines

Computing Over The Real Line Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Spaces Bristol, 4 February The real line has a central role both in mathematics and in 2020 computability theory. Transfinite Computability Surreal Numbers Because of the complex topological and combinatorial Generalised Real structure of R often other better behaved spaces such as the Line Cantor space 2 ω are used in order to study properties of R . Generalised Type 2 Computability Surreal Blum-Shub-Smale Transfer machines � 2 ω R �

Computing Over Computations Over the Reals via Tapes Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Cantor space is used to induce notions of computability over Spaces Bristol, 4 February spaces of size 2 ω . 2020 Transfinite Computability Surreal Numbers Generalised Real Line Generalised Type 2 Computability Surreal Blum-Shub-Smale machines

Computing Over Computations Over the Reals via Tapes Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Cantor space is used to induce notions of computability over Spaces Bristol, 4 February spaces of size 2 ω . 2020 Transfinite Computability Type 2 Turing Machines (T2TM) are a model of Surreal Numbers computation whose hardware is essentially the same of that Generalised Real of classical Turing machines. Contrary to classical Turing Line Machines T2TM are allowed to run for infinite ( ω ) many Generalised Type 2 Computability steps. The result of the computation is then taken to be the Surreal limit of the content of the output tape of the T2TM. Blum-Shub-Smale machines

Computing Over Computations Over the Reals via Tapes Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Cantor space is used to induce notions of computability over Spaces Bristol, 4 February spaces of size 2 ω . 2020 Transfinite Computability Type 2 Turing Machines (T2TM) are a model of Surreal Numbers computation whose hardware is essentially the same of that Generalised Real of classical Turing machines. Contrary to classical Turing Line Machines T2TM are allowed to run for infinite ( ω ) many Generalised Type 2 Computability steps. The result of the computation is then taken to be the Surreal limit of the content of the output tape of the T2TM. Blum-Shub-Smale machines These machines induce a notion of computability over Cantor space. Using coding functions one can then transfer this notion to other spaces, e.g., the real line.

Computing Over Computations Over the Reals via Registers Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire A different approach to computability over the reals is that Spaces Bristol, 4 February of register machines. 2020 Transfinite Computability Surreal Numbers Generalised Real Line Generalised Type 2 Computability Surreal Blum-Shub-Smale machines

Computing Over Computations Over the Reals via Registers Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire A different approach to computability over the reals is that Spaces Bristol, 4 February of register machines. 2020 Transfinite Computability Blum-Shub-Smale machines (BSS machines) work on Surreal Numbers registers. Each register contains a real number. The Generalised Real machine is only allowed to run for a finite amount of time. Line At each step the machine can: Generalised Type 2 Computability ◮ test the content of a register and perform a jump based Surreal Blum-Shub-Smale on the result; machines ◮ apply a rational function to registers.

Computing Over Computations Over the Reals via Registers Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire A different approach to computability over the reals is that Spaces Bristol, 4 February of register machines. 2020 Transfinite Computability Blum-Shub-Smale machines (BSS machines) work on Surreal Numbers registers. Each register contains a real number. The Generalised Real machine is only allowed to run for a finite amount of time. Line At each step the machine can: Generalised Type 2 Computability ◮ test the content of a register and perform a jump based Surreal Blum-Shub-Smale on the result; machines ◮ apply a rational function to registers. In this approach no coding is needed and the notion of computability is unique.

Computing Over Computability, Space and Time Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Spaces In classical computability theory computations are thought Bristol, 4 February 2020 as finite and discrete processes carried out by (idealised) Transfinite machines with unbounded finite data. Computability Surreal Numbers Generalised Real Line Generalised Type 2 Computability Surreal Blum-Shub-Smale machines

Computing Over Computability, Space and Time Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Spaces In classical computability theory computations are thought Bristol, 4 February 2020 as finite and discrete processes carried out by (idealised) Transfinite machines with unbounded finite data. Computability Surreal Numbers In defining notions of computability over the reals the Generalised Real Line assumptions on space and time are relaxed. Generalised Type 2 Computability Surreal Model Basic Data Space Time Blum-Shub-Smale machines T2TM 2 ω ω ω BSS ω finite R

Computing Over Classical Transfinite Computability Generalisations of the Reals Lorenzo Galeotti The idea of transfinite computability is to allow 5th Workshop on Generalised Baire computations to “go on forever”, i.e., for a transfinite Spaces Bristol, 4 February amount of time. 2020 Transfinite Computability Surreal Numbers Generalised Real Line Generalised Type 2 Computability Surreal Blum-Shub-Smale machines

Computing Over Classical Transfinite Computability Generalisations of the Reals Lorenzo Galeotti The idea of transfinite computability is to allow 5th Workshop on Generalised Baire computations to “go on forever”, i.e., for a transfinite Spaces Bristol, 4 February amount of time. 2020 ◮ Infinite Time Turing Machines (ITTM) : Introduced Transfinite Computability by Hamkins and Lewis, have the same hardware of Surreal Numbers normal Turing machine but are allowed to carry out a Generalised Real transfinite number of steps. Line Generalised Type 2 ◮ Ordinal Turing Machines (OTM) : Introduced by Computability Koepke, these machines have tapes of transfinite length Surreal Blum-Shub-Smale and are allowed to run for a transfinite number of steps. machines Model Space Time ITTM ω transfinite OTM transfinite transfinite

Computing Over Generalising Computability: The Problem Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Spaces How can the classical computability over the reals be Bristol, 4 February 2020 generalised? Transfinite Computability Surreal Numbers Generalised Real Line Generalised Type 2 Computability Surreal Blum-Shub-Smale machines

Computing Over Generalising Computability: The Problem Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Spaces How can the classical computability over the reals be Bristol, 4 February 2020 generalised? Transfinite Computability What is the right generalisation of R in this context? Surreal Numbers Generalised Real Line Generalised Type 2 Computability Surreal Blum-Shub-Smale machines

Computing Over Generalising Computability: The Problem Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Generalised Baire Spaces How can the classical computability over the reals be Bristol, 4 February 2020 generalised? Transfinite Computability What is the right generalisation of R in this context? Surreal Numbers Generalised Real In the past few years there were essentially two Line generalisations of R in the context of generalised descriptive Generalised Type 2 Computability set theory. Surreal ◮ The long reals κ - R : invented by Sikorski in the 60s and Blum-Shub-Smale machines recently studied by Asper´ o and Tsaprounis. ◮ The generalised reals R κ : used in my Master’s thesis in the context of generalised DST and generalised computable analysis.

Computing Over Surreal numbers Generalisations of the Reals Lorenzo Galeotti 5th Workshop on Surreal numbers were introduced by Conway in order to Generalised Baire Spaces formalise the abstract notion of number : Bristol, 4 February 2020 Transfinite Definition (Surreal numbers) Computability Surreal Numbers A surreal number is a function from an ordinal α ∈ On to Generalised Real { + , −} (i.e., a sequence of pluses and minuses of ordinal Line length). Generalised Type 2 Computability ◮ We will denote the class of surreal numbers by No. Surreal Blum-Shub-Smale machines ◮ Given an ordinal α we denote the set of surreals of length <α by No <α . ◮ Surreal numbers are naturally ordered using the lexicographic order.