1. ‘The Role of Algebraic Models and Type-2 Theory of Effectivity in Special Purpose Processor Design’ Gregorio de Miguel Casado Juan Manuel García Chamizo -Computability in Europe- July, 4th 2006 - University of Alicante - Specialized Processor Architectures Lab

2. introduction method application conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Contents introduction rresearch motivations bbackground method: “special purpose processor design for scientific computing calculations” CComputable Analysis  Type-2 Theory of Effectivity FFormal VLSI design  Algebraic Models of Processors application: “processor design for computable convolution operation in ” conclusions CIE 2006 introduction method application conclusions

3. introduction motivation background method application conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Research Motivation Specialized Processor Architectures Lab (UA)  research line: Scientific Computing objective  development of hardware support for some scientific computing tasks  integral transforms  Case of study: The convolution operation CIE 2006 background method application conclusions motivation

4. introduction motivation background method application conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers in interdisciplinary paradigm application” Scientific Computing rreliability demands in computer characterization of complex physical problems [Wei00] and [GoL01]  Computable Analysis: Type-2 Theory of Effectivity [Wei00]… VLSI design ccorrectness in specification and verification of processors [McT90] and [MöT98]  Formal Methods: Algebraic Models of Processors [HaT97], [FoH03]… Computer Arithmetic llimited hardware support for arithmetic precision management (IEEE 754) [Lyn95]…  signed-digit arithmetic [ErL04] Technology trends  hybrid chips (µP + ad-hoc hardware) [ANJ04]  memory integration improvements CIE 2006 background method application conclusions motivation

5. introduction method Type-2 Theory of Effectivity Algebraic Models sketch application conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Type-2 Theory of Effectivity Provides a coherent bridge between two classical disciplines: analysis/numerical analysis and computability/complexity theory Presents a realistic model of computation based on Type-2 machines Provides a concrete computability concept based on naming systems and realizations Allows the definition of computable functions on the set of all real numbers Allows a natural complexity theory  The representations based on signed-digit notation are feasible for developing ad-hoc hardware arithmetic support (precision criteria)  The amount of memory available limits the feasibility of representation implementation CIE 2006 introduction application conclusions Algebraic Models Type-2 Theory of Effectivity sketch

6. introduction method Type-2 Theory of Effectivity Algebraic Models sketch application conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Algebraic Models of Processors Formal paradigms for VLSI design Isolation of some fundamental scientific structural features of processor computation (behavior over time and of data representation and operation) Used for the specification and verification of computer architectures. Techniques: microprogramming, pipelined and superscalar processors Connection with verification tools such as Maude and HOL  Algebraic abstraction for complex computer architecture approaches  Realistic approach by levels: Programmer & Abstract Circuit CIE 2006 introduction application conclusions Type-2 Theory of Effectivity sketch Algebraic Models

7. introduction method Type-2 Theory of Effectivity Algebraic Models sketch application conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Sketch of the method CIE 2006 introduction application conclusions Type-2 Theory of Effectivity sketch Algebraic Models Test Scenarios TTE Algebraic Specification & Type-2 Theory of Effectivity S0-Defining the Problem S1-Formalizing the Problem Mathematical Expression Requirements & Restrictions Requirements & Restrictions S2-Analysing Computability Algorithms & Computable Representations Complexity Results S3-Specifying the Processor Processor Specification Proposal S4-Hardware Implementation S5-Evaluating and Verifying the Proposal

8. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Application “processor specification for computable convolution operation in ” Overview of the system architecture CIE 2006 problem formalization introduction method computability analysis conclusions specification data acquisition system control interface & scalability manager … general purpose processor memory system input/ output operating system ad-hoc applications & symbolic calculation environments application

9. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Formalization of the problem INPUT: informal problem description OUTPUTS MMathematical expression. Convolution between Lebesgue integrable functions in PProcessor requirements and restrictions Support for heterogeneous data sources ssymbolic calculation programs rreal world data series Support for scalability features by introducing several levels of parallelization of the calculation Support for variable precision capabilities in order to cover a wide range of precision requirements Support for calculation time restrictions and result quality management TTest scenarios CIE 2006 problem formalization introduction method computability analysis conclusions specification

10. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Computability Analysis (i) INPUTS MMathematical expression PPrecision requirements OUTPUTS TTTE-Computable convolution operation between Lebesgue integrable functions in spaces TTE-Representation for the set of rational step functions “Countable dense subset of “. Every integrable measurable function can be approximated by measurable step functions in the norm |·| and every measurable subset of can be approximated from above by open sets with respect to the Lebesgue measure [Klu04] CIE 2006 problem formalization conclusions introduction method computability analysis specification

11. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Computability Analysis (ii) OUTPUTS TTTE-Computable convolution operation between Lebesgue integrable functions in spaces TTE-Representation for the set of rational step functions normalized signed digit notation based on the v sd notation for the rational numbers [Wei00] CComplexity Analysis CIE 2006 problem formalization conclusions introduction method computability analysis specification

12. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Specification INPUTS rrequirements and restrictions aalgorithms based on TTE-computable representations OUTPUT: algebraic specification of the processor Functional specification   Algebraic specification CIE 2006 problem formalization conclusions introduction method computability analysis specification

13. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Functional Specification Modules Instruction set (Status_Request, Configuration Request, Configuration_Set, Halt, Convolution) Banks of registers (Configuration, Base-Adress, Status, Arithmetic) CIE 2006 problem formalization conclusions introduction method computability analysis specification

14. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Algebraic Specification Programmer’s level sstate and next state algebras mmachine algebra nnext state and output function Abstract circuit level pprogram memory ddata memory organization rrational step function arithmetic unit ccontrol unit sstate and next state algebras mmachine algebra nnext state and output function CIE 2006 problem formalization conclusions introduction method computability analysis specification

15. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Algebraic Specification. Data memory organization Mapping functions: p head_name, p addrF, p addrRSF, p headStep, p addrRangeStep, p addrLint, p addrHint, p addrA, p addrB, p addrCr, p addrCi, p RangeStep, p lInterval, p Hinterval, p a, p b, p Cr, p Ci Data memory mapping CIE 2006 problem formalization conclusions introduction method computability analysis specification

16. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Algebraic Specification. Data memory storage Normalized signed-digit representation CIE 2006 problem formalization conclusions introduction method computability analysis specification

17. introduction method application problem formalization computability analysis specification conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Algebraic Specification. Rational Step Function Unit CIE 2006 problem formalization conclusions introduction method computability analysis specification

18. introduction method application conclusions ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Conclusions Novel theoretical approach for designing a processor for computable scientific computing calculations TType-2 Theory of Effectivity AAlgebraic Models of Processors Case of study: Convolution between functions  TTE provides criteria about data precision management  TTE representations for rational step functions based on rational signed digit notation can be mapped into conventional memories  Algebraic models provide a suitable general framework for the specification of special purpose processors  Online arithmetic provides feasible circuit designs for the simple arithmetic operations involved in the calculation (addition, multiplication and comparison) Research in progress CComplete algebraic specification and verification outline PPrototype implementation and performance evaluation CIE 2006 conclusions introduction method application

19. ‘The Role of Algebraic Models and Type-2 Theory of Effectivity in Special Purpose Processor Design’ Gregorio de Miguel Casado Juan Manuel García Chamizo -Computability in Europe- July, 4th 2006 - University of Alicante - Specialized Processor Architectures Lab