The Role of Algebraic Models and TTE in Special Purpose Processor Design

- University of Alicante -Specialized Processor Architectures Lab

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

‘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 motivation background method application conclusions CIE 2006

‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ • Background “feasibility barriers in interdisciplinary paradigm application” • Scientific Computing • reliability demands in computer characterization of complex physical problems [Wei00] and [GoL01]  Computable Analysis: Type-2 Theory of Effectivity[Wei00]… • VLSI design • correctness in specification and verification of processors [McT90] and [MöT98] Formal Methods: Algebraic Models of Processors [HaT97], [FoH03]… • Computer Arithmetic • 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 motivation background method application conclusions CIE 2006

‘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 introduction Type-2 Theory of Effectivity Algebraic Models sketch application conclusions CIE 2006

‘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 introduction Type-2 Theory of Effectivity Algebraic Models sketch application conclusions CIE 2006

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

‘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 introduction method application problem formalization ad-hoc applications & symbolic calculation environments computability analysis operating system specification conclusions memory system general purpose processor input/ output data acquisition system control interface & scalability manager … CIE 2006

‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ • Formalization of the problem • INPUT: informal problem description • OUTPUTS • Mathematical expression. Convolution between Lebesgue integrable functions in • Processor requirements and restrictions • Support for heterogeneous data sources • symbolic calculation programs • 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 • Test scenarios introduction method problem formalization computability analysis specification conclusions CIE 2006

‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ • Computability Analysis (i) • INPUTS • Mathematical expression • Precision requirements • OUTPUTS • 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] introduction method problem formalization computability analysis specification conclusions CIE 2006

‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ • Computability Analysis (ii) • OUTPUTS • 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 vsd notation for the rational numbers [Wei00] • Complexity Analysis introduction method problem formalization computability analysis specification conclusions CIE 2006

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

‘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) introduction method problem formalization computability analysis specification conclusions CIE 2006

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

‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ • Algebraic Specification. Data memory organization • Mapping functions: phead_name, paddrF, paddrRSF, pheadStep, paddrRangeStep, paddrLint, paddrHint, paddrA, paddrB, paddrCr, paddrCi, pRangeStep, plInterval, pHinterval, pa, pb, pCr, pCi • Data memory mapping introduction method problem formalization computability analysis specification conclusions CIE 2006

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

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

‘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 • Type-2 Theory of Effectivity • 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 • Complete algebraic specification and verification outline • Prototype implementation and performance evaluation introduction method application conclusions CIE 2006

- University of Alicante -Specialized Processor Architectures Lab

The Role of Algebraic Models and TTE in Special Purpose Processor Design

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