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- 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”
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- 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