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- CIE 2006 Contribution - ‘The Role of Algebraic Models and Type-2 Theory of Effectivity in Special Purpose Processor Design’ Gregorio de Miguel Casado Gregorio de Miguel Casado Juan Manuel Garc Juan Manuel Garc í í a Chamizo a Chamizo - - University of Alicante University of Alicante - - Specialized Processor Architectures Lab Specialized Processor Architectures Lab

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Page 1: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

- CIE 2006 Contribution -‘The Role of Algebraic Models

and Type-2 Theory of Effectivity in Special

Purpose Processor Design’

Gregorio de Miguel CasadoGregorio de Miguel Casado

Juan Manuel GarcJuan Manuel Garcíía Chamizoa Chamizo

-- University of Alicante University of Alicante --Specialized Processor Architectures LabSpecialized Processor Architectures Lab

Page 2: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

method

processor design

conclusions

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

ContentsContents

introduction

research motivations

background

method proposal for processor design

Type-2 Theory of Effectivity

Algebraic Models of Processors

processor design for computable integral transforms

problem formalization

computability analysis

functional specification

algebraic specification

conclusions

CIE 2006 CIE 2006

introduction

method

processor design

conclusions

Page 3: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introductionmotivation

background

method

processor design

conclusions

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

Research MotivationResearch Motivation

“special purpose processor for integral transforms”

Specialized Processor Architectures Lab

Scientific Computing

Architecture overview

CIE 2006 CIE 2006

background

method

processor design

conclusions

motivation

data acquisition system

control interface & scalability manager

general purpose processor

memory system

input/output

operating system

ad-hoc applications & symbolic calculation environments

Page 4: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introductionmotivation

background

method

processor design

conclusions

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

BackgroundBackground“feasibility barriers in paradigm application”

Scientific Computingreliability demands in computer characterization of complex physical problemsComputable Analysis

VLSI designcorrectness in specification and verification of processors Formal Methods: Algebraic Models

Computer Arithmeticlimited hardware support for arithmetic precision management (IEEE 754)signed-digit arithmetic

Technology trendshybrid chips (µP + ad-hoc hardware)memory integration improvements

CIE 2006 CIE 2006

background

method

processor design

conclusions

motivation

Page 5: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

methodType-2 Theory of Effectivity

Algebraic Models

sketch

processor design

conclusions

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

TypeType--2 Theory of 2 Theory of EffectivityEffectivityProvides a coherent bridge between two classical disciplines: analysis/numerical analysis and computability/complexity theoryPresents a realistic model of computation based on Type-2 machinesProvides a concrete computability concept based on naming systems and realizationsAllows the definition of computable functions on the set of all real numbersAllows a natural complexity theoryThe 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 CIE 2006

introduction

processor design

conclusions

Algebraic Models

Type-2 Theory of Effectivity

sketch

Page 6: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

methodType-2 Theory of Effectivity

Algebraic Models

sketch

processor design

conclusions

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

Algebraic Models of ProcessorsAlgebraic Models of Processors

Formal paradigm for VLSI design

Allows the isolation some fundamental scientific structural features of processor computation (behavior over time and of data representation and operation)

Specification and verification of computer architecture techniques (microprogramming), and models (pipelined and superscalar processors)

Connection with verification tools such as Maude and HOL

Algebraic abstraction support for complex computer architectures

Realistic approach by levels: Programmer & Abstract Circuit

CIE 2006 CIE 2006

introduction

processor design

conclusions

Type-2 Theory of Effectivity

sketch

Algebraic Models

Page 7: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

methodType-2 Theory of Effectivity

Algebraic Models

sketch

processor design

conclusions

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

Sketch of the methodSketch of the method

CIE 2006 CIE 2006

introduction

processor design

conclusions

Type-2 Theory of Effectivity

sketch

Algebraic Models

Page 8: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

method

processor design

problem formalization

computability analysis

specification

conclusions

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

Formalization of RequirementsFormalization of RequirementsINPUT: informal problem description

OUTPUTS:Mathematical expression. Convolution between Lebesgue integrable functions in

Processor requirements and restrictions• Support for heterogeneous data sources (symbolic

calculation programs and real world data series)

• Support for scalability features by levels of parallelization ofthe 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

CIE 2006 CIE 2006

problemformalization

introduction

method

computability analysis

conclusions

specification

Page 9: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

method

processor design

problem formalization

computability analysis

specification

conclusions

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

Computability AnalysisComputability AnalysisINPUTS:

Mathematical expression

Precision requirements

OUTPUTS:

Computable representation of convolution operation between Lebesgue integrable functions in spaces

• rational step functions

• normalized signed digit notation

Complexity AnalysisCIE 2006 CIE 2006

problemformalization

conclusions

introduction

method

computability analysis

specification

Page 10: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

method

processor design

problem formalization

computability analysis

specification

conclusions

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

SpecificationSpecification

INPUTS:

Requirements and restrictions

Computable representations and algorithms

OUTPUT: Algebraic Specification of the processor

CIE 2006 CIE 2006

problemformalization

conclusions

introduction

method

computability analysis

specification

Page 11: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

method

processor design

problem formalization

computability analysis

specification

conclusions

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

Functional SpecificationFunctional SpecificationModules

Instruction set (Status_Request, Configuration Request, Configuration_Set, Halt, Convolution)Banks of registers (Configuration, Base-Adress, Status, Arithmetic)

CIE 2006 CIE 2006

problemformalization

conclusions

introduction

method

computability analysis

specification

Page 12: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

method

processor design

problem formalization

computability analysis

specification

conclusions

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

Algebraic SpecificationAlgebraic SpecificationProgrammer’s level

state and next state algebras

machine algebra

next state and output function

Abstract circuit levelprogram memory

data memory organization

rational step function arithmetic unit

control unit

state and next state algebras

machine algebra

next state and output function

CIE 2006 CIE 2006

problemformalization

conclusions

introduction

method

computability analysis

specification

Page 13: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

method

processor design

problem formalization

computability analysis

specification

conclusions

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

Algebraic Specification. Data memory Algebraic Specification. Data memory organizationorganization

Mapping functions: phead_name, paddrF, paddrRSF, pheadStep, paddrRangeStep, paddrLint, paddrHint, paddrA, paddrB, paddrCr, paddrCi, pRangeStep, plInterval, pHinterval, pa, pb, pCr, pCi

Data memory mapping

CIE 2006 CIE 2006

problemformalization

conclusions

introduction

method

computability analysis

specification

Page 14: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

method

processor design

problem formalization

computability analysis

specification

conclusions

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

Algebraic Specification. Data memory storageAlgebraic Specification. Data memory storage

Normalized signed-digit representation

CIE 2006 CIE 2006

problemformalization

conclusions

introduction

method

computability analysis

specification

Page 15: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

introduction

method

processor design

conclusions

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

ConclusionsConclusionsNovel 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

Research in progress

Complete algebraic specification and verification

Prototype implementation and performance evaluation

CIE 2006 CIE 2006

conclusions

introduction

method

processor design

Page 16: - CIE 2006 Contribution - ‘The Role of Algebraic Models ... · ‘The Role of Algebraic Models and TTE in Special Purpose Processor Design’ Background “feasibility barriers

- CIE 2006 Contribution -‘The Role of Algebraic Models

and Type-2 Theory of Effectivity in Special

Purpose Processor Design’

Gregorio de Miguel CasadoGregorio de Miguel Casado

Juan Manuel GarcJuan Manuel Garcíía Chamizoa Chamizo

-- University of Alicante University of Alicante --Specialized Processor Architectures LabSpecialized Processor Architectures Lab