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‘The Role of Algebraic Models and Type-2 Theory of Effectivity in Special

Purpose Processor Design’

Gregorio de Miguel Gregorio de Miguel CasadoCasado

Juan Manuel García ChamizoJuan Manuel García Chamizo-Computability in -Computability in

Europe-Europe-

July, 4th 2006July, 4th 2006

- University of Alicante -- University of Alicante -

Specialized Processor Specialized Processor Architectures LabArchitectures Lab

introduction

method

application

conclusions

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

ContentsContents

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

CIE CIE 2006 2006

introduction

method

application

conclusions

introduction

motivation

background

method

application

conclusions


Research MotivationResearch Motivation

Specialized Processor Architectures Lab (UA) research line: Scientific Computing

objectivedevelopment of hardware support for some scientific

computing tasks

integral transforms

Case of study: The convolution operation

CIE CIE 2006 2006

background

method

application

conclusions

motivation

introduction

motivation

background

method

application

conclusions


BackgroundBackground“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

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method

application

conclusions

motivation

introduction

method

Type-2 Theory of Effectivity

Algebraic Models

sketch

application

conclusions


Type-2 Theory of EffectivityType-2 Theory of EffectivityProvides 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 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 CIE 2006 2006

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application

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Algebraic Models


sketch

introduction

method


Algebraic Models

sketch

application

conclusions


Algebraic Models of ProcessorsAlgebraic 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 CIE 2006 2006

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sketch

Algebraic Models

introduction

method


Algebraic Models

sketch

application

conclusions


Sketch of the methodSketch of the method

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sketch

Algebraic Models

Test Scenarios

Test Scenarios

TTE

TTE

Algebraic Specification

&


S0-Defining the Problem S0-Defining the Problem

S1-Formalizing the Problem

S1-Formalizing the Problem

Mathematical Expression

Mathematical Expression

Requirements &

Restrictions

Requirements &

Restrictions

S2-Analysing ComputabilityS2-Analysing Computability

Algorithms & Computable Representations

Algorithms & Computable Representations

Complexity Results

Complexity Results

S3-Specifying the ProcessorS3-Specifying the Processor

Processor Specification

Processor Specification

ProposalProposal

S4-Hardware ImplementationS4-Hardware

Implementation

S5-Evaluating and Verifying the Proposal

S5-Evaluating and Verifying the Proposal

introduction

method

application

problem formalization

computability analysis

specification

conclusions


ApplicationApplication

“processor specification for computable convolution operation in ”

Overview of the system architecture

CIE CIE 2006 2006

problemformalization

introduction

method


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

introduction

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application



specification

conclusions


Formalization of the problemFormalization of the problemINPUT: informal problem description

OUTPUTSMathematical 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

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Computability Analysis (i)Computability Analysis (i)

INPUTSMathematical expression

Precision requirements

OUTPUTSTTE-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]

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Computability Analysis (ii)Computability Analysis (ii)

OUTPUTSTTE-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 AnalysisCIE CIE 2006 2006


conclusions

introduction

method


specification

introduction

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specification

conclusions


SpecificationSpecification

INPUTS requirements and restrictions

algorithms based on TTE-computable representations

OUTPUT: algebraic specification of the processor

Functional specification Algebraic specification

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specification

introduction

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specification

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Functional SpecificationFunctional SpecificationModules

Instruction set (Status_Request, Configuration Request, Configuration_Set, Halt, Convolution)

Banks of registers (Configuration, Base-Adress, Status, Arithmetic)

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specification

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Algebraic SpecificationAlgebraic Specification

Programmer’s levelstate and next state algebrasmachine algebranext state and output function

Abstract circuit levelprogram memorydata memory organization rational step function arithmetic unitcontrol unitstate and next state algebrasmachine algebra next state and output function

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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

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specification

introduction

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specification

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Algebraic Specification. Data memory Algebraic Specification. Data memory storagestorage

Normalized signed-digit representation

CIE CIE 2006 2006


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introduction

method


specification

introduction

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specification

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Algebraic Specification. Rational Step Algebraic Specification. Rational Step Function UnitFunction Unit

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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

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

CIE CIE 2006 2006

conclusions

introduction

method

application

‘The Role of Algebraic Models and Type-2 Theory of Effectivity in Special

Purpose Processor Design’

Gregorio de Miguel Gregorio de Miguel CasadoCasado

Juan Manuel García ChamizoJuan Manuel García Chamizo-Computability in -Computability in

Europe-Europe-

July, 4th 2006July, 4th 2006

- University of Alicante -- University of Alicante -

Specialized Processor Specialized Processor Architectures LabArchitectures Lab