WO2008068389A1

WO2008068389A1 - Method and system for generating a calculator of a rational polynomial function

Info

Publication number: WO2008068389A1
Application number: PCT/FR2006/002658
Authority: WO
Inventors: Yves Colombe
Original assignee: Arithmetic Components
Priority date: 2006-12-06
Filing date: 2006-12-06
Publication date: 2008-06-12

Abstract

The invention relates to a method of generating a calculator of any rational fraction having as input at least one vector of n digits, n being an integer greater than or equal to 1, to the base p, and, as output, a result vector of m digits, m being an integer strictly greater that 1, such that it comprises at least the steps of: • automatic decomposition (10) by a programmed computer of the rational fraction into m polynomial functions whose result is expressed in the form of a variable of a digit to the base p, • generation (14), from each result polynomial function, of an array of elementary logic and arithmetic gates that is suitable for carrying out the calculation of the corresponding polynomial function, • creation (16) of at least one description file for said arrays, comprising at least input parameters for computer aided circuit design tools.

Description

METHOD AND SYSTEM FOR GENERATING A CALCULATOR OF RATIONAL POLYNOMIAL FUNCTION

The present invention relates to a method and a system for generating a calculator of a rational fraction, and a computer program product for implementing the method.

Remember that a rational fraction is the quotient of two polynomials.

At present, the design of integrated circuits performing arithmetic operations uses the same tools as those of general design circuits or uses preconceived portions of circuits ("macros").

The design work follows a relatively standardized methodology. At the highest level, that is to say by analyzing the functional specifications of the circuit, it is described in a high-level language of C ++ type to simulate and validate the functionalities of the circuit. This is a simulatable specification ("Simulatable specification").

Then, manually or using a software of behavioral compiler type, this functional specification is transformed into a description RTL ("Register Transfer Logic") written in language Verilog (standard language according to IEEE 1364 reference) or VHDL ("VLSI Hardware Description Language" - language of hardware description of circuits with very high integration level, standard language according to IEEE 1076 reference). It should be noted that the RTL description which normally applies for a single clock domain can be extended to more than one clock domain.

A synthesis software, associated with a technological library, transforms the RTL description of the circuit into a gate-level description, that is to say in the form of a network of elementary logical components such as AND, OR, NAND, OR EXCLUSIVE, TILT

D, etc. The specific elements of arithmetic are currently limited essentially half-adder, full-adder, and carry-lookahead circuits.

This description is also described in VERILOG or VHDL (VITAL) language.

Finally, a placement-routing software ("P & R") transforms this "topological" description into a "geometric" description.

The geometric description, which uses the GDSII standard language, or an equivalent language, contains all the information necessary for the manufacture of photolithography masks used for the manufacture of integrated circuits.

It should be noted that in parallel with this design process, test vector files are generated to allow the production test of integrated circuits.

This methodology requires 1 to 3 years of implementation to obtain a new integrated circuit. Indeed, because of the various constraints, and in particular the management of the transmission delays of the signals inside the integrated circuit, it is often necessary to proceed by successive iterations to introduce manual modifications to the various descriptions before restarting the operations. synthesis tools.

To minimize this design time, design engineers are increasingly using block libraries that implement particular operations. Thus the manufacturers of integrated circuits provide more or less configurable function block libraries to be implemented in the manufacturer's technology. Similarly, specialized companies provide specialized function blocks such as MPEG decoders, floating-point units, and so on.

Libraries are often called IP libraries ("Intellectual Property"). Depending on its needs, the IC designer uses a predefined block and combines them to obtain; possibly with synthesizable link code, the desired functionality.

There is also a mechanism, called "DesignWare" (trademark of Synopsys Inc.), which allows the synthesis tools to automatically extract from a library predefined implementations by the IP vendor based on information placed in the library. RTL description of the product under development.

This use of predefined blocks generates a number of disadvantages. Indeed, the designer is dependent on the richness of the library of blocks at his disposal and, very often, he is obliged to develop his own function by using a set of blocks having lower level functionalities. As a result, its circuit is a sum of local optimizations, at the block level, instead of being optimized globally. Optimization, whether on the surface, in speed or in electrical consumption is therefore far from optimal.

If conversely, the designer only uses elementary components to create his circuit to optimize the parameters of it, the design time increases significantly because it then becomes necessary to recreate the entire circuit without using functions already validated, with the risks of error due to the associated complexity.

For some regular structures such as memory plans or registers, there are generators, generally called "compilers", capable, from a high-level parameterization, namely the number of bits per word and the number of words, to generate an optimized structure and thus quickly obtain all the necessary data (geometric data file, test vectors, etc.).

Such a generator does not exist for the arithmetic and logic calculation units and the designers of such a circuit use the conventional methodology described above.

However, with the development of digital signal processing in all branches of information processing such as telecommunications, image and sound processing, etc., it would be particularly advantageous to have a design tool and method capable of rapidly generating an optimized structure of any arithmetic function, or at least a wide range of class of arithmetic functions. To meet this need, according to one aspect of the invention, there is provided a method for generating a calculator of any rational fraction having as input variable at least one vector of n digits, n being an integer greater than or equal to 1 in base p, and in output variable, a result vector of m digits in said base p, m being an integer strictly greater than 1. From a data file containing the description of said rational fraction, the method has at least the steps of:

• automatic decomposition by a computer programmed of the rational fraction in m polynomial functions whose result is expressed as a variable of a digit in base p, • generation, from each polynomial function result of the decomposition, of a network of logic and arithmetic elementary gates adapted to perform the calculation of the corresponding polynomial function,

Creating at least one description file of said networks, said at least one file comprising at least one data structure constituting input parameters of computer-assisted circuit design tools.

This method thus advantageously makes it possible automatically to generate the wired network of gates, and therefore the calculation circuit, of any rational fraction, without imposing any particular property on it. According to another aspect of the invention, there is provided a calculator generation system of any rational fraction having as input variable at least one vector of n digits, n being an integer greater than or equal to 1, in base p, and, in output variable, a result vector of m digits in said base p, m being an integer strictly greater than 1. From a data file containing the description of said rational fraction, the system comprises at least: Automatic means of decomposition by a programmed computer of the rational polynomial function in m polynomial functions whose result is expressed as a variable of a digit in base p,

A generator, from each polynomial function resulting from the decomposition, of a network of logic and arithmetic elementary gates adapted to perform the calculation of the rational fraction,

Means for creating at least one description file of said networks, said at least one file comprising at least one data structure constituting input parameters of computer-assisted circuit design tools.

According to another aspect of the invention, a computer program product downloadable from a communication network and / or recorded on a computer readable medium and / or executable by a processor, includes program code instructions for the implementation of of the previous process.

Other features and particular embodiments are described in the dependent claims.

The proposed polynomial decomposition has the advantage of offering a recursive calculation method for single-digit polynomials, an easily programmable method in conventional programming languages (C, Ada, C ++, etc.).

It is advantageously usable whatever the mode of representation of numbers: integer, integer signed complement 2, fixed or floating point. The invention will be better understood on reading the description which follows, given solely by way of example, and with reference to the appended figures in which:

- Figure 1 is a schematic view of an embodiment of a computer-aided design system; FIG. 2 is a flowchart of an embodiment of the invention; and FIG. 3 is a schematic view of an embodiment of a system according to the invention.

With reference to FIG. 1, a programmed computer 1 represents a workstation of a designer 2 of circuits, in particular of integrated circuits. With this workstation, the designer 2 has at his disposal the conventional computer-aided design tools well known to those skilled in the art. It is thus able to carry out the design of a new circuit according to the conventional methodology described in the preamble above. The workstation 2 comprises in particular storage means

3 to store in the form of files the representations of the circuit at different levels of detail: functional specification, logic transfer register, logic gate network and a geometric description. It should be noted that by file is meant any form of data storage and in particular text or binary files as well as databases.

Among the mathematical functions that a designer is led to design, polynomial functions and rational fractions are very frequent.

Indeed, due to the constraints inherent in automated calculation such as the finite precision of the calculations carried out, any function is often transformed by a limited development into a rational fraction of one or more variables.

Thus, the realization of a hard function consists in designing the gate network that will execute the polynomial computation corresponding to the limited development of the function; the length of said development being adapted to the calculation accuracy required by the application.

Similarly, a calculation initially operating on floating point numbers is generally transformed by standardization operations well known to those skilled in the art into a calculation involving rational integers. When the result of the polynomial function is expressed on one bit, the polynomial function corresponds to a Boolean function for which there are tools for automatic generation of the network of arithmetic and logical gates fulfilling the function.

We will therefore consider, in the remainder of this description, that the rational fractions whose result is expressed on 2 bits and more. By generalizing from the usual binary base to any base p, p being an integer greater than or equal to 2, this amounts to considering only rational fractions whose result is expressed by 2 or more digits in this base p.

When designing the circuit, the rational fraction is, for example, described in the simulable specification file as a C ++ function. Its result is expressed as a vector of m bits, or digits, where m is an integer strictly greater than 1.

From this description file, according to one embodiment of the invention, FIG. 2, the method of generating the corresponding computer comprises a step 10 of automatic decomposition of the polynomial function into m polynomial functions whose result is expressed under form of a variable of 1 bit, that is to say in m polynomials Boolean.

This decomposition uses, in one embodiment, the properties of the polynomials in the affine rings of Witt. The ring concept of the Witt vectors of a ring A is summarized in the following references:

Nicolas Bourbaki, commutative algebra - Chapters 8 and 9, Masson, 1983;

Michel Demazure - Pierre Gabriel, Algebraic Groups, Volume 1, Chapter V: Structure of Affine Switching Groups. North Holland, 1970 (hereinafter referred to as "Demazure").

According to a first theorem, called the raising of polynomials in the affine rings of Witt, we have:

Let P (X ₀ , X ₁ , ..., X _n-1 ) ε Z [X ₀ , X ₁ , ..., X _n-1 ] be a polynomial of n variables with integer coefficients. There is one and only one sequence of polynomials

Qi (x ₀ , 0, ..., X ₀ , i, X ₁ , 0, .., X ₁ , i, ..., X _n-1 , 0, ..., χ _n-1 , i) of Z [( ^χ _m , j) (mj) ε _n χw] such that: w £ Qo, - - -, Qi) = ¹ P (W ₁ (Xo, o, ..., xo, i), - - -, ^ i (X ₁₁ - i, o, ..., x _n - 1,)) in any index. The w _t figuring in the previous relations are the Witt polynomials, namely:

• wo (x): = xo • wi (x) -. ^≈ xf + pxi

Here is a proof of the theorem. It is based on the properties of the affine ring of the Witt vectors.

Let us begin by recalling the elements of algebraic geometry necessary for this demonstration by drawing inspiration from the exhibition that is made in Michel DEMAZURE and Pierre GABRIEL, Algebraic Groups Volume I; MASSON I NORTH - HOLLAND 1970. ISBN 720420342.

In all that follows, the rings are commutative and uniform, and the homomorphisms of rings are uniform. And, according to the usage M denotes the set of natural integers (with all its structures),

denotes the ring of rational integers, etc. The objects we consider are functors and natural transformations: we call

sets the ring category functors in the set category, and the applications

natural transformations between these functors. A

set X is therefore a construct that associates a set X (A) to any ring, and an application

to any homomorphism of rings. These data are consistent in that

that is, X preserves the composition of identical homomorphisms and endomorphisms. The elements of X (A) are called the points from X to coordinates in A. An application and the data of a

application in any ring; with the condition of consistency

following: for any homomorphism of rings, the squares

are commutative. The usual set constructions, and their properties, are transposed "argument by argument" (with their usual notation) into the category of

nsets: for example the product of two ^ -sets X and F is the

set defined by

We can therefore speak, as in the category of sets, of sets in groups, rings, etc. We call them groups,

rings, etc. Similarly for morphisms:

homomorphisms of groups, rings, etc. By way of example, the ring refining Witt W vectors is a

ring: a

together said underlying and, according to the misuse designated by W, provided with two laws of internal compositions, namely an addition and a multiplication, that is to say two ^ -applications

for which, strictly speaking, the triplets (W ( _^ 4), + W, A, * W, A) are rings and the W (φ): W (A) → W (B) rings homomorphisms . In other words, a

nneaux is a functor of the category of rings in itself, a morphism of

nneaux is a natural transformation between two such functors. Likewise, a

group is a ring category functor in the group category, etc.

Explain the word "affine": consider a set X; we the

say affine when it is representable. That is, when it has a representation, namely a pair (R, x) composed of a ring R and a generic point xe X (R) (a point of X to coordinates in R) , To say that this pair is a representation means that every point α from X to coordinates in a ring A is a specialization of the generic point, in the sense that there exists then a unique ring homomorphism α _# :

called specialization, such that α = X (α #) (x). We show that the representation of an affine set is unique to a single isomorphism. We draw a first consequence: consider a

application

from source one

affine set X; and with no assumptions about its purpose. Let X be by (R, x); so

is biunivocally determined by its generic value flx) = f _R (x) e Y (R). Indeed, the value f _A (ά) of the component

in a point α GX (A) is then f _A (α) = Y (α _# ) (f (x)). And it is enough, to conclude to verify that the applications thus defined constitute an application.

Before proceeding, it should be noted that the notion of

together affine is one of the ways to present that of affine schema; the two expressions "affine" and "affine schema" are thus used to designate such an object. It is a generalization of the notion of affine algebraic variety.

Some useful examples for understanding the following: the affine line A is the functor who "forgets" the ring structures. As a result A (A) is the set underlying the ring A and is the application

who wears ring homomorphism

It is an affine scheme represented by the ring

polynomials with integer coefficients in an indeterminate x, with x for generic point. This statement is immediately justified by the universal property characteristic of polynomial rings. More generally, for any set /, the affine space A ¹ of dimension / is represented by the ring

polynomials with integer coefficients in the indeterminate family x = (χ,) _{/ e /} , with x = (χj) iei for generic point. By construction, A \ A) is the / -th power A ¹ of the ring A; the generic point χ = (x,) _ie j is therefore an element of / -th power

of the ring of polynomials

. More generally, for two sets / and J, the affine scheme A ^I xA ^J , naturally isomorphic to A ^IκJ , is represented by the ring

polynomials with integer coefficients in the families of the indeterminates {χ, y) = ((χi) isi, (ydieù, with ( ^χ , y) = (( ^χ ,) isi, (ydisi) for a generic point, likewise for A ^I xA ^J xA ^κ , etc.

Consider one of an affine space of

dimension a set / in an affine space of dimension J. Represent A ¹ by the ring

and the generic point x = (x,) _ieI ; as we have seen, it is biunivocally determined by its generic value: a point of A ^J with coordinates in, that is to say a

family of polynomial fa) = (Mxuisihj.

So the component

is defined by

or

denotes the value of the polynomial ZK ( ^χ diei), calculated in the ring A, after respective substitutions of a _t to indeterminates x _t . Similarly for

applications etc.

We thus show that a ring structure on the affine space A ¹ of dimension a set / is biunivocally determined by the generic values of its addition and multiplication.

that is, two families of polynomials s (x, y) = (Sj {(x /) _te /. (v /) / _e /) and pix, y) = (p / (xi) iei, (ydiei) of two such families constituting a ring structure si, and

only if, axioms of structures are verified generically, that is, s (x, s (y, z)) = s (x, $ (y, z)) for associativity, etc. Two remarks are useful for the understanding of the continuation.

First, consider a polynomial P (X ₀ , Xi, ..., X _n -i) of Panels Z [X ₀ , Xi, - .., X _n -i] of polynomials with integer coefficients following indeterminates ( X ₀ , Xi, ..., X _n -I), and an affine ring Λ = (4 ^/ , Λ: + _{Λ <} y, xx _Λ .y)) carried by the affine space of dimension a set /. We associate him with a

component application the polynomial application

in any ring A (P ( _ao , at ..., a _n -i) denotes the value of the polynomial P (X ₀ , Xi, ..., X _n - ₁ ) calculated in the ring R (A) after substitutions of α _, to indeterminates X _t ). The generic value of this

application is the family of polynomials

P (X ₀ , Xu ... JC _n -i) = (Qj ((X ₀ , /) / ₆ /, (X1, /) "= /, • • •> (Xn-1, /) te / )) jel of polynomials of the ring. This ring

represents R "(carried by the affine space (A) ' ¹ ), with

(Xθ, Xh ... - X / i-f) - ((Xθ, /) / e /, (r r, die!, - - -, (Xn-1, diel) for generic point.

Secondly, as universal algebra teaches us, it is equivalent, to define the ring structure of R on the affine space A ¹ , to give its sum and its product, namely x + _R y and Λ: x _R y, or give yourself ^ -applications

, associated in all n with the polynomials of

, in a manner compatible with the respective compositions of Z-applications and polynomials. It is easy to see that the sum and the product correspond to the polynomials χ + y and xy; while, reciprocally, it suffices for the sum and the product to calculate the values of all these polynomials. There is, of course, a natural ring structure on the affine line A: the corresponding ring is designated by O and called the ring refines regular functions. Seen as a functor of the category of rings in itself, it is simply the identical endomorphism of this category; representing _^ 4 x _^ 4 by the ring of polynomials in

two indeterminates x and y, and the generic point (χ, y generic values of its sum and its product are the polynomials χ + y and xy.) Similarly, the affine space A ¹ of dimension a set / carries the affine ring & 7-th ring power refines regular functions With the preceding representations, the generic values of the sum + _o and the product x ₀ of this power of affine rings are respectively x + oy ^≈ fa + ydiei and ^χ xoy - The affine ring of the Witt W vectors is carried by the affine space A ^m of dimension M. The construct thus consists in constructing the sequences of the polynomials

and

of ui are the generic values of the sum and vector product of Witt. This construction is based on the following result:

Consider the -application

of the affine space of dimension N in the N-th power of the ring refines regular functions. Its generic value is the following of Witt's polynomials. It's called the -Witt app.

Theorem says that there exists a ring structure, and only one, on the affine space of dimension N for which the Witt application is a homomorphism of affine rings. We denote by W and we call affine ring Witt vectors the corresponding affine ring.

Specifically, the following five assertions verified:

(1) There exists a unique affine ring W, carried by the affine space of dimension N, for which the application of Witt

is a morphism of affine rings.

(2) There are two applications

and only two, for which squares

and

are com (3) There are two sets of polynomials

and x xwy = Pw ( ^χ , y) = (pφdie "0 <W)) W of

_> and only two, for which and

in all

nneau

polynomials with integer coefficients and in the following indeterminate n (Xo ₎ X ₁ , ..., X * -i), there exists a ^ ^ - application

and only one, called a raising of the polynomial P, for which the square

is commutative. any polynomial P (X ₀ , Xi, ..., X _n -i), of the ring

polynomials with integer coefficients following n indeterminates (XQ, X _I , ..., X _n -i), there exists a single sequence

from polyn

(in posa

Let's start by establishing equ equivalence

and

affine source applications and values

almost immediate: it is enough to show that the sums and products obtained by recovery verify the axioms of the structure of the rings. Gold

since an obvious recurrence proves that is an isomorphism as soon as A is an algebra s

along with the implication of Witt associativity, commutativity,

distributivity, existence of a zero and a unit in the polynomial ring

^e * is concluded by noting that the foregoing relates only polynomials

..,

• The implication

is immediate: the recovery

of

P (Xo, Xi X _n -i) is obtained simply by calculating this polynomial in the rings of Witt since (2) is a particular case of (4).

To demonstrate that they are verified, it is sufficient to demonstrate that one of them is verified, this is what is proposed, by a method based on polynomial calculations in the reference "Bourbaki" cited above. We will also find another, more conceptual, proof based on the theorem called symmetrical elementary polynomials in the reference "M. Demazure and P. Gabriel" quoted above.

Let's complete this reminder by presenting the properties of the affine ring of the Witt vectors that are useful to us. Easy recurrences make it possible to establish them. And anyway they are demonstrated in the works cited in references. They are related to the section of Teichmuller

(its generic value φ) is the vector of Witt whose coordinates are null, except that of index 0 which is equal to x). This is a section of the 0 index component of Witt's homomorphism (ie

Clearly, it is multiplicative (ie φy) = ΦMy)) and uniferal (ie <i) = i), but not additive (ie φ + ^) ≠ φ) + τ (y)). And they are related to so-called offset endomorphisms V: W → W and Frobenius

. The first is defined by its generic value

It is not an endomorphism of rings: it is additive, but neither multiplicative nor uniferal. And the Frobenius

is an endomorphism of rings whose generic value

is defined by equalities W _n (Fc (X), F ₁ (X), ..., F _n (x)) = w, ₁₊ feo, xi,:., X _{n +} i). These operations check the generic relations τ (x) x _w (yi) _iew = (/ yi) _teΛr

V (x + _W y) = V (x) + w VQ) V (x) x _w y = V (xx _w F (y))

V '(x) x _w V (y) = p "x _w V (xx _w y) and

FV (x) = px _w x.

For any natural integer n and any vector of Witt (generic) x: = (xj) _teΛr> n XψX - x + ψx + w • •• + wx is the sum of Witt- ôe n copies of x!

These relationships show that, for any natural whole number, the image

W (beware, here n is a superscript index, not a power) of the endomorphisms is an ideal W c W of the affine ring of the Witt vectors. The set W (A) of the points of this ideal are the coordinate Witt vectors in the rings A whose n index components strictly smaller than n are zero. It is carried by the closed affine subschema of generic point V (x) (where x = (x,) _iew designates a generic point of W). The sequence of ideals (W) _n≡M is a decreasing filtration of the ring of Witt vectors. It is exhaustive (ie of meeting equal to W), separated (of intersection reduced to the null ideal) and complete: for any ring

A, the CAUCHY sequences of vectors of W (A) are convergent. A suite

(a _n ) nεsr '_' ^≈ (( ^a n, diew) nes _f of such vectors is of CAUCHY as soon as in any natural number n, there exists a natural integer N such that a _v - _w a _u & W (A) for all natural numbers N <u and N <v. And to say this convergent sequence means that it has a limit: a vector b = (bdiew of W (A) such that, for any natural number n, there exists a natural integer N such that h - _w a _u ≡ W (A) in all natural numbers N <u The separation condition ensures the uniqueness of this limit! We immediately deduce that a sequence (μ _n ) _neM : = ((a _n j) _lem ) _new of vector of W (A) is summable, namely that the sequence of partial sums (Σi _in βi Xejir has a limit, if and only if, the sequence (a _n ) _new converges to zero (which means that for every natural number n, the ideal W \ A) contains almost all the _u , that is to say all but a finite number of them). Under these conditions, the limit of the sequence of the partial sums is unique, it is called the sum of the sequence (a _n ) _mM , and it is designated by Σ _{/ 6Λf} α ₇ -. The preceding topological considerations, and the remark (x ₀ , xu -, Xi, Xi + h- - •) = (xo, Xu • -, Xi, 0, ...) + w (β, 0, ..., 0, Xi + u ...) (the second term of the right-hand member starts with / zeros), allows to write (generically)

For any natural number; the affine ring W _n WITT vectors of length n is carried by the affine space A "of dimension n, it is represented by the ring of polynomials and the generic point

x: = (xo, χi, ..., χ _n -i) - The set W _n (A) of these coordinate points in a ring A are the finite sequences a - ~ (μo, a- _t ,. .., a _n -i) of n elements of, 4. The ring structure of W _n is generically defined by "truncating to order n" that of W \ so its sum and and product are generically defined by (X ₀ , xf, ..., x _n - i) + w (yo, yu ---, y "- ύ = (sdx, y), s ^ x, y), ..., $" - i (x, y)) and

(xo, ^χ u ---, Xn-i) ^' Xfv (yo, yu -, yn ~ i) = (pd ^χ , y), pi ( ^χ , y), ..., $ _n -i ( ^χ , y)) - The polynomials s _t (x, y) and p, {x, y) being the polynomials of the corresponding indices used in the descriptions of the operations of W; which makes sense because these index polynomials i only depend on index coordinates at most equal to /! Similarly, with a small obvious "truncation", and the same notations for the Tθichmϋller sections and the offset and FROBENIUS endomorphisms. Let's conclude on the WITT vectors of length n noticing that the natural projection

is a homomorphism of rings compatible in a clear sense with Teichmϋller sections, offsets and Frobenius.

Coordinate Witt vectors in characteristic rings p (i.e. verifying pa = 0), they verify the following relations in addition to previous relations:

F (X) = {χP% _M px _w x = VF (x) ≈ FV (x) = (0, _{X (} f, xf, ..., xf, ...) (in particular shift and Frobenius switch to feature p) and

V "(x) x _w V (y) =

XwF "(y)) It follows immediately that, on any ring of characteristic p, the topologies associated with the filtrations (W (A)) _mm and /" - adic (ie the filtration by the powers of the principal ideal p _y.w W (A)) coincide.

For any perfect ring of characteristic p, that is to say a ring A in which pa ~ 0 and on which the endomorphism of Frobenius A → A -. a → cf is an isomorphism, the previous result is more precise: for all natural numbers the ideals

W "(A) = p ⁿ x _w W (A) = (W ¹ (A) f are equal and when, in addition, A is integral, so is W (A) and natural homomorphism

is injective. Here, as for any ring, 1 _{W (A} ) denotes the unit of the ring W (A), so n Xw Iw (A) represents the sum, in W (A), of n times this unit. Let us write the integer n in the form of their p-adic development n = n _o + nip + ... + n _d p ^d ; then the preceding relations make it possible to write u _{W (A)} (n) = u _W (A) (no) + u _W (A) (n 1) XWPW (A) + ••• + uw (A) (n) ) Xwp ^d w (A)

= φo) + τ (m) Xw VF (Iw (A)) + ... + τ (n _d ) Xw V ⁱ F ⁱ ( _W (A)) = τ (nό) + V (F (T ( H ₁ ))) + ... + F ^ τfa,))) = <"o) + V (τ (nf)) + ... + J ^ (τ (n /)). The homomorphism u _{w (A)} : factorizes naturally

in inclusions n

aturelles (the first body obviously integrates, is perfect) and

or

is the natural inclusion of the first body

in .4; it is deduced from a passage to the quotient. Since W (A) is the ideal

principal generated by p "in W (A), we have

p Uw ( _A ^ (W (A)), we deduce, by passage to the quotient, an injective homomorphism

It induces an isomorphism

as we see immediately by simple enumeration. We also deduce from it (for example by passing to the separated complete? -Adic) an isomorphism

of the ring of p-adic integers on W (F _P ). Let's not forget, cf = a

in ! Finally, by inverting p in these two rings (p and its powers

p ⁿ are the only non-invertible integers in and in,

we are building an isomorphism

(the symbol

means that almost all, ie all but a finite number, negative index coordinates are null). To invert p in W (A), we construct the group of bi vectors of Witt, namely the union (strictly the strict inductive limit)

where the morphisms of transitions are offsets (trivially

injectives!). This construction is done to extend the V: WW shift in an affine group isomorphism: the shift As it is obvious that the Frobenius component is a

isomorphism into any perfect ring of characteristic p, we deduce that

is, as we wished, an isomorphism for any such ring. It remains to be seen that, by translation, we can calculate sum and product of Witt bi vectors to conclude. Thus, by truncating the Witt vectors, we have a unified mathematical model of signed and unsigned integers, and fixed and floating point numbers by keeping only bi vectors whose non-zero coordinates are negative indices, with an implicit or explicit representation in the second case, of the exponent of the base! In feature 2, the presentation of W as the meeting

where inclusions

correspond to the prolongation by the sign, shows that our model is an algebraic representation of arithmetic in addition to 2 usual in computer science, both in the hardware ("hardware"), as in the software ("software"). As we have seen in the p-base variants, it covers representations of unsigned, signed, fixed-point and floating-point numbers.

Here is how we use this representation: for any polynomial P (X ₀ , X ₁ X _n -i), of the rings

polynomials with integer coefficients following the indeterminates (Xo, Xi, ..., X _n -i) _> we calculate the first polynomials of the sequence

Q (XO, X ₁ , .-), Xn-V ⁼ (Qj ((Xθ, die M '( ^X 1, di≡lNi - -> ( ^χ n-1, die WJ) jem> of its bearings, into polynomials of the ring% [( ^χ o, diei, ( ^χ i, diei, ---, (x _n -i, disi], then, for any sequence (x _α -, ^χ _n ~ i) of whole numbers write their representations in base p: either

as well as

P (X ₀ , ..., X _n -O = y = Σy _i 2 _> of the value in this sequence (x _a ..., x _n -i) of the polynomial P [X ₀ , X ₁ X ,, - i). Then, yt ⁼ Qfaθ, O ,. - -, Xθ, h Xi, 0, - - -, Xi, b _' - -, Xn-1,0, - - -> ^x n-1, d in any index.

In other words, the polynomial P is decomposed into polynomial Q ₁ , the result of the computation of each polynomial g, corresponding to the i-th digit y _t of the result of the computation of P.

Take, for example, a polynomial of two variables P (X, Y) of By construction, the polynomials Q fa, y) which reads P (χ, j) satisfies

equality W ₁ (Q ₀ , ..., Qu = P (wix \ wi (y)) ( ^* )

Where Wi (Witt polynomials) check the following recurrence relation (immediate translation of their definitions):

• w _o {z) = Z ₀ and • Wi (z) = Wj (zo, Zi, ..., Z _r ) = W _i -i (Z (f, zf, ..., Z _i -iF) + p'Zi for all indices, hence the recurrence:

• Qd ^χ , y) = P ( ^χ o, yo) and

•

to calculate the Qi. So :

• Qd. ^χ , y) = P ( ^χ o, yό)

• Q < ^χ > y) = Vp \ P ( ^χ <? + P ^χ u yf + pyύ - P ( ^χ o, yόf] m

An effective algorithm is thus obtained for successively calculating the polynomials Qfa, y). It will be noted that calculations involving divisions by powers of p give a result without denominator after simplification (this is the main consequence of the bearing theorem in this application); which allows to calculate the figures in their representation modulop. It can be seen that a mode of decomposition which applies to any rational fraction has been determined in a particularly advantageous manner, without requiring any particular properties thereof.

The recursion calculation thus makes it possible to determine the polynomial for calculating each digit of the output vector.

When using a binary base, which is currently the case for the vast majority of integrated circuits, we obtain as many Boolean polynomials as there are output bits to the polynomial function.

Each boolean polynomial QiX, Y) is simplified, in step 12, to a boolean function QiX, Y) using the Boolean optimization techniques well known to those skilled in the art.

By using a logic synthesis tool, logic and arithmetic gate arrays are generated at 14 from the optimized Boolean polynomials.

These gate networks are described at 16 in files in a data format adapted to serve as input parameters to the usual design tools. For example, these files are in VHDL format or

VERILOG, at the level of logic gates and are used for placement-routing tools, generation of test vectors and logic simulation.

It should be noted that this polynomial decomposition applies to any numerical representation such as integers, integers signed in complement to 2, fixed or floating point numbers. Thus, the skilled person understands that any number can be represented in a form of number vectors in the base considered, with or without a representation of the exponent. The method described can be implemented in the form of a computer program. Executed on the workstation 2 of FIG. 1, it thus constitutes, for example, a system, FIG. 3, of generating a calculator of any rational fraction having as input variable at least one vector of n digits, n being a an integer greater than or equal to 1, in base p, and, in output variable, a result vector of m digits in said base p, m being an integer strictly greater than 1. This system, from a data file containing the description of said rational fraction comprises at least: Means 30 for automatic decomposition by a programmed computer of the rational polynomial function in m polynomial functions whose result is expressed as a variable of a digit in base p, • a generator 32, from each polynomial function result of the decomposition, a network of logical and arithmetic elementary gates adapted to perform the calculation of the rational fraction,

Means for creating at least one file for describing said networks, said at least one file comprising at least one data structure constituting input parameters for computer-assisted circuit design tools. Although the above description is based, for example, on the design of logical integrated circuits in binary logic, it is conceivable that it is possible to apply the method to circuits made in different technologies such as, for example, for example, optical circuits or biocircuits.

Similarly, the polynomial decomposition applies to circuits using ternary or other logic, the demonstrations in support of the mathematical properties used being valid for any base p. The described method thus makes it possible to automatically generate a calculator of rational fractions.

Claims

A method of generating a calculator of any rational fraction having as input variable at least one vector of n digits, n being an integer greater than or equal to 1, in base p, and, as an output variable, a vector result of m digits in said base p, m being an integer strictly greater than 1, characterized in that, from a data file containing the description of said rational fraction, it comprises at least the steps of: • automatic decomposition (10) by a programmed computer of the rational fraction in m polynomial functions whose result is expressed as a variable of a digit in base p,

Generating (14), from each polynomial function resulting from the decomposition, a network of logic and arithmetic elementary gates adapted to perform the calculation of the corresponding polynomial function,

• creating (16) at least one description file of said networks, said at least one file comprising at least one data structure constituting input parameters of computer-assisted circuit design tools.

2. Method according to claim 1, characterized in that the decomposition of the rational fraction comprises a generation by recurrence of the Witt polynomials related to said rational fraction and m polynomial functions whose result is expressed in the form of a variable of a p-base number from the Witt polynomials and the rational fraction.

3. Method according to claim 2, characterized in that the Witt Wi polynomials are calculated by the following recurrence relation • W ₀ (X) = Xo •

and the m polynomial functions O / are calculated by the recurrence relation

• •

or x, ... represent the at least one vector (x _Ol Xi, ... x _n ) and P the rational fraction.

4. Method according to claim 1, 2 or 3, characterized in that the rational fraction is an entire polynomial function.

5. Method according to any one of the preceding claims, characterized in that the base p is a binary base.

6. Method according to claim 5, characterized in that the polynomial functions results of the decomposition are Boolean functions.

7. Method according to claim 5 or 6, characterized in that the input vectors represent integers in unsigned binary form or in binary form signed in addition to 2, or fixed-point or floating-point numbers.

8. Computer generating system of any rational fraction having as input variable at least one vector of n digits, n being an integer greater than or equal to 1, in base p, and, as an output variable, a result vector of m digits in said base p, m being an integer strictly greater than 1, characterized in that, from a data file containing the description of said rational fraction, it comprises at least:

Means (30) for automatic decomposition by a programmed computer of the rational polynomial function in m functions polynomials whose result is expressed as a variable of a digit in base p,

A generator (32), from each polynomial function resulting from the decomposition, of a network of logic and arithmetic elementary gates adapted to perform the calculation of the rational fraction,

Means (34) for creating at least one description file of said networks, said at least one file (36) comprising at least one data structure constituting input parameters of computer-assisted circuit design tools; .

9. Computer program product downloadable from a communication network and / or recorded on a computer readable medium and / or executable by a processor, characterized in that it comprises program code instructions for the implementation of process steps according to any one of claims 1 to 7.