A computational definition of the notion of vectorial space

A computational definition of the noti o n of v ectorial spa ce P ablo Arrighi ∗ Gilles Dow e k † Abstract W e usually define an algebraic structure by a set, some op erations defined on th is set and some prop ositions that the algebraic structure must v alidate. In some cases, we can replace these propositions by an algorithm on terms constructed up on these op erations that t h e alge braic structure m ust v alidate. W e sho w in this note that th is is the case for the notions of vectorial space and b ilinear op eration. An algorithm defined b y a confluent and terminating rewrite sys tem R on terms of a languag e L is sa id to b e valid in a structure M on the languag e L if for each rule l − → r and assig nment φ , w e hav e J l K φ = J r K φ . Thus, algorithms and theories play the sa me r ole with resp ect to the notion of model: like a theor y , an algorithm may or may not be v alid in a mo del. This notion of v alidity of an algo rithm, like the notion of v a lidit y o f a theory , can be used in tw o ways: to study the algorithms or to define algebraic structures as models of some algorithm. When a cla s s o f algebr aic structures — such as the class of g roups or that o f rings — can b e defined as the class of mo dels o f some equationa l theory T and this equatio nal theor y can be transformed into a rewrite system R , w e ha ve the following equiv alence • A is a member of the class (i.e. is a group, a ring, ...), • A is a mo del o f the theory T , • A is a mo del o f the algorithm R . In this case, w e say that the class of algebraic structure s has a c omputational definition . The goal of this no te is to show tha t the cla ss of vectorial spaces has such a computational definition, i.e. that the axioms of vectorial spaces can be or iented as a r e write system. Moreover, the alg orithm o btained this way is a well-known algorithm in linear alg ebra: it is an a lg orithm trans forming any term expres s ing a v ector in to a linear com bination of the unknowns. This algorithm is a lso central to the op erational seman tic of our functional programming language ∗ Institut Gaspard Monge, 5 Bd Descartes, Champs-sur-M arne, 77574 Marne-l a-V all´ ee Cedex 2, F rance, arrighi@univ-m lv.fr. † ´ Ecole p olytec hnique and INRIA, LIX, ´ Ecole p olytec hnique, 91128 Palaiseau Cedex, F rance, Gilles.Dowek@pol ytechnique.fr. for quantum computing Linea l [1], b ecaus e in such languages a pr ogram and its input v a lue for m a term ex pressing a vector whose v a lue, the o utput, is a linear combination of the base vectors. More gener a lly , several algorithms used in linear algebr a, s uch as matr ix multiplication algor ithms, tra nsform a ter m expressing a vector with v ario us constructs int o a linea r c o mbin atio n of ba s e vectors. This algorithm is v a lid in all v ectorial spaces and we s how that it moreov er completely defines the notion of vectorial spa ce. The main difficulty to or ie n t the theory of vectorial space s is that this theory has a sort for vectors and a sort fo r scala rs and that the s calars must form a field. The theory of fields is already difficult to orient, b ecause divis io n is a partial op eration. Ho wev er, there are many fields, for insta nce the field Q o f rational nu mbers, whose a ddition and m ultiplication can b e pres ent ed by a terminating and gr ound confluent rewrite system. Th us, w e sha ll not c o nsider a n ar bitr ary vectorial space o ver an arbitrar y field. Instead, we consider a giv en field K defined by a terminating and ground confluent rewrite system S and fo c us on K -vectorial spaces. Our rewr ite system for vectors will thus be parametr iz ed by a r ewrite system for sca lars and we will hav e to pr ovide pr o ofs o f c o nfluence and termination using minimal requirements on the scalar rewr ite system. This leads to a new metho d to prov e the confluence of a rewrite system built as the union of tw o systems. Moreov er, this computationa l de finitio n of the notion o f vectorial spa ce can be extended to define o ther algebra ic notions such a s bilinear ope rations. 1 Rewrite systems Definition 1. 1 (Rewriting) L et L b e a first-or der language and R b e a r ewrite system on L . We say that a t erm t R - rewrites in one step to a term u if and only if ther e is an o c cu rr enc e α in t he term t , a r ewrite rule l − → r in R , and a substitution σ such that t | α = σ l and u = t [ σ r ] α . Definition 1. 2 (Asso ciative-Comm utativ e Rewriti ng) L et L b e a fi rst- or der language c ontaining binary fun ct ion symb ols f 1 , ..., f n and R b e a r ewrite system on L . We sa y that a term t R / AC ( f 1 , ..., f n )-rewrites in one step t o a term u if and only if ther e is a term t ′ , an o c curr enc e α in the term t ′ , a r ewrite rule l − → r in R , and a su bstitution σ such that t ′ = AC t , t ′ | α = σ l and u = AC t ′ [ σ r ] α . R emark: This notion must b e distinguished fro m that of R,AC-r ewriting [3] where a term t rewrites to a term u only when it has a subterm A C-equiv alent to an insta nce of the left hand side of a rewrite rule . F or instance with the rule x + x − → 2 .x the term t + ( u + t ) R/ AC -r e w r ites to 2 .t + u but is R , AC -normal. 2 2 Mo d els Definition 2. 1 (Algebra) Le t L b e a first- or der language. A n L -algebra is a family forme d by a set M and for e ach symb ol f of L of arity n , a function ˆ f fr om M n to M . The denotation J t K φ of a t erm t for an assignment φ mapp ing variables to elements of M is de fine d as usual. Definition 2. 2 (Mo del of a rewrite system) L et L b e a first- or der language and R an algorithm define d by a r ewrite system on t erms of the language L . An L -algebra M is a mo del of the algorithm R , or the algorithm R is v alid in the mo del M , ( M | = R ) if for al l re write rules l − → r of the r ewrite system and valuations φ , J l K φ = J r K φ . Definition 2. 3 (Mo del of an AC-rewrite system) L et L b e a first-or der language c ontaining binary function symb ols f 1 , ..., f n , and R an algorithm de- fine d by an AC ( f 1 , ..., f n ) -r ewrite system o n t erms of t he language L . An L - algebra M is a mo del of the algorithm R ( M | = R ) if • for al l r ewrite rules l − → r of R and valuations φ , J l K φ = J r K φ , • for al l valuations φ and indic es i J f i ( x, f i ( y , z )) K φ = J f i ( f i ( x, y ) , z ) K φ J f i ( x, y ) K φ = J f i ( y , x ) K φ Example: Consider the language L formed b y tw o binar y symbols + and × and the algo rithm R defined by the rules ( x + y ) × z − → ( x × z ) + ( y × z ) x × ( y + z ) − → ( x × y ) + ( x × z ) transforming for insta nc e , the ter m ( a + a ) × a to the term a × a + a × a . The structure h{ 0 , 1 } , min , max i is a mo del of this algo rithm. R emark: This de finitio n of the v a lidit y of a n alg orithm in a mo del extends some definitions of the semantics of a pro g ramming lang uage where a seman tic is defined by a set M , a function [ ] mapping v alues of the language to elements o f M and n -ary programs to functions from M n to M , such that the progra m P taking the v alues v 1 , ..., v n as input produces the v alue w as o utput if and only if [ w ] = [ P ]([ v 1 ] , ..., [ v n ]). Indeed, let us cons ider a prog ramming languag e where the set of v alues is defined b y a fir st-order language, whose symbo ls are called c onstructors . Con- sider a n extensio n of this lang uage with a function symbo l p and p os s ibly other function symbols. A program P in this lang uage is given by a terminating and confluent rew r ite system on the extended lang ua ge, s uc h that for a ny n -uple of v alues v 1 , ..., v n the progra m P taking the v alues v 1 , ..., v n as input pr o duces the 3 v alue w as output if and only if the normal for m o f the term p ( v 1 , ..., v n ) is w . Then, a mo del o f this rewr ite system is for med b y a set M , fo r each constructo r c of arity m , a function ˆ c from M m to M , a function ˆ p from M n to M , and po ssibly other functions, such that for all rules l − → r of the rewrite system and v alua tions φ , J l K φ = J r K φ . The denotations of the constructor s define the function [ ] ab ov e mapping v alues to elemen ts of M and the function ˆ p is the function [ P ]. F or an y n -uple of v a lues v 1 , ..., v n , if the nor mal for m of the term p ( v 1 , ..., v n ) is the v alue w then J w K = ˆ p ( J v 1 K , ..., J v n K ) and thus [ w ] = [ P ]([ v 1 ] , ..., [ v n ]). 3 Computing linear com binations of the unkno wns 3.1 An algorithm Let L be a 2- sorted language with a sort K for s c a lars and a sort E for vectors containing tw o binar y symbols + and × o f rank h K , K , K i , t wo constants 0 and 1 of sor t K , a binary symbol, also written +, of r ank h E , E , E i , a binary symbol . of ra nk h K, E , E i and a constant 0 of sort E . T o tra nsform a term o f so rt E into a linear combination o f the unknows, we wan t to develop s ums of vectors λ. ( u + v ) − → λ. u + λ. v but factor sums o f scala rs a nd nested products λ. u + µ. u − → ( λ + µ ) . u λ. ( µ. u ) − → ( λ × µ ) . u we als o need the trivial rules u + 0 − → u 0 . u − → 0 1 . u − → u and, finally , three mor e rules for co nfluence λ. 0 − → 0 λ. u + u − → ( λ + 1) . u u + u − → (1 + 1) . u As we wan t to b e a ble to apply the factor ization rule to a term of the form (3 . x + 4 . y ) + 2 . x , reductio ns in the ab ov e r e write system m ust b e defined mo dulo the ass o ciativity and commutativit y of +. This leads to the following definitio n. 4 Definition 3. 1 (The rewri te system R ) The re write system R is the AC(+)- r ewrite system u + 0 − → u 0 . u − → 0 1 . u − → u λ. 0 − → 0 λ. ( µ. u ) − → ( λ.µ ) . u λ. u + µ. u − → ( λ + µ ) . u λ. u + u − → ( λ + 1) . u u + u − → (1 + 1) . u λ. ( u + v ) − → λ. u + λ. v Definition 3. 2 (Scalar rewrite system) A sca lar rewrite sy stem is a r ewrite system on a language c ont aining at le ast the symb ols + , × , 0 and 1 such t hat: • S is terminating and gr ound c onfluent, • for al l close d terms λ , µ and ν , the p air of terms – 0 + λ and λ , – 0 × λ and 0 , – 1 × λ and λ , – λ × ( µ + ν ) and ( λ × µ ) + ( λ × ν ) , – ( λ + µ ) + ν and λ + ( µ + ν ) , – λ + µ and µ + λ , – ( λ × µ ) × ν and λ × ( µ × ν ) , – λ × µ and µ × λ have the same normal forms, • 0 and 1 ar e normal terms. W e no w wan t to prove that the for an y scalar rewr ite system S , the system R ∪ S is ter minating and confluent. 5 3.2 T ermination Prop ositio n 3.1 The system R terminates. Pr o of: Co ns ider the following in terpreta tion (compatible with A C) | u + v | = 2 + | u | + | v | | λ. u | = 1 + 2 | u | | 0 | = 0 Each time a term t r ewrites to a term t ′ we have | t | > | t ′ | . Hence, the system terminates. ✷ Prop ositio n 3.2 F or any sc alar re write system S , t he system R ∪ S terminates. Pr o of: By definition of the function | | , if a term t S -reduces to a term t ′ then | t | = | t ′ | . Consider a ( R ∪ S )-re ductio n sequence. A t each R -reduction step, the measure of the term stric tly decrease s and at each S -r eduction step it re ma ins the same. Thu s ther e are only a finite n umber of R -reductio n steps in the sequence and, as S termina tes, the sequence is finite. ✷ 3.3 Confluence Definition 3. 3 (The rewri te system S 0 ) The system S 0 is forme d by the rules 0 + λ − → λ 0 × λ − → 0 1 × λ − → λ λ × ( µ + ν ) − → ( λ × µ ) + ( λ × ν ) wher e + and × ar e AC symb ols. Prop ositio n 3.3 The r ewrite system S 0 terminates. Pr o of: Co ns ider the following in terpreta tion (compatible with A C) || λ + µ || = || λ || + || µ || + 1 || λ × µ || = || λ |||| µ || || 0 || = || 1 || = 2 Notice that all ter ms ar e worth at leas t 2 and th us that each time a term t rewrites to a term t ′ we ha ve || t || > || t ′ || . Hence, the system terminates. ✷ 6 Prop ositio n 3.4 The system R ∪ S 0 terminates. Pr o of: B y definition o f the function | | , if a term t S 0 -reduces to a ter m t ′ then | t | = | t ′ | . Consider a ( R ∪ S 0 )-reduction sequence. A t each R -reduction step, the measure o f the ter m strictly decreases and at each S 0 -reduction step, it r emains the same. Thus there are only a finite n umber of R -reduction steps in the sequenc e and, a s S 0 terminates, by Pro po sition 3.3, the sequence is finite. ✷ Prop ositio n 3.5 The r ewrite system R ∪ S 0 is c onfluent. Pr o of: As the system terminates b y Prop osition 3 .4, it is sufficient to prov e the all cr itica l pair c lose. This can be mec hanica lly c heck ed, for instance using the system CIME 1 . ✷ Definition 3. 4 (Subsumption) A t erminating and c onfluent r elation S sub- sumes a r elation S 0 if whe never t S 0 u , t and u have the same S -normal form. Definition 3. 5 (Commut ation) The r elation R commutes with the r elation R ′ , if whenever t R u 1 and t R ′ u 2 , t her e exists a term w su ch that u 1 R ′ w and u 2 R w . Prop ositio n 3.6 L et S b e a sc alar r ewrite system, then R c ommutes with the r eflex ive-tr ansit ive closur e S ∗ of S . Pr o of: W e c heck this for eac h rule of R , using the fac t that in the left member of a rule, eac h subterms of sort scalar is either a v ar ia bles or 0 or 1, whic h are normal forms. ✷ Prop ositio n 3.7 (Key Lem ma) L et R , S and S 0 b e t hr e e r elations define d on a set such that S is terminating and c onfluent, R ∪ S terminates, R ∪ S 0 is c onflu ent, S subsumes S 0 ans the r elation R c ommutes with S ∗ . Then, the r elation R ∪ S is c onfluent. Pr o of: W e write t ↓ for the S -normal form of t . W e define the relatio n S ↓ by t S ↓ u if u is the S -normal form of t and the relatio n R ; S ↓ by t ( R ; S ↓ ) u if there exists a term v s uch that t R v S ↓ u . First notice that, if t R u then t ↓ ( R ; S ↓ ) u ↓ using the commutation of R and S ∗ and the unicit y o f S -no rmal forms. Thus if t ( R ∪ S ) ∗ u then t ↓ ( R ; S ↓ ) ∗ u ↓ simulating ea ch R -reduction step by a ( R ; S ↓ )-reduction step on no rmal forms. In a similar wa y , if t ( R ∪ S 0 ) ∗ u then t ↓ ( R ; S ↓ ) ∗ u ↓ , simulating each R - reduction step by a ( R ; S ↓ )-reduction step on norma l forms and using the subsumption of S 0 by S for S 0 -steps. 1 http://c ime.lri.fr/ 7 W e then chec k that R ; S ↓ is lo cally c o nfluent . If t ( R ; S ↓ ) v 1 and t ( R ; S ↓ ) v 2 then there e x ist terms u 1 and u 2 such that t R u 1 S ↓ v 1 and t R u 2 S ↓ v 2 . Thus, by confluence, of R ∪ S 0 , there exists a term w such that u 1 ( R ∪ S 0 ) ∗ w and u 2 ( R ∪ S 0 ) ∗ w . Thus u 1 ↓ ( R ; S ↓ ) ∗ w ↓ and u 2 ↓ ( R ; S ↓ ) ∗ w ↓ i.e. v 1 ( R ; S ↓ ) ∗ w ↓ and v 2 ( R ; S ↓ ) ∗ w ↓ . As the r elation R ; S ↓ is lo cally confluent and terminating, it is confluent. Finally , if w e hav e t ( R ∪ S ) ∗ u 1 and t ( R ∪ S ) ∗ u 2 then w e hav e t ↓ ( R ; S ↓ ) ∗ u 1 ↓ and t ↓ ( R ; S ↓ ) ∗ u 2 ↓ . Thus, there exists a term w such that u 1 ↓ ( R ; S ↓ ) ∗ w and and u 2 ↓ ( R ; S ↓ ) ∗ w . Th us u 1 ( R ∪ S ) ∗ w a nd u 2 ( R ∪ S ) ∗ w . ✷ Prop ositio n 3.8 L et S b e a sc alar r ewrite system. The r ewrite syst em R ∪ S is c onfluent on terms c ontaining variables of sort E but no variables of sort K . Pr o of: W e use the Key Lemma on the set o f semi-op en terms, i.e. ter ms with v ar iables o f sort E but no v ariables of sort K . As S is ground confluent and terminating, it is co nfluen t and ter minating o n semi-op en terms, by Pro po sition 3.2, the system R ∪ S terminates, by P rop osition 3 .5, the system R ∪ S 0 is confluent, the system S subsumes S 0 bec ause S is a s calar r e write system, and by Pr op osition 3.6, the s ystem R commutes with S ∗ . ✷ R emark: Confluence on semi-o pen terms implies ground confluence in a n y ex- tension of the language with constants for v ectors, typically base vectors. 3.4 Normal for ms Prop ositio n 3.9 L et t b e a normal term whose variables ar e among x 1 , ..., x n . The t erm t is 0 or a term of the form λ 1 . x i 1 + ... + λ k . x i k + x i k +1 + ... + x i k + l wher e the indic es i 1 , ..., i k + l ar e distinct and λ 1 , ..., λ k ar e neither 0 nor 1 . Pr o of: The term t is a sum u 1 + ... + u n of nor mal terms that ar e not sums (we take n = 1 if t is no t a sum). A nor ma l term that is not a sum is either 0 , a v ariable, or a ter m of the form λ. v . In this case, λ is neither 0 nor 1 and v is neither 0 , nor a s um o f tw o vectors no r a pro duct of a scalar by a v ector , thus it is a v ariable. As the term t is normal, if n > 1 then none of the u i is 0 . Hence, the ter m t is either 0 or a term of the form λ 1 . x i 1 + ... + λ k . x i k + x i k +1 + ... + x i k + l where λ 1 , ..., λ k are neither 0 nor 1. As the term t is nor mal, the indices i 1 , ..., i k + l are distinct. ✷ 8 4 V ectorial spaces Given a field K = h K , + , × , 0 , 1 i the class of K -vectorial spa c e s can b e defined as follows. Definition 4. 1 (V ectorial s pace) The structur e h E , + , ., 0 i is a K -ve ctorial sp ac e if and only if the structur e h K , + , × , 0 , 1 , E , + , ., 0 i is a mo del of the 2- sorte d the ory. ∀ u ∀ v ∀ w (( u + v ) + w = u + ( v + w )) ∀ u ∀ v ( u + v = v + u ) ∀ u ( u + 0 = u ) ∀ u ∃ u ′ ( u + u ′ = 0 ) ∀ u (1 . u = u ) ∀ λ ∀ µ ∀ u ( λ. ( µ. u ) = ( λ.µ ) . u ) ∀ λ ∀ µ ∀ u (( λ + µ ) . u = λ. u + µ. u ) ∀ λ ∀ u ∀ v ( λ. ( u + v ) = λ. u + λ. v ) W e no w prov e that, the cla s s o f K -vectorial space s can b e defined a s the class of mo dels o f the rew r ite system R . Prop ositio n 4.1 L et K = h K , + , × , 0 , 1 i b e a field. The structu re h E , + , ., 0 i is a K -ve ctorial sp ac e if and only if the stru ctur e h K, + , × , 0 , 1 , E , + , ., 0 i is a mo del of the r ewrite system R . Pr o of: W e first check tha t a ll the rules of R are v alid in all vectorial s paces, i.e . that the pr op ositions ( u + v ) + w = u + ( v + w ) u + v = v + u u + 0 = u 0 . u = 0 1 . u = u λ. 0 = 0 λ. ( µ. u ) = ( λ.µ ) . u λ. u + µ. u = ( λ + µ ) . u λ. u + u = ( λ + 1) . u u + u = (1 + 1) . u λ. ( u + v ) = λ. u + λ. v 9 are theorems of the theory of vectorial spaces. Seven of them are axioms of the theo ry of vectorial spaces , the prop os itions λ. u + u = ( λ + 1 ) . u and u + u = (1 + 1) . u a re consequence of 1 . u = u and λ. u + µ. u = ( λ + µ ) . u . Let us prove tha t 0 . u = 0 . Let u ′ be such that u + u ′ = 0 . Then 0 . u = 0 . u + 0 = 0 . u + u + u ′ = 0 . u + 1 . u + u ′ = 1 . u + u ′ = u + u ′ = 0 . Finally λ. 0 = 0 is a consequence of 0 . u = 0 and λ. ( µ. u ) = ( λ.µ ) . u . Conv ersely , we pr ov e that all axioms of v ector ial spaces are v alid in a ll mo dels of R . The v alidity of eac h of them is a consequence of the v alidity of a rewrite rule, except ∀ u ∃ u ′ ( u + u ′ = 0 ) that is a consequence of u + ( − 1) . u = 0 itself being a consequence of λ. u + µ. u = ( λ + µ ) . u and 0 . u = 0 . ✷ Prop ositio n 4.2 (Univ ersality) L et t and u b e two t erms wh ose variables ar e among x 1 , ..., x n . The fol lowing pr op ositions ar e e quivalent: 1. the normal forms of t and u ar e identic al mo dulo A C, 2. the e quation t = u is valid in al l K - ve ctorial sp ac es, 3. and the denotation of t and u in K n for t he assignment φ = e 1 / x 1 , ..., e n / x n , wher e e 1 , ..., e n is the c anonic al b ase of K n , ar e identic al. Pr o of: Pro po sition (i) implies pro p o sition (ii) and pr o p o sition (ii) implies prop o- sition (iii). Let us prov e that prop osition (iii) implies prop osition (i). Let t b e a normal term whose v aria ble s a r e among x 1 , ..., x n . The de c om- p osition of t along x 1 , ..., x n is the se q uence α 1 , ..., α n such that if ther e is a subterm of the form λ. x i in t , then α i = λ , if there is a subterm o f the form x i in t , then α i = 1 , and α i = 0 other wise. Assume J t K φ = J u K φ . Let e 1 , ..., e n be the ca no nical ba se of K n and φ = e 1 / x 1 , ..., e n / x n . Call α 1 , ..., α n the coo rdinates of J t K φ in e 1 , ..., e n . Then the decomp ositions o f the norma l forms o f t and u a re b oth α 1 , ..., α n and thus they are identical modulo A C. ✷ 5 Bilinearit y 5.1 An algorithm Definition 5. 1 (The rewri te system R ′ ) Consider a language with four sorts: K for sc alars and E , F , and G for the ve ctors of t hr e e ve ctor sp ac es, t he symb ols + , × , 0 , 1 for sc alars, thr e e c opies of t he symb ols + , . and 0 for e ach sort E , F , and G and a symb ol ⊗ of r ank h E , F, G i . The system R ′ is the r ewrite system forme d by thr e e c opies of the ru les of the system R and the rules ( u + v ) ⊗ w − → ( u ⊗ w ) + ( v ⊗ w ) ( λ. u ) ⊗ v − → λ. ( u ⊗ v ) 10 u ⊗ ( v + w ) − → ( u ⊗ v ) + ( u ⊗ w ) u ⊗ ( λ. v ) − → λ. ( u ⊗ v ) 0 ⊗ u − → 0 u ⊗ 0 − → 0 Prop ositio n 5.1 The r ewrite system R ′ terminates. Pr o of: W e extend the in terpre ta tion of Definition 3.1 with | u ⊗ v | = (3 | u | + 2)(3 | v | + 2) ✷ Prop ositio n 5.2 F or any s c alar r ewrite system S , the system R ′ ∪ S termi- nates. Pr o of: As in Prop ositio n 3.2. ✷ Prop ositio n 5.3 The system R ′ ∪ S 0 terminates. Pr o of: As in Prop ositio n 3.4. ✷ Prop ositio n 5.4 The r ewrite system R ′ ∪ S 0 is c onfluent. Pr o of: As in the pro of o f Prop osition 3.5, we prov e lo cal confluence by chec king that all cr itica l pair close . ✷ Prop ositio n 5.5 L et S b e a sc alar rew rite system, then R ′ c ommu t es with S ∗ . Pr o of: As in the pro of of P rop osition 3.6. ✷ Prop ositio n 5.6 L et S b e a sc alar r ewrite system. The r ewrite system R ′ ∪ S is c onfluent on terms c ont aining variabl es of sort E , F , and G but no variable s of sort K . Pr o of: Using the Key Lemma. ✷ 11 Prop ositio n 5.7 L et t b e a normal term whose variables of sort E ar e among x 1 , ..., x n , whose variables of sort F ar e among y 1 , ..., y p , and that has no vari- ables of sort G and K . If t has sort E or F , t hen it has t he same form as in Pr op osition 3.9. If it has sort G , t hen it has the form λ 1 . ( x i 1 ⊗ y j 1 ) + ... + λ k . ( x i k ⊗ y j k ) + ( x i k +1 ⊗ y j k +1 ) + ... + ( x i k + l ⊗ y j k + l ) wher e the p airs of indic es h i 1 , j 1 i , ..., h i k + l , j k + l i ar e distinct and λ 1 , ..., λ k ar e neither 0 nor 1 . Pr o of: The term t is a sum u 1 + ... + u n of nor mal terms that ar e not sums (we take n = 1 if t is no t a sum). A no rmal term that is not a sum is either 0 , a term o f the form v ⊗ w , or o f the form λ. v . In this case, λ is neither 0 nor 1 a nd v is neither 0 , nor a sum of t wo vectors nor a pro duct of a sca lar b y a vector, thus it is of the form v ⊗ w . In a term o f the form v ⊗ w , ne ither v nor w is a sum, a pr o duct o f a scalar by a vector or 0 . Th us b oth v and w are v ar iables. As the term t is normal, if n > 1 then none of the u i is 0 . Hence, the ter m t is either 0 or a term of the form λ 1 . ( x i 1 ⊗ y j 1 ) + ... + λ k . ( x i k ⊗ y j k ) + ( x i k +1 ⊗ y j k +1 ) + ... + ( x i k + l ⊗ y j k + l ) where λ 1 , ..., λ k are neither 0 no r 1. As the term t is normal, the pairs of indices ar e distinct. ✷ 5.2 Bilinearit y Definition 5. 2 (Bilinear op eration) L et E , F , and G b e thr e e ve ctorial sp ac es on the same field. A n op er ation ⊗ fr om E × F t o G is said to b e bilinear if ( u + v ) ⊗ w = ( u ⊗ w ) + ( v ⊗ w ) ( λ. u ) ⊗ v = λ. ( u ⊗ v ) u ⊗ ( v + w ) = ( u ⊗ v ) + ( u ⊗ w ) u ⊗ ( λ. v ) = λ. ( u ⊗ v ) Prop ositio n 5.8 L et K = h K , + , × , 0 , 1 i b e a field. The structu r es h E , + , ., 0 i , h F, + , ., 0 i , h G, + , ., 0 i ar e K -ve ctorial sp ac es and ⊗ is a biline ar op er ation fr om E × F to G if and only if h K , + , × , 0 , 1 , E , + , ., 0 , F, + , ., 0 , G, + , ., 0 , ⊗i is a mo del of the system R ′ . Pr o of: The v alidity of the r ule s of the three copies of the system R , ex pr ess that h E , + , ., 0 i , h F, + , ., 0 i , h G, + , ., 0 i ar e K -vectorial spac e s. The v alidity of the six other rules is the v a lidit y of the axioms of Definition 5.2 plus the t wo extra prop ositions 0 ⊗ u = 0 and u ⊗ 0 = 0 that are conseq uences of these axioms. ✷ 12 Definition 5. 3 (T ensorial pro duct) L et E and F b e t wo ve ctorial sp ac es, the p air forme d by the ve ctorial sp ac e G and the biline ar op er ation fr om E × F to G is a tensorial pro duct of E and F if for al l b ases ( e i ) i ∈ I of E and ( e ′ j ) j ∈ J of F the family ( e i ⊗ e ′ j ) h i,j i is a b ase of G . Example: Let ⊗ b e the unique bilinea r o pe r ation such that e i ⊗ e ′ j = e ′′ p ( i − 1)+ j where e 1 , ..., e n is the ca nonical base of K n , e ′ 1 , ..., e ′ p that of o f K p , and e ′′ 1 , ..., e ′′ np that of K np . Then K np together with ⊗ is the tensor ial pro duct of K n and K p . Prop ositio n 5.9 (Univ ersality) L et t and u b e two terms whose variables of sort E ar e among x 1 , ..., x n , whose variables of sort F ar e among y 1 , ..., y p , and that have n o variables of sort G and K . The f ol lowing pr op ositions ar e e quivalent: 1. the normal forms of t and u ar e identic al mo dulo A C, 2. the e quation t = u is valid in al l st r u ctur es forme d by thr e e ve ctorial sp ac es and a bili ne ar op er ation, 3. the e quation t = u is valid in al l st ructur es forme d by two ve ctorial sp ac es and their tensorial pr o duct, 4. and the denotation of t and u in K np for the assignment φ = e 1 / x 1 , ..., e n / x n , e ′ 1 / y 1 , ..., e ′ p / y p wher e e 1 , ..., e n is the c anonic al b ase of K n , e ′ 1 , ..., e ′ p that o f K p and ⊗ is the unique biline ar op er ation s uch that e i ⊗ e ′ j = e ′′ p ( i − 1)+ j wher e e ′′ 1 , ..., e ′′ np is the c anonic al b ase of K np . Pr o of: Prop o s ition (i) implies prop os ition (ii), prop osition (ii) implies prop o- sition (iii) and prop osition (iii) implies pr op osition (iv). Let us prov e that prop osition (iv) implies propo sition (i). Let t b e a normal ter m of s o rt G with v aria bles of so r t E among x 1 , ..., x n , v ar iables of sort F a mong y 1 , ..., y p , and no v a r iables of sort G and K . The de c omp osition of t along x 1 , ..., x n , y 1 , ..., y p , is the sequence α 1 , ..., α np such that if there is a subterm of the form λ. ( x i ⊗ y j ) in t , then α p ( i − 1)+ j = λ , if there is a subterm of the form x i ⊗ y j in t , then α p ( i − 1)+ j = 1, a nd α p ( i − 1)+ j = 0 otherwise. Assume J t K φ = J u K φ . Call α 1 , ..., α np the co or dinates o f J t K φ in e ′′ 1 , ..., e ′′ np . Then the decomp ositions of the normal forms of t and u are both α 1 , ..., α np and thus they are identical mo dulo A C. ✷ 13 Conclusion W e us ually define an algebraic structure by three comp onents: a set, some op erations defined on this set and so me prop o s itions that m ust be v a lid in the structure. F o r instance a K -vectorial space is defined b y a set E , the ope rations 0 , + and . and the equations o f Definition 4.1. W e can, in a more computation-or ient ed w ay , define an algebraic structure by a set, op eratio ns on this set a nd an algorithm on ter ms constructed up on these op er ations tha t must b e v alid in the structure. F or instance a K -vectorial space is defined b y a set E , the op erations 0 , + and . and the algorithm R . This algor ithm is a w ell-k nown a lgorithm in linear algebra: it is the algor ithm that transfor ms any linear expr ession int o a linear combination of the unknowns. This alg orithm is, at a first lo ok, only one among the ma n y algorithms used in linear a lgebra, but it completely defines the notio n of vectorial spa ce: a vectorial space is any structure where this algorithm is v a lid, it is any structure whe r e linear expressions can be transformed this w ay into linear com binations of the unknowns. Ac kno wledgemen ts The autho rs w ant to thank ´ Evelyne Contejean, Claude Kir chner and Claude March ´ e for c o mment s on a previous draft o f this paper. References [1] P . Arrighi and G. Dow ek, Op eratio nal semantics for formal tensorial cal- culus, 2nd International Workshop on Quantu m Pr o gr amming L anguages , Helsinki, 20 04. [2] N. Dershowitz a nd J.-P . Joua nna ud, R ewrite systems , Handb o ok of theo- retical co mputer science (vol. B): formal mo dels a nd semantics, MIT Pres s, 1991. [3] G.E . Peterson a nd M.E. Stick el, Complete sets of r e ductions for some e qu a- tional the ories , Journal of the ACM, 28, 2, p.233 -264, 1981. 14