A  relational  calculus  is  a  formal  system  in  which  relation  is  the  fundamental  concept.  The  simplest  relational  calcu1us,  that  of  ordinary  binary  relations,  was  introduced  by  Tarski  in  [4].  Tarski's  system  is  essentially  an  algebra  in  which  the  operations  are  the  usual  Boolean  operations  (on  sets  of  ordered pairs) together with the two special operations converse (denoted by u)  and  composition (denoted by ";" or by juxtaposition)

Wadge, William W.

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http://wrap.warwick.ac.uk/            Original citation: Wadge, W. W. (1975) A complete natural deduction system for the relational calculus. Coventry, UK: Department of Computer Science. (Theory of Computation Report).  CS-RR-005  Permanent WRAP url: http://wrap.warwick.ac.uk/59404    Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions.  Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners.  To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.  Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge.  Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.  A note on versions: The version presented in WRAP is the published version or, version of record, and may be cited as it appears here.  For more information, please contact the WRAP Team at: publications@warwick.ac.uk  The Univensity of WarwickTHEORY OFCOIVP[J-TA,TICNREPORTN0" 5A Complete llatural lleduction Systemfor theRel ati ona'l Cal cul ushlI LLIAM l'l.!^IADGEO. IntroductionA relational calculus ie a formal system in which relation is the fund-amental concept. The simplest relational calcu1us, that of ordinary binaryrelations, wa6 introduced by Tarst<i in [4]. Tarskits system is essentially analgebra in which the operations are the usual tsoolean operations (on sets oforcieredpairs)to6etherwiththetwospeciaIoperation'ry(denotedby")and composition (denoted by ";tt or by Juxtaposition). For any binary relationsPandaF=[(x,y) l(y,D€P]P;Q = ( (xry) I <*rt) € P and (vrz) €'l for some v ).The unusual fea.ture of the relational calculi is the manner in which it isoossible to make assertions about relations without referring to points. Forexample, the inclusion R;R = R holds iff R is transitive, and R = il iffR is symmetric. In a sense the relational calculj. are to the concept ofrelation as category theory is to tite concept of functj-on.Interest in relational calculi has been spurred recently by the search forsuitable mathematical systems in which to express the semantics of programmingIanguages. Given a machine of the appropriate type we can associate with eachprogran or part of a program the binary relation which characterizes its input-output behaviour: as de Roever says in [Jrp.1]"the collection of all pairs [consiotingJ of aun initial state ofthe memory, for which the prograrn terminates, and its correspondingfinal state of the nemory."For example, if relations Rl arrd Ra represent prograns Pt and PZ respec-tively, then the relation Rf iRa represents the program rrfirst Pt thenPr'rr and the equation(n.,;nr);u = u(where U is the universal relation) asserts that the resulting progran haltsfor every initial state.l.or this approach to work it is necesaary to consider not just simplebinary relations (between points and points) but also typed binary relations(between sequences of points and sequencee of points)' For'example' the systemdescribed by l{itchcock and Park in [1] has for each natural numbers rt and mrelations of type n-rm which un<ler a standarci interpretation woul'd denotesubsets of D\Dm.Certain difficultiee ariae when itrelational calculi. The usual approachrelational al5ebra. 'Iarski gave as hiscome6 to constructing proof systems foris to axiomatize the notion of aaxions the usual Boolean identities plusT'l: (XrY);'l = X;(Y;Z)9T2:X-XI3z (X;I)-= f ;fT4: X;!J = XT5z (x;Y)nz = n '+ (Y;I)ni = Iwhere u is the universal relation, O is its comprement and E is equality'De Roeverre axioms are basically a typed version of Tarskirs with sone addedrules for projections. Neither system has any non-logical rules of inferenceti.e.anyrulesotherthangubstitirtion,replacement,modusponensetc..Amodeloftheseaxiomsi""Is,I@algebra,andarelationalgcbraisstandardiffitisisomorphictoanalgebraobtainedbytakingacollectionofrelationsonsome domain and interpretin8 the relation constants and operations in the usualway (there are nonstandard relation algebras)'The trouble with at least Tarskirs axions is that they are incomplete: Lyndonin [2] showed that the inclusionx 1 x2 n Y 1 Y 2 nz 1z 2c x,l ( X,1 r,1 nxrY, n ( i,l z' {x 2i z) ti' .t .nz 2i' ?-) )Y 2'though true in every standard relation algebra, does not follow from Tarskirsaxioms.Thetheoryofstandardre}ationalgebrasisnevertheles.;tlxiomatizablesimplybecauseitisenunerable:everyfornulagoftherelationalcalculus-2-can be translated into a formula gt of the predicate calculus in euch a waythat models of cpr correspond to standard models of I and vice ver6a. Thedifficulty lies in findlng a sufficiently elegant axiomatization. It is pos-sible that de Roeverrs exioms are enough, for for he has provedlyndours formulaby using projections, but the question has so far not been resolved.The most serious drawback of the algebraic approach is ttre tendeney ofproofs of even simple assertions to be complex and unnatural. Consider, forexample, tbe proof on L1rrt7) of the inportant lemmaX;YflZ = X;(f;ZnY) nZ.the proof is nontrivial, uses the somewhat cryptic axiom 15 and has as asubstep the verification of(x;x;z) nz = O.In this work we attempt to remedy this defect by constructing a system ofnatural deduction for the relational calcul-us (typed or untyped) which is con-plete for standard relation algebr&s1 i.€. in which a formula is provable iffit is true in every standard relation algebra. The system is natural in thesenae that rules used in proving an assertion are determined by the structureof the assertion. The only disadvantege of the method is that it expands theformalism by adding point variables.1 - Informal ExPlanationTo understand how the system workar consider how one might prove thelemma given above in a nonalgebraic mannero The equality is actually two j"n-clusions, the most difficult beingX;YnZ cx;(f;ZnY)nz.In terms of pointsr we must show that for any(a,b) e X;Y n z ) then ax;(f ;znY )nzb. Thealent to the assertion that d(;Yb and thatis in turn equival-ent to the assertion thata and b, if aX;YflZb (i.e. ifassertion that d(;YflZb is err.uiv-aZb, and the assertion that aX;YbaXc and that cYb for some co-5-Thus we nust derive aX;([;ZnI)nZU from the assumption that aXc, cYb andaZb hold for some c. Now the former assertion is equi"valent to the assertionthat both aX;(i;ZnY)b and aZb and since aZb ie am aasumptionl we needonly deri-ve aX;(X;ZnY)b. This is in turn equivalent to aXv and vf ;ZnYbfor some v. Since we have aXc aa an aseumption it will be enough to prove9r/'cl;Zflyb. Next we break cI;ZnYb down into cYb (an assumption) and cX;Zbwhich in turn follows from "i" and wZb for any wr tsut both aXc and aZbare assumptions and so the proof is complete.The important point is that in formalizing this type of proof it is notnecessary to add the quantifiers and logical connectives of first order logic.For examplel although in connection with fr'tr we used the phraeesrrfor somertand t'for any'r in fact only the following two rules are needed: (1) to deriveaR;Sv derive aRv and vSb for some vr and (2) to to derive a result fromsome assumptions plue aR;Sb derive the result from the same assumptions plusaRv and vSb for any variable v not appearing in the result or in the otherassumpti ons.2. FormqliqqOur formal system consists of (1) a language whose formulas are of theform xRy with R a relation expreasion and x and y point variables;(2) a colLection of interpretation6 corresponding to standard relation algebras;and (l) a relatiol rr[-,r of deducibility generated by a set of natural deductionrules. The completeness reBUlt states that a formula r can be derived frona set f of formulas iff every interpretatlon which makes every formula inf true also makes F true.2.1 SptaxThe alphabet of the fornal language contains the symbols(i) a, b, c1... a set V of point variables;(ii) A1 811Ce... relation variablesl.(iii) O' U, E relation constants;(iv) U, n, ir -r- relation oPerations.AternrforexamplettA;BflCrrrisanexpreesionfornedfrom the relation conetants, variables and expressions' Aanple 'tdAlB0Caf', is a word of the form xRy with R a term and x and ypointvariables.Thenegationoftheformula:iRyistheStrictly speaking the set of formulas depends on the set Vand at times we will indicate this dependence by referringNote that the symbol ttst' is not in the alphabet; insteadtto be the formula "sg ffie".2.2 SemanticsFor any nonvoid set D and any set v of point variables a (v'D)-inter-pretation A is a map which assigns to each tern a subset of D<D and toeach point variable an element of D. The map a must be consistent with theusual interpretation of the relation operations and constantse €r.$' 4(RnS) mustbe the intersection of A(R) and 4(S), ffid An) must be the enpty set'Note that an interpretation is therefore determined by its restriction to theooint and relation variables.If A is an interPretation, x andterm, then a E fiY @ eatisfies xRYty are point variables and R is aor xRy is true in A itt <dG) 'd(Y)>lsin a(R). If r isasetofformulasand F i-saformulathen dFriffa Eg for every formula G in r., and r f r iff a Er iqplies A EF forevery interPretation Q.in the usual waYfo{rnglat for ex-formula *f;y.of point variablesto fV-formulas!.we define tB = S"2.3 Rulee of InferenceThe rules of inference of our system are the introduction and eliminationrules given on the next page together with the rule F l-F. In these rules Fancl G are arbitrary formulasl xr, y anrl v are arbitrary variables; I is anarbitrary set of formulas and R and S are arbitrary terns' In (;E) trvrris assumed not to occur in any formula in f and in (=T) rrxl and rryrr alreassumed not to occur in any formula in f . Formally speaking the relation tt l- t'between sets of formulae and formulas is the least one for vlbich these ruleeare valid.-5-( rrr ) xRY IsY\ ur '' xRUSY xRUSYr, nr \ xRY xSY\ | rr .,, xRf,lSy. -r \ rrxnY l- -afla\ r/ r F xRy/nr \ *RY *frY\r14./ afh(AI)(uI)(-r )(;r)(gr )(nr )rl-.rr,GFFxRv vSyxR; SYRules of Inference(AE)(ne)(-E)(rru)(uE)(E)xR0Sy xRf'lSYxRy xSYWs0tI( un)&xRy(;E)(=E) L=h:lr(En) %F1- ExarnplesFor our first example we verify tyndOnte inclusion (see p2)' the proofillustrates the manner in which a natural deduction systen alLows one to workbackwerd from the statement of a problem. It is for this reason that naturaldeduction systens are easier to mechanlze than axiomatic-algebraic ones'For convenience letYtr = xtztn xanzt, = it,,n'*I = i Y,,,n x2Y2n t1$2so that we wish to showx1x2n rlY2n zfz2 x19Y2(note we have omitted ri'rrru). . Then by (=)r;utxRv+yxxE;f .xRrFR-5-it is enough to show thataXlx2nYlY zn'z.Z.b l- aX19Y2b .tt follows from the (08) and the (AE) rules that it is enough to establishaxlx2br aYrYnbr a?rZrb l- aXlPY2band it follows from the (;E) rule in turn that it is enough to showaXrpr PX2b, aY,Qr QYZb, allrrt tZrb F &\1PY2b .Let us caII the set of formulas on the left hand side f. Then we will haver F d(1cpY2biff from f we can derive aX,,,vr rfpw and wYrb for some v and lr. The factthat dtp and qYZb are in f suggest v=P and q=w. Thus it remainsto derive SPq from f. SinceVVP = X1Y1n XtIrn t1$2we must derive nf,,Yrl, nxrirl ana nt.,'tre from l. The first two are immcd-iate: for example, since dtP and aY.,Q are in f, we have pfrt by (-P)jand then px.,v.,e by (;I). To derive pr!1$2q we need p$1t and t$eq foraome t. From the definition of {r it follows we must derive firort and$rirt, and this follows easily if we take t=r. Since rtea also followuin the sane vrayr the proof is complete.For our second example we verify Tarski's fifth axiom, i.e. show thatx;Y o 'z=a l- Y;z n I=O.The proof is presented in straight line form as in first order logic, eachassertion being derived from thoee above and not to the left.(1) X;Y fl Z =Q assumPtion(2) av;Z n Iu assumPtion$t aY;Zu (nn) ana (a)(4) a.[u r'(5) bxa ("8)(?) aYv assumPtion(B) "Zu assunPtion(g) bzv (-E) and (B)(10) bxiYv (;I) using (5) and (7)(11) bX;Yftzv (nr)using(9)*16(10)-7-(12)( tl)( 14)( 15)bClvaflUaflbY;ZflX=O4. CompletenessIn thie sectionrFF iff rFFstraight forward.@I. Forany V-formula F:(gp) usine (1)(frE) using (12)(;E) using (7)(g|) usin6 (14)and (11 )and (l) cancellins (7)'(B)cancelling (2)we show that l- and F are equivalentr i'e' thatfor all f and F. The rifr direction, soundneost isany set V of variablesl any set f of V-formulas andif f l-F then f FF.Ilroof. Since l- waa defined to be the least relation for which our rulesare valid, it is sufficient to verify that each of the rul-es is sound (ttris isomitted).our proof of the 'only ifr direction is an adaptation of the Henkin con-pleteness proof for firet order }ogic - we show that every consistent theoryhas an interpretation. By a ll-i!gY. (V a set of variables) we mean a set Tof V-tormulas such tlrat T l- F inplies F € T for every V-formula F' AV-theory T is consistent iff T does not contain the negation of any formulain T, and T is ggilglg iff T contains the negation of every formula notin T. Fina1ly, T is Henkin-complete (H-complete) iff for any variables xand y and any terms R and sr if xR;Sy e T then there is a v in Tsuch that xRv € T and vSy € T. The strategy of the proof is to show thatevery consistent theory has a con.sistent, complete, H-complete extension, andthat ev.:ry such theory has an interpretation'The first step is to show that every consistent theory has a eonsistentcomplete extension.-8-Theorem 11. I'or any countable set v of variables and any v-theoryif T is consistent there is a consistent complete v-theory Tlextending T.proof. The standard proof carries over. Let 4'i I ieN) be an enumerationof all V-formulas. Define a sequenc" {i I iep of V-theories extending Tby setting Tn = T and Ti*1 = Ti if Fre f , otherwise Ti*1 = the closureof T U (fi). It io easily verified that U(f. I ieNi is a consistent com-plete extension of T.Next we show that every consistent theory has a consistent H-completeextension.Theorern III. For any countable set V of variables and any V-theory T:if T is consistent there is a countable set Vr of variables ex-tending Vr ancl a consistent H-complete Vt-theory Tr extending T.pr-oof . Define a aequenc" {i I iep of sets of variables iind a sequence<T. I i€N> of theories, each T- a V.-theory' as follows:i' I r/'r\ rF - 'F.\ |/ ^o _ ^'(2) for anY irV. a new variable v andresulting set of vard"ablesset of fornulars is T. ..Then the u(vil i€N)-theory rIr (= utT. I ieul) is a consistent H-completeextension of T.To see this note first that an easy induction on i shows that each Tiis consistent. In the caee i = O the result is immediate. Now suppose thatfor some \ Ti is consietent but Ti*1 is not. Then Ti*t l- af,)a and soby the definition of Ti*, there is a finite set of the form(*j*j'j, ojsjxjl i Sn)such thatTi, *oRouo, vosovor ..o , *oRnvrr, trrsnJn F af)a.for each formula in Ti of the form xR;Sy add toadd to T. the formulas xRv and vSy. fhe1is V. . and the closure under l- of the resultin61+l-9 -But then n applications of the (;E) rute give uo Ti l- ufta which contra-di"cts the consistencY of Ti"Since each Ti is consistent, Tr must be consistent'Next, note that Tr muet be Fl-complete. For if xR;Sy is in Tt, itmust be in Ti for some i' But then there is a v in Vi*, such thatxRv and vsy are in Ti*1 and therefore in Tr'Finallyweputthelasttwotheorenstogethertogetthedegiredresulton extensions.Theorem IV. For any countable set V of variables and any V-theory t:if T is consistent there is a set vf of variables extending vand a consistent, complete, H-complete theory T' extending T'j!g!. Define a sequence 4* | iep of theories extenrling T as follows:(t) Tn = TiQ) tAL*1 is a complete consistent extension of Tet;(t) T2i*2 is an H-complete consistent r:xtension of T2i*1'It is easily verified that U[Til i€N) is a consistent, completetH-complete extension of T.Theorem V. For any countable eet V of variables and any V-theory T:ifTisconsistent,completeanclH-completethenthereisaninterpretation A such that A I f for every f in T'@|. Foranyvaribles x and y in V let x3Y iff xEy€T'It is errsily verified that : is an equivalence relation' Therefore for anyv in V 1et V be the equivalence class of vr and let f = (t I v€V)'Then for any variable v in V let O(v) = t, and for any V-formula xRylet a(xRy) = ( (1rI) I xRy € T ). The nap a so defined is then an inter-pretation of T.v,/e must first show that a is in fact an interpretationl i'e' thata is consistent with the intended meanings of the relation operations andconstants. We give only the proofs fortt;tt andItU"r the others being similar.(t) union. !/e muet show that 4nus) is the union of AG) and A$)for any terms R and S. This anounts to showing that for any variables xand y in V' xRUSy € T iff xRy € T or xSy € T.If either xRy or xsy is in f the (UI) rule gives us xRUSy € T.Now suppose that xRUSy € T but that neither xlly nor xsy is in T.Since T is complete, both xf;y and xSy must be in T. But thie meansthat xFnEy e T and therefore (omitting the proof) ttrat xfri3y Q T, whichcontradicts the consistency of T.(1) Composition. We must show that (irf) e A(RiS)and (lr$) € d(S) for sone Vo This amounts to showingxRv € T and vSy € T for sone vr and this is just theH-completeness.Finall)rr the fact that A F F for every F in Tiff (*,t) € aG)that xR;Sy€T iffstatement ofis immediate.Corollary. Every consistentProof. Extend the theory ofH-complete theory.of formulas has an interpretation.set to a consistent, completetsetthefhis gives us the nain result.Theorem V. For any countable set V of variables, any set f ofV-formulas and any V-fornulaif f FF then f FI'.Proof. Suppose that F does not follow from i. Then f and the negationof f are consistent and so have an interpretationt impossible.5. Tvpad Relational Calculi- 4This natural deduction system can be carried over quite easily to a typedsysten such as that of Hitchcock and Park. Fornulas would be of the form*..x....xnRy,ly2...f, with R aterm of type n-+m. The rules of inference givenon page 6 wouLd remain valid if xr y and are interpreted as se,llences of- 11 -variables of appropriate lengths. The rules for the extra operations are com-pletely obvioust e.g.xr....x'.Ryr',,. ' '|p Vn" 'V'SWO'''Wn.ulw^v\ lr'/ x^...x.vo...t.Rlsyo'Xnwo"'wr.tuJL',tltvXn. . .x ivo. . .V*R l syo, . .ykwo. . .wn xo. - .x..,vo. .'v*R l Syo"'ynwo"'wn(lE) ffi.",,VJVThe completeness proof for this typed system goes through with Iittle change'The extra operations offer no difficulty because their rules are purelydefinitional, like the ruleo for -'Acknowledgement This work was supportcd by a fellowahip from the ScienceResearch Council of the United Kingdon'-12-Referencest1] p. Hitchcock and D.M.R. Park. "Induction rules and terrnination proofs",in Automata, Languages and hogranming, ed. Nivat, North-liolland (1972).t2] R.C. Lyndon. rrThe representation of relational algebras", Annals ofIulathenatics 51 (1950)r pp. 7o7 - 729.lil !r/.p. de Roever. rtOperational , mathematical and axiomatized senantics forrecursive proceduree and data structuresr', Mathcmatical Centre reportTD 1/?4, Amoterdam, 1974.t4] A. Tareki. "On the calculus of relations", J. Symbolic Log;ic 6 (941)pp- ?t - 89.-13-

Induction rules and terrnination proofs&quot;,

rrThe representation of relational algebras&quot;,

rtOperational , mathematical and axiomatized senantics for recursive proceduree and data structuresr',

A complete natural deduction system for the relational calculus

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A complete natural deduction system for the relational calculus

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