A Qualitative Modal Representation of Quantum Register Transformations

A Qualitativ e Modal Repr esentation of Quantum Register T ransf ormations (Extended V ersion) Andrea Masini Luca V igan ` o Mar gherita Zorzi Department of Computer Science Univ ersity of V erona, Italy { andrea.masini | luca.vigano | margher ita.zorzi } @univr .it Abstract W e intr o duce tw o modal natural ded uction systems that ar e suitab le to r epr esent a nd r eason abo ut transformations of quantu m r egister s in an abstr act, qua litative, way . Quantum r e gisters r epr esent qua ntum systems, and can be viewed a s the structur e of quan tum data for quan tum operations. Our systems pr ovide a mo dal framework for r ea soning abou t operations on quantum r egisters ( unitary transformations and measur ements) in terms o f possible worlds (as abstractions of quantum r e gister s) and a ccessibility r elations between these world s. W e give a Kripke–style semantics tha t formally describes qu antum r e gister transformations, and pr ove the sound ness and completeness of our systems with r espect to this semantics. 1. Intr oduction Quantum compu ting defines an alternative co mputation al p aradigm, based on a quantum model [4] rather than a classical one. The basic units of th e quantum model are the qua ntum b its , or qu bits for sh ort (mathem atically , norma lized vectors of the Hilbert Sp ace C 2 ). Qubits r epresent infor mational units an d can assum e bo th classical values 0 and 1 , an d all th eir superpo sitional values. A qu antum r e g ister is a g eneralization of the qubit: a ge neric quantu m register is the representation of a quantum state of n qub its (mathematically , it is a norm alized vector of th e Hilbert space C 2 n ). In this p aper , we are not interested in the structure o f qu antum registers, but rather in the way qu antum registers are transfor med. Hence , we will abstract away fro m the in ternals of quantum registers and represent th em in a generic way in ord er to describe how operatio ns transform a re gister into another one. It is possible to modify a q uantum register in two ways: by app lying a unitary transfo rmation o r by m easuring. Unitary transform ations (co rrespond ing to the so-called u nitary operato rs o f the Hilbert space ) model the internal ev olution of a quantum system, wher eas m easurements co rrespond to the results of the interaction between a quantum system and an observer . Th e outcome o f an ob servation can be either the reduction to a quan tum state or the reduction to a classical (non quantum ) state. In particular, in this p aper, we say that a quantum register w is classical iff w is idemp otent with respect to measurement, i.e. each measur ement of w has w as o utcome. W e call a mea surement to tal w hen the outcome of the measuremen t is a classical re gister . W e propose to mo del measurement and un itary transformations by mean s of suitable mod al operators. More specifically , the main contribution of this paper is the formalization of a modal natura l deduction system [12, 14] in ord er to rep resent (in an abstract, qualitati ve, way) the fu ndamenta l o perations on quantum r egisters: unitary tran sformation s an d total measurements. W e c all this s ystem MSQR . W e also formalize a variant of this system, called MSpQR , to represent the case of generic (not necessarily total) measurements. It is imp ortant to observe that our logical system s are not a quantum log ic. Since 1936 [5], various logics have b een in vestigated as a means to formalize reasonin g a bout pro positions tak ing into acco unt the pr inciples of qu antum theory , e.g. [7, 8]. In general, it is possible to vie w quantum logic as a logical axiomatization of quan tum theory , which pr ovides an adequate foundation for a theory of rev ersible quantum processes, e.g. [1, 2, 3, 10]. Our w ork m oves fro m qu ite a dif ferent po int of vie w: we d o n ot aim to pr opose a general log ical fo rmalization o f qu antum theory , ra ther we describe how it is possible to use mod al logic to reason in a simple way about quantum register transfo r- mations. Infor mally , i n our pr oposal, a mod al w orld repr esents (an abstraction of) a quantu m re gister . Th e discrete temporal ev olution of a q uantum r egister is controlled and determined by a sequenc e of u nitary transforma tions and measurem ents that ca n ch ange the descriptio n of a q uantum state into o ther descriptions. So, the evolution of a q uantum register can be viewed as a graph, where the no des are the (abstract) quantum r egisters an d the arr ows represent quantum tr ansformatio ns. The arr ows gi ve us the so- called accessibility relations o f Kripke mo dels and two nodes lin ked by an arrow represent two related quantum states: the target n ode is obtained f rom the source node by means of the oper ation specified in the decoration of the arrow . Modal log ic, as a logic of possible worlds, is thus a natural way to represent this description of a quantu m system: th e worlds model the quantum registers and the relations of accessibility between worlds m odel the din amical behavior of the system, as a consequence of the ap plication of measurem ents and unitary transfor mations. T o em phasize this semantic view of mod al logic, we give our deductio n system in the style of labelled deduction [9, 13, 15], a framework for gi ving uniform presentation s of dif ferent non-classical log ics. The intuition b ehind labelled deduction is that the lab elling ( sometimes also called prefixing , ann otating or sub scripting) allows one to explicitly encode in the synta x ad ditional inf ormation, of a semantic or proo f-theoretical natu re, that is other wise implicit in the logic one wants to capture. M ost notably , in the case of mod al logic, this additional information co mes from the un derlying Kripke semantics: the labelled formula x : A intuitively means that A ho lds at the w orld denoted by the label x within the underlying Krip ke stru cture ( i.e. model), and labels also allo w one to spec ify at th e syntactic le vel how the d if ferent world s are related in the Kripke structu res (e.g . the for mula xRy specifies that the world denoted by y is accessible from that denoted by x ). W e proceed as fo llows. I n Section 2, we d efine the labe lled modal n atural deductio n system MSQR , wh ich contains two modal operators s uitable to represent and reason ab out unitary transformations and total measurements of quantum registers. In Section 3, we giv e a possible world s semantics that fo rmally describes these q uantum register transfor mations, and prove the sound ness and co mpleteness of MSQR with resp ect to this sem antics. In Section 4, we f ormalize MSpQR , a variant of MSQR that provides a modal system representing all the possible (thus not necessarily total) measurements. W e con clude in Section 5 with a brief summary and a d iscussion of future work. Full proofs of the technical r esults a re giv en in the a ppendix . 2 The deduction system MSQR Our labelled mod al n atural deduc tion system MSQR , which f ormally represents unitary transfo rmations an d total mea- surements of quan tum r egisters, co mprises o f rules that d eriv e form ulas of two k inds: mo dal form ulas and r elational formulas. W e th us define a modal language and a relational language. The alphabet of the r elational language consists of: • the binary symbols U and M , • a denum erable set x 0 , x 1 , . . . of labels . Metav ariables x, y , z , possibly ann otated with subscripts and sup erscripts, r ange over th e set of labels. For brevity , we will sometimes speak of a “world” x meaning that th e label x stands for a world I ( x ) , where I is an interpretatio n fu nction mapping labels into worlds as formalized in Definition 2 below . The set of r elational formulas ( r –formulas ) is given by e xpressions of the form x U y and x M y . The alphabet of the modal langu age con sists of: • a denum erable set r , r 0 , r 1 , . . . of pr o positional symbols , • the standard pr o positional connectives ⊥ an d ⊃ , • the unary modal operators  and  . The set of moda l formulas ( m–formulas ) is the least set that contains ⊥ and the proposition al symbols, and is closed u nder the propo sitional co nnectives and the m odal o perators. M eta variables A , B , C , possibly indexed, ran ge over modal formulas. Other connectives can b e defin ed in the usual manner, e.g. ¬ A ≡ A ⊃ ⊥ , A ∧ B ≡ ¬ ( A ⊃ ¬ B ) , A ↔ B ≡ ( A ⊃ B ) ∧ ( B ⊃ A ) , ♦ A ≡ ¬  ¬ A ,  A ≡ ¬  ¬ A , etc. Let us giv e, in a rather informal way , the intuiti ve meaning of the modal oper ators of our langua ge: 2 [ x : A ] . . . . x : B x : A ⊃ B ⊃ I x : A ⊃ B x : B x : B ⊃ E [ x : ¬ A ] . . . . y : ⊥ x : A RAA x : ⊥ α ⊥ E [ xRy ] . . . . y : A x : ⋆ A ⋆ I ∗ x : ⋆ A xRy y : A ⋆ E x U x U r efl x U y y U x U symm x U y y U z x U z U tr ans x M y x U y U I [ x M y ] . . . . α α M ser ∗ x M y y M y M sr efl α ( x ) x M x x M y α ( y /x ) M sub1 α ( y ) x M x x M y α ( x/y ) M sub2 In ⋆ I , y is fresh: it is different f rom x and does not occur in any assumption on which y : A depends other than xRy . In M ser , y is fresh: it is different from x and does not occur in α n or i n any assumption on which α dep ends o ther t han x M y . Figure 1. The rules of MSQR •  A means: A is true after the application of any unitary transformation. •  A means: A is true in each quantum register obtained by a total measuremen t. A labelled formula ( l–fo rmula ) is an exp ression x : A , where x is a label and A is an m–f ormula. A formula is either an r–formula or an l–f ormula. Th e metav ariable α , possibly indexed, ranges over f ormulas. W e write α ( x ) to denote that the label x o ccurs in the formula α , so that α ( y /x ) denotes the substitution of the label y fo r all occurences of x in α . Figure 1 shows the rules o f MS QR , where the notion of dischar ged/open a ssumption is standa rd [12, 14], e.g. the formula [ x : A ] is d ischarged in the rule ⊃ I : Propositional rules : The rules ⊃ I , ⊃ E a nd RAA are just the labelled version of the standard ([12, 14]) natural deduction rules for imp lication introdu ction and elimination and f or r eductio ad ab sur dum , where we do not enforce Prawitz’ s side condition th at A 6 = ⊥ . 1 The “mixed” rule ⊥ E allows us to deri ve a generic formu la α whenev er we have o btained a contrad iction ⊥ at a world x . Modal rules: W e give the rules for a generic modal op erator ⋆ , with a correspond ing generic accessibility relation R , since all the modal oper ators sha re the stru cture of these basic in troduction /elimination rules; this ho lds because, fo r instance, we express x :  A as the metalevel implication x U y = ⇒ y : A for an arbitrary y accessible f rom x . In particula r: • if ⋆ is  then R is U , • if ⋆ is  then R is M . Other rules: • In order to axiomatize  , we add rules U r efl , U symm , and U tr ans , formalizing that U is an eq uiv alence relation. • In order to axiomatize  , we add rules formalizing the following properties: – If x M y then there is specific unitary tr ansformation (d ependin g on x and y ) t hat generates y fro m x : ru le U I . – The to tal m easurement p rocess is serial: rule M ser says that if f rom th e assump tion x M y we can derive α for a fresh y (i.e. y is different fro m x and does n ot occur in α nor in any assum ption on which α depen ds other than x M y ), then we can discharge th e a ssumption (since there always is so me y such that x M y ) an d conclud e α . – The total measurement process is shift-reflexive: rule M sr efl . 1 See [15] for a deta iled discussion on the rule RAA , which in parti cular explains how , in order to maintai n the duality of modal operat ors like  and ♦ , the rule must allo w one to de ri ve x : A from a contradicti on ⊥ at a possibly differe nt world y , and thereby discha rge the assumption x : ¬ A . 3 – In variance with respect to classical worlds: rules M sub1 and M sub2 say that, if x M x an d x M y , then y must be equal to x a nd so we can substitute the one for the other in any formula α . Definition 1 (Deri vations and proof s) . A deriv ation o f a formula α fr om a s et of formulas Γ in MSQR is a tr ee formed using the rules in MSQR , end ing with α and d epending only on a finite subset of Γ ; we then write Γ ⊢ α . A deriva tion o f α in MSQR de pending on the empty set, ⊢ α , is a pro of of α in MSQR and we then say that α is a theo r em of M SQR . For instan ce, the following lab elled formula schemata are all provable in MSQR (where, in parentheses, we give the intuitive mean ing of each formula in terms of quantum register transformatio ns): 1. x :  A ⊃ A (the identity transformatio n is unitary). 2. x : A ⊃ ♦ A (each unitary transfor mation is in vertible). 3. x :  A ⊃  A (unitary transfor mations are compo sable). 4. x :  A ⊃  A (it is always possible to perform a total measurement of a quantum re gister). 5. x :  ( A ↔  A ) (it is always possible to per form a total me asurement with a co mplete reductio n of a quan tum r egister to a classical one). 6. x :  A ⊃  A (total measuremen ts are composab le). As c oncrete examples, Figur e 2 co ntains the pr oofs of the formu las 5 and 6, where, for simplicity , h ere and in the following (cf. Figure 5), we employ the rules for eq uiv alence ( ↔ I ) and for negation ( ¬ I and ¬ E ), which are deriv ed from the propo sitional rules as is standard. For instance, [ x : A ] 1 . . . . y : ⊥ x : ¬ A ¬ I 1 abbreviates [ x : A ] 1 . . . . y : ⊥ x : ⊥ ⊥ E (or RAA ) x : A ⊃ ⊥ ⊃ I 1 W e can similarly derive rules about r –form ulas. For in stance, we can derive a rule for the tran siti vity o f M as sh own at the top of the proof of the formula 6 in Figure 2: x M y y M z x M z M tr ans abbreviates y M z y M z z M z M sr efl x M y x M z M sub1 3. A semantics f or unitary transf ormations and total measur ements W e giv e a semantics that forma lly describes un itary transfo rmations an d to tal measurem ents of qua ntum registers, a nd then prove that MSQR is sound and complete with respe ct to this semantics. T ogeth er with the co rrespond ing result fo r generic measur ements in Section 4, this me ans that our mo dal systems indee d provide a represen tation of quantum registers and operations on them, which was the main goal of the paper . Definition 2 (Frames, models, structures) . A frame is a tuple F = h W, U, M i , wher e: 4 [ y : A ] 2 [ x M y ] 1 y M y M sr efl [ y M z ] 3 z : A M sub1 y :  A  I 3 y : A ⊃  A ⊃ I 2 [ y :  A ] 4 [ x M y ] 1 y M y M sr efl y : A M sub1 y :  A ⊃ A ⊃ I 4 y : A ↔  A ↔ I x :  ( A ↔  A )  I 1 [ x :  A ] 1 [ y M z ] 3 [ y M z ] 3 z M z M sr efl [ x M y ] 2 x M z M sub1 z : A  E y :  A  I 3 x :  A  I 2 x :  A ⊃  A ⊃ I 1 Figure 2. Examples of pr oofs in MSQR v M / / w v M / / w M   v M   M   ? U   v M / / w M   M   ? U (ii) (iii) (iv) (iii) and (iv) Figure 3. Some pr oper ties of the relation M • W is a non-empty set of world s (r epr esenting abstr actly the quantu m re g isters ); • U ⊆ W × W is an equivalence r elation ( v U w means that w is o btained by app lying a unitary transformation to v ; U is an equivalenc e r elation since id entity is a unitary transformation, each u nitary transformation mu st be in vertible, and u nitary transformations a r e compos- able); • M ⊆ W × W ( v M w means that w is obtain ed by means of a total measur emen t of v ); with the following pr o perties: (i) ∀ v , w . v M w = ⇒ v U w (ii) ∀ v . ∃ w. v M w (iii) ∀ v , w . v M w = ⇒ w M w (iv) ∀ v , w . v M v & v M w = ⇒ v = w (i) mean s that although it is no t true th at measurement is a unitary transformation, locally for each v , if v M w then ther e is a pa rticular unitary transformation, d epending o n v and w , that generates w fr o m v ; th e vice versa c annot h old, since in q uantum th eory measu r ements can not be used to o btain the unitary evolution of a quantu m system. (ii) mea ns tha t ea ch quantu m r e gister is totally measurable. (iii) an d (i v) together mean that after a to tal measur ement we o btain a classical world. F igur e 3 shows pr ope rties (ii) , (iii) and (iv) , r espectively , as well as the combination of ( iii) and (iv) . 2 2 Note tha t while (i v) says that v is idempotent with respect to M , a unitary transformati on U could still be appli ed to v (and hence the dotted arro w decorat ed with a “?” for U ). 5 ⊃ I , ⊃ E , RAA , ⊥ E , ⋆ I ∗ , ⋆ E , U r efl , U symm , U tr ans , x P y x U y PU I x P y y P z x P z P tr ans [ x P y ] [ y P y ] . . . . α α class ∗ α ( x ) x P x x P y α ( y /x ) P sub1 α ( y ) x P x x P y α ( x/y ) P sub2 In ⋆ I , y is fresh: it is different f rom x and does not occur in any assumption on which y : A depends other than xRy . In class , y is f resh: it is d ifferent from x and does not occur in α nor in a ny assumption on which α depe nds other than x P y and y P y . Figure 4. The rules of MSpQR A mo del is a pair M = h F , V i , where F is a frame an d V : W → 2 Pr o p is an in terpr etation function ma pping worlds into sets of formulas. A stru cture is a pair S = h M , I i , wher e M is a model and I : V ar → W is an in terpr etation func tion mappin g variables ( labels) into worlds in W , and mappin g a r elation symbol R ∈ { U , M } into the corr esponding frame relation I ( R ) ∈ { U, M } . W e e xtend I to formulas and sets of formulas in th e ob vious wa y: I ( x : A ) = I ( x ): A , I ( xRy ) = I ( x ) I ( R ) I ( y ) , a nd I ( { α 1 , . . . , α n } ) = { I ( α 1 ) , . . . , I ( α n ) } . Giv en this semantics, we can define what it means for formulas to be true, and the n pr ove the soundn ess and completeness of MSQR . Definition 3 (T ruth) . Truth for an m–formula in a model M = h W , U, M , V i is the smallest r elation  sa tisfying: M , w  r iff r ∈ V ( w ) M , w  A ⊃ B iff M , w  A = ⇒ M , w  B M , w   A iff ∀ w ′ . wU w ′ = ⇒ M , w ′  A M , w   A iff ∀ w ′ . wM w ′ = ⇒ M , w ′  A Thus, for an m–formula A , we write M  A iff M , w  A for all w . T ruth for a formula α in a structure S = h M , I i is then the smallest r elation  satisfying: M , I  x M y if f I ( x ) M I ( y ) M , I  x U y iff I ( x ) U I ( y ) M , I  x : A iff M , I ( x )  A W e will o mit M when it is not r elevant, and we will den ote I  x : A also by  I ( x ): A or even  w : A for I ( x ) = w . By extension, M , I  Γ iff M , I  α for all α in the set of formulas Γ . Thus, for a set of fo rmulas Γ and a formula α , Γ  α iff ∀ M , I . M  I (Γ) = ⇒ M  I ( α ) iff ∀ M , I . M , I  Γ = ⇒ M , I  α By adapting standard proo fs (see, e.g., [9, 12, 13, 14, 15] and the proo fs in the app endix), we ha ve: Theorem 1 (Soundness and completen ess of MSQR ) . Γ ⊢ α iff Γ  α . 4. Generic measur ements In quantu m com puting, not all measur ements ar e required to be total: th ink, for example, of th e case of ob serving on ly the first qubit of a qu antum register . T o this en d, in this section , we formalize MSpQR , a variant o f MSQR that p rovides a modal system representing all th e possible (th us n ot n ecessarily total) me asurements. W e obtain MSpQR from MSQR by means of the following changes: 6 [ x :  ¬ ( A ⊃  A )] 2 [ x P y ] 1 y : ¬ ( A ⊃  A )  E [ y : A ] 3 [ y P y ] 1 [ y P z ] 4 z : A P sub1 y :  A  I 4 y : A ⊃  A ⊃ I 3 y : ⊥ ¬ E x : ¬  ¬ ( A ⊃  A ) ¬ I 2 x : ¬  ¬ ( A ⊃  A ) class 1 Figure 5. An example pr oof in MSpQR • The a lphabet of the mo dal langu age contain s the un ary mod al op erator  in stead of  , with correspon ding  , whe re  A intuiti vely means that A is true in each quantum register obtained by a measurem ent. • The set of relational formulas contains expressions of the form x P y instead of x M y . • The r ules of MSpQR are given in Figu re 4 . In p articular, ⋆ is either  (as b efore) o r  , for wh ich the n R is P , and whose prope rties are formalized by the following additional rules: – If x P y th en there is a specific unitary transformation (dependin g on x and y ) that gen erates y from x : ru le PU I . – The measuremen t p rocess is transitiv e: rule P tr ans . – There are (always reachab le) classical worlds: class say s that y is a classical world reach able from world x by a measuremen t. – In variance w ith respect to classical worlds for measurement: r ules P sub1 an d P sub2 . Deriv ations and proofs in MSpQR are defin ed as for MSQR . For in stance, in addition to the fo rmulas f or  already listed fo r MSQR , the following labelled fo rmula schemata are all p rovable in MSpQR (as sh own, e.g., f or fo rmula 3 in Figure 5): 1. x :  A ⊃  A (it is alw ays possible to perfor m a measuremen t of a quantum register). 2. x :  A ⊃   A (measurem ents a re composable) . 3. x :  ( A ⊃  A ) , i.e. x : ¬  ¬ ( A ⊃  A ) (it is always po ssible to per form a measu rement with a complete red uction of a quantum register to a classical one). The seman tics is also ob tained by simple c hanges with respe ct to the definitio ns of Section 3. A frame is a tu ple F = h W , U, P i , wh ere P ⊆ W × W and v P w means th at w is o btained by means o f a measure ment of v , with th e following proper ties: (i) ∀ v , w . v P w = ⇒ vU w (as for (i) in Section 3). (ii) ∀ v , w ′ , w ′′ . vP w ′ & w ′ P w ′′ = ⇒ v P w ′′ (measurem ents are composable). (iii) ∀ v . ∃ w. v P w & wP w (each quan tum register v can be redu ced to a classical one w by m eans of a measurement). (iv) ∀ v , w . v P v & v P w = ⇒ v = w (each measurem ent of a classical register v has v as outcome). Models and structur es are defined as before, with I ( P ) = P , while the truth relation now comprises the clauses M , w   A iff ∀ w ′ . wP w ′ = ⇒ M , w ′  A M , I  x P y if f I ( x ) P I ( y ) 7 Finally , MSpQR is also sound and complete. Theorem 2 (Soundness and completen ess of MSpQR ) . Γ ⊢ α if f Γ  α . 5. Conclusions and futur e work W e have shown that our m odal natural dedu ction systems MSQR and MSpQR provide suitable representation s of quan- tum register transfor mations. As futu re work, we plan to investigate the pro of th eory of o ur systems (e.g . normalizatio n, subform ula pr operty , (u n)decidab ility), in view o f a po ssible mechaniza tion o f reasoning in MSQR and MSpQR (e.g. en- coding them into a logical framework [11]) . W e are also working at extending o ur approac h to rep resent an d reason about further quantum notions, such as entanglement. Refer ences [1] S. Abramsky and R. Duncan. A ca tegorical quantum logic. Math. Structur es Comput. Sci. , 16(3):469 –489, 20 06. [2] A. Baltag and S. Smets. The logic of q uantum pro grams. In Pr oceedings of the 2nd Internationa l W orksho p on Quantum Pr ogramming Langu ages QPL , 20 04. [3] A. Baltag and S. Smets. LQP: the d ynamic lo gic of quan tum infor mation. Math. Stru ctur es Compu t. Sci. , 16( 3):491 – 525, 2006. [4] J.-L. Basde vant and J. Dalibard. Quantum mechanics . Sp ringer-V erlag, 2005. [5] G. Birkhoff and J. von Neumann. The log ic of quantum mechanics. An n. of Math. (2) , 37(4):82 3–843 , 19 36. [6] B. F . Ch ellas. Modal Logic . Cambridge Uni versity Press, 1980 . [7] M. L. Dalla Chiara. Quantum lo gic a nd ph ysical modalities. J . Philo s. Logic , 6 (4):391 –404, 1 977. Special issue: Symposium on Quantu m Logic (Bad Homburg, 1976). [8] M. L. Dalla Chiara. Quan tum logic. In D. Gabbay an d F . Guenthn er , editors, Handbo ok o f P hilosophica l Logic: V olume III: Alternatives to Classical Logic , pages 427–4 69. Reidel, 1986. [9] D. M. Gabbay . Labelled Deductive Systems , volume 1. Clarendo n Press, 1996. [10] P . Mittelstaedt. Th e modal logic of quantum logic. J. Philos. Logic , 8(4):479 –504, 19 79. [11] F . Pfenning. Logical frameworks. In A. Robinson and A. V oronkov , editor s, Ha ndboo k of Automated Reasonin g , chapter 17, pages 1063 –1147 . Elsevier Science and MIT P ress, 2001 . [12] D. Prawitz. Natural deduction , a pr o of-theoretical study . Almqv ist and W iksell, 196 5. [13] A. Simpson. The pr o of theo ry a nd semantics of in tuitionistic mo dal logic . PhD thesis, Un i versity of Edinburgh, UK, 1993. [14] A. S. T roelstra and H. Schwichtenberg. Basic pr o of theory . Cambridge Univ ersity Press, 1996. [15] L. V igan ` o. Labelled Non- Classical Logics . Kluwer Academic Publishers, 2000. 8 A Pr oof of soundness and completeness Theorem 1 follows from Theorems 3 and 4 belo w . Theorem 3 (Soundness of MSQR ) . Γ ⊢ α implies Γ  α . Pr o of. W e let M be an arbitrary m odel a nd p rove th at if Γ ⊢ α then  I (Γ) implies  I ( α ) f or any I . The proof p roceeds by induction on the structure of the deriv ation of α from Γ . The base case, where α ∈ Γ , is trivial. There is one step case for each rule of MSQR . Consider an application of the rule RAA , [ x : ¬ A ] . . . . y : ⊥ x : A RAA where Γ ′ ⊢ y : ⊥ with Γ ′ = Γ ∪ { x : ¬ A } . By the inductio n hyp othesis, Γ ′ ⊢ y : ⊥ implies I (Γ ′ )  I ( y ): ⊥ fo r any I . W e assume  I (Γ) and prove  I ( x ): A . Since 2 w : ⊥ f or any world w , from the induction h ypothesis we obtain 2 I (Γ ′ ) , and thus 2 I ( x ): ¬ A , i.e.  I ( x ): A and 2 I ( x ): ⊥ . Consider an application of the rule ⊥ E , x : ⊥ α ⊥ E with Γ ⊢ x : ⊥ . By the indu ction hy pothesis, Γ ⊢ x : ⊥ implies I (Γ)  I ( x ): ⊥ for a ny I . W e assum e  I (Γ) and prove  I ( α ) fo r an arb itrary formula α . I f  I (Γ) then  I ( x ): ⊥ by th e induction hyp othesis. But sin ce 2 w : ⊥ for any world w , then 2 I (Γ) an d thus  I ( α ) for any α . Consider an application of the rule ⋆ I [ xRy ] . . . . y : A x : ⋆ A ⋆ I where Γ ′ ⊢ y : A with y fresh an d with Γ ′ = Γ ∪ { xRy } . By the inductio n hypothesis, fo r all interpretatio ns I , if  I (Γ) then  I ( y ): A . W e let I be any interpretatio n such that  I (Γ) , and sho w th at  I ( x ): ⋆ A . L et w be an y world such t hat I ( x ) I ( R ) w wher e I ( R ) ∈ { U , M } depend ing o n ⋆ . Since I can be trivially extended to anoth er interpretation (still called I for s implicity) by s etting I ( y ) = w , the induction hy pothesis y ields  I ( y ): A , i.e.  w : A , and thus  I ( x ): ⋆ A . Consider an application of the rule ⋆ E x : ⋆ A xRy y : A ⋆ E with Γ 1 ⊢ x : ⋆ A and Γ 2 ⊢ xRy , and Γ ⊇ Γ 1 ∪ Γ 2 . W e assum e  I (Γ) and p rove  I ( y ): A . By the ind uction hypothesis, for all inter pretations I , if  I (Γ 1 ) then  I ( x ): ⋆ A and if  I (Γ 2 ) then  I ( x ) I ( R ) I ( y ) , where I ( R ) ∈ { U, M } depend ing o n ⋆ . If  I (Γ) , th en  I ( x ): ⋆ A and  I ( x ) I ( R ) I ( y ) , and thus  I ( y ): A . The rules U r efl , U symm , and U tr ans ar e sound by the properties of U . The rule U I is sound by prop erty (i) in Definition 2. Consider an application of the rule M ser [ x M y ] . . . . α α M ser with Γ ′ = Γ ∪ { x M y } , fo r y fresh. By the ind uction hypothe sis, Γ ′ ⊢ α im plies I (Γ ′ )  I ( α ) fo r any I . Let us supp ose that the re is an I ′ such th at  I ′ (Γ ′ ) a nd 2 I ′ ( α ) . Let us consider an I ′′ such th at I ′′ ( z ) = I ′ ( z ) for all z suc h that z 6 = y an d I ′′ ( y ) is the world w such th at I ′′ ( y ) M w , wh ich exists by pro perty (ii) in Definition 2. Since y does n ot occur in Γ nor in α , we th en have that  I ′′ (Γ ′ ) and 2 I ′′ ( α ) , con tradicting the universality of the co nsequence of the ind uction hypoth esis. Hen ce, M ser is sou nd. The rule M sr efl is sound by pro perty (iii) in Definition 2. 9 Consider an application of the rule M sub1 α ( x ) x M x x M y α ( y /x ) M sub1 with Γ 1 ⊢ α ( x ) , Γ 2 ⊢ x M x , Γ 3 ⊢ x M y , and Γ ⊇ Γ 1 ∪ Γ 2 ∪ Γ 3 . W e assume  I (Γ) and p rove  I ( α ( y /x )) . By the induction hy pothesis, Γ 1 ⊢ α ( x ) implies I (Γ 1 )  I ( α ( x )) , Γ 2 ⊢ x M x implies I (Γ 2 )  I ( x ) M I ( x ) , and Γ 3 ⊢ x M y implies I (Γ 3 )  I ( x ) M I ( y ) . By pro perty (iv) in Definition 2, we then have I ( x ) = I ( y ) and thus  I ( α ( y /x )): A . The case for rule M sub2 fo llows analogou sly . T o p rove completeness (Theo rem 4), we give some pre liminary definition s and r esults. For simp licity , we will split each set of f ormulas Γ into a p air ( LF , RF ) of the subsets of l–formu las and r–formulas of Γ , and then prove ( LF , RF )  α implies ( LF , RF ) ⊢ α . W e call ( LF , RF ) a context and, slightly abusing notation, we wr ite α ∈ ( LF , RF ) whenever α ∈ LF or α ∈ RF , and wr ite x ∈ ( LF , RF ) whenever the label x occu rs in some α ∈ ( LF , RF ) . W e say that a context ( LF , RF ) is consistent iff ( LF , R F ) 0 x : ⊥ for e very x , so that we hav e: Fact 1. If ( LF , RF ) is co nsistent, then for every x a nd every A , eith er ( LF ∪{ x : A } , RF ) is consistent or ( LF ∪{ x : ¬ A } , RF ) is consistent. Let ( LF , RF ) be the deductive closur e of ( LF , RF ) for r –formu las under the rules of MSQR , i.e. ( LF , RF ) ≡ { xRy | ( LF , R F ) ⊢ xRy } for R ∈ { U , M } . W e say that a context ( LF , RF ) is maximally consistent iff 1. it is consistent, 2. it is deductively clo sed for r –formu las, i .e. ( LF , RF ) = ( LF , R F ) , and 3. for every x an d e very A , either x : A ∈ ( LF , RF ) or x : ¬ A ∈ ( LF , RF ) . Let us wr ite ( LF , RF )  S c α when S c  ( LF , R F ) implies S c  α . Completeness follows by a Hen kin–style proo f, where a canonical structure S c = h M c , I c i = h W c , U c , M c , V c , I c i is built to show that ( LF , RF ) 0 α implies ( LF , RF ) 2 S c α , i.e. S c  ( LF , RF ) and S c 2 α . In standard pr oofs fo r unlab elled modal logics (e.g. [ 6]) an d for other non-classical lo gics, the set W c is obtained by progr essi vely building max imally co nsistent sets of formulas, wher e c onsistency is loca lly checked within each set. In our case, gi ven the presence of l–formu las and r–formulas, we modify the Lind enbaum lemma to e xtend ( LF , RF ) to one single maximally consistent con text ( LF ∗ , RF ∗ ) , wher e consistency is “globally” checked also against the ad ditional assumptions in RF . 3 The elements of W c are then built by partitioning LF ∗ and RF ∗ with respect to the labels, and the relation s R between the worlds are defined by e xploiting the inform ation in RF ∗ . In the Linde nbaum lemma for predicate logic, a maximally consistent and ω - complete set of formulas is induc ti vely b uilt by add ing for every for mula ¬∀ x.A a witness to its tr uth, namely a fo rmula ¬ A [ c/x ] for som e new ind i vidual con stant c . This ensures th at the resulting set is ω -complete, i.e. that if, fo r every closed term t , A [ t/x ] is co ntained in the set, th en so is ∀ x.A . A similar pro cedure applies h ere in the case of l– formulas o f the form x : ¬ ⋆ A . That is, tog ether with x : ¬ ⋆ A we consistently ad d y : ¬ A and xRy for some new y , wh ich acts as a witness world to the tr uth of x : ¬ ⋆ A . T his ensu res that the maximally c onsistent context ( LF ∗ , RF ∗ ) is such that if xRz ∈ ( LF ∗ , RF ∗ ) implies z : B ∈ ( LF ∗ , RF ∗ ) f or ev ery z , then x : ⋆ B ∈ ( LF ∗ , RF ∗ ) , as shown in Lem ma 2 belo w . Note that in the standard completeness proof for unlabelled modal logics, on e instead con siders a canon ical model M c and shows that if w ∈ W c and M c , w  ¬ ⋆ A , th en W c also co ntains a world w ′ accessible from w that serves as a witness world to the truth of ¬ ⋆ A at w , i.e. M c , w ′  ¬ A . Lemma 1. Every consistent context ( LF , RF ) can be e xtended to a maximally consistent con te xt ( LF ∗ , RF ∗ ) . 3 W e consider only consistent cont ext s. If ( LF , RF ) is inconsi stent, then LF , RF ⊢ x : A for all x : A , and thu s completen ess immediately holds for l–formulas. O ur language does not allow us to define inconsistenc y for a set of r–formulas, but, whene ver ( LF , RF ) is inconsist ent, the canonical model buil t in the follo wing is nonetheless a counter -model to non-deri vable r–formulas. 10 Pr o of. W e first extend the lang uage of MSQR with infinitely many n ew co nstants for witn ess worlds. Systematically let b range over labe ls, c range over th e new constants for witness worlds, and a range over both. All these may be subscrip ted. Let l 1 , l 2 , . . . be an enumeration of all l– formulas in the extended language; whe n l i is a : A , we write ¬ l i for a : ¬ A . Starting from ( LF 0 , RF 0 ) = ( LF , RF ) , we inductively build a sequen ce of consistent contexts by defining ( LF i +1 , RF i +1 ) to be: • ( LF i , RF i ) , if ( LF i ∪ { l i +1 } , RF i ) is inconsistent; else • ( LF i ∪ { l i +1 } , RF i ) , if l i +1 is not a : ¬ ⋆ A ; else • ( LF i ∪ { a : ¬ ⋆ A, c : ¬ A } , RF i ∪ { aRc } ) for a c 6∈ ( LF i ∪ { a : ¬ ⋆ A } , RF i ) , if l i +1 is a : ¬ ⋆ A . Every ( LF i , RF i ) is consistent. T o show this we show th at if ( LF i ∪ { a : ¬ ⋆ A } , RF i ) is consistent, then so is ( LF i ∪ { a : ¬ ⋆ A, c : ¬ A } , RF i ∪ { aRc } ) fo r a c 6∈ ( LF i ∪ { a : ¬ ⋆ A } , RF i ) ; the other cases fo llow by construction . W e pr oceed by contrapo sition. Su ppose that ( LF i ∪ { a : ¬ ⋆ A, c : ¬ A } , RF i ∪ { aRc } ) ⊢ a j : ⊥ where c 6∈ ( LF i ∪ { a : ¬ ⋆ A } , RF i ) . Th en, by RAA , ( LF i ∪ { a : ¬ ⋆ A } , RF i ∪ { aRc } ) ⊢ c : A , and ⋆ I y ields ( LF i ∪ { a : ¬ ⋆ A } , RF i ) ⊢ a : ⋆ A . Since also ( LF i ∪ { a : ¬ ⋆ A } , RF i ) ⊢ a : ¬ ⋆ A , by ¬ E we have ( LF i ∪ { a : ¬ ⋆ A } , RF i ) ⊢ a : ⊥ , i.e. ( LF i ∪ { a : ¬ ⋆ A } , RF i ) is inconsistent. Contradiction . Now define ( LF ∗ , RF ∗ ) = ( [ i ≥ 0 LF i , [ i ≥ 0 RF i ) W e show that ( LF ∗ , RF ∗ ) is maximally c onsistent, by sho wing that it satisfies the three conditions in th e d efinition of maximal consistency . For the first condition, note that if ( [ i ≥ 0 LF i , [ i ≥ 0 RF i ) is consistent, then so is ( [ i ≥ 0 LF i , [ i ≥ 0 RF i ) . Now suppo se that ( LF ∗ , RF ∗ ) is in consistent. Then for some finite ( LF ′ , RF ′ ) in cluded in ( LF ∗ , RF ∗ ) there exists an a such th at ( LF ′ , RF ′ ) ⊢ a : ⊥ . Every l– formula l ∈ ( LF ′ , RF ′ ) is in some ( LF j , RF j ) . For each l ∈ ( LF ′ , RF ′ ) , let i l be the least j such that l ∈ ( LF j , RF j ) , and let i = max { i l | l ∈ ( LF ′ , RF ′ ) } . Then ( LF ′ , RF ′ ) ⊆ ( LF i , RF i ) , and ( LF i , RF i ) is inconsistent, which is not the case. The second condition is satisfied by definition of ( LF ∗ , RF ∗ ) . For the third condition, su ppose that l i +1 6∈ ( LF ∗ , RF ∗ ) . Then l i +1 6∈ ( LF i +1 , RF i +1 ) and ( LF i ∪ { l i +1 } , RF i ) is inconsistent. Thus, by Fact 1, ( LF i ∪ { ¬ l i +1 } , RF i ) is consistent, an d ¬ l i +1 is consistently added to some ( LF j , RF j ) during the construction, and therefore ¬ l i +1 ∈ ( LF ∗ , RF ∗ ) . The following lemma states some pro perties of maximally consistent contexts. Lemma 2. Let ( LF ∗ , RF ∗ ) be a maximally consistent context. Th en 1. ( LF ∗ , RF ∗ ) ⊢ a i Ra j iff a i Ra j ∈ ( LF ∗ , RF ∗ ) . 2. ( LF ∗ , RF ∗ ) ⊢ u : A iff a : A ∈ ( LF ∗ , RF ∗ ) . 11 3. a : B ⊃ C ∈ ( LF ∗ , RF ∗ ) if f a : B ∈ ( LF ∗ , RF ∗ ) implies a : C ∈ ( LF ∗ , RF ∗ ) . 4. a i : ⋆ B ∈ ( LF ∗ , RF ∗ ) if f a i Ra j ∈ ( LF ∗ , RF ∗ ) implies a j : B ∈ ( LF ∗ , RF ∗ ) for all a j . Pr o of. 1 an d 2 follow immediately by defin ition. W e only treat 4 as 3 f ollows an alogously . For th e left-to-r ight d irection, suppose th at a i : ⋆ B ∈ ( LF ∗ , RF ∗ ) . Then, b y (ii), ( LF ∗ , RF ∗ ) ⊢ a i : ⋆ B , and, by ⋆ E , we have ( LF ∗ , RF ∗ ) ⊢ a i Ra j implies ( LF ∗ , RF ∗ ) ⊢ a j : B for all a j . By 1 and 2 , conclud e a i Ra j ∈ ( LF ∗ , RF ∗ ) implies a j : B ∈ ( LF ∗ , RF ∗ ) for all a j . For the conv erse, suppose th at a i : ⋆ B 6∈ ( LF ∗ , RF ∗ ) . Then a i : ¬ ⋆ B ∈ ( LF ∗ , RF ∗ ) , and, by the constru ction of ( LF ∗ , RF ∗ ) , there exists an a j such that a i Ra j ∈ ( LF ∗ , RF ∗ ) and a j : B 6∈ ( LF ∗ , RF ∗ ) . W e ca n now d efine the canonical structure S c = h M c , I c i = h W c , U c , M c , V c , I c i Definition 4. Given a maximal consistent context ( LF ∗ , RF ∗ ) , we define the canonical structure S c as follows: • W c = { a | a ∈ ( LF ∗ , RF ∗ ) } , • ( a i , a j ) ∈ U c iff a i U a j ∈ ( LF ∗ , RF ∗ ) , • ( a i , a j ) ∈ M c iff a i M a j ∈ ( LF ∗ , RF ∗ ) , • V c ( r ) = a if f a : r ∈ ( LF ∗ , RF ∗ ) , • I c ( a ) = a . Note that the standard definition of R c adopted for unlabelled modal logics, i.e. ( a i , a j ) ∈ R c iff { A |  A ∈ a i } ⊆ a j , is not app licable in our setting, since { A |  A ∈ a i } ⊆ a j does not imply ⊢ a i Ra j . W e would theref ore be unable to pr ove completen ess fo r r–formu las, sinc e there would be cases, e.g. when RF = { } , where 0 a i Ra j but ( a i , a j ) ∈ R c and thus S c  a i Ra j . Hence, we in stead defin e ( a i , a j ) ∈ R c iff a i Ra j ∈ ( LF ∗ , RF ∗ ) ; note that ther efore a i Ra j ∈ ( LF ∗ , RF ∗ ) implies { A |  A ∈ a i } ⊆ a j . As a fu rther compa rison with the standar d definition, n ote that in the ca nonical mode l th e label a can be iden tified with the set of formu las { A | a : A ∈ ( LF ∗ , RF ∗ ) } . More over , we immediately ha ve: Fact 2. a i Ra j ∈ ( LF ∗ , RF ∗ ) if f ( LF ∗ , RF ∗ )  S c a i Ra j . The deductive closure of ( LF ∗ , RF ∗ ) f or r–formulas ensures not only co mpleteness for r–formula, as shown in Th eorem 4 below , b ut also that the conditions on R c are satisfied, so that S c is really a structure for MSQR . Mor e concretely: • U c is an e quiv alence relation by construction an d ru les U r efl , U symm , an d U t r ans . For instance, fo r transiti vity , consider an arb itrary c ontext ( LF , RF ) from wh ich we build S c . Assume ( a i , a j ) ∈ U c and ( a j , a k ) ∈ U c . Then a i U a j ∈ ( LF ∗ , RF ∗ ) and a j U a k ∈ ( LF ∗ , RF ∗ ) . Since ( LF ∗ , RF ∗ ) is deductiv ely closed, by 1 in Lemma 2 and rule U tr ans , we have a i U a k ∈ ( LF ∗ , RF ∗ ) . Thus, ( a i , u k ) ∈ U c and U c is indeed transitiv e. • ∀ v , w ∈ W c . vM w = ⇒ v U w h olds by construction and rule U I . • ∀ v ∈ W c . ∃ w ∈ W c . vM w holds by con struction and rule M ser . For the sake o f contrad iction, consider an arbitr ary a i and a variable a ′ j that do not satisfy the prop erty . Define ( LF ′ , RF ′ ) = ( LF ∗ , RF ∗ ) ∪ { a i M a ′ j } . Then it canno t be the case tha t ( LF ′ , RF ′ ) ⊢ α , for otherwise ( LF ∗ , RF ∗ ) ⊢ α would be deriv able b y an application of the rule M ser . Thus, ( LF ′ , RF ′ ) 0 α . But then ( LF ′ , RF ′ ) mu st be in the chain of co ntexts built in Lemma 2. So , by the maximality of ( LF ∗ , RF ∗ ) , we have that ( LF ′ , RF ′ ) = ( LF ∗ , RF ∗ ) , contrad icting our assumption . Hence, for some a j , the r–formula a i M a j is in ( LF ∗ , RF ∗ ) , which is what we had to show . • ∀ v , w ∈ W c . vM w = ⇒ w M w holds by con struction and rule M sr efl . 12 • ∀ v , w ∈ W c . v M v & v M w = ⇒ v = w hold s by constru ction and rules M sub1 and M sub2 sin ce v is a classical world. Consider an arbitrar y co ntext ( LF , R F ) fro m which we build S c and assume ( a i , a i ) ∈ M c and ( a i , a j ) ∈ M c . Then a i M a i ∈ ( LF ∗ , RF ∗ ) an d a i M a j ∈ ( LF ∗ , RF ∗ ) . Thu s, fo r each a i : A ∈ ( LF ∗ , RF ∗ ) , we also hav e a j : A ∈ ( LF ∗ , RF ∗ ) ; otherwise, since ( LF ∗ , RF ∗ ) is ded uctiv ely closed, w e would hav e a j : ¬ A ∈ ( LF ∗ , RF ∗ ) a nd also a j : A ∈ ( LF ∗ , RF ∗ ) b y 1 in Lemm a 2 and rule M s ub1 , and thus a contrad iction. Sim ilarly , if a j : A ∈ ( LF ∗ , RF ∗ ) then a i : A ∈ ( LF ∗ , RF ∗ ) b y rule M sub2 . Hen ce, for each m–form ula A , we ha ve that a i : A ∈ ( LF ∗ , RF ∗ ) iff a j : A ∈ ( LF ∗ , RF ∗ ) , which means that a i and a j are equal with respect to m–form ulas. Under the same assumptions, we can similarly show that a i and a j are equal with respect to r –form ulas, i.e. that when- ev er ( LF ∗ , RF ∗ ) contains an r–form ula that includes a i then it also con tains the same r–formula with a j substituted for a i , and vice versa. T o this end, we must consider 8 dif ferent cases correspon ding to 8 different r–formulas. – If a k U a i ∈ ( LF ∗ , RF ∗ ) for som e a k , th en from th e assum ption that a i M a j ∈ ( LF ∗ , RF ∗ ) we h av e a i U a j ∈ ( LF ∗ , RF ∗ ) , by 1 in Lemm a 2 and rule U I . T herefor e, a k U a j ∈ ( LF ∗ , RF ∗ ) by rule U tr ans . – W e can reason similarly for a j U a k ∈ ( LF ∗ , RF ∗ ) and also apply r ules U I and U tr ans to conclude that then also a i U a k ∈ ( LF ∗ , RF ∗ ) . – If a i U a k ∈ ( LF ∗ , RF ∗ ) for som e a k , th en from th e assum ption that a i M a j ∈ ( LF ∗ , RF ∗ ) we h av e a i U a j ∈ ( LF ∗ , RF ∗ ) , by 1 in Lemma 2 and rule U I , and thus a j U a i ∈ ( LF ∗ , RF ∗ ) , b y rule U symm . Theref ore, a j U a k ∈ ( LF ∗ , RF ∗ ) by rule U tr ans . – W e can reason similarly for a k U a j ∈ ( LF ∗ , RF ∗ ) and also ap ply ru les U I , U symm , and U t r ans to conclude that then also a k U a i ∈ ( LF ∗ , RF ∗ ) . – If a k M a i ∈ ( LF ∗ , RF ∗ ) for some a k , then f rom the assum ption that a i M a j ∈ ( LF ∗ , RF ∗ ) we h a ve a k M a j ∈ ( LF ∗ , RF ∗ ) , by 1 in Lemm a 2 and the derived rule M tr ans . – W e can reason similarly for a j M a k ∈ ( LF ∗ , RF ∗ ) and also ap ply rule M tr ans to conc lude that th en also a i U a k ∈ ( LF ∗ , RF ∗ ) . – If a i M a k ∈ ( LF ∗ , RF ∗ ) for some a k , then from the assumptions that a i M a i ∈ ( LF ∗ , RF ∗ ) and a i M a j ∈ ( LF ∗ , RF ∗ ) we hav e a j M a k ∈ ( LF ∗ , RF ∗ ) , by 1 in Lemma 2 and rule M sub1 . – W e c an reason similarly for a k M a j ∈ ( LF ∗ , RF ∗ ) and apply rule M sub2 to con clude that th en also a k M a i ∈ ( LF ∗ , RF ∗ ) . Hence, a i and a j are eq ual also with r espect to r –form ulas, and th us a i = a j whenever ( a i , a i ) ∈ M c and ( a i , a j ) ∈ M c , which is what we had to show . By Lemma 2 and Fact 2, it follows that: Lemma 3. a : A ∈ ( LF ∗ , RF ∗ ) if f ( LF ∗ , RF ∗ )  S c a : A . Pr o of. W e proceed by in duction o n th e g rade of a : A , a nd we treat only the step case where a : A is a i : ⋆ B ; the other cases follow analog ously . For the l eft-to-r ight directio n, assume a i : ⋆ B ∈ ( LF ∗ , RF ∗ ) . Then, by Lemma 2, a i Ra j ∈ ( LF ∗ , RF ∗ ) implies a j : B ∈ ( LF ∗ , RF ∗ ) , for all a j . F act 2 and th e in duction hyp othesis yield that ( LF ∗ , RF ∗ )  S c a j : B for a ll a j such that ( LF ∗ , RF ∗ )  S c a i I c ( R ) a j , i.e. ( LF ∗ , RF ∗ )  S c a i : ⋆ B by Definition 3. For the converse, assum e a i : ¬ ⋆ B ∈ ( LF ∗ , RF ∗ ) . T hen, by Lemma 2, a i Ra j ∈ ( LF ∗ , RF ∗ ) and a j : ¬ B ∈ ( LF ∗ , RF ∗ ) , fo r some a j . Fact 2 and the induction hypoth esis yield ( LF ∗ , RF ∗ )  S c a i Ra j and ( LF ∗ , RF ∗ )  S c a j : ¬ B , i.e. ( LF ∗ , RF ∗ )  S c a i : ¬ ⋆ B by Definition 3. W e ca n now fina lly sho w: Theorem 4 (Completeness of M SQR ) . Γ  α implies Γ ⊢ α . Pr o of. If ( LF , RF ) 0 b i Rb j , then b i Rb j 6∈ ( LF ∗ , RF ∗ ) , and thus ( LF ∗ , RF ∗ ) 2 S c b i Rb j by Fact 2. If ( LF , RF ) 0 b : A , then ( LF ∪ { b : ¬ A } , RF ) is c onsistent; o therwise the re exists a b i such th at ( LF ∪ { b : ¬ A } , R F ) ⊢ b i : ⊥ , and then ( LF , RF ) ⊢ b : A . There fore, by Lemma 1, ( LF ∪ { b : ¬ A } , RF ) is includ ed in a maximally consistent co ntext (( LF ∪ { b : ¬ A } ) ∗ , RF ∗ ) . Then, by Lemma 3, (( LF ∪ { b : ¬ A } ) ∗ , RF ∗ )  M C b : ¬ A , i.e. ( ( LF ∪ { w : ¬ A } ) ∗ , RF ∗ ) 2 S c b : A , and thus ( LF , RF ) 2 S c w : A . 13 W e can r eason similarly t o s how the so undness and completeness of MSpQR with respect to the correspo nding sem antics: Theorem 2 follows from Theorems 5 and 6 below . Theorem 5 (Soundness of MSpQR ) . Γ ⊢ α implies Γ  α . Pr o of. W e let M be an arbitrary m odel a nd p rove th at if Γ ⊢ α then  I (Γ) implies  I ( α ) f or any I . The proof p roceeds by induction on the structure of the deriv ation of α from Γ . The base case, where α ∈ Γ , is trivial. There is one step case for each rule of MSpQR , where the so undness of the ru les ⊃ I , ⊃ E , RAA , ⊥ E , U r efl , U s ymm , U tr ans fo llows exactly like in the proof of Theorem 3. The s ound ness of the rules ⋆ I and ⋆ E follows exactly like in the proof of Theo rem 3, with the only difference th at when ⋆ is  then R is P . The rule PU I is sound by prop erty (i) i n the definition of the semantics for MSpQR . The rule P tr ans is sou nd by property (ii) in the definition of the semantics for MSpQR . The sou ndness of the rule class follows like for th e sound ness of the rule M ser in the pro of of Theorem 3, this time exploiting property (iii) in the definition of the semantics for MSpQR . The soun dness of the ru les P sub1 and P su b2 follows lik e for the soun dness of th e rules M sub1 and M sub2 in the proof of Theorem 3, this time exploiting property (iv) i n the definition of the semantics for MSpQR . T o prove completen ess (Theorem 4) , we p roceed like fo r the c ase o f MSQR , m utatis mutan dis in th e con struction of the canonical mo del. In particular, given a maxima l con sistent context ( LF ∗ , RF ∗ ) , we define the can onical structure S c = h W c , U c , P c , V c , I c i by setting • ( a i , a j ) ∈ P c iff a i P a j ∈ ( LF ∗ , RF ∗ ) . T o show that the cond itions on R c are satisfied, so tha t S c is really a structu re for MSpQR , we re use the results pr oved for MSQR an d in addition sho w the following: • ∀ v , w ∈ W c . vP w = ⇒ v U w h olds by construction and rule PU I . • ∀ v , w ′ , w ′′ ∈ W c . v P w ′ & w ′ P w ′′ = ⇒ v P w ′′ holds by constru ction and rule P tr ans . • ∀ v ∈ W c . ∃ w ∈ W c . v P w & w P w holds by construction and rule class . For the sake of co ntradiction , consider a n arbitrary a i and a variable a ′ j that do n ot satisfy the prop erty . Define ( LF ′ , RF ′ ) = ( LF ∗ , RF ∗ ) ∪ { a i P a ′ j , a ′ j P a ′ j } . Then it can not be the case th at ( LF ′ , RF ′ ) ⊢ α , fo r otherwise ( LF ∗ , RF ∗ ) ⊢ α wou ld be der i vable by an ap plication of the rule class . Thu s, ( LF ′ , RF ′ ) 0 α . But then ( LF ′ , RF ′ ) must be in the c hain of con texts b uilt in Lemma 2. So, by the maximality of ( LF ∗ , RF ∗ ) , we have th at ( LF ′ , RF ′ ) = ( LF ∗ , RF ∗ ) , contradictin g ou r assumption. He nce, for some a j , the r –formu las a i M a j and a j M a j are both in ( LF ∗ , RF ∗ ) , which is what we had to show . • ∀ v , w ∈ W c . v P v & vP w = ⇒ v = w holds by con struction and rules P sub1 an d P s u b2 since v is a classical world. Consider an arbitrary context ( LF , R F ) from which we build S c and ass ume ( a i , a i ) ∈ P c and ( a i , a j ) ∈ P c . Then a i P a i ∈ ( LF ∗ , RF ∗ ) and a i P a j ∈ ( LF ∗ , RF ∗ ) . Thus, for each a i : A ∈ ( LF ∗ , RF ∗ ) , we also h a ve a j : A ∈ ( LF ∗ , RF ∗ ) ; otherwise, sin ce ( LF ∗ , RF ∗ ) is de ductively closed, we w ould have a j : ¬ A ∈ ( LF ∗ , RF ∗ ) and also a j : A ∈ ( LF ∗ , RF ∗ ) by 1 in Lemma 2 and ru le P sub1 , and th us a con tradiction. Similarly , if a j : A ∈ ( LF ∗ , RF ∗ ) then a i : A ∈ ( LF ∗ , RF ∗ ) b y rule P sub2 . Hence , for each m–formula A , we hav e that a i : A ∈ ( LF ∗ , RF ∗ ) iff a j : A ∈ ( LF ∗ , RF ∗ ) , which means that a i and a j are equal with respect to m–form ulas. Under the same assumptions, we can similarly show that a i and a j are equal with respect to r –form ulas, i.e. that when- ev er ( LF ∗ , RF ∗ ) contains an r–form ula that includes a i then it also con tains the same r–formula with a j substituted for a i , and vice versa. T o this end, we must consider 8 dif ferent cases correspon ding to 8 different r–formulas. – If a k U a i ∈ ( LF ∗ , RF ∗ ) for some a k , then fr om the assumption that a i P a j ∈ ( LF ∗ , RF ∗ ) we have a i U a j ∈ ( LF ∗ , RF ∗ ) , by 1 in Lemm a 2 and rule PU I . T herefor e, a k U a j ∈ ( LF ∗ , RF ∗ ) by rule U tr ans . – W e c an reason similarly for a j U a k ∈ ( LF ∗ , RF ∗ ) and also apply rules PU I a nd U tr ans to conclude that then also a i U a k ∈ ( LF ∗ , RF ∗ ) . – If a i U a k ∈ ( LF ∗ , RF ∗ ) for some a k , then fr om the assumption that a i P a j ∈ ( LF ∗ , RF ∗ ) we have a i U a j ∈ ( LF ∗ , RF ∗ ) , by 1 in Le mma 2 an d rule PU I , a nd thus a j U a i ∈ ( LF ∗ , RF ∗ ) , by rule U symm . Ther efore, a j U a k ∈ ( LF ∗ , RF ∗ ) by rule U tr ans . 14 – W e can r eason similar ly fo r a k U a j ∈ ( LF ∗ , RF ∗ ) a nd also apply rules PU I , U symm , a nd U t r ans to conclu de that then also a k U a i ∈ ( LF ∗ , RF ∗ ) . – If a k P a i ∈ ( LF ∗ , RF ∗ ) for some a k , then fr om the a ssumption that a i P a j ∈ ( LF ∗ , RF ∗ ) we have a k P a j ∈ ( LF ∗ , RF ∗ ) , by 1 in Lemm a 2 and the rule P tr ans . – W e can reason similarly for a j P a k ∈ ( LF ∗ , RF ∗ ) and also apply rule P tr ans to conclud e that then also a i U a k ∈ ( LF ∗ , RF ∗ ) . – If a i P a k ∈ ( LF ∗ , RF ∗ ) for some a k , th en from th e assumptions th at a i P a i ∈ ( LF ∗ , RF ∗ ) and a i P a j ∈ ( LF ∗ , RF ∗ ) we hav e a j P a k ∈ ( LF ∗ , RF ∗ ) , by 1 in Lemm a 2 and rule P sub1 . – W e can reason similarly for a k P a j ∈ ( LF ∗ , RF ∗ ) and apply rule P sub2 to conclud e that then also a k P a i ∈ ( LF ∗ , RF ∗ ) . Hence, a i and a j are eq ual also with respect to r–formulas, and thus a i = a j whenever ( a i , a i ) ∈ P c and ( a i , a j ) ∈ P c , which is what we had to show . Proceeding like for MSQR , we then ha ve: Theorem 6 (Completeness of M SpQR ) . Γ  α implies Γ ⊢ α . △ 15