QUASIPOLYNOMIAL NORMALISATION IN DEEP INFERENCE VIA ATOMIC FLOWS AND THRESHOLD FORMULAE

Transcription

1 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE VI TOMIC FLOWS ND THRESHOLD FORMULE POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT BSTRCT Jeřábek showed ha cus in classical proposiional logic proos in deep inerence can be eliminaed in quasipolynomial ime The proo is indirec and i relies on a resul o serias Galesi and Pudlák abou monoone sequen calculus and a correspondence beween ha sysem and cu-ree deep-inerence proos In his paper we give a direc proo o Jeřábek s resul: we give a quasipolynomial-ime cu-eliminaion procedure or classical proposiional logic in deep inerence The main new ingredien is he use o a compuaional race o deep-inerence proos called aomic lows which are boh very simple (hey only race srucural rules and orge logical rules) and srong enough o aihully represen he cu-eliminaion procedure INTRODUCTION Deep inerence is a proo-heoreic mehodology where proos can be reely composed by he logical operaors insead o having a rigid ormula-direced ree srucure as in Genzen proo heory [Gug07 BT0 Brü04 GGP0] s a resul inerence rules apply arbirarily deep inside ormulae conrary o radiional proo sysems such as naural deducion and he sequen calculus where inerence rules only deal wih he ouermos srucure o ormulae This induces a new symmery which can be exploied or achieving localiy o inerence rules and which is no available wih Genzen mehods Localiy in urn makes i possible o use new mehods oen wih a geomeric lavour in he normalisaion heory o proo sysems The greaer reedom in composing proos o deep inerence is boh a source o immediae echnical diiculy and o new powerul proo-heoreic mehods general mehodology allows us o design deep-inerence proo sysems having more symmeries and iner srucural properies han he sequen calculus does For insance cu and ideniy become really dual o each oher whereas hey only are morally so in he sequen calculus and all he srucural rules can be reduced o heir aomic orm whereas his does no hold in he sequen calculus or example in he case o he conracion inerence rule In deep inerence he cu rule is more general han is counerpar in he sequen calculus and makes i possible o obain a broader range o dynamics in normalisaion procedures However despie he sequen calculus sysems and heir normalisaion procedures being special cases o deep inerence sysems and procedures cu eliminaion in deep inerence sill guaranees consisency and he rivial urning o proo sysems ino algorihms or proo search Dae: February Bruscoli and Guglielmi have been suppored by EPSRC grans EP/E042805/ Complexiy and Nondeerminism in Deep Inerence and EP/K08868/ Eicien and Naural Proo Sysems and by an NR Senior Chaire d Excellence iled Ideniy and Geomeric Essence o Proos Gundersen has been suppored by an Overseas Research Sudenship and a Research Sudenship boh o he Universiy o Bah and by an NR Senior Chaire d Excellence iled Ideniy and Geomeric Essence o Proos Parigo has been suppored by Projec INFER Theory and pplicaion o Deep Inerence o he gence Naionale de la Recherche Gundersen and Parigo are suppored by Projec STRUCTURL Srucural and Compuaional Proo Theory o he gence Naionale de la Recherche

2 2 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT Sixeen years o research have guaraneed ha all usual logics have deep-inerence proo sysems enjoying cu eliminaion (see [Gug] or a complee overview) The radiional mehods o cu eliminaion o he sequen calculus can be adaped o a large exen o deep inerence despie having o cope wih a higher generaliy bu new mehods are also achievable The sandard proo sysem or proposiional classical logic in deep inerence is sysem SKS and is cu eliminaion has been achieved in several dieren ways [BT0 Brü04 GG08] all requiring a wors exponenial ime in he size o he proo o be normalised ew years ago Jeřábek showed ha cu eliminaion in SKS proos can be done in quasipolynomial ime [Jeř09] more speciically in ime n O(log n) The resul is surprising because all known cu-eliminaion mehods or classical-logic proo sysems require exponenial ime in paricular or Genzen s sequen calculus Jeřábek obained his resul by relying on a consrucion over hreshold uncions by serias Galesi and Pudlák in he monoone sequen calculus [GP02] Jeřábek s echnique is indirec because normalisaion is perormed over proos in he sequen calculus which are in urn relaed o deep-inerence ones by polynomial simulaions originally sudied in [Brü06] and [BG09] In his paper we give a direc proo o Jeřábek s resul: ha is we give a quasipolynomial-ime cu-eliminaion procedure in proposiional-logic deep inerence which in addiion o being inernal has a srong compuaional lavour Our proo uses wo ingrediens: () an adapaion o serias-galesi-pudlák echnique o deep inerence which simpliies he echnicaliies associaed wih he use o hreshold uncions; in paricular he ormulae and derivaions ha we adop are smaller and srucurally simpler han hose in [GP02]; (2) a recenly inroduced graphic ormalism racing aoms in deep-inerence proos called aomic lows [GG08] omic lows which can be considered specialised Buss low graphs [Bus9] play a major role in designing and conrolling he cu eliminaion procedure presened in his paper omic lows are very simple (hey only race srucural rules and orge logical rules) bu hey are srong enough o aihully represen cu eliminaion [GG08 GGS0] and o relae several ormalisms regarding heir proo complexiy [Das2 Das4] omic lows conribue o he overall clariicaion o his highly echnical maer by reducing our dependency on synax The echniques developed via aomic lows olerae variaions in he proo sysem speciicaion In ac heir geomeric naure makes hem largely independen o synax provided ha cerain lineariy condiions are respeced (and his is usually achievable in deep inerence) The paper is sel-conained Secions 2 and 3 are devoed respecively o he necessary background on deep inerence and aomic lows Threshold uncions and ormulae are inroduced in Secion 5 We normalise proos in wo seps each o which has a dedicaed secion in he paper: () We ransorm any given proo ino wha we call is simple orm No use is made o hreshold ormulae and no signiican proo complexiy is inroduced This is presened in Secion 4 mosly an exercise on deep inerence and aomic lows (2) In Secion 6 we show he cu eliminaion sep saring rom proos in simple orm Here hreshold ormulae play a major role Normalisaion can be aken one sep urher by removing he insances o he only inerence rule le ha is no analyic in he deep-inerence sense viz coweakening This is perormed by a simple and sandard deep-inerence procedure in Secion 7 Secion 8 concludes he paper wih commens on uure research direcions Pars o his paper were presened a LPR 6 [BGGP0] and some appear in [Gun09] Recenly hreshold uncions have been used in [Das4] o build quasipolynomial size

3 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 3 cu-ree deep-inerence proos o he proposiional pigeonhole principle ha crucially do no use coconracion which is a orm o dagness 2 PROPOSITIONL LOGIC IN DEEP INFERENCE Inside he deep-inerence mehodology we can deine several ormalisms ie general prescripions on how o design proo sysems in he same sense as he sequen calculus and naural deducion are ormalisms in Genzen-syle proo heory (where he srucure o proos is deermined by he ree srucure o he ormulae hey prove) The irs and concepually simpler ormalism ha has been deined in deep inerence is called he calculus o srucures or CoS [Gug07] noher deep-inerence ormalism has laer been inroduced in [GGP0] called open deducion Open deducion is more general han CoS in he sense ha every CoS derivaion is also an open-deducion derivaion On he oher hand every open-deducion derivaion can be ransormed ino a CoS derivaion by a sraighorward ransormaion ha essenially amouns o inerleaving derivaions The cos o his ransormaion is a mos quadraic in he size o he original open-deducion derivaion; hereore rom he poin o view o complexiy CoS and open deducion are equivalen CoS and open deducion are equivalen also rom he poin o view o proo heory because he wo ormalisms are jus wo dieren noaions or derivaions o he same naure and so every derivaion ransormaion ha can be perormed in one ormalism can also be perormed in he oher In his paper we will adop he open-deducion noaion especially because i is more eicien or he reader However given ha mos o he lieraure in deep inerence adops he CoS noaion which is more similar o he radiional Genzen synax we will presen boh syles in his secion The sandard proo sysem o proposiional logic in deep inerence is called SKS The basic proo-complexiy properies o SKS and so o proposiional logic in deep inerence have been sudied in [BG09] (which also could be used as an inroducion o SKS) Those properies are: SKS is polynomially equivalen o Frege proo sysems SKS can be exended wih Tseiin s exension and subsiuion and he proo sysems so obained are polynomially equivalen o Frege proo sysems augmened wih exension and subsiuion Cu-ree SKS polynomially simulaes cu-ree Genzen proo sysems or proposiional logic bu he converse does no hold: in ac Saman s auologies admi polynomial proos in cu-ree SKS bu only exponenial ones in cu-ree Genzen [Sa78] We now quickly inroduce all he necessary noions n excellen and more relaxed inroducion o SKS in CoS and is basic properies is [Brü04] Formulae denoed by B C and D are reely buil rom: unis (alse) (rue); aoms denoed by a b c d and e; disjuncion and conjuncion [ B] and ( B) The dieren brackes have he only purpose o improving legibiliy; we usually omi exernal brackes o ormulae and someimes we omi superluous brackes under associaiviy On he se o aoms a (non-idenical) involuion is deined and called negaion; a and ā are dual aoms We denoe conexs ie ormulae wih a hole by K{ } and H{ }; or example i K{a} is b [a c] hen K{ } is b [{ } c] K{b} is b [b c] and K{a d} is b [(a d) c] Noe ha negaion is only deined or aoms and his is no a limiaion because hanks o De Morgan laws negaion can always be pushed o aoms lso noe ha here are no negaive or posiive aoms in an absolue sense; we can only say ha i we arbirarily consider ā posiive hen a mus be negaive or example

4 4 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT For boh CoS and open deducion an (inerence) rule ρ is an expression ρ where he B ormulae and B are called premiss and conclusion respecively; an insance o ha rule is an expression C ρ where C and D are insances o and B D Sysem SKS is a proo sysem common o CoS and open deducion deined by he ollowing srucural inerence rules: ai a ā aw a a a ac a ideniy weakening conracion a ā ai a aw a ac a a cu coweakening coconracion and by he ollowing wo logical inerence rules: [B C ] s ( B) C swich ( B) (C D) m [ C ] [B D] medial cu-ree derivaion is a derivaion where ai is no used ie a derivaion in SKS \ {ai } C In addiion o hese rules here is a rule = such ha C and D are opposie sides in one o he ollowing equaions: D () B = B = B = B = [ B] C = [B C ] = ( B) C = (B C ) = We do no always show he insances o rule = and when we do show hem we gaher several coniguous insances ino one We consider he = rule as implicily presen in all sysems The equaliy relaion = on ormulae is deined by closing he equaions in () by relexiviy symmery ransiiviy and by sipulaing ha = B implies K{} = K{B}; o indicae lieral equaliy o he ormulae and B we adop he noaion B We now deine boh syles o derivaions CoS and open deducion The dierence is in he way we compose insances o rules: in CoS we only allow inerences o compose verically in chains similar o sequen calculus proos made only o one-premiss rule insances In open deducion insead derivaions can be composed by he same connecives ha ormulae are made o For simpliciy we give here a deiniion o open-deducion derivaion ha is limied o our purposes in his paper and is no he mos general C In CoS a rule insance ρ generaes an (inerence) sep K{C } ρ or each conex K{ } D K{D} CoS derivaion rom (premiss) o (conclusion) B is a chain o inerence seps wih a he op and B a he boom derivaion can be denoed by Φ B where is he name o he proo sysem or a se o inerence rules (we migh omi Φ and ); a proo oen denoed by Π is a derivaion wih premiss ; besides Φ we denoe derivaions wih Ψ Someimes we group n 0 inerence seps o he same rule ρ ogeher ino one sep and we label he sep wih n ρ

5 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 5 In open deducion and jus or he speciic case o proposiional logic wih and negaion on aoms we deine he noions o derivaion premiss and conclusion inducively as ollows: i Φ is a uni or an aom hen Φ is a derivaion wih premiss Φ and conclusion Φ; i Φ is a derivaion wih premiss and conclusion B and i Ψ is a derivaion wih premiss C and conclusion D hen [Φ Ψ] is a derivaion wih premiss [ C ] and conclusion [B D] (Φ Ψ) is a derivaion wih premiss ( C ) and conclusion (B D) B i ρ is an insance o an inerence rule hen Φ ρ is a derivaion wih premiss C Ψ and conclusion D We adop he same convenions as or CoS o denoe derivaions in open deducion We omi srucural rule names in open-deducion noaion The irs wo rows in Figure 2 illusrae wih examples all he conceps inroduced above The irs row shows hree example CoS derivaions and below each o hem here is an equivalen derivaion in open deducion n open deducion derivaion can be obained rom a CoS one by sharing he conexs in inerence seps Vice versa a CoS derivaion can be obained rom an open deducion one by choosing an order or he chain o inerence seps Besides SKS anoher sandard deep-inerence sysem is SKSg which is he same as SKS excep ha i does no conain medial and is srucural rules are no resriced o aoms In paricular we use in his paper he rules w w c and c Clearly a derivaion in SKS is also a derivaion in SKSg I can easily be proved ha SKS and all is ragmens conaining he logical and = rules polynomially simulae respecively SKSg and is corresponding ragmens [BG09] For example {s m = ac } polynomially simulaes {s = c } This allows us o ranser properies rom SKS o SKSg; in paricular he main resul in his paper ie ha SKS proos can be ransormed ino cu-ree ones in quasipolynomial ime holds also or SKSg proos One reason o work wih SKS insead o SKSg as we do in his paper is ha aomiciy o rules allows us o use aomic lows more convenienly noable cu-ree sysem is KS = {s m= ai aw ac } which is complee or proposiional logic [BT0 Brü04]; his o course enails compleeness or all he sysems ha conain KS such as SKS We can replace insances o nonaomic srucural rules by derivaions wih he same premiss and conclusion and ha only conain aomic srucural rules The price o pay is a quadraic growh in size This is saed by he ollowing rouine proposiion (keep in mind ha rom now on we consider he = rule as implicily presen in all sysems) n example is he righmos upper derivaion in Figure 2 which sands or a nonaomic coconracion Proposiion Rule insances o w w c and c can be derived in quadraic ime by derivaions in {aw } {aw } {m ac } and {m ac } respecively Someimes we use a nonaomic rule insance o sand or some derivaion in SKS ha derives ha insance as per Proposiion By {a /B a h /B h } we denoe he operaion o simulaneously subsiuing ormulae B B h ino all he occurrences o he aoms a a h in he ormula respecively; noe ha he occurrences o ā ā h are no auomaically subsiued Oen we only subsiue cerain occurrences o aoms and hese are indicaed wih superscrips ha esablish a relaion wih aomic lows s a maer o ac we exend he noion o

6 6 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT subsiuion o derivaions in he naural way bu his requires a cerain care The issue is clariied in Secion 3 (see in paricular Noaions 2 and 4 and Proposiion 3) The size o a ormula and he size Φ o a derivaion Φ is he number o uni and aom occurrences appearing in i The size o CoS derivaions is obviously a mos quadraic in he size o he corresponding open-deducion derivaions We use his ac implicily hroughou he paper and we always measure he CoS size o derivaions even i we show hem in open-deducion noaion 3 TOMIC FLOWS omic lows which have been inroduced in [GG08] are essenially specialised Buss low graphs [Bus9] They are paricular direced graphs associaed wih SKS derivaions: every derivaion yields one aomic low obained by racing he aom occurrences in he derivaion Ininiely many derivaions correspond o each aomic low; his suggess ha much o he inormaion in a derivaion is los in is associaed aomic low; in paricular here is no inormaion abou insances o logical rules only srucural rules play a role s shown in [GG08 GGS0] i urns ou ha aomic lows conain suicien srucure o conrol normalisaion procedures providing in paricular inducion measures ha can be used o ensure erminaion Such normalisaion procedures require exponenial ime on he size o he derivaion o be normalised In he presen work we lower he complexiy o proo normalisaion o quasipolynomial ime bu an essenial role is played by he complex logical relaions o hreshold ormulae which are exernal and independen rom he given proo This means ha aomic lows are no suicien o deine he normalisaion procedure; however hey sill are a very convenien ool or deining and undersanding several o is aspecs We can single ou hree eaures o aomic lows ha in general and no jus in his work help in designing normalisaion procedures: () omic lows convenienly express he opological srucure o aom occurrences in a proo This is especially useul or deining a cerain simple orm o proos (Deiniion 7) (2) omic lows provide or an eicien way o conrol subsiuions or aom occurrences in derivaions This helps us o deine he cu-ree orm o proos (Deiniion 23) (3) We can deine graph rewriing sysems over aomic lows ha conrol normalisaion procedures on derivaions This can be used o conrol a urher reinemen o he normalisaion procedure (Theorem 27) Our aim now is o quickly and inormally provide he necessary noions abou aomic lows especially concerning aspecs () and (2) above lhough he eaure (3) o aomic lows namely graph rewriing sysems o lows did help us in obaining proos in normal orm we esimae ha ormally inroducing he necessary machinery is unjusiied in his paper In ac given our limied needs here we can operae direcly on derivaions wihou he inermediae suppor o aomic lows Noneheless being aware o he underlying aomic-low mehods is useul or he reader who wishes o urher invesigae his maer So we inormally provide in Secion 7 enough maerial o make he connecion wih he aomic-low echniques ha are ully developed in [GG08] We obain one aomic low rom each derivaion by racing all is aom occurrences and by keeping rack o heir creaion and desrucion (in ideniy/cu and weakening/coweakening insances) heir duplicaion (in conracion/coconracion) and heir dualiy (in ideniy/cu) Technically aomic lows are direced graphs o a special kind bu i is more inuiive o consider hem as diagrams generaed by composing elemenary aomic lows ha belong o one o seven kinds

7 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 7 ai a ā 2 2 aw a 2 ac a a a a ā 2 ai 2 a aw a 3 3 ac a a 2 2 FIGURE Verices o aomic lows The irs kind o elemenary aomic low is he edge which corresponds o one or more occurrences o he same aom in a given derivaion all o which are no acive in any srucural rule insance ie hey are no he aom occurrences ha insaniae a srucural rule The oher six kinds o elemenary diagrams are associaed wih he six srucural inerence rules as shown in Figure and hey are called verices; each verex has some inciden edges he le o each arrow we see an insance o a srucural rule where he aom occurrences are labelled by small numerals; a he righ o he arrow we see he verex corresponding o he rule insance whose inciden edges are labelled in accord wih he aom occurrences hey correspond o We qualiy each verex according o he rule i corresponds o; or example in a given aomic low we migh alk abou a conracion verex or a cu verex and so on Insead o small numerals someimes we use ε or ι or colour o label edges (as well as aom occurrences) bu we do no always use labels ll edges are direced bu we do no explicily show he orienaion Insead we consider i as implicily given by he way we draw hem namely edges are oriened along he verical direcion So he verices corresponding o dual rules in Figure are disinc or example an ideniy verex and a cu verex are dieren because he orienaion o heir edges is dieren On he oher hand he horizonal direcion plays no role in disinguishing aomic lows; his corresponds o commuaiviy o logical relaions We can deine (aomic) lows as he smalles se o diagrams conaining elemenary aomic lows and closed under he composiion operaion consising in ideniying zero or more edges such ha no cycle is creaed In addiion or a diagram o be an aomic low i mus be possible o assign i a polariy according o he ollowing deiniion polariy assignmen is a mapping o each edge o an elemen o { +} such ha he wo edges o each ideniy or cu verex map o dieren values and he hree edges o each conracion or coconracion verex map o he same value We denoe aomic lows by φ and ψ Le us see some examples The low (2) is obained by juxaposing (ie composing by ideniying zero edges): hree edges a low obained by composing a cu verex wih a coconracion verex and a low obained by composing an ideniy verex wih a cu verex Noe ha here are no cycles in he low and ha we can ind 32 dieren polariy assignmens ie wo or each o he ive conneced componens o he low (his is a general rule)

8 8 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT Le us see anoher example These are hree dieren represenaions o he same low: and 3 4 where we label edges o show heir correspondence In he wo righmos lows we indicae he wo dieren polariy assignmens ha are possible The ollowing wo diagrams are no aomic lows: and The le one is no a low because i conains a cycle and he righ one because here is no possible polariy assignmen Le us see how o exrac aomic lows rom derivaions Given an SKS derivaion Φ we obain by he ollowing prescripions a unique aomic low φ such ha here is a surjecive map beween aom occurrences in Φ and edges o φ: Each srucural inerence sep in Φ is associaed wih one and only one verex in φ such ha acive aom occurrences in he rule insance map o edges inciden wih he verex The correspondence is indicaed in Figure For example he low associaed wih he inerence sep a he le is indicaed a he righ: a b 2 a 3 a 4 b 2 a 5 and ac a Noe ha he nonacive aoms are raced by associaing each race wih one K{a a} edge; his corresponds well o abbreviaing say he inerence sep ac by K{a} a a K a For each oher inerence sep in Φ all he aom occurrences in he premiss are respecively mapped o he same edges o φ as he aom occurrences in he conclusion For example he low associaed wih he inerence sep a b 2 c 3 d 4 e 5 m a is b 2 d 4 c 3 e 5 The low φ so obained is called he aomic low associaed wih he derivaion Φ We show hree examples in Figure 2: in he op row we see hree SKS derivaions in he sandard CoS synax; in he row below we show he same derivaions in he open deducion noaion; in he boom row we see he hree corresponding aomic lows Perhaps surprisingly i can be proved ha every low is associaed wih ininiely many SKS derivaions (see [GG08]) We inroduce now some graphical shorcus When cerain deails o a low are no imporan bu only he verex kinds and is upper and lower edges are we can use boxes labelled wih all he verex kinds ha can appear in he low hey represen For example he ollowing le and cenre lows could represen he previously seen low (2) whereas 5

9 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 9 ai a ā = (a ) ( ā) m [a ] [ ā] = [a ] [ā ] s ([a ] ā) = (ā [a ]) s [(ā a) ] = (a ā) ai = a ā m [a ] [ ā] [a ] ā s s a ā (a [ā ]) ā ai (a [ā [ā a]]) ā = (a [[ā ā] a]) ā s [(a [ā ā]) a] ā ac [(a ā) a] ā ai [ a] ā = a ā ac (a a) ā = a (a ā) ai a a ā ā a s a ā ā ā a a a ā = a a ā [a b] a ac [(a a) b] a ac [(a a) (b b)] a ac [(a a) (b b)] (a a) m ([a b] [a b]) (a a) = ([a b] a) ([a b] a) a b a a b b m [a b] [a b] a a a FIGURE 2 Examples o derivaions in CoS and open deducion noaion and associaed aomic lows he righ low canno: φ ψ ψ and The low a he righ canno represen low (2) because i has he wrong number o lower edges and because a necessary cu verex is no allowed by he labelling o he boxes s jus shown we someimes label boxes wih he name o he low hey represen For example low φ above could represen low (2) and i he cenre low sands or (2) hen lows ψ and ψ are respecively and When no verex labels appear on a box we assume ha he verices in he corresponding low can be any (so i does no mean ha here are no verices in he low)

10 0 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT We someimes use a double line noaion or represening muliple edges For example he ollowing diagrams represen he same low: ε ε l ψ ι ι m and ε l ψ ι m where l m 0; noe ha we use ε l o denoe he vecor (ε ε l ) We migh label muliple edges wih one o he ormulae ha he associaed aom occurrences orm in a derivaion We exend he double line noaion o collecions o isomorphic lows For example or m 0 he ollowing diagrams represen he same low: ι ι m and ι m We observe ha he low o every SKS derivaion can always be represened as a collecion o m 0 conneced componens as ollows: φ ψ φ m ψ m such ha each edge in low φ i is associaed wih some occurrence o some aom a i and each edge in low ψ i is associaed wih some occurrence o aom ā i Noe ha i migh happen ha or i j we have a i a j I we do no insis on dealing wih conneced componens we can adop he same represenaion as above and sipulae ha i j implies a i a j ā j This would mean ha he derivaion only conains occurrences o aoms a a m such ha hese aoms and heir dual are all muually disinc Noe ha no maer how we assign a polariy all he edges in φ i and all hose in ψ i are respecively mapped o dual polariy values Given a polariy assignmen we alk abou negaive and posiive rule insances o (co)weakening and (co)conracion rules according o wheher he edges inciden wih he associaed verices map o or + respecively In he ollowing when inormally dealing wih derivaions we reely ranser o hem noions deined or heir lows For example we can say ha an aom occurrence is negaive or a given polariy assignmen (i he edge associaed wih he aom occurrence maps o ) or ha wo aom occurrences are conneced (i he associaed edges belong o he same conneced componen) In ac one o he advanages o working wih lows is ha hey provide us wih convenien geomerical noions s we menion a he beginning o his secion aomic lows help in selecively subsiuing or aom occurrences In ac given a derivaion and is associaed low we can use edges and boxes o individuae aom occurrences in he derivaion and hen possibly subsiue or hem For example le us suppose ha we are given he ollowing associaed derivaion and low: (a ) a ā m Φ = a a ā and = a ā ā

11 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE We can hen disinguish beween he hree occurrences o ā ha are mapped o edge and he one ha is no as in (a ) a ā m Φ = a a a ā ā ; = ā we can also subsiue or hese occurrences or example by {ā /}; such a siuaion occurs in he proo o Theorem Noe ha simply subsiuing or ā would invalidae his derivaion because i would break he cu and weakening insances; however he proo o Theorem speciies how o ix such broken insances We generalise his labelling mechanism o boxes For example we can use a dieren represenaion o he low o Φ o individuae wo classes a φ and ā φ o aom occurrences as ollows: (a ) a āφ m Φ = a a a φ āφ ā φ and φ = ā φ In order o deine he noion o cu-ree orm (Deiniion 23) we need he ollowing proposiion which we sae here because i consiues a good exercise abou aomic lows Noe ha in he ollowing we use several boxes labelled by φ: his means ha we are dealing wih several copies o he same low φ Noaion 2 Given a ormula in a derivaion whose associaed aomic low conains a low φ we indicae wih a φ every occurrence o he aom a in whose associaed edge is in φ So as in he ollowing Proposiion 3 {a φ /B ā φ / B} sands or he ormula where he aom occurrences o a and is dual whose associaed edges are in φ are subsiued wih ormula B and is dual respecively Proposiion 3 Given a derivaion Φ SKS le is associaed low have shape φ ψ such ha φ is a conneced componen each o whose edges is associaed wih aom a or ā; hen or any ormula B here exiss a derivaion {a φ /B ā φ / B} Ψ SKS {a φ /B ā φ / B}

12 2 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT whose associaed low is φ φ ψ }{{} m where m is he number o aom occurrences in B; moreover he size o Ψ depends a mos linearly on he size o Φ and quadraically on he size o B Proo We can proceed by srucural inducion on B and hen on φ For he wo cases o B C D and B C D we have o consider or each verex o φ one o he ollowing siuaions: C C D D s C C D D s C D C D and heir dual ones C D C D C C C D D D (C D) (C D) m C C C D D D Noaion 4 In he hypoheses o Proposiion 3 we can describe Ψ as Φ{a φ /B ā φ / B}; one o a φ /B or ā φ / B migh be missing when no ideniy or cu verices are presen in φ 4 NORMLISTION STEP : SIMPLE FORM The irs sep in our normalisaion procedure deined here consiss in rouine deepinerence manipulaions which are bes undersood in conjuncion wih aomic lows For his reason his secion is a useul exercise or a reader who is no amiliar wih deep inerence and aomic lows In his secion we deine proos in simple orm in Deiniion 7 and we show ha every proo can be ransormed ino simple orm in Theorem Le us esablish he ollowing convenions (hey are especially useul o simpliy our dealing wih hreshold ormulae in he nex secions o he paper) Noaion 5 We use a n m o denoe he vecor (a m a m+ a n ) Convenion 6 When we alk abou a se o disinc aoms we mean ha no wo aoms are he same or dual Deiniion 7 Given a proo Π o in SKS i here exis n 0 disinc aoms a a n such ha he proo and is aomic low have shape respecively a ā φ Ψ a n ā φ n n a āφ a n āφ n n and we say ha Π is in simple orm (over a n ) and ha Ψ is a simple core o Π Proos in simple orm are such ha all he cu insances are conneced o ideniy insances via lows he φ i ones above ha only have one lower edge The idea is ha in a proo in simple orm we can subsiue ormulae or all he occurrences o aoms ā i ha map o some edge in φ i wihou alering he conclusion o he proo O course doing φ φ n

13 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 3 his would invalidae ideniy and cu insances bu we acually only need he simple core o he proo Our normalisaion procedure essenially relies on gluing ogeher simple cores where we subsiue he a i aom occurrences ha map o edges in φ i wih cerain ormulae called pseudocomplemens (see Secion 5 and Deiniion 23) Remark 8 proo in simple orm over a 0 is cu-ree In order o prove Theorem we need wo acs Proposiion 9 and Lemma 0 In he ollowing (rouine) proposiion we use he swich rule s o push ouside or pull inside a ormula relaive o a conex K{ } Proposiion 9 For any conex K{ } and ormula here exis derivaions whose size is less han K{} 2 and have shape K{} {s} K{} and K{} {s} K{} Proo We only build he derivaion a he le in he claim he consrucion being dual or he one a he righ We reason by inducion on he number n o - alernaions in he ormula-ree branch o { } in K{ } I n = 0 hen K{} = K{} I n > 0 consider H{} {s} B H{} C s (H{} B) or some conex H{ } and ormulae B and C such ha K{ } = (H{ } B) C and he number o - alernaions in he ormula-ree branch o { } in H{ } is n The number o s insances is n and we have ha n K{} Noe ha he aomic lows o he derivaions in he previous proposiion only consis o edges because no srucural rules appear To prove Theorem we could now proceed as ollows Given a proo we assign i (and is low) an arbirary polariy under cerain assumpions ha we can always easily saisy We hen ocus on he negaive pahs connecing ideniy and cu verices I coconracion verices lie along hese pahs we have a poenial problem because some aoms in he conclusion o he proo migh be conneced o aoms in some ideniy insances This would preven us rom subsiuing pseudocomplemens as previously menioned because by doing so we would aler he conclusion o he proo However we can solve he problem by replacing each coconracion verex by an appropriae low involving ideniy cu and conracion verices in such a way ha he only conracion verex so inroduced is posiive cually he lemma below akes a more radical approach which simpliies exposiion and also has broader applicaion: we replace all negaive conracion and coconracion insances This unnecessarily bloas he proo bu sill says well inside polynomial bounds Lemma 0 Given any derivaion Φ SKS B we can in linear ime in he size o Φ consruc a derivaion SKS B

14 4 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT such ha is aomic low has shape ε l and such ha no wo aoms associaed wih ε ε l are dual or some l 0 Proo ssign a polariy o he low o Φ such ha no wo dual aoms are boh associaed wih negaive edges; hen replace each negaive conracion insance as ollows: K ΨSKS ā ā ā Ψ SKS B becomes ΨSKS [ā ā] a ā a K [ā ā] a a s a [(a ā) ā] s a ā a ā s ā Ψ SKS This corresponds in he low o replacing each negaive conracion verex as ollows: B becomes 3 Proceed analogously wih negaive coconracion insances We are now ready o prove he main resul o his secion Theorem Given any proo Π o in SKS we can in cubic ime in he size o Π consruc a proo o in simple orm Proo We proceed in hree seps () By Lemma 0 we can ransorm Π in linear ime in is size ino a proo whose low has shape Π SKS ε l ι m where l m 0 and such ha no wo aoms associaed wih ε ε l are dual For i m we successively ransorm Π as ollows or some Π Φ Φ K{ }

15 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 5 and H{ }: H Π K āι i Φ a ā ι i Φ becomes Π K{} Φ{ā ι i /} a H Φ This way we obain in linear ime a proo Π SKS whose low is ε l and whose size is smaller han Π (2) Thanks o Proposiion 9 or i l we successively ransorm Π as ollows or some Ψ Ψ and K { }: Ψ K a ā ε i Ψ becomes a ā ε i [a ā] Ψ [a ā ε i ] K {} {s} K {a ā ε i } Ψ ; we also apply he dual ransormaion or each ai insance This way we obain a proo a ā ε a l ā ε l l a āε Ψ a l ā ε l l whose low is he same as ha o Π because each ransormaion conserves he low I Π = n and given ha n > 2l he size o each derivaion inroduced by virue o Proposiion 9 is a mos 4n 2 So each o he 2l ransormaions increases he size o he proo by O(n 2 ) which makes or a oal complexiy o O(n 3 )

16 6 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT (3) Consider b n such ha b b n are disinc and {a a l } = {b b n } We can build in linear ime he proo b b b b n n {c } [a ā ε ] [a l āε l ] Ψ (a ā ε ) (a l āε l ) l {c } b b b n b n which is in simple orm over b n We can hen obain a proo in SKS in ime O(n 2 ) because o Proposiion The ransormaion in Sep () in he previous proo is a case o weakening reducion or aomic lows sudied in [GG08] In Secion 7 we commen more on his Remark 2 In general given a proo Π and by he consrucion in he proo o Theorem we can obain several dieren simple orms rom Π In ac apar rom permuaions o rule insances commuaiviy and associaiviy he simple orms depend on he choice o a polariy assignmen (Lemma 0) l 5 THRESHOLD FORMULE We presen here he main consrucion o his paper ie a class o derivaions Γ ha only depend on a given se o aoms and ha allow us o normalise any proo conaining hose aoms The complexiy o he Γ derivaions dominaes he complexiy o he normal proo and is due o he complexiy o cerain hreshold ormulae on which he Γ derivaions are based The Γ derivaions are consruced in Deiniion 9; his direcly leads o Theorem 2 which saes a crucial propery o he Γ derivaions and which is he main resul o his secion Threshold ormulae realise boolean hreshold uncions which are deined as boolean uncions ha are rue i and only i a leas k o n inpus are rue (see [Weg87] or a horough reerence on hreshold uncions) In he ollowing x denoes he maximum ineger n such ha n x There are several ways o encoding hreshold uncions ino ormulae and he problem is o ind among hem an encoding ha allows us o obain Theorem 2 Eicienly obaining he propery saed in Theorem 2 crucially depends also on he proo sysem we adop The ollowing class o hreshold ormulae which we ound o work or sysem SKS is a simpliicaion o he one adoped in [GP02] Deiniion 3 Consider n > 0 disinc aoms a a n and le p = n/2 and q = n p; or k 0 we deine he hreshold ormulae θ n k an as ollows: or any n > 0 le θ n 0 an ; or any n > 0 and k > n le θ n k an ; θ (a ) a ; or any n > and 0 < k n le θ n k an i+ j =k 0i p 0 j q θ p i a p θq j an p+

17 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 7 θ 2 0 (a b) θ 2 (a b) (θ (a) θ 0 (b)) (θ 0 (a) θ (b)) (a ) ( b) = a b θ 2 2 (a b) θ (a) θ (b) a b θ 3 0 (a b c) θ 3 (a b c) (θ (a) θ2 0 (b c)) (θ 0 (a) θ2 (b c)) (a ) ( [(b ) ( c)]) = a b c θ 3 2 (a b c) (θ (a) θ2 (b c)) (θ 0 (a) θ2 2 (b c)) = (a [b c]) (b c) θ 3 3 (a b c) θ (a) θ2 2 (b c) (a (b c)) = a b c θ 5 0 (a b c d e) θ 5 (a b c d e) (θ2 (a b) θ3 0 (c d e)) (θ2 0 (a b) θ3 (c d e)) = a b c d e θ 5 2 (a b c d e) (θ2 2 (a b) θ3 0 (c d e)) (θ2 (a b) θ3 (c d e)) (θ2 0 (a b) θ3 2 (c d e)) = (a b) ([a b] [c d e]) (c [d e]) (d e) θ 5 3 (a b c d e) (θ2 2 (a b) θ3 (c d e)) (θ2 (a b) θ3 2 (c d e)) (θ2 0 (a b) θ3 3 (c d e)) = (a b [c d e]) ([a b] [(c [d e]) (d e)]) (c d e) θ 5 4 (a b c d e) (θ2 2 (a b) θ3 2 (c d e)) (θ2 (a b) θ3 3 (c d e)) = (a b [(c [d e]) (d e)]) ([a b] c d e) θ 5 5 (a b c d e) θ2 2 (a b) θ3 3 (c d e) = a b c d e θ 5 6 (a b c d e) FIGURE 3 Examples o hreshold ormulae Compared o he deiniion in [GP02] we require i + j = k insead o i + j k The semanics o he ormulae does no change bu heir size is smaller and heir srucure is much simpler arguably beneiing urher research See in Figure 3 some examples o hreshold ormulae The only reason why we require aoms o be disinc in hreshold ormulae is o avoid cerain echnical problems wih subsiuions in he deiniion o cu-ree orm laer on However here is no subsanial diiculy in relaxing his deiniion o any se o aoms The ormulae or hreshold uncions adoped in [GP02] correspond or each choice o k and n o ik θn i an We presume ha [GP02] employs hese more complicaed ormulae because he ormalism adoped here he sequen calculus is less lexible han deep inerence requiring more inormaion in hreshold ormulae in order o consruc suiable derivaions Remark 4 For n > 0 we have θ n an = a a n and θn n an = a a n

18 8 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT The size o he hreshold ormulae dominaes he cos o he normalisaion procedure so we evaluae heir size We leave as an exercise he proo o he ollowing proposiion Proposiion 5 For any n > 0 and k 0 θ n k an θ n n/2 + an Lemma 6 The size o θ n n/2 + an is no(log n) Proo Observe ha θ n k an θ n+ a n+ k Le p = n/2 and q = n p and consider: θ n p+ an = θ p i+ j = p+ i a p + θ q j an p+ 0i p 0 j q (3) θ q i+ j = p+ + 0i j q 2(q + ) i aq θ q q/2 + aq θ q j aq where we use Proposiion 5 We show ha or h = 2/(log 3 log 2) and or any n > 0 we have n h log n We reason by inducion on n; he case n = rivially θ n n/2 + an holds By he inequaliy (3) and or n > we have 2(n n/2 + )(n n/2 ) h log(n n/2 ) θ n n/2 + an n 2 n h log(2n/3) = n h log n h(log3 log2)+2 = n h log n Theorem 7 For any k 0 he size o θ n k an is no(log n) Proo I immediaely ollows rom Proposiion 5 and Lemma 6 Given a hreshold ormula θ n k an we can consider or each a l such ha l n he ormulae (θ n k an ){a l /} and (θn k+ an ){a l /}: we call boh o hem inormally pseudocomplemens o a l The reason or his name is ha we can manage o replace in a given proo all occurrences o hose ā l ha appear in cu insances wih he pseudocomplemens o a l The cu insances and heir corresponding ideniy insances are hen removed leaving us wih derivaions whose premiss and conclusion conain each a hreshold ormula Moreover he k-level o he hreshold ormula in he premiss is one less han he k-level o he hreshold ormula in he conclusion This way we obain several derivaions corresponding o increasing values o k ha we are able o sich ogeher unil we ge a normalised proo ll his o course needs clariicaion and or many i migh be helpul only aer having grasped he ull proo in is echnical orm However we hink ha i is convenien here o provide a summary o he main consrucions allowing or his siching operaion Le us read derivaions op-down; he ollowing are he seps ha we need o perorm or 0 k n () Build θ n k an a l (θ n k an ){a l /} ie creae rom a k-level hreshold ormula a disjuncion beween a l and is pseudocomplemen (θ n k an ){a l /} (Proposiion 22); hen replace he pseudocomplemen ino ā l or each ideniy insance (2) Increase he k-level by using he derivaions (θ n k an ){a l /} (θ n k+ an ){a l /}

19 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 9 (Theorem 2); hese are he Γ derivaions menioned in he inroducion o his secion (3) For each cu insance collec he conjuncion beween a l and is pseudocomplemen (θ n k+ an ){a l /}; hen build a l (θ n k+ an ){a l /} θ n k+ an ie creae a (k + )-level hreshold ormula (Proposiion 22) The derivaions menioned above do no require any use o ideniy and cu and allow us o move in n + seps rom θ n 0 an o θn n+ an which is he secre o success The consrucions in and 3 are deep-inerence rouine and inroduce low complexiy We deal now wih he crucial sep 2 by designing Deiniion 9 and hen checking i careully so as o ge he propery saed in Theorem 2 Deiniion 9 is echnical bu is philosophy is simple; all one has o do o build he derivaions required by Theorem 2 is: ideniy he aom occurrences ha mus occur in he premiss and ha mus no occur in he conclusion and remove hem using coweakening and ideniy he aom occurrences ha mus occur in he conclusion and ha mus no occur in he premiss and add hem using weakening We have implemened Deiniion 9 as a program [Gug09] I can be useul o read he deiniion ogeher wih he examples in Figures 4 and 3 which have been generaed by he program Remark 8 Given n > le p = n/2 and q = n p For 0 k q and l p he ollowing derivaion is well deined: (θ p p a p ){a l /} θq k an p+ = a a l a l + a p θq k an p+ w w nalogously or 0 k p and p + l n we can deine he ollowing derivaion: θ p a p k (θq q a n p+ ){a l /} = θ p a p k a p+ a l a l + a n w w Boh classes o derivaions are used in Deiniion 9 Deiniion 9 Consider n > 0 disinc aoms a a n and le p = n/2 and q = n p For n > and l n we deine he derivaions Υ n kl an and n kl an as ollows: (θ p p a p ){a l /} θq k p an p+ w i p k n and l p Υ n kl an = θ p a p k q (θq q a n p+ ){a l /} w i q k n and p < l oherwise and n kl an = w θ q k an p+ w θ p k a p i 0 < k q and l p i 0 < k p and p < l oherwise

20 20 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT Γ 5 0 a = b c d e Γ 5 a = b [c d e] b (c [d e]) (d e) Γ 5 2 a = (b [c d e]) [(c [d e]) (d e)] b b Γ 5 3 a = (b [(c [d e]) (d e)]) b Γ 5 4 a = (b c d e) b [(c [d e]) (d e)] Γ 5 5 a = b c d e Γ 5 03 a = d e a b Γ 5 3 [a a = b] d e d e d e a b Γ 5 23 a a = b [a b] d e d e d e Γ 5 33 a a = b d e [a b] (d e) d e Γ 5 43 a a = b (d e) Γ 5 53 a = a b d e Γ 5 05 a = d Γ 5 5 a = [a b] Γ 5 25 a = a b Γ 5 35 a = a b Γ 5 45 a = a b c a b [d e] d c c Γ 5 55 a = a b c d d c d (c d) d c [a b] c d e b [c d e] c d e [a b] d e d c d d [a b] [a b] c d [d e] (d e) [d e] a b d d (c d) d d e (c d) d c d FIGURE 4 Examples o Γ 5 a where a = (a b c d e) kl

21 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 2 For k 0 and l n we deine he derivaions Γ n kl an recursively on n as ollows: Γ 0 (a ) = ; or k > 0 Γ (a k ) = ; or k > n Γ n kl an = ; or n > and k n le Γ n kl an = i+ j =k 0i< p 0 j q i+ j =k 0i p 0 j <q Γ p il a p θq j an p+ Υ n kl an n k+l an p θ i a p Γq j l p an p+ Υ n kl an n k+l an i l p i p < l Example 20 See in Figure 4 some example o derivaions Γ n kl an Noe ha or clariy we removed all insances o he rivial derivaions Υ 2 a2 = Υ2 2 a2 = Υ3 a3 = w We can do so because hese derivaion insances appear as disjuncs Theorem 2 For any n > 0 k 0 and l n he derivaion Γ n kl an has shape and Γ n kl an is n O(log n) (θ n k an ){a l /} {aw aw } (θ n k+ an ){a l /} Proo The shape o Γ n kl an can be veriied by inspecing Deiniion 9 For example his is he case when n > and l p k < q where p = n/2 and q = n p: (θ n k an ){a l /} Γ n kl an (θ n k+ an ){a l /} = (θ p i a p ){a l /} Γ p i+ j =k a p il θ q (θ p 0i< p (θ p 0 j q i+ a p ){a l /} j an p a p ){a l /} θq k p an p+ p+ w (Remember ha θ n k an i+ j =k 0i p 0 j q p θ i a p θq j an p+ w θ q k+ an p+ and θ p 0 a p ) General (co)weakening rule insances can be replaced by aomic ones because o Proposiion The size bound on Γ n kl an ollows rom Proposiion and Theorem 7 6 NORMLISTION STEP 2: CUT-FREE FORM In his secion we deine he cu-ree orm o proos based on proos in simple orm Proos in cu-ree orm have no cu insances bu can have coweakening ones which preven hese proos rom being analyic (in he sense ha aoms appear in premisses only i hey do so in conclusions) Theorem 24 he main resul o he secion shows how o obain a cu-ree proo rom any proo Mos o he ingenuiy o quasipolynomially normalising an SKS proo ino one in analyic SKS resides in going rom a simple orm o a cu-ree one Removing coweakening insances rom a cu-ree orm is easy; we dedicae Secion 7 o his

22 22 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT Beore deining he cu-ree orm we need o esablish he ollowing ac Proposiion 22 For any ormula and aom a here exis derivaions whose size is cubic in and ha have shape {aw ac s} a {a/} and a {a/} {aw ac s} Proo I here are no occurrences o a in he desired derivaions are a a (h ) ac a = a {a/} and and a = I here are h > 0 occurrences o a in obain by repeaedly applying Proposiion 9 he ollowing derivaions: a (h ) ac {s} a a {a/} I = n he size o he desired derivaions is O(n 3 ) because we have o apply Proposiion 9 a mos O(n) imes Deiniion 23 For n > 0 le Π be a proo in simple orm over a n such ha i and is aomic low have shape {s} a ā φ Ψ a n ā φ n n and φ φ n a āφ a n āφ n n or some derivaion Ψ For 0 i n + le θ i θ n i an For 0 k n we deine he derivaions θ k Φ k SKS\{ai } θ k+ as θ k (n ) c θ k θ k {aw ac s} {aw ac s} a θ k {a /} a n θ k {a n /} a θ k {a /} Γ n k an {aw aw } θ k+ {a /} {aw ac s} Ψ k SKS\{ai } θ k+ θ k+ (n ) c θ k+ a n θ k {a n /} Γ n kn an {aw aw } θ k+ {a n /} {aw ac s}

23 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 23 φ 0 φ θ θ k φ k φ ψ k θ k+ θ n φ n φ FIGURE 5 omic low o a proo in cu-ree orm where Ψ k = Ψ{ā φ /θ k {a /}āφ n n /θ k {a n /}} and where we use Proposiion 22 We deine he cu-ree orm o Π as he ollowing proo in SKS \ {ai }: θ 0 Φ 0 θ Φ θ 2 θ n Φ n θn+ n c

24 24 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT (We recall ha θ 0 and θ n+ ) Theorem 24 Given any proo Π o in SKS we can consruc a proo o in SKS \ {ai } in ime quasipolynomial in he size o Π Proo By Theorem we can consruc rom Π in polynomial ime a proo Π o in simple orm We can hen proceed wih he consrucion o Deiniion 23 o which we reer here For 0 k n consrucing Φ k requires quasipolynomial ime because o Proposiions 3 and 22 and Theorems 7 and 2 and because obaining Ψ k rom Ψ requires quasipolynomial ime Consrucing he cu-ree rom o Π rom Φ 0 Φ n is done in polynomial ime Remark 25 In Figure 5 we show he aomic low o he cu-ree orm obained rom a proo Π in simple orm We reer o Deiniion 23 Le he ollowing be he low o he simple core Ψ o Π: φ ψ where ψ is he union o lows φ φ n and where we denoe by he edges corresponding o he aom occurrences appearing in he conclusion o Π We hen have ha or 0 < k < n he low o Φ k is φ as in Figure 5 where ψ k k is he low o he derivaion Ψ k The lows o Φ 0 and Φ n are respecively φ 0 and φ n 7 NORMLISTION STEP 3: NLYTIC FORM O special imporance in his paper is he ollowing proo sysem: Deiniion 26 nalyic SKS is he sysem asks = SKS \ {ai aw } For example he sysem {s m= ac } polynomially simulaes he sysem {s= c } and asks = {s m= ai aw ac ac } polynomially simulaes {s= i w c c } (where i is he nonaomic ideniy) In his secion we show ha we can ge proos in analyic SKS ie sysem asks in quasipolynomial ime rom proos in SKS Transorming a proo in cu-ree orm ino an analyic one requires eliminaing coweakening rule insances This can be done by ransormaions ha are he dual o hose over weakening insances employed in Sep () o he proo o Theorem Theorem 27 (Jeřábek [Jeř09]) Given any proo Π o in SKS we can consruc a proo o in asks in ime quasipolynomial in he size o Π Proo By Theorem 24 we can obain rom Π a cu-ree proo Π o he same ormula in quasipolynomial ime in he size o Π We associae Π wih is aomic low φ so ha we have a way o ideniy he aom occurrences in Π associaed wih each edge o φ and subsiue over hem We repeaedly examine each coweakening insance aw in Π or some edge ε o φ and we perorm one ransormaion ou o he ollowing exhausive lis o cases or some Π Φ Ψ K{ } and H{ }: a ε

25 QUSIPOLYNOMIL NORMLISTION IN DEEP INFERENCE 25 () (2) (3) (4) K Π H Π K H K a ε ā Φ a ε Ψ a ε Φ a ε Ψ Π H a a a ε Φ a ε Ψ K Π H a a ε a Φ a ε Ψ becomes becomes becomes becomes K K Π ā Φ{a ε /} H{} Ψ Π [ ] s ( ) Φ{a ε /} H{} Ψ Π K a a Φ{a ε /} H{} Ψ Π K{a} Φ{a ε /} H{} Ψ The process erminaes in linear ime on he size o Π because each ransormaion eliminaes some aom occurrences The inal proo is in asks The ransormaions described in he proo o Theorem 27 are he minimal ones necessary o produce a proo in asks However i is possible o urher reduce he proo so obained The ransormaions in he proo o Theorem 27 ogeher wih he one menioned in Sep () in he proo o Theorem all belong o he class o weakening and ; ; ;

26 26 POL BRUSCOLI LESSIO GUGLIELMI TOM GUNDERSEN ND MICHEL PRIGOT aw -ac : 2 2 ac -aw : 2 2 aw -ai : ai -aw : aw -aw : aw -ac : 2 2 ac -aw : 2 2 FIGURE 6 Weakening and coweakening aomic-low reducions coweakening reducions sudied in [GG08] In he res o his secion we quickly ouline a possible urher ransormaion o he analyic orm produced by hose reducions and reer he reader o [GG08] or a more horough explanaion I is advanageous o describe he weakening and coweakening ransormaions direcly as aomic-low reducion rules These are special graph rewriing rules or aomic lows ha are known o correspond o sound derivaion ransormaions in he ollowing sense I Φ is a derivaion wih low φ and φ can be ransormed ino ψ by one o he aomic-low reducion rules hen here exiss a derivaion Ψ whose low is ψ and such ha i has he same premiss and conclusion as Φ Moreover Ψ can be obained rom Φ by insaniaing some aoms and changing some rule insances in linear ime The weakening and coweakening aomic-low reducion rules are shown in Figure 6 The reducion rule labelled aw -ai is employed in Sep () in he proo o Theorem The reducion rules labelled ac -aw ai -aw aw -aw and ac -aw are employed in he proo o Theorem 27 respecively as Case (4) () (2) and (3) I we apply he ull se o weakening and coweakening reducions unil possible saring rom a proo in cu-ree orm we obain a proo o he same ormula and whose low has shape Noe ha he graph rewriing sysem consising o he reducions in Figure 6 is conluen 8 FINL COMMENTS ND FUTURE WORK Sysem asks is no a minimal complee sysem or proposiional logic because he aomic coconracion rule ac is admissible (via ac s ai and cu eliminaion) Removing ac rom asks yields sysem KS naural quesion is wheher quasipolynomial-ime normalisaion holds or KS as well We would guess ha coconracion plays an essenial role in keeping he complexiy low For example one can noe in Figure 5 how coconracion limis he size o he n pieces o derivaion below each θ k I we had o expand hose coconracion insances ino a ree we would have an exponenial blow-up On he oher hand an encouraging resul in he opposie direcion is conained in [Das4] where he auhor obains n O(loglog n) -size proos o he weak pigeonhole principle using deep-inerence echniques o improve he previous bound or monoone proos There is reason o believe ha polynomial normalisaion is achievable because i is possible o compue hreshold uncions wih polynomial ormulae However he hardes problem seems o be obaining polynomial Γ-like (cu-ree) derivaions wih he propery o Theorem 2 We end o hink ha polynomialiy ough o be possible and deep