A General View of Normalisation through Atomic Flows

Transcription

1 A General View o Normalisaion hrough Aomic Flows Tom Gundersen To cie his version: Tom Gundersen A General View o Normalisaion hrough Aomic Flows Mahemaics [mah] The Universiy o Bah 2009 English <el v3> HAL Id: el hps://elarchives-ouveresr/el v3 Submied on 11 Aug 2010 HAL is a muli-disciplinary open access archive or he deposi and disseminaion o scieniic research documens wheher hey are published or no The documens may come rom eaching and research insiuions in France or abroad or rom public or privae research ceners L archive ouvere pluridisciplinaire HAL es desinée au dépô e à la diusion de documens scieniiques de niveau recherche publiés ou non émanan des éablissemens d enseignemen e de recherche rançais ou érangers des laboraoires publics ou privés

2 A General View o Normalisaion hrough Aomic Flows submied by Tom Erik Gundersen or he degree o Docor o Philosophy o he Universiy o Bah Deparmen o Compuer Science Augus 2009 COPYRIGHT cbea Aenion is drawn o he ac ha copyrigh o his hesis ress wih is auhor This hesis is licensed under he Creaive Commons Aribuion-NonCommercial-ShareAlike 30 Unpored License To view a copy o his license visi hp://creaivecommonsorg/ licenses/by-nc-sa/30/ or send a leer o Creaive Commons 171 Second Sree Suie 300 San Francisco Caliornia USA

3 A General View o Normalisaion hrough Aomic Flows Tom Erik Gundersen i

4 ABSTRACT Aomic lows are a geomeric invarian o classical proposiional proos in deep inerence In his hesis we use aomic lows o describe new normal orms o proos o which he radiional normal orms are special cases we also give several normalisaion procedures or obaining he normal orms We deine and use o presen our resuls a new deep-inerence ormalism called he uncorial calculus which is more lexible han he radiional calculus o srucures To our surprise we are able o 1) normalise proos wihou looking a heir logical connecives or logical rules; and 2) normalise proos in less han exponenial ime ii

5 ACKNOWLEDGEMENTS I was irs inroduced o proo heory by my supervisor Alessio Guglielmi and or his I suppose I ough o be graeul Should anyone reading his be considering a PhD in proo heory hen I urge you o give Alessio a call as he ruly is he mos wonderul supervisor Furhermore I wish o hank Alessio and Paola Bruscoli or looking aer me when I moved o France You have boh been like mohers o me I would also like o express my graiude o all my colleagues oo many o name who have been inluenial in he shaping o his hesis Two people deserve special menion: Alessio or eaching me wha I know and my examiner François Lamarche or his horough and insighul commens Finally I wish o hank my riends and amily or providing me wih an endless source o enerainmen hroughou he course o my sudies In paricular I wan o hank my husband John or his love and suppor I some o my saniy remains i is hanks o you iii

6 Conens 1 Inroducion 1 I Derivaions 5 2 Proposiional Classical Logic 6 21 The Funcorial Calculus 7 22 The Calculus o Srucures Sysem SKS 14 II Aomic Flows 21 3 Aomic Flows Pahs and Cycles Sublows 27 4 Aomic Flows and Derivaions Exracing Flows rom Derivaions A Normal Form o Derivaion 34 5 Normal Forms 36 III Normalisaion 41 6 Global Reducions 42 iv

7 61 Simpliier Isolaed Sublow Removal Pah Breaker Muliple Isolaed Sublows Removal Threshold Formulae 65 7 Local Reducions Soundness Terminaion and Conluence Complexiy 78 8 Main Resul 80 Index 81 Bibliography 84 v

8 Chaper 1 Inroducion Srucural proo heory is he subdiscipline o logic ha sudies ormal represenaion and manipulaion o mahemaical proos A language or represening proos is called a ormalism Tradiionally ormalisms are variaions o Genzen s naural deducion and sequen calculus [Gen69] Essenially a ormalism ollowing Genzen s mehodology represens a proo as a ree obained by recursively breaking ormulae apar a heir main connecive The rules by which proos are consruced are called inerence rules A logic is represened in a given ormalism by a se o inerence rules called a logical sysem Deep inerence [Gug07] is a mehodology ha allows generalisaions o Genzen s ormalisms The sandard deep-inerence ormalism he calculus o srucures generalises he sequen calculus by allowing deducion a any place in a ormula raher han resricing i o he main connecive As a consequence i is possible or all inerence rules o be unary In oher words proos are represened as liss o ormulae raher han as rees o sequens In his hesis a new deep-inerence ormalism named he uncorial calculus is presened While in he sequen calculus he juxaposiion o wo proos denoes ha hey are composed by a conjuncion in he uncorial calculus his horizonal composiion is generalised o allow boh disjuncions and conjuncions In oher words proos are represened as direced acyclic graphs o ormulae raher han as rees o sequens The calculus o srucures and he uncorial calculus are closely relaed and ranslaions beween he wo are given The relaionship beween he wo ormalisms is explored urher in [GGP10] where a generalisaion called open deducion is presened I is shown here ha a uncorial calculus proo corresponds o an equivalence class o calculus o srucures proos The uncorial calculus was chosen or his hesis raher han he calculus o srucures or wo reasons Firsly he smaller proos and ewer arbirary choices required by he uncorial calculus simpliies he presenaion o he resuls Secondly some o he resuls o his hesis have been presened elsewhere in erms o he calculus o srucures [GG08 BGGP10 1

9 GGS10] so using he uncorial calculus illusraes he ac ha he resuls are no ighly coupled o a speciic ormalism The ocus o his hesis is proposiional classical logic By exploiing he symmery available in deep-inerence i is possible o represen proposiional classical logic in a sysem where every inerence rule belongs o one o wo kinds: aomic or linear [BT01] An inerence rule is linear i or every insance o he rule here is a one-o-one correspondence beween he aom occurrences in he premiss and he aom occurrences in he conclusion Linear inerence rules increases he lexibiliy o proos as oher inerence rule insances can in mos cases rivially be permued hrough he linear ones The aomic inerence rules are rules where only a given aom or is dual occur in every insance By replacing a generic inerence rule wih several aomic ones he lexibiliy o he proo is increased as he dieren aomic rules can be permued independenly rom each oher The possibiliy which is no presen in he sequen calculus [Brü03b] o having only linear and aomic inerence rules allows represenaions o proos which are exremely malleable The irs par o his hesis will inroduce classical logic in he uncorial calculus show he relaionship beween he uncorial calculus and he calculus o srucures and presen some sandard deep-inerence resuls A ormalism usually comes wih a normalisaion heory ie a noion o normal orm o proos as well as a procedure describing how o manipulae proos in order o obain heir normal orm In naural deducion a proo is in normal orm i no eliminaion rule ollows an inroducion rule ; and in he sequen calculus a proo is in normal orm i i does no conain he cu rule The cu rule also known as modus ponens is a he hear o proo heory The cu rule allows an auxiliary resul o be proven only once bu used many imes When viewing proos as programs he cu is he applicaion o a uncion o an argumen and normalisaion is compuaion As in he sequen calculus he cu rule is admissible rom deep-inerence proos wihou a premiss In [Brü04] Brünnler presens a cu-eliminaion procedure or he calculus o srucures and sudies he connecion beween proos wih and wihou cu in he calculus o srucures and in he sequen calculus The ac ha he sequen calculus represen proos as rees makes i inherenly asymmeric in he horizonal axis This asymmery is no presen in he calculus o srucures or he uncorial calculus In ac an asymmery has o be enorced or he cu rule o be admissible The symmery ha is possible in deep-inerence ormalisms allows more noions o normal orms han jus cu eliminaion In paricular he dual o cu eliminaion also holds: axioms can be eliminaed rom proos o alsehood In his hesis a new noion o normal orm o proposiional classical logic proos called 2

10 sreamlining is inroduced Unlike cu or axiom eliminaion sreamlining applies o all deepinerence proos and in he asymmeric case where cu or axiom eliminaion is applicable he noions coincide Unlike normal orms based on he order o inerence rule insances sreamlining is invarian under rule permuaions Furhermore sreamlining is a largely synax independen noion in he sense ha i is no ied o a speciic ormalism or a speciic logical sysem In order o describe he noion o sreamlining and he relaed normal orms we inroduce a proo invarian ha we call aomic lows Aomic lows are cerain kinds o direced acyclic graphs ha capure he srucural inormaion o proos Inuiively an aomic low is obained rom a proo by reaining he causal dependencies beween creaion duplicaion and desrucion o aoms and discarding all inormaion abou logical connecives unis and linear inerence rules A proo is sreamlined i here is no pah in is aomic low rom he creaion o he desrucion o an aom The second par o his hesis is devoed o aomic lows heir relaionship wih proos and he deiniion o normal orms in erms o aomic lows Aomic lows were designed o describe normal orm o proos However i urns ou ha aomic lows are also a very convenien ool or designing and arguing abou normalisaion procedures In he hird par o his hesis wo kinds o normalisaion procedures are given All he procedures are irs presened in erms o aomic lows beore hey are lied o derivaions The global procedures work by making several copies o an enire aomic low pruning each copy and siching hem ogeher Three dieren global procedures are presened all producing derivaions in he same normal orm I appears ha here is grea lexibiliy in he design o he global procedures and here is a lo o room or uure invesigaions especially wih respec o complexiy We show ha he global procedures can have less han exponenial cos However hey are all inherenly non-conluen Whereas he global procedures consider he whole aomic low he local procedures work on one pair o adjacen verices These procedures are conluen bu heir cos is inherenly exponenial I is expeced ha proposiional classical logic normalisaion is inherenly exponenial and non-conluen and in ac we observe boh hese phenomena However hey are separaed ino wo disinc phases which can be sudied independenly I is worh noing ha cu eliminaion is achieved wih less han exponenial cos The main conribuion o his hesis is he use o aomic lows or arguing abou normalisaion While i is rue ha all he resuls could be reormulaed in erms o derivaions his would only serve o obuscae wha is going on I should be noed ha all he imporan properies o normalisaion can be proven in erms o aomic lows alone In paricular resuls abou complexiy erminaion conluence and correcness can be proven wihou reerence o derivaions The challenge in designing normalisaion procedures is inding he correc aomic low ransormaion veriying ha a ransormaion can be lied o derivaions is always sraigh orward 3

11 There are wo reasons o consider lows o describe he essence o proos rom he poin o view o normalisaion: Firsly he low o a proo deermines how he proo can be normalised Secondly isomorphisms beween aomic lows are preserved by normalisaion Tha is he resuls o normalising wo proos wih isomorphic aomic lows have isomorphic aomic lows Wih respec o uure work wo aspecs o normalisaion are especially relevan o his hesis: bureaucracy and complexiy The complexiy o cu eliminaion in he sequen calculus is known o be exponenial [Sa78] and i is known ha cu eliminaion has less han exponenial cos in deep inerence [Jeř09] however no lower bound exiss Furhermore his hesis presens normal orms or which only exponenial cos normalisaion procedures are known A possible direcion o uure work is o esablish aomic lows as a ool or sudying complexiy and o discover new normalisaion procedures wih lower complexiy bounds The erm bureaucracy was coined by Girard o denoe arbirary synacic dependencies in proos The presence o bureaucracy means ha proos ha are essenially he same do no have a common canonical represenaion Since all known ormalisms have some degree o bureaucracy an imporan aspec o any normalisaion procedures is how i behaves wih respec o bureaucracy A desirable propery is ha i wo proos are he same modulo bureaucracy hey have he same normal orms modulo bureaucracy For he procedures presened in his hesis his propery always holds or noions o bureaucracy capured by aomic lows Hence anoher possible direcion o uure work is o show wha noions o bureaucracy aomic lows capure and o adap aomic lows o capure more noions o bureaucracy 4

12 Par I Derivaions 5

13 Chaper 2 Proposiional Classical Logic The radiional ormalism in deep inerence is he calculus o srucures [Gug07] The idea o a new ormalism named ormalism A based on he calculus o srucures bu where derivaions conain less bureaucracy was proposed by Guglielmi in [Gug04] and laer Brünnler and Lengrand developed a erm calculus around hese ideas [BL05] In his chaper I deine a ormalism based on he ideas o ormalism A and call i (as suggesed by François Lamarche) he uncorial calculus The reason o inroduce a new ormalism is ha i grealy simpliies he presenaion o some o he more echnical resuls in his hesis (in paricular Secion 641 on page 65) Aer presening he uncorial calculus we compare i briely wih he calculus o srucures beore we inroduce he sandard deducive sysem or classical logic in deep inerence and show some preliminary resuls We now deine ormulae and inerence rules which are in common beween boh he uncorial calculus and he calculus o srucures Deiniions 201 o 204 on pages 6 7 are based on deiniions given in [BG09] The ocus o his hesis is classical proposiional logic and he ollowing deiniions relec his However i is worh noing ha he deiniions can be generalised o oher unis and connecives i one wans o presen oher logics Deiniion 201 We deine a se o ormulae denoed by γ δ o be: aoms denoed by a b c d and ā b c d; ormula variables denoed by A B C D; unis (alse) and (rue); and he disjuncion and conjuncion o ormulae and denoed by [ ] and ( ) respecively A ormula is ground i i conains no variables We usually omi exernal brackes o ormulae and someimes we omi dispensable brackes under associaiviy We use o denoe lieral 6

14 equaliy o ormulae The size o a ormula is he number o uni aom and variable occurrences appearing in i On he se o aoms here is an involuion called negaion (ie is a bijecion rom he se o aoms o isel such ha ā a); we require ha ā a or every a; when boh a and ā appear in a ormula we mean ha aom a is mapped o by ā by A conex is a ormula where one hole { } appears in he place o a subormula; or example a (b { }) is a conex; he generic conex is denoed by ξ { } The hole can be illed wih ormulae; or example i ξ { } b [{ } c] hen ξ {a} b [a c] ξ {b} b [b c] and ξ {a b} b [(a b) c] The size o ξ { } is deined as ξ { } = ξ {a} 1 Deiniion 202 A renaming is a map rom he se o aoms o isel and i is denoed by {a 1 /b 1 a 2 /b 2 } A renaming o by {a 1 /b 1 a 2 /b 2 } is indicaed by {a 1 /b 1 a 2 /b 2 } and is obained by simulaneously subsiuing every occurrence o a i in by b i and every occurrence o ā i by b i ; or example i a [b (a [ā c])] hen {a/ b b/c} b c b [b c] A subsiuion is a map rom he se o ormula variables o he se o ormulae denoed by {A 1 / 1 A 2 / 2 } An insance o by {A 1 / 1 A 2 / 2 } is indicaed by {A 1 / 1 A 2 / 2 } and is obained by simulaneously subsiuing every occurrence o variable A i in by ormula i ; or example i A (b c) hen {A/ c b } c b (b c) Convenion 203 By he above deiniion ormula variables will only be used o deine inerence rules and will never appear in derivaions However when we perorm normalisaion we will someimes single ou aom occurrences (by decoraing hem) and subsiue on hem as i hey were ormula variables Deiniion 204 An inerence rule ρ is an expression ρ where he ormulae and are called premiss and conclusion respecively A (deducive) sysem is a inie se o inerence rules An inerence rule insance ρ o ρ is such ha γ and δ are ground and γ δ γ {a 1 /b 1 a 2 /b 2 }{A 1 / 1 A 2 / 2 } and δ {a 1 /b 1 a 2 /b 2 }{A 1 / 1 A 2 / 2 } or some renaming {a 1 /b 1 a 2 /b 2 } and subsiuion {A 1 / 1 A 2 / 2 } 21 The Funcorial Calculus We now presen he uncorial calculus in he conex o classical proposiional logic and give some basic resuls The inuiion behind derivaions in he uncorial calculus is ha we can compose derivaions by he same connecives we can compose ormulae Deiniion 211 Given a deducive sysem and ormulae and ; a (uncorial calculus) derivaion Ψ in rom o denoed is deined o be Ψ 7

15 1 a ormula: Ψ = ; 2 a verical composiion: Ψ = Φ 1 ρ Φ 2 where are derivaions; or ρ is an insance o an inerence rule rom and 3 a horizonal composiion: Φ 1 and Φ 2 Ψ = 1 Φ Φ 2 or Ψ = 2 1 Φ Φ Φ 1 where and 1 and 1 2 respecively 2 Φ 2 2 are derivaions and 1 2 and 1 2 or 1 2 A derivaion wih premiss is rom now on called a proo The size o a derivaion Ψ denoed Ψ is deined o be he sum o he size o he ormulae appearing in Ψ Convenion 212 Given derivaions and 2 ρ 2 3 we consider 1 Φ Φ 2 2 and 3 Φ 3 3 and inerence rule insances 1 ρ Φ 1 1 ρ 1 2 Φ 2 2 ρ 2 3 Φ 3 3 and 8 1 Φ 1 1 ρ 1 2 Φ 2 2 ρ 2 3 Φ 3 3

16 o be equal and we denoe hem boh by 1 Φ 1 1 ρ 1 2 Φ 2 2 ρ 2 3 Φ 3 3 Remark 213 I desireable Convenion 212 on he preceding page could be made redundan by orcing associaiviy o horizonal composiion in Deiniion 211 on page 7 The only reason we did no do his was or he sake o breviy o he ollowing resuls Lemma 214 Given a derivaion ξ { } can be consruced ξ {} Φ and a conex ξ { } a derivaion Ψ wih size Φ + ξ {} Proo We proceed by srucural inducion on ξ { } The base case ξ { } { } is rivial For he inducive case le ξ { } ξ { } γ ξ { } γ ξ { } ξ { } ξ { } γ or ξ { } γ ξ { } or some ormula γ and a conex ξ { } By he inducive hypohesis we can consruc he ξ {} derivaion Ψ ξ {} so he resul ollows by case (3) o Deiniion 211 on page 7 ξ {} Noaion 215 Given a derivaion Φ and a conex ξ { } he derivaion consruced ξ {} in he proo o Lemma 214 is denoed ξ {Φ} Lemma 216 Given wo derivaions can be consruced Φ 1 and a derivaion Φ 2 γ wih size Φ 1 + Φ 2 Ψ γ Proo We argue by srucural inducion on Φ 1 1 i Φ 1 = hen Ψ = Φ 2 wih size Φ 1 + Φ 2 ; 9

17 2 i Φ 1 = Φ 1 ρ Φ 1 hen by he inducive hypohesis we can consruc can hen build wih size Φ 1 + Ψ = Φ 1 + Φ 1 Ψ = Φ 1 ρ Ψ γ Ψ γ wih size + Φ 2 = Φ 1 + Φ 2 ; Φ 1 + Φ 2 we 3 i Φ 1 = 1 Φ Φ 12 we argue by srucural inducion on Φ 2 : 2 or Φ 1 = 1 Φ Φ 12 2 (a) i Φ 2 is a ormula (resp a verical composiion) he resul ollow by a symmeric argumen o case 1 (resp 2) above (b) i Φ 2 = 1 Φ 21 γ 1 2 Φ 22 γ 2 or Φ 2 = 1 Φ 21 hen by he irs inducive hypohesis we can consruc wih size build 1 Ψ 1 γ 1 Φ 11 + Φ21 1 and Ψ = 1 Ψ 1 γ 1 and 2 Ψ 2 γ 2 γ 1 2 Φ 22 Φ 12 + Φ22 2 respecively we can hen 2 Ψ 2 or Ψ = γ 2 1 Ψ 1 γ 1 2 Ψ 2 γ 2 + Φ12+ Φ21+ Φ22 ( 1 wih size Ψ 1 + Ψ 2 = Φ ) = Φ 1 + Φ 2 10 γ 2

18 Φ 1 Noaion 217 Given derivaions and o Lemma 216 on page 9 is denoed: he derivaion Φ 2 γ Φ 1 Φ 2 γ Ψ consruced in he proo γ 22 The Calculus o Srucures We now presen he calculus o srucures and in Theorem 222 and Theorem 226 on page 13 we show ha he uncorial calculus and he calculus o srucures polynomially simulae each oher The inuiion behind derivaions in he calculus o srucures is ha we rewrie ormulae by applying inerence rules inside a conex Deiniion 221 Given a deducive sysem a se o ormulae and and rom ; a calculus o srucures derivaion Ψ in rom o denoed is deined o be Ψ 1 a ormula: Ψ = ; or 2 a verical composiion: Ψ = Φ 1 ξ { } ρ ξ { } Φ 2 Φ 1 where ρ is an insance o an inerence rule rom and and ξ { } o srucures derivaions ξ { } Φ 2 are calculus The size o a calculus o srucures derivaion Ψ denoed Ψ is deined o be he sum o he size o he ormulae appearing in Ψ 11

19 Theorem 222 A calculus o srucures derivaion Φ can be ransormed ino a uncorial calculus derivaion Ψ such ha Ψ Φ Proo We argue by srucural inducion on Φ The base case is rivial; Φ = = Ψ For he inducive case consider he ollowing calculus o srucures derivaion: Φ = Φ 1 ξ { } ρ ξ { } Φ 2 ξ { } By he inducive hypohesis here are uncorial calculus derivaions and Ψ 2 such ξ { } ha Ψ 1 Φ 1 and Ψ 2 Φ 2 By Lemma 214 on page 9 here is a uncorial calculus derivaion ξ wih size ξ { } + + By Lemma 216 on page 9 we can ρ combine he hree uncorial calculus derivaions o creae Ψ wih size Ψ 1 + Ψ 2 + ξ { } + + ξ { } ξ { } = Ψ 1 + Ψ 2 ξ { } Φ 1 + Φ 2 = Φ Example 223 Figure 4-1 on page 30 has hree examples o calculus o srucures derivaions ransormed ino uncorial calculus derivaions Lemma 224 Given a calculus o srucures derivaion Φ and a conex ξ { } a calculus o srucures derivaion ξ {} Ψ ξ {} can be consruced such ha he number o inerence rule insances in Ψ is he same as he number o inerence rule insances in Φ and he size o he larges ormula in Ψ is he sum o he larges ormula in Φ and ξ { } Proo The saemens ollows by srucural inducion on Φ Lemma 225 Given wo calculus o srucures derivaions Φ 1 and Φ 2 a calculus o srucures γ derivaion can be consruced such ha he number o inerence rule insances in Ψ is he sum Ψ γ 12 Ψ 1

20 o he number o inerence rule insances in Φ 1 and Φ 2 combined and he larges ormula in Ψ is he larges ormula o Φ 1 or he larges ormula o Φ 2 Proo The saemen ollows by srucural inducion on Φ 1 Theorem 226 A uncorial calculus derivaion Φ can be ransormed ino a calculus o srucures derivaion Ψ such ha he size o Ψ depends a mos quadraically on he size o Φ Proo We irs show how o consruc Ψ: The base cases when Φ is a ormula or a verical composiion are rivial For he inducive case consider a conjuncion o uncorial calculus derivaions (he argumen or he disjuncion is similar): Φ = 1 Φ Φ 2 2 By he inducive hypohesis and Lemma 224 on he preceding page here are calculus o srucures derivaions 1 1 Ψ and 1 2 Ψ and by Lemma 225 here exiss a calculus o srucures derivaion 1 2 Ψ 1 2 To ind an upper bound on he size o Ψ we observe ha i depends a mos linearly on he number o inerence rule insances in Ψ muliplied by he size o he larges ormula in Ψ Furhermore by he above Lemmaa he number o inerence rules in Ψ is he same as he number o inerence rules in Φ and he size o he larges inerence rule depends a mos linearly on he size o Φ so he size o Ψ depends a mos quadraically on he size o Φ The calculus o srucures is now well developed or classical [Brü03a Brü06a Brü06d BT01 Brü06b] inuiionisic [Tiu06a] linear [Sr02 Sr03b] modal [Brü06c GT07 So07] and commuaive/non-commuaive logics [Gug07 Tiu06b Sr03a Bru02 DG04 GS01 GS02 GS09 Kah06 Kah07] The basic proo complexiy properies o he calculus o srucures are known [BG09] The calculus o srucures promoed he discovery o a new class o proo nes or classical and linear logic [LS05a LS05b LS06 SL04] (see also [Gui06]) There exis implemenaions in Maude o deep-inerence proo sysems [Kah08] 13

21 23 Sysem SKS We now deine he sandard deducive sysem SKS or classical proposiional logic in deep inerence [Brü03a Brü06a Brü06d BT01] For an excellen reerence o previous work on normalisaion in SKS see [Brü04] Subsysems o SKS are used hroughou his hesis The resuls presened in his secion wih he excepion o Theorem 2314 on page 18 are sandard resuls which can be ound in he lieraure We include he proos or compleeness and as means or giving examples o he uncorial calculus Deiniion 231 Sysem SKS is deined by he ollowing srucural inerence rules: ai a ā aw a a a ac a a ā ai a aw a ac a a he logical inerence rules: and he inverible (logical) rules: A [B C ] s (A B) C (A B) (C D) m [A C ] [B D] A B = c B A A B = c B A A [B C ] = a [A B] C (A B) C = a A (B C ) A = A A = A A = A A = A = = = = The calculus o srucures and sysem SKS were originally deined in erms o equivalence classes o ormulae called srucures and wihou he above inverible logical rules However we ind i more convenien o use ormulae insead since i makes i simpler o race aom occurrences which we will see in Secion 41 on page 28 We now show ha he wo approaches are morally he same Deiniion 232 We deine he relaion = such ha given ormulae and = i here is a derivaion Φ {= c = c = a = a = = = = = = = = } Noaion 233 I = we oen wrie = 14

22 Remark 234 By Noaion 233 on he acing page and Lemma 214 on page 9 or any ormulae and and conex ξ { } we have ha = implies ξ {} = ξ {} Proposiion 235 The relaion = deined in Deiniion 232 on he acing page is an equivalence relaion I urns ou ha he equivalence class induced by = is he same as he srucures used in [Brü04] Remark 236 I = hen (as remarked in [BG09]) here exiss a derivaion Φ {= c = c = a = a = = = = = = = = } whose size depends a mos quadraically on he sum o he sizes o and Noaion 237 When we work in (subsysems o) SKS we oen omi menioning he inverible rules Given be a subsysem o SKS hen unless speciied oherwise when we wrie we mean {= c = c = a = a = = = = = = = = } Furhermore i ρ SKS and here is a derivaion = ρ = we someimes wrie ρ Eg insead o he derivaion we wrie = c γ = c γ [ ] s (γ ) = c γ = c γ [ ] γ s ( γ) See he proos o Theorems 632 o 644 on pages or more examples o implici equaions We now give some sandard resuls which will also serve as examples o sysem SKS and he uncorial calculus 15

23 Lemma 238 Given a conex ξ { } and a ormula here exis derivaions ξ {} {s} ; boh o whose size depend a mos quadraically on he size o ξ {} ξ {} ξ {} {s} ξ {} and Proo We show how o consruc he irs derivaion he second one can be done symmerically We argue by inducion on he number o aom occurrences in ξ { } The base case ξ { } = { } is rivial and he inducive cases are: ξ {} = ξ {} s ξ {} Ψ {s} ξ {} = ξ {} and ξ {} = ξ {} Ψ {s} ξ {} = ξ {} or some ξ { } and where is no a uni and Ψ and Ψ exis by he inducive hypohesis ξ {} ξ {} Noaion 239 We oen wrie ss and ss insead o respecively he ξ {} ξ {} ξ {} ξ {} derivaions and as deined in he proo o Lemma 238 Insead o he ξ {} ξ {} derivaion ζ {} ss ζ {} ξ {} we wrie ζ {} ξ {} ss ζ {} ξ {} s ζ {} ξ {} ss ξ {} We now show a consequenc o he previous Lemma which will be very useul in Subsecion 641 on page 65 Lemma 2310 Given a ormula and an aom a here exis derivaions a {a/} {ac s} and {ac s} ; boh o whose size depend a mos quadraically on he size o {a/} a 16

24 Proo We show how o consruc he irs derivaion he second one can be done symmerically The resul ollows by inducion on he number o occurrences o a in and Lemma 238 on he preceding page The base case is rivial Le ξ { } be some conex such ha = ξ {a} hen he inducive case is: a (ξ {a/}){} a a {s} a (ξ {a/}){a} For an example o he use o Lemma 2310 on he acing page see Remark 2316 on page 19 Lemma 2311 Given a ormula here exis derivaions size depend a mos quadraically on he size o {aw s} {aw s} and ; boh o whose Proo We show how o consruc he irs derivaion he second one can be done symmerically Le a 1 a n be he aoms appearing in hen here exiss a derivaion {a 1 /a n /} {aw } Since {a 1 /a n /} conains no aom occurrences here exiss a derivaion {= = = = } {a 1 /a n /} or = [ ] s ( ) = {= = = = } {a 1 /a n /} Lemma 2312 Given a ormula here exis derivaions whose size depend a mos quadraically on he size o {ac m} and {ac m} ; boh o Proo We show how o consruc he irs derivaion he second one can be done symmerically We argue by inducion on he size o We have o consider he ollowing hree base cases = = 17 and a a a

25 and wo inducive cases: ( ) ( ) m {ac m} {ac m} and [ ] [ ] = {ac m} {ac m} Noaion 2313 In he non-aomic version o sysem SKS he derivaions shown in he proos o Lemma 2311 on he preceding page and Lemma 2312 on he previous page correspond o (co)weakening and (co)conracions respecively For his reason we someimes wrie he inerence rules {aw s} {ac m} and w {ac m} w c respecively and c insead o he derivaions {aw s} To give an example o he noions deined so ar we now show a compleeness proo o sysem SKS Theorem 2314 Sysem SKS is complee or proposiional classical logic Proo Consider a auology We show by inducion on he number o aoms appearing in ha here exiss a proo o in SKS For he base case le consis only o unis Then since is a auology we can build {= = = = } For he inducive case le be a auology conaining insances o he aom a We consider wo cases: i does no conain an insance o ā hen {a/} is a auology so by he inducive hypohesis we can build {a/} {aw } ; 18

26 oherwise boh {a/ ā/} and {a/ ā/} are auologies so by he inducive hypohesis we can build Φ = {a/ ā/} {ai aw } {a/[a ā]} {ss } ā ā {ac } ā Using Φ and he inducive hypohesis we can build he desired derivaion: {a/ ā/} {aw } {ā/} {ā/[ ā]} {ss } c Remark 2315 Given any ormulae and and any conex ξ { } hen by a consrucion similar o he one in he proo o Lemma 238 on page 16 we can build a derivaion ξ { } {s= c = c = a } I we use his derivaion insead o he rule ss in he proo o Theorem 2314 on he preceding page i ollows ha he ξ {} sysem {ai ac aw s m= c = c = a = = = = } is complee or classical logic This jusiies he naming o he inverible rules as he radiion is in deep inerence o label admissible rules wih an Remark 2316 I we do no resric ourselves o he downragmen o SKS we can build a more compac proo han wha we do in Theorem 2314 on he acing page by using he 19

27 ollowing as he inducive case: {a/ ā/} a ā {a/ ā/} s (a {a/ ā/}) ā {a/ ā/} {a/ ā/} a {aw } ā {aw } {a/} {ā/} s {ac s} {ac s} c where we have used he derivaions consruced in he proo o Lemma 2310 on page 16 20

28 Par II Aomic Flows 21

29 Chaper 3 Aomic Flows In his chaper we inroduce he main ool used in his hesis a geomeric proo invarian called aomic lows An aomic low is a direced graph obained rom a derivaion by only reaining inormaion abou he creaion and desrucion o aom occurrences Noably he aomic low o a derivaion compleely disregards all he logical relaions and linear inerence rule insances; so an aomic low is no a derivaion Aomic lows can be seen as eiher specialised Buss low graphs [Bus91 Car97] or a variaion o he kind o proo nes developed in [Sr05 Sr09] based on work done in [LS05b] The only dierence beween aomic lows and hese proo nes is ha he proo nes implemen (co)associaiviy o (co)conracion and dinauraliy o ineracion and cu while aomic lows do no For a more deailed comparison see [Sr09] Despie heir similariies he moivaion and use o aomic lows dier rom ha o proo nes We can hink o aomic lows as composie diagrams ha are reely generaed rom a se o six elemenary diagrams Technically aomic lows are special kinds o labelled direced acyclic graphs and he properies o heir verices are dicaed by heir labels which we deine as ollows Deiniion 301 We call he ollowing six diagrams (aomic-low) labels: ai or ineracion aw or weakening ac or conracion ai or cu aw or coweakening ac or coconracion Deiniion 302 An (aomic) low is a uple (V Eηuplo) such ha: 1 V is a inie se o verices denoed by ν; 2 E is a inie se o edges denoed by ε ι or small numerals 1 2 ; 22

30 3 η: V {ai ai aw aw ac ac } maps verices o heir labels; 4 up: E V { } and lo: E V { } are respecively he upper and lower maps and and are special verices no belonging o V ; we deine or every ν V { } he se L ν = {ε up(ε) = ν } o lower edges o ν he se U ν = {ε lo(ε) = ν } o upper edges o ν and he se E ν = L ν U ν o edges o ν; 5 i S denoes he cardinaliy o se S we have ha i η(ν) = ai hen L ν = 2 and U ν = 0 i η(ν) = ai hen L ν = 0 and U ν = 2 i η(ν) = aw hen L ν = 1 and U ν = 0 i η(ν) = aw hen L ν = 0 and U ν = 1 i η(ν) = ac hen L ν = 1 and U ν = 2 i η(ν) = ac hen L ν = 2 and U ν = 1; 6 here is no sequence ε 1 ε h o edges o V such ha up(ε i ) = lo(ε i+1 (mod h) ) or 1 i h; 7 here is a polariy assignmen π: E { +} such ha or every ν V (a) i η(ν) {ac ac } hen π(e ν ) = { } or π(e ν ) = {+}; (b) i η(ν) {ai ai } hen π(e ν ) = { +} Given an aomic low φ we say ha he ses L = {ε 1 ε h } and U = {ι 1 ι k } conain respecively he upper and lower edges o φ Noaion 303 We will use he leers φ and ψ someimes wih sandard addiional decoraions o denoe aomic lows An aomic low is a direced graph whose edges are associaed o aom occurrences in derivaions and he direcion o he edges corresponds o he up-down direcion in a derivaion Verices are associaed o poins in he derivaion where aom occurrences are creaed or desroyed and he naure o each verex is described by is label Naurally hese graphs are acyclic (condiion 6) The wo special verices and represen he op and boom o a derivaion: we can consider he verex ha creaes all he aom occurrences in he premiss and he verex ha desroys all aom occurrences in he conclusion The polariy assignmen condiion (7) ensures ha aoms in (co)conracions have he same polariy and hose in ineracions and cus have dual polariies (as happens in derivaions) Every aomic low has 2 n polariy assignmens where n is he number o conneced componens in he graph We should no be worried abou he apparen complexiy o he polariy assignmen condiion: in ac we could equivalenly consider wo sors o (co)conracion and (co)weakening labels he negaive and he posiive ones and ask or verices o be joined by respecing heir polariies This is clearly a locally checkable propery much simpler han or example some global correcness crierion or proo nes 23

31 Noaion 304 Le φ be a low wih upper edges ε = ε 1 ε n and lower edges ι = ι 1 ι m we hen represen i as ε 1 ε n ε φ ι 1 ι m or ι φ We someimes use low labels o indicae wha kind o verices a low migh conain Eg he ollowing lows and do no conain ai ai aw aw verices and in addiion he low o he righ does no conain ac verices In general we represen aomic lows as direced-graph diagrams excep ha he special verices and are no shown and he labels o he verices are explicily shown as graphical elemens When we reer o he verices o an aomic low we do no include and Someimes we ideniy verices wih heir labels Example 305 Consider he low A = ({ ν 1 ν 2 ν 3 } { } { ν 1 ai ν 2 ac ν 3 ai } { ν 2 4 ν 2 5 } { 1 ν 1 2 ν 2 3 ν 1 4 ν 3 5 ν 3 }) ; he ollowing are hree o is possible represenaions: and 3 4 in he las wo diagrams we also indicaed each o he wo possible polariy assignmens This low has one coconracion and wo coineracion verices; i has hree upper edges 1 2 and 5 and no lower edges Example 306 The low is obained by juxaposing (ie aking he disjoin union o): hree edges 24

32 a low obained by composing a cu verex wih a coconracion verex and a low obained by composing an ineracion 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 Example 307 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 Deiniion 308 Given wo lows φ 1 = (V 1 E 1 η 1 up 1 lo 1 ) and φ 2 = (V 2 E 2 η 2 up 2 lo 2 ) an isomorphism beween φ 1 and φ 2 is a pair o uncions ( V E ) such ha V is a bijecion rom V 1 o V 2 ; and E is a bijecion rom E 1 o E 2 such ha or every ε in E 1 or every ν in V 1 up 1 (ε) = ν (resp lo 1 (ε) = ν) i and only i up 2 ( E (ε)) = V (ν) (resp lo 2 ( E (ε)) = V (ν)); and up 1 (ε) = (resp lo 1 (ε) = ) i and only i up 2 ( E (ε)) = (resp lo 2 ( E (ε)) = ) Noaion 309 We exend he double-line noaion o collecions o isomorphic lows For example or n 0; ε = ε 1 ε n ; ε = ε 1 ε n ; and ε = ε 1 ε he ollowing diagrams n represen he same low: ε ε ε and ε 1 ε 1 ε 1 ε n ε n ε n Noaion 3010 Given a low ε φ ι and a low ψ which is isomorphic o φ whenever we wrie (ε) ψ = (φ) we mean ha is a given isomorphism beween φ and ψ (ι) 25

33 Noaion 3011 Given a low φ and a polariy assignmen π or φ whenever we wrie + φ or φ respecively we mean ha all he edges in φ have polariy assignmen + or respecively I we label a low wih a polariy assignmen i can no conain any ineracion or cu verices duo o propery 7 o Deiniion 302 on page 22 Deiniion 3012 Given a low φ and a polariy assignmen π or φ he polariy assignmen π or φ is deined o be or every ε in φ: i π(ε) = + π(ε) = + oherwise 31 Pahs and Cycles We now deine he noions o pah ai-pah and ai-cycle in aomic lows Pahs are sequences o adjacen edges ha only go down or only go up ; ai-pahs are ormed by joining pahs a ineracion or coineracion verices; ai-cycles are circular ai-pahs Deiniion 311 Given an aomic low (V Eηuplo) and ε 1 ε h E such ha or 1 i < h we have lo(ε i ) = up(ε i+1 ) up(ε 1 ) = ν and lo(ε h ) = ν we say ha ε 1 ε h is a pah rom ν o ν and ha ε h ε 1 is a pah rom ν o ν; boh pahs have lengh h An ai-pah rom ν o ν o lengh h is eiher a pah rom ν o ν o lengh h or a sequence o edges ε 1 ε k ε k+1 ε h such ha ε k ε k+1 and or some ν V wih η(ν ) {ai ai } we have ha ε 1 ε k is an ai-pah rom ν o ν and ε k+1 ε h is an ai-pah rom ν o ν An ai-pah o lengh h is maximal i no ai-pah conaining is edges has lengh greaer han h An ai-pah rom (resp o) ν o lengh h is a maximal ai-pah rom (resp o) ν i no ai-pah rom (resp o) ν conaining is edges has lengh greaer han h Example 312 The low on he le has he ai-pahs on he righ and he pahs are marked wih an aserisk: In addiion he low has he pahs and ai-pahs obained rom he shown ones by invering he order o edges or example is an ai-pah The ai-pahs rom he ineracion verex are 1 and 2 and 2 4 and 2 4 5; he ai-pahs o he conracion verex are 1 2 and 2 and 3 and 4 and 5 4 The maximal ai-pahs are and and heir inverses The maximal ai-pahs rom he coineracion verex are and 4 3 and 5; he maximal ai-pahs o he conracion verex are 1 2 and 3 and

34 32 Sublows Deiniion 321 Given wo lows φ 1 = (V 1 E 1 η 1 up 1 lo 1 ) and φ 2 = (V 2 E 2 η 2 up 2 lo 2 ) we say ha φ 1 is a sublow o φ 2 i V 1 V 2 ; E 1 E 2 ; η 1 = η 2 V1 ; or every ε in E 1 up 1 (ε) = up2 (ε) i up 2 (ε) V 1 oherwise and lo 1 (ε) = lo2 (ε) i lo 2 (ε) V 1 oherwise ; and i ν 1 and ν 2 are verices in φ 1 and here is a verex ν in φ 2 such ha here are pahs rom ν 1 o ν and rom ν o ν 2 in φ 2 hen ν is a verex in φ 1 Deiniion 322 Given wo lows φ and ψ such ha φ is a sublow o ψ we say ha φ is an isolaed sublow o ψ i here is no pah in ψ rom a verex in φ o or Example 323 In he ollowing low φ is an isolaed sublow o ψ: ψ= φ For oher examples o isolaed sublows see Deiniion 621 on page 47 and Deiniion 641 on page 59 Deiniion 324 Given wo lows φ and ψ such ha φ is a sublow o ψ we say ha φ is a conneced componen o ψ i or any wo polariy assignmens π and π or ψ and or any wo edges ε and ε in φ π(ε) = π(ε ) i and only i π (ε) = π (ε ) 27

35 Chaper 4 Aomic Flows and Derivaions 41 Exracing Flows rom Derivaions We now deine he mapping rom derivaions o lows As we said he idea is ha srucural rule insances map o he respecive aomic-low verices and he edges race he aom occurrences beween rule insances Deiniion 411 Given a derivaion Φ we deine he low φ associaed wih Φ: i Φ is a uni hen φ is he empy low; i Φ is an aom hen φ is a low conaining only he edge ε and no verices; we say ha Φ is mapped o ε; i Φ = Ψ 1 Ψ 2 or Φ = Ψ 1 Ψ 2 and ψ 1 and ψ 2 are he low associaed wih Ψ 1 and Ψ 2 respecively hen φ is he disjoin union o ψ 1 and ψ 2 ; and i Φ = A Φ 1 A ρ B Φ 2 B where ψ 1 (resp ψ 2 ) is he low associaed wih Ψ 1 (resp Ψ 2 ) hen φ is obained by modiying he disjoin union o φ 1 and φ 2 in he ollowing way: i ρ is a srucural inerence rule φ also conains a new verex ν ha is labelled wih he name o ρ Furhermore he lower (resp upper) map o φ maps each o he lower (resp upper) edges o φ 1 (resp φ 2 ) o ν; we say ha ρ is mapped o ν and 28

36 i ρ is a linear inerence rule hen he lower edges o φ 1 are pairwise ideniied wih he upper edges o φ 2 in such a way ha an aom occurrence in he premiss o ρ is mapped o he same edge as he corresponding aom occurrence in he conclusion o ρ Remark 412 Given a derivaion Φ one can associae several aomic lows wih i because we have o choose names or he verices and edges However his is a raher rivial orm o non-deerminism since he posiion o aom occurrences and inerence rule insances can be locaed in a derivaion wihou any ambiguiy Thus given wo aomic lows φ and φ associaed wih he same derivaion Φ here is a unique low isomorphism beween hem ha makes he verices correspond o heir posiion in Φ Furhermore i φ is associaed wih Φ and i : φ φ is an aomic low isomorphism hen one can immediaely urn φ ino an associaed low or Φ in he ollowing way: or every aom occurrence a (resp srucural inerence rule insance ρ) in Φ and edge ε (resp verex ν) in φ we le a (resp ρ) map o ε (resp ν) i and only i a (resp ρ) maps o (ε) (resp (ρ)) Remark 413 I should be noed ha he mapping rom aom occurrences (resp rule insances) in Φ o edges (resp verices) in φ is no uniquely deined In oher words φ migh have non-rivial auomorphisms However his will no cause us any problems in his hesis as in he cases where he mapping is ambiguous (Secion 71) we only rely on is exisence Example 414 The ollowing low has an auomorphism ha maps 1 o 2 and 2 o i can hereore be associaed wih he ollowing derivaion in wo dieren ways a ac a a = (a a) ( ) m [a ] [a ] s a ( [a ]) = a a ac a Example 415 Figure 4-1 on he nex page has hree examples o derivaions and heir associaed lows where colours are used o indicae he mapping rom aom occurrences o edges Deiniion 416 Given a derivaion Φ wih low φ and an aom a he resricion o φ o a is he larges sublow ψ o φ such ha every edge o ψ is mapped o rom occurrences o a or ā 29

37 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 4-1: Examples o derivaions in he calculus o srucures (op row) heir ranslaion ino he uncorial calculus (middle row) and he lows associaed wih he laer (boom row) 30

38 Example 417 Consider he righmos derivaion and is associaed low in Figure 4-1 on he acing page The resricion o his low o a is: We now show ha he process o associaion o a low o a derivaion is surjecive modulo renaming in he sense ha every low is associaed wih some derivaion I should be noed ha he ollowing resul relies on he ac ha boh he ormula srucures o he premiss and conclusion as well as all unis occuring in he derivaion are ignored when exracing a low In paricular he derivaion we consruc in he ollowing proo is rivial in he sense ha i proves rue rom rue An example o his kind o consrucion can be seen in he irs derivaion o Figure 4-1 on he preceding page Theorem 418 Every aomic low is associaed wih some derivaion Proo Firs we show ha or any aom a and ormula conexs ξ { } and ζ { } here exiss a derivaion (ξ {} ζ {a}) {sm} (ξ {a} ζ {}) in oher words we can move he aom a rom he conex ξ { } o he conex ζ { } by using a derivaion whose low conains no verices: ξ {} ζ {a} ss ξ {a} ζ {} = (ξ {a} ) ( ζ {}) m [ξ {a} ] [ ζ {}] = [ξ {a} ] [ζ {} ] s ([ξ {a} ] ζ {}) = ζ {} [ξ {a} ] s (ζ {} ξ {a}) = (ξ {a} ζ {}) This consrucion can be used repeaedly o build he derivaion Ψ or h 0: ξ {} {} ζ {a1 } {a h } Ψ {sm} ξ {a1 } {a h } ζ {} {} We can now prove he heorem by inducion on he number o verices o a given low φ The cases where φ only has zero or one verex are rivial Le us hen suppose ha φ has 31

39 more han one verex; hen φ can be considered as composed o wo lows φ 1 and φ 2 each wih ewer verices han φ as ollows: φ 1 φ= ε 1 ε h φ 2 where h 0 (his can possibly be done in many dieren ways) By he inducive hypohesis γ ξ {a ε 1 1 } {aε h } here exis derivaions Φ h 1 and ζ {a ε Φ 1 1 } {aε h } 2 whose lows are respecively φ 1 h δ and φ 2 Using hese we can build whose low is φ γ Φ 1 ξ {} {} ζ {a ε 1 1 } {aε h } h Ψ ξ {a ε 1 1 } {aε h } h ζ {} {} Φ 2 δ Remark 419 From Proposiion 412 on page 29 and Theorem 418 on he previous page we can conclude ha: Given a derivaion Φ and a low φ deciding i φ is associaed wih Φ is equivalen o deciding i wo lows are isomorphic This will never be an issue in his hesis as we all he lows we will consider are associaed wih he relevan derivaions by consrucion Noaion 4110 Given a derivaion Φ an aom occurrence a in Φ and he low φ o Φ hen whenever we wrie a ε or a ψ we mean ha here is a sublow ψ o φ conaining he edge ε such ha a is mapped o ε We will now see how his noaion migh be useul when selecively subsiuing or aom occurrences For example le us suppose ha we are given he ollowing associaed derivaion and low: (a ) a ā Φ = m a a a = ā ā ā 32 and 1

40 We can hen disinguish beween he hree occurrences o ā ha are mapped o edge 1 and he one ha is no as in (a ) a ā 1 m Φ = a a a = ā1 ā ; ā 1 we can also subsiue or hese occurrences or example by {ā 1 /}; such a siuaion occurs in he proo o Theorem 623 on page 48 Noe ha simply subsiuing or ā 1 would invalidae his derivaion because i would break he cu and weakening insances; however he proo o Theorem 623 speciies how o ix he broken cu insance and Proposiion 4111 speciies how o ix he broken weakening 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 Φ = m a a a φ = āφ ā φ ā φ and φ This noaion is used in Proposiion 4111 where we deine how we can in cerain cases subsiue ormulae in place o aom occurrences This echnique is used in Theorem 613 on page 44 Theorem 623 on page 48 and Theorem 644 on page 61 Proposiion 4111 Given a derivaion Φ SKS le is associaed low have shape φ ψ such ha φ is a conneced componen whose edges are each associaed wih occurrences o he aom a; hen or any ormula γ here exiss a derivaion whose associaed low is {a φ /γ} Ψ SKS {a φ /γ} 1 (φ) n (φ) ψ 33

41 where n is he number o aom occurrences in γ ; moreover he size o Ψ depends linearly on he size o Φ and quadraically on he size o γ Proo We can proceed by srucural inducion on Φ For every ormula in Φ we subsiue a φ wih γ Since all he edges in φ are mapped o rom a (and no ā) we know ha all he verices in φ are mapped o rom insances o ac ac aw and aw We subsiue every insance o ac ac aw and aw where a φ appears by c c w w respecively wih γ in he place o a φ The resul hen ollows by Lemma 2311 on page 17 and Lemma 2312 on page 17 Noaion 4112 The derivaion Ψ obained in he proo o Proposiion 4111 on he preceding page is denoed Φ{a φ /γ} Remark 4113 The noion o subsiuion can be exended o allow φ o conain ineracion and cu verices bu we shall no need ha in his hesis 42 A Normal Form o Derivaion In his secion we inroduce he ai-decomposed orm o a derivaion The reason or inroducing his normal orm is ha we will oen ind i convenien o assume ha ineracion insances appear a he op and cu insances appear a he boom o a derivaion The imporan eaures o his normal orm is ha a derivaion can be ransormed ino ai-decomposed orm wihou changing is aomic low and wihou signiicanly changing is size Deiniion 421 Given wo derivaions Φ and Ψ = a 1 ā 1 b m b m a n ā n SKS\{ai ai } b 1 b 1 or some aoms a 1 a n b 1 b m such ha Φ and Ψ have isomorphic lows we say ha Ψ is an ai-decomposed orm o Φ Convenion 422 Given a derivaion Φ and an ai-decomposed orm o Φ: a 1 ā 1 d l d l a n ā n d 1 d 1 c 1 c 1 SKS\{ai ai } b m b m 34 c k c k b 1 b 1

42 we someimes wan o single ou only some o he ineracion or cu insances We hereore also call he ollowing derivaion an ai-decomposed orm o Φ: a 1 ā 1 a n ā n = [a 1 ā 1 ] a n ā n d l d l d 1 d 1 c 1 c 1 SKS\{ai ai } c k c k bm b m b1 b 1 = b m b m b 1 b 1 Theorem 423 Given a derivaion Φ an ai-decomposed orm o Φ whose size depends a mos cubically on he size o Φ can be consruced Proo Using Lemma 238 on page 16 apply rom op-o-boom and le-o-righ he ollowing ransormaions o each o he ineracion and cu insances in Φ: ξ Ψ a ā Ψ a ā Ψ ξ {} ss ξ [a ā] Ψ and ξ Ψ a ā Ψ Ψ ξ (a ā) ξ {} Ψ a ā ss o obain an ai-decomposed orm o Φ The size o he ai-decomposed orm obained in his way depends a mos cubically on he size o Φ since by Lemma 238 on page 16 each o he ransormaions increase he size o he derivaion a mos quadraically and he number o ransormaions is bounded by he size o Φ Remark 424 The only reason o insis on perorming he ransormaions in he proo o Theorem 423 in a cerain order is o ensure ha he resuling derivaion is unique The uniqueness is useul in he ollowing deiniion Deiniion 425 Given a derivaion Φ he ai-decomposed orm o Φ obained as described in he proo o Theorem 423 is called he (canonical) ai-decomposed orm o Φ 35

43 Chaper 5 Normal Forms In his chaper we see he irs use o aomic lows namely o deine normal orms o derivaions Tradiionally in Genzen-syle ormalisms a derivaion in normal orm is a cu-ree derivaion The noion o cu-reeness is a synacic noion which does no ranslae nicely o he more general deep-inerence ormalisms In boh Genzen-syle ormalisms and deep-inerence ormalisms he cu can be considered horizonal composiion o wo proos We make wo observaions: 1) deep-inerence ormalisms are symmeric in he verical axis whereas Genzen-syle ormalisms are no; and 2) in order or he cu o be admissible rom deep-inerence derivaions he symmery mus be broken o correspond o he asymmery o Genzen-syle ormalisms In paricular he cu is only admissible rom proos and no derivaions These observaions promped us o look or a generalisaion o cu eliminaion ha work or all deep-inerence derivaions Furhermore since we are in he business o designing new ormalisms we waned normal orms based on geomeric noions which would be as synax independen as possible We deined normal orms based on he causal dependency beween srucural inerence rule insances Aomic lows conain (by design) exacly he inormaion needed in order o deine normal orms in his way We call our generalisaion o cu eliminaion sreamlining and we describe i in erms o aomic lows Inuiively i we consider ideniies and weakenings o be he creaors o aom occurrences and cus and coweakening as he desroyers o aom occurrences hen an aomic low is sreamlined i no aom is irs creaed and hen desroyed The shape o a sreamlined aomic low is given in case (4) o Deiniion 501 on he nex page The mos challenging aspec o sreamlining is he eliminaion o pahs rom ineracion o cu verices For his reason we deine he noion o a weakly sreamlined aomic low in case (3) o Deiniion 501 An aomic low is weakly sreamlined i i conains no pah rom an ineracion o a cu verex This is he opic o Chaper 6 on page 42 A pah can be eliminaed by removing he edges ha make up he pah However we 36