as in the cassica case. As far as we know essentiay no work has been done in this area or in the broader area of cacui for genera PSPACE-compete sets,

Transcription

1 Compexity of the intuitionistic sequent cacuus Andreas Goerdt TU Chemnitz Theoretische Informatik D Chemnitz Germany Abstract. We initiate the study of proof compexity for intuitionistic propositiona proof systems: It is known that the set of intuitionistic tautoogies is PSPACE-compete (as opposed to conp-compete in the cassica case). We show that formuas derived from the "cique tautoogies" used in cassica proof compexity have ony exponentia-size proofs in intuitionistic sequent cacui or Frege systems (without substitution). This is in contrast to the cassica case where the compexity of Frege systems is sti open. Introduction The theory of cassica propositiona proof systems is motivated by the conjecture NP 6= co-np. Moreover, recenty there is increased interest in the impact of this theory on questions of a more agorithmic nature,for exampe [Bo et a.97],[be et a. 97]. As the set of cassica tautoogies is co-np compete and propositiona proof systems are just non-deterministic agorithms having at east 1 accepting computation for each tautoogy, there shoud be tautoogies having ony proofs of non-poynomia size for each propositiona proof system. The obvious research program is to nd something out about NP 6= co-np by presenting such tautoogies for proof systems of increasing strength. There has been a considerabe amount of iterature pubished since some breakthrough resuts concerning ower bounds in the mid 80's, some recent exampes are [Kr 98], [Pu 97], [BoPiRa 97]. However, the question as to whether a non-poynomia bound can be proved for propositiona proof systems prominent in many ogic textbooks ike Frege or Hibert systems or (equivaenty) Gentzen's sequent cacuus is sti wide open. There are even quite strong indications that the current techniques are not appicabe to Frege systems with substitution [KrPu 98] and even without substitution [BoPiRa 97]. The current boundary of exponentia ower bounds is determined by bounded depth Frege systems with parity gates [Kr 98] and by cutting pane systems [Pu 97], [BoPiRa 97]. The set of tautoogies being intuitionisticay vaid, where intuitionistic vaidity is determined by derivabiity in some intuitionistic propositiona proof cacuus is PSPACE-compete [St 79]. Thus the PSPACE 6= NP conjecture (which is apparenty weaker than NP 6= co-np) motivates an anaogous research program

2 as in the cassica case. As far as we know essentiay no work has been done in this area or in the broader area of cacui for genera PSPACE-compete sets, except perhaps [KrPu 90].In sharp contrast to the cassica case we can show that the usua textbook systems of intuitionistic propositiona ogic known as Gentzens's sequent cacuus LI [Ta 87] or (equivaenty) Frege or Hibert stye systems (without substitution) have an exponentia ower bound for their proof size. The hard exampes we use are derived from the "cique tautoogies" used in the cassica case. The proof method we appy is the interpoation method as used most expicity in [Pu 97]. The conceptuay new feature here is the use of a suitaby modied cut eimination procedure for the intuitionistic sequent cacuus.that cut eimination may be hepfu in the present context can be seen in [BuMi 98]. After the basic denitions we give a more detaied outine of our argument at the end of section 1. The detais foow in section 2 and 3. 1 Denitions and outine We consider usua propositiona formuas as buit up from an innite set of propositiona variabes with the connectives _; ^; :; ( for impication). The reason to incude is that in the intuitionistic context :x _ y is not equivaent to x y. For more on intuitionistic basics we refer to [Ta 87]. The intuitionistic sequent cacuus uses sequents of the form A 1 ; A 2 ; : : : A n ) A where the A i are formuas, n 0, and A is either a formua or may be empty. The meaning of such a sequent is that A can be derived from the A i. For A being empty this means that a contradiction foows. The A i are the assumptions of the sequent, A is sometimes caed succedent. We impicity view the A i as set. The axioms and rues of the intuitionistic cacuus LI are those of the cassica cacuus LK restricted in that we have ony 1 or 0 formua as succedent. The axioms are A ) A for any formua A. The rues of LI are:? ) A Weakening eft:?; B ) A? )? ) C C; ) A Weakening right: Cut:? ) A?; ) A where?; are sets of assumptions. The rues for the ogica connectives are: ^-eft: A;? ) B A;? ) B C ^ A;? ) B A ^ C;? ) B ^-right:? ) A? ) B? ) A ^ B _-eft: A;? ) B C;? ) B? ) B? ) B _-right: A _ C;? ) B? ) A _ B? ) B _ A -eft:? ) A B; ) C A;? ) B -right:?; ; A B ) C? ) A B :-eft:? ) A A;? ) :-right: :A;? )? ) :A Note, that the succedent may be empty in a rues where this does not ead to any obvious syntactic dicuties. It is not as straightforward to dene semantica vaidity as in the cassica case.as usua we say, a sequent is intuitionisticay vaid 2

3 i it is derivabe in LI. A formua A is intuitionisticay derivabe i the sequent ) A is derivabe. For subsequent usage some basic intuitionistic facts: Fact 1. (1)If? ) A is derivabe in LI, then? impies A cassicay. (2)If? impies A cassicay, then? ) ::A is derivabe in LI. If? impies :A cassicay, then? ) :A is derivabe in LI [K 52], p (3)Each derivation in LI can be transformed into one without the cut rue [Ta 87]. In [BuMi 98] a dierent intuitionistic sequent cacuus is given. It is not dicut to see that LI and the system of Buss,Mints can simuate each other ecienty. Note that the negation :A must be expressed in their system as A?. Frege or Hibert systems make ony use of formuas instead of sequents. They consist of a set of tautoogica axioms that are usuay more compex that the axioms of LI. Often the ony rue is the modus ponens. Simpe transation techniques [CoRe 79] show that LI and intuitionistic Frege systems simuate each other ecienty. For us the size of a proof is measured as usua in propositiona proof compexity: It is the ength of a string representing the proof. In case of cassica systems a propositiona formua encoding the fact: Any graph on n nodes with an m-cique cannot have an (m? 1)-coouring of its vertices, is used in connection with the interpoation method [Pu 97],[BoPiRa 97]. This fact is encoded by a cassicay unsatisabe formua, being the conjunction of the foowing disjunctions: q i;1 _ q i;2 _ : : : _ q i;n for 1 i m; :q i1;j _ :q i2;j for 1 i 1 < i 2 m; 1 j n: With q i;j = 1 i i is mapped to j, these causes say essentiay, that the q i;j for 1 i m and 1 j n represent an injective embedding of an m-eement set into the set of n vertices. p i;j _ :q i1;i _ :q i2;j for 1 i < j n; 1 i 1 < i 2 m: With p i;j = 1 i there is the edge (i; j) present, these causes say that the m vertices hit by the q's are a connected by edges, hence represent an m-cique. r i;1 r i;m?1 for 1 i n; :p i;j _ :r i; _ :r j; for 1 m? 1; 1 i < j n: With r i; saying that node i has coour, the ast set of disjunctions says the the r's describe a correct (m? 1)-coouring of the graph. By "cique tautoogy" we mean the negation of the formua above represented as a disjunction of conjunctions. For usage ater on we x aready here the foowing abbreviations: F = F (p; q) is the rst part of the cique tautoogy, saying that the q's do not present an injection or we have no m-cique. E = E(p; r) is the second part saying that we have no (m? 1)-coouring. For each 0? 1-vector b for p the formua E( b; r) or F (q; b) (or both) is a cassica tautoogy. For the rest of this paper et us x as the set of assumptions consisting of the formuas p i;j _ :p i;j for 3

4 1 i j n. It is not directy cear whether the cique tautoogy is intuitionisticay derivabe.but we have as aready observed in [Pu 97]: Lemma 2. The sequent ) ::F _ ::E is derivabe in LI. Proof: We give a derivation of the sequent in LI. Let a 1 ; : : : ; a 2 f0; 1g be given where =? n 2. We assume the variabes pi;j are numbered from 1 to and et p 0 = :p and p 1 = p. Then we have that the sequent p a1 ) F is a cassica tautoogy (if the p's or a's do not give an m-cique) or we have that p a1 ) E is a cassica tautoogy (if the p's do not aow for an (m? 1)- coouring). It may we be that both conditions are satised by the p's. According to fact 1(2) p a1 ) ::F or p a1 ) ::E is derivabe in LI. Appying the rue _-right and _-eft we get a derivation of ) ::F _ ::E in LI. The ower bound resut used to make the interpoation method work is the ower bound of Razborov for monotone circuits computing the cique function. The version we appy is theorem 6 of [Pu 97]: Fact 3. Let C be a circuit having gates computing monotone booean functions such that C receives inputs p i;j where 1 i < j n to represent a graph on n nodes. The function computed by C is specied as: C gives a 1 if the graph represented has an m-cique and 0 if the graph is (m? 1)-coourabe. (The behaviour of C on non-(m? 1)-coourabe graphs without an m-cique is not specied.) We have: If m = $ 1 n 8 og n 2=3 % then the size of C is 2 ((n= og n)1=3) : We now give a short outine of our argument: Given an intuitionistic proof of the sequent ) :: F _ :: E and a vector p a1 ; a i 2 f0; 1g; =? n 2, we see, waking backwards through a proof which we assume to be cut free for the moment (cf. fact 1(3)), that a sequent essentiay ike p a1 ) ::E must occur in this proof if the p's represent a graph with an m-cique. This sequent cannot occur if the graph is (m? 1)-coourabe. Again we have no information if the graph presented has no m-cique and is not (m? 1)-coourabe. Thus a function as specied in fact 3 is ecienty computabe from a cut free proof. However, our proof needs by no means be cut free and the fact that cut eimination arguments invove exponentia growth in proof size is we-known. Fortunatey we can show by a suitaby modied version of the usua cut eimination argument as presented for exampe in [Ta 87] that the sequent p a1 ) ::E must be a cassica consequence of the sequents of the given proof and the p's as "inputs" if it occurs in the proof obtained after cut eimination. A simiar observation has aso been made in [BuMi 98] in the case that the sequent proved is simpy a disjunction ) A _ B: It is known, that in this case A or B must occur in a cut free proof. The authors show that A or B must be a cassica consequence of the sequents any proof of ) A _ B. In our case we get that E 4

5 must be a cassica consequence of the sequents of the proof if the p's represent a graph with an m-cique. E cannot be a cassica consequence, if the graph is m? 1-coourabe. The formuas of the sequents are simpy viewed as propositiona atoms in the present context. We can thus consider sequents as Horn causes (that is disjunctions with 1 or 0 positive iteras). Simpe agorithms for Horn cause satisabiity [Sch 89] can determine ecienty if the sequent p a1 ) ::E is a cassica consequence of the p's and the sequents of the proof or not. Hence, we have what is known ecient interpoation property of proofs in LI. The Horn cause satisabiity agorithm we use can be transformed ecienty into a monotone circuit, as the formua E is monotone in the p's. Fact 3 impies that we must have exponentiay many of these Horn causes and hence exponentia proof size to begin with. 2 Ecient interpoation In this section we present and anayze our restricted cut eimination procedure. It is crucia that the sequents of a proof after this procedure sti beong to the "Horn cosure" of the initia proof. Denition 4. Let P be a proof in LI and et be a consistent set of formuas. (That means that the sequent ) is not derivabe. This hods in particuar for a cassicay consistent sets.) The Horn cosure of P with respect to, C(P; ), is dened by: Each formua of and each sequent of P beongs to C(P; ). If ) A 2 C(P; ) and A;? ) B 2 C(P; ) then ;? ) B 2 C(P; ). Note that in denition 4 the formuas of the sequents are simpy viewed as propositiona variabes and the sequents as Horn causes. A sequents in C(P; ) are cassicay impied by. We wi not eiminate a cuts but ony those "with respect to a given set of assumptions": Denition 5. Given a proof P and a consistent set of formuas. A cut with respect to is a cut with premisses 1 ) C and 2 ; C ) A. The concusion is 1 ; 2 ) A where 1 ; 2 C(P; ). The use of the Horn cosure with respect to in the preceding denition instead of ony itsef is necessary to make the subsequent induction go through. Note, as 1 C(P; ) and 1 ) C is a sequent of P we have C 2 C(P; ). The foowing theorem is the technica heart of this section: Theorem 6. Let P be a proof in LI and et be a consistent set of formuas. We can transform P into a proof R which is cut free with respect to and a sequents of R beong to C(P; ). Before proving the theorem we show how it is appied to the cique tautoogies. Reca the notation we have xed just before emma 2. 5

6 Coroary 7. Let = fp a1 g; =? n 2, where the ai are 0 or 1 and p 0 = :p; p 1 = p. Let = [. (a) Let. Any proof of ) :: F _ :: E which is cut free with respect to contains a sequent? ) :: F or? ) :: E for a suitabe?. (b) For an arbitrary proof P of ) :: F _ :: E we have :: F 2 C(P; ) or :: E 2 C(P; ). We can decide in time poynomia in the size of P if :: E 2 C(P; ): Proof: (a) We proceed inductivey on the size of the given proof of ) :: F _ :: E viewed as a tree. The induction base hods triviay as the sequent is no axiom of LI. For the induction step we observe that the ast rue of the proof can ony be: weakening-eft, weakening-right, _-eft, or _-right. It cannot be the cut rue as this woud be a cut with respect to. The remaining rues cannot occur due to the syntactica structure of the formuas invoved. In case of weakening-eft, we appy the induction hypothesis to the proof of the premise, weakening-right cannot occur as is consistent, and in the case of _-right we have a sequent as needed. In the case that the ast rue is _-eft we have 2 premisses: p; 0 ) :: F _ :: E and : p; 0 ) :: F _ :: E where 0 = n fp _ : pg. If p 2 we appy the induction hypothesis to the proof of the sequent p; 0 ) :: F _ :: E. If : p 2 we proceed to :p; 0 ) :: F _ :: E. (b) We appy theorem 6 to the proof P with =. We get a proof R having a its sequents in C(P; ) such that R is cut free with respect to. Then (a) impies the rst caim. To determine in poynomia time if :: E 2 C(P; ) we view a occurring formuas as propositiona atoms and appy a standard Horn cause satisabiity agorithm (e. g. [Sch 89]). This agorithm works by marking the formuas of P and by "executing" the sequents: First a occurrences of formuas from and of formuas A where the sequent ) A is in P are marked. Then we repeat unti no change occurs: For any sequent A 1 ; : : : ; A r ) A of P mark a occurrences of A if A 1 ; : : : ; A r are a marked. We have A 2 C(P; ) i A is marked after executing this agorithm. It remains to give the Proof of theorem 2.3: Let P and be xed as in the theorem. We abbreviate C = C(P; ). Let? be the set of assumption free sequents which occur in C. We show by induction on the size of the proof tree of the proof Q: Any proof Q with a its sequents beonging to C can be transformed into a proof which is cut free with respect to? and a the sequents of the new proof sti beong to C. The proof of this resut adapts the usua argument famiiar from cut eimination proofs [Ta 87]. Induction base: Q is an axiom, in this case the proof Q is cut free. For the induction step we assume that the proof tree Q has 2 nodes. Depending on the ast rue of Q severa cases need to be distinguished. If the ast rue is not a cut with respect to? it stays as it is and we appy the induction hypothesis to the proofs of the premisses. The interesting case is that the ast rue of Q 6

7 is a cut with respect to?. Let? 1 ) C and? 2 ; C ) A be the premisses of the concusion? 1 ;? 2 ) A where? 1 ;? 2?. We eiminate this cut. By induction hypothesis we assume that we have proofs of the premisses? 1 ) C and? 2 ; C ) A which are cut free with respect to? and have a their sequents in C. We proceed inductivey on the syntax of the cut formua C.But rst, 2 specia cases: If? 1 ) C is an axiom we have that the concusion is just the other premise and we simpy omit the ast cut. If on the other hand? 2 ; C ) A is an axiom the concusion reads? 1 ) C which again is the other premise of the cut. Thus in the seque we can assume that none of the premisses of the cut is an axiom. For the induction base of the induction on C et C = x. The eft premise of the cut now reads? 1 ) x and the right one is x;? 2 ) A. We proceed by induction on the rank of the cut. The rank is the sum of the eft rank and the right rank. The eft rank is the maxima ength (= number of edges) of a path up the proof tree starting with the eft premise of the cut, such that each sequent of this path has the form ) x for a suitabe. The right rank is the ength of an anaogous maxima path starting with the right premise of the cut, x;? 2 ) A. That means each sequent on this path has the form ; x ) D for suitabe and D. For the induction base of the rank induction the rank is 0. The ony possibe rue giving the eft premise? 1 ) x, can be weakening-right. The premise in this case is? 1 ). This cannot be as the consistency of impies the same for?? 1. Now assume that the rank of the cut is > 0.First, et the eft rank > 0 and the right rank = 0. The ast rue eading to the right premise x;? 2 ) A is weakening-eft, the premise of x;? 2 ) A is? 2 ) A and we obtain the concusion? 1 ;? 2 ) A directy by severa weakenings-eft. A new sequents beong to C, as? 1? and? 1 ;? 2 ) A 2 C. Now et the right rank > 0 (and the eft 0). Depending on the ast rue which gives the right premise x;? 2 ) A we distinguish severa cases. Weakening-eft: As the right rank is > 0 the premise of x;? 2 ) A is x;? 0 ) 2 A where? 2 0 =? 1 n fbg. We shift the cut rue up the proof 1 step owering the rank by 1: Derive? 1 ;? 0 ) 2 A with the cut rue from? 1 ) x; x;? 0 2 ) A. The concusion? 1 ;? 2 ) A foows with the weakening rue. To check that the new sequent x;? 0 2 ) A is in C observe that the formua B is in?. Weakening-right: The premise of? 2 ; x ) A is? 2 ; x ). using the cut rue with? 1 ) x and? 2 ; x ) we derive? 1 ;? 2 ). As? is consistent, this cannot occur. Cut: This case cannot occur because we assume by induction hypothesis (on the tree size) that the proof of x;? 2 ) A has no cuts with respect to?. Observe that x;? 2 ;? as? 1 ) x 2 C. ^-eft: The premise of x;? 2 ) A is D;? 0 2 ; x ) A and? 2 ; = D ^ B;? 0 2 for a suitabe B. We exchange the cut and the ^-eft: First we derive? 1 ; D;? 2 ) A using the cut rue and then? 1 ;? 0 2 ) A with ^-eft. The new sequent? 1 ; D;? 2 ) A beongs to C as D;? 2 0 ; x ) A and? 1 ) x beong 7

8 to C. If D does not any more beong to? we are nished, otherwise the caim foows by the induction on the rank. ^-right: In this case A = A 1 ^ A 2 and the premisses of x;? 2 ) A are x;? 2 ) A 1 and x;? 2 ) A 2. We shift the cut upwards in the proof tree, deriving? 1 ;? 2 ) A 1 and? 1 ;? 2 ) A 2, then we get? 1 ;? 2 ) A with ^-right. The new sequents beong to C as? 1 ) x is in C. The new cuts are sti cuts with respect to?, but their rank has decreased by 1 and we proceed with the induction hypothesis on the rank. _-right: The premise of? 2 ; x ) A is? 2 ; x ) A 1 and A = A 1 _ A 2. We shift the cut upwards, deriving? 1 ;? 2 ) A 1 and then? 1 ;? 2 ) A with an _-right. The new sequent beongs to C, as? 1 ) x is in C. The rank of the cut has decreased by 1 and we proceed with te induction hypothesis. -eft: The premisses of? 2 ; x ) A are 1 ) B and D; 2 ) A where? 2 ; x = 1 ; 2 ; B D. If 1 = 0 1 ; x we derive 0 1 ) B with the cut rue, if 2 = 0 2 ; x we get D; 0 2 ; 1 ) A with the cut rue and an appication of -eft yieds 0 1 ; 0 2 ;? 1; B D ) A, which is? 1 ;? 2 ) A. The new sequents are in C as? 1 ) x is, the rank of the cuts decrease by 1 due to the transformation. -right: We have A = A 1 A 2 and the premise of? 2 ; x ) A is? 2 ; x; A 1 ) A 2. Appying the cut rue with? 1 ) x gives? 1 ;? 2 ; A 1 ) A 2 from which the required concusion foows with an -right. Again the new sequent beongs to C as? 1 ) x beongs to C. :-eft: This case cannot occur, as? 2 ; x? and? is consistent. :-right: We have A = : A 1 and the premise of? 2 ; x ) A is A 1? 2 x ). We appy the cut with? 1 ) x and A 1 ;? 2 ; x ) giving A 1 ;? 1 ;? 2 ).As? is consistent we have that A 1 =2?. Therefore the new cut is no cut with respect to? and we are done. The new sequent beongs to C, as A 1 ;? 2 ; x ) and? 1 ) x beongs to C. This brings the induction base of the induction on the syntax of the cut formua C to an end. For the induction step we assume that C is non-atomic. Our proof tree Q now ends with a cut where the premisses are? 1 ) C and? 2 ; C ) A where? 1 ;? 2?. Again we perform an induction on the rank of this cut (which is dened totay anaogous as before, just substitute C for x). For the induction base et the rank of the cut be 0. We distinguish cases according to which rue derives the eft premise of the cut? 1 ) C. Weakening-right: As before this case cannot occur because? is consistent. ^-right: We have C = C 1 ^ C 2 and the premisses of this sequent are? 1 ) C 1? 1 ) C 2. We now have to ook at the other premise of the cut,? 2 ; C ) A. We proceed by case distinction according to the ast rue which derives the right premise. Weakening-eft: As the rank sti is 0 the premise of? 2 ; C ) A must be? 2 ) A. We derive the na concusion? 1 ;? 2 ) A by severa weakeningseft. As? 1 ;? 2 ) A is in C and? 1? C a intermediate sequents beong to C. ^?eft: The premise of? 2 ; C ) A must be? 2 ; C 1 ) A (or? 2 ; C 2 ) A, 8

9 note that C = C 1 ^ C 2 as we are in the case ^-right for the eft premise). We appy the cut to the premisses? 1 ) C 1 and? 2 ; C 1 ) A giving the resut. The compexity of the cut formua has decreased and we proceed with the induction hypothesis on C. Note that additiona cases cannot occur as the rank is 0 and C = C 1 ^ C 2. _-right: We now have C = C 1 _ C 2 and the premise of? 1 ) C is? 1 ) C 1 (or? 1 ) C 2 ). Again we distinguish severa cases as to which rue is the ast one to derive the right premise of the cut. Weakening-eft: Once again we get? 1 ;? 2 ) A by severa weakenings-eft from? 2 ) A. _-eft: The premisses of? 2 ; C ) A are? 2 ; C 1 ) A and? 2 ; C 2 ) A. We derive? 1 ;? 2 ) A from? 1 ) C 1 and? 2 ; C 1 ) A with a cut. This cut is sti one with respect to? but the syntactic compexity has decreased and this case is nished. -right: We have C = C 1 C 2 and the premise of? 1 ) C is? 1 ; C 1 ) C 2. If the right premise of our na cut is derived with the weakening eft rue, we proceed as before. If it is derived derived by -eft, we have the premisses 1 ) C 1 ; and C 2 ; 2 ) A where? 2 = 1 ; 2. We rst derive? 1 ; 1 ) C 2 with the cut rue and then? 1 ; 1 ; 2 ) A again with the cut rue. The new sequents beong to C and the compexity of the cut formua has decreased. :-right: We have C = :C 1 and the premise of? 1 ) C is? 1 ; C 1 ). In case the right premise? 2 ; C ) A of our cut is obtained by a weakening-eft we proceed as above. If it is introduced by :?eft, the formua A stands for the empty sequent contradicting the consistency of?. This brings the induction base of the induction on the rank to an end. Finay et the rank of the na cut be > 0. First, we assume that the right rank is 0 and we ower the eft rank. According to the rue giving the eft premise we distinguish severa cases. Weakening-eft: We have? 1 =? 0 1 ; B and the premise of? 1 ) C is? 0 1 ) C. We appy the cut to? 0 ) 1 C and? 2; C ) A to get? 0 1 ;? 2 ) A from which we get the concusion with the weakening rue. The rank has decreased by 1 and the new sequent? 0 1 ;? 2 ) A is in C as B 2? 1?. Cut: As? 1? this woud be a cut with respect to?. This case does not occur according to our assumption on the proof Q. ^-eft: We have? 1 =? 0 1 ; B ^ D and the premise of? 1 ) C is? 0 1 ; B ) C. With the cut rue we derive? 0 1 ; B;? 2 ) A and from this we get? 1 ;? 2 ) A with ^-eft. As we have? 0 1 ; B ) C and? 2; C ) A in C, we get that the new sequent? 0 1 ; B;? 2 ) A is in C. If B 2? and hence? 0 1 ; B;? 2? we have a cut with respect to? which can be eiminated by induction hypothesis as the rank has decreased by 1. If B =2? we have not a cut with respect to? and are therefore nished. _-eft: We have? 1 =? 1 0 ; D _ B and the premisses of? 1 ) C are? 1 0 ; D ) C and? 0 1 ; B ) C. Appying the cut to these premisses and C;? 2 ) A gives us? 0 1 ;? 2; D ) A and? 0 1 ;? 2; B ) A from which? 1 ;? 2 ) A foows with _-eft. The new sequents beong to C. Depending whether D; B 2? the new cuts are cuts with respect to? or not. In the rst case we proceed with the induction hypotesis and in the second case we are nished. 9

10 -eft: We have? 1 =? 0 1 ; B D and the premisses are 1 ) B; and D; 2 ) C, and 1 ; 2 =? 1. 0 We appy the cut with D; 2 ) C and C;? 2 ) A to get D; 2 ;? 2 ) A. An appication of -eft with the preceding sequent and 1 ) B gives us 1 ; 2 ; B D;? 2 ) A which is? 1 ;? 2 ) A. By denition of C the new sequent beongs to C. If D 2? we proceed with the induction hypothesis on C otherwise we are done. :-eft: This case cannot occur as the eft rank is > 0. Finay, we have to treat the situation where C is non-atomic and the right rank of the cut is > 0 (the eft is 0.) Again some cases need to be distinguished according to the rue which gives us the right premise of the na cut,? 2 ; C ) A. Weakening-eft: The premise of? 2 ; C ) A is? 0 2 ; C ) A; where? 0 2? 2. We shift the cut upwards 1 step to get? 0 2 ;? 1 ) A. Then? 2 ;? 1 ) A foows with weakening-eft. The new sequent is in C as? 2?. Weakening-right: This case cannot occur as? is consistent. Cut: This case cannot occur because? 2 ; C? as? 1? and? 1 ) C 2 C. We thus woud have a cut with respect to?, contradicting our assumptions concerning the proof Q. ^-eft: The premise of? 2 ; C ) A is D;? 0 2 ; C ) A and? 2 = D ^ B;? 0 2 for a suitabe B. We exchange the cut and the ^-eft: First we get D;? 0 2 ;? 1 ) A and then? 2 ;? 1 ) A. The new sequent is in C. If D 2? the resut foows by induction hypothesis of the induction on the rank, otherwise we are done. ^-right: Here we have A = A 1 ^ A 2 and the premisses of? 2 ; C ) A are? 2 ; C ) A 1 and? 2 ; C ) A 2. We push the cut up the proof 1 step to get? 1 ;? 2 ) A. An appication of ^-right gives the concusion. The new cuts are cuts with respect to? and we appy ths induction hypothesis on the rank to nish this case. _-eft: The premisses of? 2 ; C ) A are D;? 0 2 ; C ) A and B;? 0 2 ; C ) A. Pushing the cut up 1 step we get D;? 0 2 ;? 1 ) A and B;? 0 2 ;? 1 ) A. The sequent? 2 ;? 1 ) A foows with a subsequent _-eft. The new sequents beong to C. If D 2? or B 2? we proceed with the induction hypothesis on the rank. Otherwise we are done. _-right: The premise of? 2 ; C ) A is? 2 ; C ) A 1 and A = A 1 _ A 2. Shifting the cut upwards, we get? 2 ;? 1 ) A 1 and an _-right gives us? 2 ;? 1 ) A 1 _ A 2. The new sequent beongs to C, the cut is with respect to? but the rank has decreased by 1 aowing for an appication of the induction hypothesis. -eft: The premisses of? 2 ; C ) A are: 1 ) B and D; 2 ) A where? 2 ; C = 1 ; 2 ; B D. If 1 = 0 1 ; C we derive? 1; 0 1 ) B with a cut with? 1 ) C. If 2 = 0 2 ; C we get? 1; D; 0 2 ) A with another cut with? 1 ) C. Now we appy -eft to get? 1 ; 0 2 ; 0 1 ; B D ) A which is? 1 ;? 2 ) A. As the ranks of the new cuts have decreased by 1 we are done. -right: We have A = A 1 A 2 and the premise of? 2 ; C ) A is? 2 ; C; A 1 ) A 2. Appying the cut to this premise we get? 2 ;? 1 ; A 1 ) A and -right gives the concusion. If A 1 2? the caim foows by induction on the rank, otherwise we are done. :-eft: In this case A stands for the empty succedent. This cannot be as C;? 2 10

11 ? and? is consistent. :-right: We have A = :A 1 and the premise of C;? 2 ) A is C;? 2 ; A 1 ). We appy the cut with? 1 ) C and C;? 2 ; A 1 ) to get? 1 ;? 2 ; A 1 ). From this we get? 1 ;? 2 ) :A 1 with :-right. The new cut is not a cut with respect to? as? is consistent and we are done. 3 Ecient monotone interpoation Reca the notation from just before emma 2. Theorem 8. Let P be a proof in LI of the cique tautoogy, ) ::F _ :: E. We can construct a monotone booean circuit C which receives as inputs p 1 ; : : : ; p ; =? n 2, and gives 1 as output if the graph presented by the input bits for the p's has an m-cique and 0 if this graph is (m? 1)-coourabe. The size of C is at most the square of the proof size. Proof: First we construct a not necessariy monotone circuit, afterwards we show how to make it into a monotone circuit. Our circuit simpy simuates the marking agorithm for the satisabiity probem of Horn causes 7(b). Let A 1 ; : : : ; A r be the set of syntacticay dierent formuas which occur in the sequents of P except of the formuas p 1 ; : : : ; p ; :p 1 ; : : : ; :p ; p 1 _ :p 1 ; : : : ; p _ :p which are treated separatey. The circuit consists of r + 1 eves, eve 0, eve 1,...,eve r. Leve i consists of r unary gates A i;1 ; : : : ; A i;r (at which just the identity function is computed). The eve-0-gates A 0;j receive the constant 1 as input if ) A j is a sequent of the proof. The other eve-0-gates receive 0 as input. The gates from eve i are connected to those of eve i + 1 as prescribed by the sequents of the proof P : The gate A i+1;j is set to 1 if there exists a sequent A j1 ; : : : ; A jk ) A j in P such that the gates A i;j1 and : : : and A i;jk give a 1. Accordingy, the input wire to the gate A i+1;j is the output wire of an _{gate. The inputs to this _{gate are a direct wire from A i;j and wires which are exits of ^{gates. Each ^-gate corresponds to a sequent of the proof with succedent A j. The ^{gate corresponding to the sequent A j1 ; : : : ; A jk ) A j has the outputs from gates A i;j1 ; : : : ; A i;jk as input wires. If we have a variabe p among the A jr we simpy take the corresponding input wire. If we have : p we simpy negate it. If we have an A jr = p _ :p we just set it to 1 (or omit it). The output of the circuit is the wire eaving the gate of ::E at the ast eve. This circuit outputs 1 if :: E is in the Horn cosure of the proof with respect a to p 1 a 1 ; : : : ; p if it gets as inputs a 1 ; : : : ; a. Hence it gives a 1 if the graph represented by the input-bits has an m-cique. It cannot give a 1 if the graph is m? 1-coourabe. In order to make the circuit above monotone, we ony have to get rid of the :p's, which may occur. We simpy substitute the :p's by the constant 1. As ::E is monotone in the p's one sees going backwards through the circuit ike through an and-or-tree that the function computed is not changed by this. 11

12 4 Concusion Our ower bound proof depends strongy on the disjunction property of LI: If ) A _ B is vaid in LI where is without _ then we have that ) A or ) B is vaid in LI. Take a ook at a dierent presentation of the cique tautoogies: C 1 ; : : : ; C k ) where the C i are a causes of the unsatisabe version of the cique tautoogies. This version of the cique tautoogies is derivabe in LI, as the negation is cassicay vaid and hence derivabe in LI. Appying cut eimination to the proof in LI shows the derivabiity of the sequent in question. We do not see how to appy our ower bound proof to this representation of the cique tautoogies. A ower bound for the present version woud have the interesting consequence that the pigeonhoe principe represented as a sequent: "Each pigeon sits in a hoe. " ) "There are 2 in 1 hoe." woud ony have exponentiay ong proofs in LI. It is not dicut to derive the cique tautoogies in the contradictory form above from the pigeonhoe principe. See [Pu 98] for further consequences of a ower bound for the pigeonhoe principe in LI. The major dicuty in getting a ower bound for cique tautoogies here is that the dua of the disjunction property does not hod, that is ) :(A _ B) in LI does not impy ) : A or ) : B is derivabe in LI. References [BuMi 98] Sam Buss and Grigori Mints. The compexity of the disjunction and existentia properties in intuitionistic ogic. Manuscript (1998). [Bo et a.97] Maria Luisa Bonet, Juan Luis Esteban, Nicoa Gaesi, and Jan Johannsen. Exponentia separations between restricted resoution and cutting pane systems.stoc [Be et a. 97] Pau Beame, Richard Karp, Toniann Pitassi, and Michae Saks. On the compexity of unsatisabiity proofs for random k-cnf-formuas. FoCS [BoPiRa 97] Maria Bonet, Toniann Pitassi, and Ran Raz. Lower bound proofs for cutting pane proofs with sma coecients. The Journa of Symboic Logic. 62 (1997) [BoPiRa 97] Maria Luisa Bonet, Toniann Pitassi, and Ran Raz. No feasibe interpoation for T C 0 -Frege proofs. In Proceedings 38th FoCS (1997) [CoRe 79] Stephen A. Cook and Robert A. Reckhow. The reative eciency of propositiona proof systems. The Journa of Symboic Logic 44 (1979) [Kr 98] Jan Krajicek. Bounded-depth Frege systems with parity gates. Manuscript (1998). [KrPu 98] Jan Krajicek and Pave Pudak. Some consequences of cryptographica conjectures for S 1 2 and EF. Information and Computation 140 (1998) [KrPu 90] Jan Krajicek and Pave Pudak. Quantied propositiona cacui and fragments of bounded arithmetic. Zeitschrift fur mathematische Logik und Grundagen der Mathematik 36 (1990) [K 52] Stephen C. Keene. Introduction to Metamathematics Van Nostrand, New York (1952). [Pu 97] Pave Pudak. Lower bounds for resoution and cutting pane proofs and monotone computations. The Journa of Symboic Logic 62(3) (1997)

13 [Pu 98] Pave Pudak. On the compexity of the propositiona cacuus. Manuscript (1998). [St 79] Richard Statman. Intuitionistic propositiona ogic is poynomia-space compete. Theoretica Computer Science 9 (1979) [Ta 87] Gaisi Takeuti. Proof theory, 2nd edition. North Hoand (1987). [Sch 89] Uwe Schoning. Logic for Computer Science. Birkhauser (1989). 13