Proof complexity of positive branching programs
aa r X i v : . [ c s . CC ] F e b PROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS
ANUPAM DAS ˚ AND AVGERINOS DELKOS ˚ Abstract.
We investigate the proof complexity of systems based on positivebranching programs, i.e. non-deterministic branching programs (NBPs) where,for any 0-transition between two nodes, there is also a 1-transition. PositiveNBPs compute monotone Boolean functions, just like negation-free circuits orformulas, but constitute a positive version of (non-uniform) NL , rather than P or NC , respectively.The proof complexity of NBPs was investigated in previous work by Buss,Das and Knop, using extension variables to represent the dag-structure, overa language of (non-deterministic) decision trees, yielding the system eLNDT .Our system eLNDT ` is obtained by restricting their systems to a positivesyntax, similarly to how the ‘monotone sequent calculus’ MLK is obtainedfrom the usual sequent calculus LK by restricting to negation-free formulas.Our main result is that eLNDT ` polynomially simulates eLNDT over pos-itive sequents. Our proof method is inspired by a similar result for MLK byAtserias, Galesi and Pudl´ak, that was recently improved to a bona fide poly-nomial simulation via works of Jeˇr´abek and Buss, Kabanets, Kolokolova andKouck´y. Along the way we formalise several properties of counting functionswithin eLNDT ` by polynomial-size proofs and, as a case study, give explicitpolynomial-size poofs of the propositional pigeonhole principle. Introduction
Proof complexity is the study of the size of formal proofs. This pursuit is fun-damentally tied to open problems in computational complexity, in particular duethe Cook-Rechow theorem [10]: co NP “ NP if and only if there is a propositionalproof system (suitably defined) that has polynomial-size proofs of each proposi-tional tautology. This has led to what is known as ‘Cook’s program’ for separating P and NP : find superpolynomial lower bounds for stronger and stronger systemsuntil we have found a general method (see, e.g., [7, 20]).Systems of interest in proof complexity are typically motivated by analogousresults from circuit complexity and other non-uniform models of computation. Forinstance bounded depth systems restrict proofs to formulas with a limit on thenumber of alternations between _ and ^ in its formula tree, i.e. AC concepts.Indeed, H˚astad’s famous lower bound techniques for AC [14] have been lifted to thesetting of proof complexity, yielding lower bounds for a propositional formulationof the pigeonhole principle [4] via a refined version of the switching lemma. Monotone proof complexity is motivated by another famous lower bound result,namely Razborov’s lower bounds on the size of -free circuits [27, 28] (and similarones for formulas [18]). In this regard, there has been much investigation intothe negation-free fragment of Gentzen’s sequent calculus, called
MLK [2, 3, 16, ˚ University of Birmingham, UK
Date : February 15, 2021. LK by MLK on -free sequents byformalising an elegant counting argument using quasipolynomial-size negation-freecounting formulae. This has recently been improved to a polynomial simulationby an intricate series of results [3, 16, 6], solving a question first posed in [26].However, note the contrast with bounded depth systems: restricting negation hasdifferent effects on computational complexity and proof complexity.In this work we address a similar question for the setting of branching programs .These are (presumably) more expressive than Boolean formulas, in that they arethe non-uniform counterpart of log-space ( L ), as opposed to NC . They haverecently been given a proof theoretic treatment in [5], in particular addressingproof complexity. We work within that framework, only restricting ourselves toformulas (with extension) representing positive branching programs.Positive (or ‘monotone’) branching programs have been considered several timesin the literature, e.g. [13, 17]. They are identical to Markov’s ‘relay-diode bipoles’from [25]. [13, 12] give a general way of making a non-deterministic model of compu-tation ‘positive’; in particular, a non-deterministic branching program is positive if,whenever there is a 0-transition from a node u to a node v , there is also a 1-transitionfrom u to v . As in the earlier work [5] we implement such a criterion by using dis-junction to model nondeterminism. As far as we are aware, there is no other workinvestigating the proof complexity of systems based on positive/monotone branch-ing programs.1.1. Contribution.
We present a formal proof calculus eLNDT ` , reasoning withformula-based representations of positive branching programs, by restricting thecalculus eLNDT from [5] appropriately. We consider the ‘positive closures’ of well-known polynomial-size ‘ordered’ BPs (OBDDs) for counting functions, and showthat their characteristic properties admit polynomial-size proofs in eLNDT ` .As a case study, we show that these properties can be used to obtain polynomial-size proofs of the propositional pigeonhole principle, by adapting an approach of [2]for MLK . Our main result is that eLNDT ` in fact polynomially simulates eLNDT over positive sequents. For this we again use representations of positive NBPs forcounting and small proofs of their characteristic properties. At a high level weadapt the approach of [3], but there are several additional technicalities specificto our setting. In particular, we require bespoke treatments of negative literals in eLNDT and (iterated) substitutions of representations of positive NBPs into otherpositive NBPs.1.2. A terminological convention.
Throughout this work, we shall reserve thewords ‘monotone’, ‘monotonicity’ etc. for semantic notions, i.e. as a property ofBoolean functions. For (non-uniform) models of computation such as formulas,branching programs, circuits etc., we shall say ‘positive’ for the associated syntactic constraints, e.g. negation-freeness for the case of formulas or circuits. While manyworks simply say ‘monotone’ always, in particular [12, 13], let us note that thedistinction we make is employed by several other authors too, e.g. [1, 23, 22, 11].2.
Preliminaries on proof complexity and branching programs
In this section we will recall some of the content from [5]. The reader familiarwith that work can safely omit this section, though they should take note of ourconventions in the definition of the system eLNDT , cf. Remark 2.10.
ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 3
Throughout this work we will use a countable set of propositional variables ,written p, q etc., and
Boolean constants assignment is just a map α from propositional variables to t , u . For allintents and purposes we may assume that they have finite support, e.g. nonzeroonly on variables occurring in a formula or proof. We extend an assignment α tothe constants in the natural way, setting α p q “ α p q “ Boolean function is just a map from (finitely supported) assignments to t , u .2.1. Proof complexity.
In proof complexity, a (formal) propositional proof system is just a polynomial-time function P from Σ ˚ to the set of propositional tautologies,where Σ is some finite alphabet. Intuitively, the elements σ P Σ ˚ code proofs in thesystem, while P itself is a (efficient) ‘proof-checking’ algorithm that verifies that σ is indeed a correctly written proof, and if so returns its conclusion, i.e. the theoremit proves. If not, it just returns 1, by convention.The significance of this definition is due to the following result from [10]: Theorem 2.1 (Cook-Reckhow) . There is a propositional proof system with polynomial-size proofs of each tautology if and only if co NP “ NP . In practice, this ‘Cook-Reckhow definition’ of a propositional proof system covers allwell-studied proof systems for propositional logic, under suitable codings. We shallrefrain from giving any of these codings explicitly in this work, as is standard forproof complexity. However, let us point out that the systems we consider routinelyadmit polynomial-time proof checking in the way described above, and so indeedconstitute formal propositional proof systems.See [19, 8, 21] for more comprehensive introductions to proof complexity.2.2.
Non-deterministic branching programs.
A (non-deterministic) branchingprogram (NBP) is a (rooted) directed acyclic graph G with two distinguished sink nodes, 0 and 1, such that: ‚ G has a unique root node, i.e. a unique node with in-degree 0. ‚ Each non-sink node v of G is labelled by a propositional variable. ‚ Each edge e of G is labelled by a constant 0 or 1.A run of a NBP G on an assignment α is a maximal path beginning at the rootof G consistent with α . I.e., at a node labelled by a propositional variable p therun must follow an edge labelled by α p p q P t , u .We say that G accepts α if there is a run on α reaching the sink 1. We mayextend α to a map from all NBPs to t , u by setting α p G q “ G accepts α . In this way, each NBP computes a unique Boolean function α ÞÑ α p G q .A comprehensive introduction to (variants of) branching programs and theirunderlying theory can be found in, e.g., [30]. Example 2.2 (OBDD for 2-out-of-4 Exact) . The 2-out-of-4 Exact function, whichreturns 1 just if precisely two of its four arguments are 1, is computed by thebranching program in Fig. 1. 0-edges are indicated dotted and 1-edges are indicatedsolid, a convention that we adopt throughout this work. Formally, each 0-leafcorresponds to the same sink.Note that this program is deterministic: there is exactly one 0-edge and one 1-edge outgoing from each non-sink node. It is also ordered : all the variables appearin the same order in each path. Thus its semantics may be verified by checkingthat every path leading to the 1-sink has exactly two 1-edges and vice versa.
PROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS p p p p p p p p p p Figure 1.
An OBDD computing the 2-out-of-4 Exact function.0-edges are indicated dotted, and 1 edges are indicated solid.2.3.
Representation of NBPs by extended formulas.
Since we will be work-ing in formal proof systems, we shall use a natural representation of NBPs by‘formulas with extension’, just like in [5]. For this, we shall make use of extensionvariables e , e , e , . . . in our language.An extended non-deterministic decision tree formula, or eNDT formula, written A, B etc., is generated from propositional variables, extension variables and con-stants by disjunction, _ , and decisions : for formulas A, B and p a propositionalvariable, p ApB q is a formula. Intuitively, p ApB q expresses “if p then B else A ”.As usual, we often omit external brackets of formulas and write long disjunctionswithout internal brackets, under associativity. The size of a formula A , written | A | ,is the number of symbols occurring in A . Remark 2.3 (Distinguishing extension variables) . Note that we formally distin-guish extension variables from propositional variables. This is for the same technicalreasons as in [5] we must not allow extension variables to be decision variables, i.e.we forbid formulas of the form Ae i B . If we did allow this then we would be ableto express all Boolean circuits succinctly, whereas the current convention ensuresthat we only express NBPs.The semantics of (non-extended) NDT formulas under an assignment will bestandard. With extension variables, however, the interpretation is parametrised bya set of extension axioms , allowing extension variables to ‘abbreviate’ more complexformulas. Definition 2.4 (Extension axioms) . A set of extension axioms A is a set of theform t e i Ø A i u i ă n , where each A i may only contain extension variables among e , . . . , e i ´ . Definition 2.5 (Semantics of eNDT formulas) . Satisfaction with respect to a setof extension axioms A “ t e i Ø A i u i ă n , written ( A , is a (infix) binary relationbetween assignments and formulas over e , . . . , e n ´ defined as follows: ‚ α * A α ( A ‚ α ( A p if α p p q “ ‚ α ( A A _ B if α ( A A or α ( A B . ‚ α ( A ApB if either α p p q “ α ( A A , or α p p q “ α ( A B . ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 5 ‚ α ( A e i if α ( A A i . Example 2.6 (2-out-of-4 Exact, revisited) . Recall Example 2.2 and the branchingprogram from Fig. 1. Under the semantics above, we may represent this branchingprogram as e under the following extension axioms: e Ø e p e e Ø e p e e Ø e p e e Ø e p e e Ø e p e e Ø e p e e Ø p e Ø p e Ø p e Ø p e ij represents the j th node (left to right) on the i th row (top to bottom).Note that, in order to strictly comply with the subscripting condition on extensionaxioms, we may identify e ij with e i ´ j .Note that the notion ( A is indeed well-defined, thanks to the subscripting con-ditions on sets of extension axioms: intuitively, each e i abbreviates a formula con-taining only extension variables among e , . . . , e i ´ , and so on. More generally: Remark 2.7 ( A -induction) . Given a set of extension axioms A “ t e i Ø A i u i ă n we may define a strict partial order ă A on formulas over e , . . . , e n ´ by: ‚ p ă A ApB and A ă A ApB and B ă A ApB . ‚ A ă A A _ B and B ă A A _ B . ‚ A i ă A e i , for each i ă n .Notice that ă A is indeed well-founded by the condition that each A i must containonly extension variables among e , . . . , e i ´ . Thus we may carry out argumentsand make definitions by induction on ă A , which we shall simply refer to as ‘ A -induction’.We can now see Definition 2.5 above of ( A as just a definition by A -induction.In this way, fixing some set of extension axioms A “ t e i Ø A i u i ă n , each eNDTformula A over e , . . . , e n ´ computes a unique Boolean function f : α ÞÑ α ( A A . In this case, we may say that A computes f with respect to A .Since many of our arguments will be based on A -induction, let us make thefollowing observation for complexity matters: Observation 2.8 (Complexity of A -induction) . Let A “ t e i Ø A i u i ă n be a set ofextension axioms and A contain only extension variables among e , . . . , e n ´ . Then |t B ă A A u| ď | A | ` ř i ă n | A i | and, if B ă A A , then | B | ď max p| A | , | A | , . . . , | A n ´ |q . The system eLNDT . We now recall the system for NBPs introduced in [5].The language of the system eLNDT comprises of just the eNDT formulas. A sequent is an expression Γ Ñ ∆, where Γ and ∆ are multisets of eNDT formulas (‘ Ñ ’ is justa syntactic delimiter). Semantically, such a sequent is interpreted as a judgement“some formula of Γ is false or some formula of ∆ is true”.Notice that the semantic interpretation of eNDT formulas we gave in Defini-tion 2.5 means that ApB is logically equivalent to both p p ^ A q _ p p ^ B q and p p Ą A q ^ p p Ą B q . It is this observation which naturally yields the followingsystem for eNDT sequents from [5]: Definition 2.9 (Systems
LNDT and eLNDT ) . The system
LNDT is given by therules in Fig. 2. An
LNDT derivation of Γ Ñ ∆ from hypotheses H “ t Γ i Ñ ∆ i u i P I PROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS
Initial sequents and cut: Ñ Ñ id p Ñ p Γ Ñ ∆ , A Γ , A Ñ ∆ cut Γ Ñ ∆ Structural rules: Γ Ñ ∆ w - l Γ , A Ñ ∆ Γ Ñ ∆ w - r Γ Ñ ∆ , A Γ , A, A Ñ ∆ c - l Γ , A Ñ ∆ Γ Ñ ∆ , A, A c - r Γ Ñ ∆ , A Logical rules: Γ , A Ñ ∆ , p Γ , p, B Ñ ∆ p - l Γ , ApB Ñ ∆ Γ Ñ ∆ , A, p Γ , p Ñ ∆ , B p - r Γ Ñ ∆ , ApB Γ , A Ñ ∆ Γ , B Ñ ∆ _ - l Γ , A _ B Ñ ∆ Γ Ñ ∆ , A, B _ - r Γ Ñ ∆ , A _ B Figure 2.
Rules for system p e q LNDT .is defined as expected: it is a finite list of sequents, each either some Γ i Ñ ∆ i from H or following from previous ones by rules of LNDT , ending with Γ Ñ ∆.An eLNDT proof is just an LNDT derivation from hypotheses that are a set ofextension axioms A “ t e i Ø A i p e j q j ă i u i ă n ; here we construe A Ø B as an abbre-viation for the pair of sequents A Ñ B and B Ñ A . We require that the conclusionof an eLNDT proof is free of extension variables.The size of a proof or derivation P , written | P | , is just the number of symbolsoccurring in it.Note that, despite the final condition that conclusions of eLNDT proofs are freeof extension variables, we may sometimes consider intermediate ‘proofs’ with ex-tension variables in the conclusions. In these cases we will always make explicit theunderlying set of extension axioms. Remark 2.10 (Differences from original eLNDT ) . In order to ease the exposition,we have slightly adjusted the definition of eLNDT . The variations are minor and,in particular, the current presentation is polynomially equivalent to that of [5].Nonetheless, let us survey these differences here: ‚ We admit constants 0 and 1 within the language. As mentioned in [5], thisdoes not significantly affect proof size, since 0 can be encoded as ppp and1 as ppp , for an arbitrary propositional variable p . ‚ We do not have symbols for negative literals, to facilitate our later definitionof ‘positivity’. Note, however, that p is equivalent to the formula 1 p ‚ More generally, we admit decisions on only positive literals, not negativeones, for the same reason. Again, a formula
ApB may be replaced by theequivalent one
BpA .As shown in [5], the system eLNDT is adequate for reasoning about non-deterministicdecision trees.
Proposition 2.11 (Soundness and completeness, [5]) . eLNDT proves a sequent Γ Ñ ∆ (without extension variables) if and only if Ź Γ Ą Ž ∆ is valid. ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 7
One sanity check here is that the set of valid extension-free sequents is indeed co NP -complete, and so comprises an adequate logic for proof complexity. This isshown explicitly in [5], but is also subsumed by the analogous statement for the‘positive’ fragment of this language that we consider in the next section, namelyProposition 3.13.3. Monotone functions and positive proofs
In this section we shall recall monotone Boolean functions and positive NBPsthat compute them, and introduce a restriction of the system eLNDT that reasonsonly with such positive NBPs.3.1.
Monotone Boolean functions and positive programs.
A Boolean func-tion f : t , u n Ñ t , u is usually called ‘monotone’ if, whenever c P t , u n isobtained from b P t , u n by flipping 0s to 1s, we have f p b q ď f p c q . Rephrasingthis into our setting we have: Definition 3.1 (Monotonicity) . Given assignments α, β , we write α ď β if, for allpropositional variables p , we have α p p q ď β p p q , i.e. if α p p q “ β p p q “ f is monotone if α ď β ùñ f p α q ď f p β q .There are several known non-uniform ‘positive’ models for computing monotonefunctions, e.g. -free circuits or formulas, monotone span programs [17], and, inour setting, positive NBPs: Definition 3.2 (Positive NBPs, e.g. [13]) . A NBP is positive if, for every 0-edgefrom a node u to a node v , there is also a 1-edge from u to v . Fact 3.3.
A positive NBP computes a monotone Boolean function.Proof sketch.
Suppose α ď β and α p G q “
1. Let v “ p v , . . . , v n q be an acceptingrun, where v n “ i ă n , each v i is labelled by some propositional variable p i . We argue that v is also an accepting run of β . The critical case is when α p p i q “ β p p i q “
1; in which case the positivity condition on G ensures that there isnonetheless a 1-edge from v i to v i ` . (cid:3) A digression on monotone complexity and closures.
NBPs are a non-uniform version of non-deterministic logspace ( NL ): each NL language is acceptedby a polynomial-size family of NBPs and, conversely, the evaluation problem forNBPs is complete for NL . In particular, NL is precisely the class of languagesaccepted by, say, L -uniform families of NBPs.Naturally, our positive NBPs correspond to a positive version of NL too, called mNL by Grigni and Sipser in [13, 12]. Those works present a comprehensive devel-opment of positive models of computation and their underlying theory. In particularthere is a well-behaved notion of positive non-deterministic Turing machine basedon a similar idea to that of positive NBPs: roughly speaking, whenever a transi-tion is available when reading a 0, the same transition is available when reading a1. It turns out that the class mNL , induced by this machine model restricted tologarithmic size work tapes, is equivalent to the class of languages recognised by L -uniform families of positive NBPs.One natural construction that is available in the NBP setting (as opposed to,say, Boolean formulas or circuits) is the notion of a ‘positive closure’, which weshall work with later. PROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS p p p p p p p p p p Figure 3.
Positive closure of the OBDD for 2-out-of-4 Exact from Fig. 1.
Definition 3.4 (Positive closure of a NBP) . For a NBP G with 0-edges E and1-edges E , we write G ` for the NBP with the same vertex set and 0-edges E and1-edges E Y E . I.e., G ` is obtained from G by adding, for every 0-edge from anode u to a node v , a 1-edge from u to v (if there is not already one). Example 3.5.
The positive closure of the OBDD for 2-out-of-4 Exact from Fig. 1is given in Fig. 3.Note that G ` is always a positive NBP. Thus this construction gives us a ‘canon-ical’ positive version of a NBP. In many (but not all) cases, we can precisely charac-terise the semantic effect of taking positive closures, thanks to the following notion: Definition 3.6 (Monotone closure) . For a Boolean function f , we define its mono-tone closure m p f q , by m p f qp α q “ D β ď α.f p β q “ f is that it is the ‘least’ monotonefunction that dominates f , i.e. such that f ď m p f q . There is also a dual notionof the ‘greatest’ monotone function dominated by f which is similarly related to‘positive co-NBPs’, but we shall not make use of it in this work.In certain cases, the positive closure of a NBP G computes precisely the mono-tone closure of (the function computed by) G . Call a NBP read-once if, on eachpath, each propositional variable appears at most once. We have: Proposition 3.7 ([13]) . Let G be a read-once NBP computing a Boolean function f . Then G ` computes m p f q .Proof sketch. Suppose m p f qp α q “
1, and let v be an accepting run of G on some β ď α . Notice that v is also accepting for G ` on α : at any node v i labelled bysome p on which α and β differ, i.e. β p p q “ α p p q “
1, by positivity we alsohave a 1-edge v i to v i ` .Now suppose that G ` accepts α by a run v of nodes labelled by p respectively.Define β ď α by β p p i q “ v i to v i ` in G , otherwise β p p i q “
0. Note that β is indeed well-defined, by the read-once property, and indeed β ď α by definition of G ` . (cid:3) In particular, the above result holds when G is ‘ordered’ (an ‘OBDD’), i.e. propo-sitional variables occur in the same relative order in each path through G . We willsee an example of this with counting functions later in Section 4. ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 9
Example 3.8 (Monotone closure of Exact) . The monotone closure of the 2-out-of-4 Exact function is the 2-out-of-4 Threshold function, returning 1 if at least two ofits four inputs are 1. Since the branching programs from Fig. 1 were read-once, wehave by Proposition 3.7 above that the branching programs from Fig. 3 computethe 2-out-of-4 Threshold function.Note, however, that the result does not hold for arbitrary NBPs. In fact, there isno feasible notion of ‘positive closure’ on NBPs that always computes the monotoneclosure, due to the following result:
Theorem 3.9 ([13]) . There are monotone functions computed by polynomial-sizefamilies of NBPs, but no polynomial-size family of positive NBPs.
Note that this result is the analogue for the NBP model to Razborov’s seminalresults for the circuit model [27, 28]. This result follows by establishing a non-uniform version of ‘ mNL ‰ co mNL ’; in particular there is a monotone co NL lan-guage (namely non-reachability in a graph) computed by no polynomial-size familyof positive NBPs. The result above now follows by the Immerman-Szelepcs´enyitheorem that NL “ co NL [15, 29] and NL -completeness of NBP evaluation.3.3. Representations of positive branching programs.
Let us now return toour representation of NBPs by extended formulas. Recall that we implement non-determinism using disjunction, so we may duly define the corresponding notion ofpositivity at the level of eNDT formulas themselves:
Definition 3.10 (Positive formulas) . An eNDT formula is positive if, for eachsubformula of the form
ApB , we have B “ A _ C for some C .A set of extension axioms A “ t e i Ø A i u i ă n is positive if each A i is positive.Positive eNDT formulas, under positive extension axioms, are just representa-tions of positive NBPs. Notice in particular that a positive decision Ap p A _ B q issemantically equivalent to A _ p p ^ B q , which is monotone in A , p and B . Sinceevery other symbol/connective also computes a monotone function we may thusdirectly obtain the analogue of Fact 3.3: Proposition 3.11.
Suppose A “ t e i Ø A i u i ă n is a set of positive extension ax-ioms. Each positive eNDT formula A over e , . . . , e n ´ computes a monotoneBoolean function with respect to A . This argument proceeds by A -induction on A and is routine. Other than thefact that all connectives are monotone, we also rely on the fact that we have nonegative literals in our language.3.4. A system for positive branching programs.
The semantic equivalence of Ap p A _ B q and A _ p p ^ B q motivates the following ‘positive decision’ rules:(1) Γ , A Ñ ∆ Γ , p, B Ñ ∆ p ` - l Γ , Ap p A _ B q Ñ ∆ Γ Ñ ∆ , A, p Γ Ñ ∆ , A, B p ` - r Γ Ñ ∆ , Ap p A _ B q Naturally, these rules satisfy the subformula property and, moreover, are derivablein eLNDT by small proofs. As expected, none of the arguments A , p or B above‘change sides’ by the rules above. This is due to the fact that positive decisionsare, indeed, monotone, unlike general decisions, for which the x may change sides. Note, however, that the two rules above are not ‘dual’, in the logical sense, unlikegeneral decisions. This is because positive decisions are no longer self-dual.
Definition 3.12 (System eLNDT ` ) . The system eLNDT ` is defined just like eLNDT , except replacing the p - l and p - r rules by the positive ones above in Equa-tion (1). Moreover, all extension axioms and formulas occurring in a proof (inparticular cut-formulas) must be positive.As for eLNDT , we should verify that the set of valid positive sequents (withoutextension variables) is actually sufficiently expressive to be meaningful for proofcomplexity, i.e. that they are co NP -complete. While this is fairly immediate forother positive systems, such as MLK , it is not so clear here so we give a self-containedargument.
Proposition 3.13.
The set of valid positive sequents (without extension variables)is co NP -complete.Proof. By the Cook-Levin theorem [9, 24], we know that the validity problem forDNFs is co NP -complete, so we will show how to encode a DNF as an equi-validpositive sequent.First, note that we may express positive terms (i.e. conjunctions of propositionalvariables) as positive NDT formulas by exploiting the equivalence 0 x p _ B q ðñ _ p x ^ B q ðñ x ^ B . Recursively applying this equivalence we obtain:(2) m ľ i “ p i ðñ p p _ p p _ p¨ ¨ ¨ p m ´ p _ p m q ¨ ¨ ¨ qqq Let us write Conj p p , . . . , p m q for the positive NDT formula on the right above.Now, fix a DNF instance A over propositional variables p , . . . , p k and let A beobtained by replacing each negative literal p i by a fresh (positive) propositionalvariable p i . Write A “ n Ž i “ Ź p i , where each p i is a sequence of propositionalvariables (among p , . . . , p k , p , . . . , p k ). Now we have that the following positiveNDT sequent is equi-valid with A , as required: p _ p , . . . , p k _ p k Ñ Conj p p q , . . . , Conj p p n q (cid:3) Finally, our calculus eLNDT ` is indeed adequate for reasoning about positivesequents: Proposition 3.14 (Soundness and completeness) . eLNDT ` proves a positive se-quent Γ Ñ ∆ (without extension variables) if and only if Ź Γ Ą Ž ∆ .Proof sketch. Similarly to the argument in [BDK20] for eLNDT , we may proceed bycut-free proof search, and will not make use of any extension variables. Notice thateach logical rule is invertible , i.e. the validity of the conclusion implies the validityof each premiss. Moreover, each premiss (of a logical rule) has fewer connectivesthan the conclusion. Thus, bottom-up, we may simply repeatedly apply the logicalsteps until we reach sequents of only atomic formulae. Such a sequent is valid ifand only if there is a 0 on the LHS, a 1 on the RHS, or some propositional variable p on both sides. Each of these cases may be derived from initial sequents using theweakening rules, w - l and w - r . (cid:3) ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 11
Remark 3.15 (Completeness with respect to extension axioms) . While it is stan-dard to only consider extended proofs over extension-free theorems, let us pointout that we also have a stronger version of completeness with respect to sets of(positive) extension axioms.Given a set of (positive) extension axioms A “ t e i Ø A i u i ă n , a (positive) for-mula A over e , . . . , e n ´ is A -valid if, for every assignment α , we have α ( A A (cf. Definition 2.5). The same argument as in Proposition 3.14 can now be appliedto show completeness for A -valid positive sequents by proofs using the extensionaxioms A . The only difference is that, when we reach a connective-free sequent(bottom-up), extension variables may also occur, not just propositional variablesand constants. In this case we must use the extension axioms to unwind the exten-sion variables and continue the proof search algorithm. Termination of this processnow follows by appealing to A -induction (cf. Remark 2.7).3.5. Some basic theorems.
Let us now present some basic theorems of eLNDT ` ,which will all have polynomial-size proofs. These will be useful for our later ar-guments and, at the same time, exemplify how we will conduct proof complexitytheoretic reasoning in what follows.Let us first point out an expected property, that we can polynomially derive ageneral identity rule from the atomic version included in the definition of eLNDT ` .Albeit a simple observation, it has the consequence that applying substitutions offormulas for variables in proofs has only polynomial overhead in proof size. Proposition 3.16 (General identity) . Let A “ t e i Ø A i u i ă n be a set of positiveextension axioms. There are polynomial-size eLNDT ` proofs of A Ñ A , for positiveformulas A containing only extension variables among e , . . . , e n ´ .Proof. We construct a (dag-like) proof of the required sequent by A -induction.More precisely, for each such A , we construct a polynomial-size proof containingsequents B Ñ B for each B ď A A by A -induction on A . ‚ If A is a propositional variable then we are done by the rule id . ‚ If A “ Ñ w - r Ñ ‚ If A “ Ñ w - l Ñ ‚ If A “ e i for some i ă n , then we extend the proof obtained by the inductivehypothesis as follows, e i Ñ A i IH A i Ñ A i A i Ñ e i cut e i Ñ e i where the sequent marked IH is obtained by the inductive hypothesis, andthe other premisses are extension axioms from A . ‚ If A “ B _ C then we extend the proof obtained by the inductive hypothesisas follows, IH B Ñ B w - r B Ñ B, C IH C Ñ C w - r C Ñ B, C _ - l B _ C Ñ B, C _ - r B _ C Ñ B _ C where sequents marked IH are obtained by the inductive hypothesis. ‚ If A “ Bx p B _ C q then we extend the proof obtained by the inductivehypothesis as follows: IH B Ñ B w - r B Ñ B IH B Ñ B w - r B Ñ B, C p ` - r B Ñ Bp p B _ C q id p Ñ p w - l, w - r p, C Ñ B, p IH C Ñ C w - l, w - r p, C Ñ B, C p ` - r p, C Ñ Bp p B _ C q p ` - l Bp p B _ C q Ñ Bp p B _ C q where sequents marked IH are obtained by the inductive hypothesis.To evaluate proof size note that, at each step of the argument above, we add aconstant number of lines of polynomial size in A and A . Thus a polynomial boundfollows by Observation 2.8. (cid:3) Notice, in the final step above, that we do not formally ‘duplicate’ the subprooffor B Ñ B as this, recursively applied, could cause an exponential blowup. Thisis why the construction by A -induction is phrased as constructing a single proofthat contains all ‘smaller’ instances of identity already, with inductive steps justextending that proof. In what follows we shall be less rigorous when constructingformal proofs in this way, simply saying that we ‘construct them by A -induction’.We shall also typically leave proof complexity analysis like the one above implicit.For our later simulations, the following ‘truth conditions’ for positive decisionswill prove useful: Proposition 3.17 (Truth conditions) . Let A “ t e i Ø A i u i ă n be a set of posi-tive extension axioms and let A and B be formulas over e , . . . , e n ´ . There arepolynomial-size eLNDT ` proofs of the following sequents with respect to A : (1) Ap p A _ B q Ñ A, p (2) Ap p A _ B q Ñ A, B (3) A Ñ Ap p A _ B q (4) p, B Ñ Ap p A _ B q Proof.
We give the proofs explicitly: p q : id A Ñ A w - r A Ñ A, p id p Ñ p w - l, w - r p, B Ñ A, p p ` - l Ap p A _ B q Ñ A, p p q : id A Ñ A w - r A Ñ A, B id B Ñ B w - l, w - r p, B Ñ A, B p ` - l Ap p A _ B q Ñ A, B
ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 13 p q : id A Ñ A w - r A Ñ A, p id A Ñ A w - r A Ñ A, B p ` - r A Ñ Ap p A _ B q p q : id p Ñ p w - l, w - r p, B Ñ A, p id B Ñ B w - l, w - r p, B Ñ A, B p ` - r p, B Ñ Ap p A _ B q where the steps marked id are derivable by Proposition 3.16. (cid:3) Notice that, given that we have polynomial-size proofs for general identity,Proposition 3.16, the result above also just follows immediately from semantic va-lidity of the sequents (1)-(4) and completeness of eLNDT ` , Proposition 3.14, bysimply substituting the formulas A and B for appropriate constant-size instancesof (1)-(4). We gave the argument explicitly to exemplify formal proofs of the system eLNDT ` . We shall, however, make use of the aforementioned observation in theremainder of this work. Example 3.18 (A positive ‘medial’) . Branching programs enjoy elegant symme-tries. For instance, in our eNDT notation, we have validity of the following pair ofsequents, p AqB q p p CqD q Ø p ApC q q p BpD q corresponding to a certain permutations of nodes in NBPs.We also have a positive version of the law above, namely:(3) p Aq p A _ B qq p pp Aq p A _ B qq _ p Cq p C _ D qqq Ø p Ap p A _ C qq q pp Ap p A _ C qq _ p Bp p B _ D qqq The validity of this equivalence can be seen by noticing that each side is equivalentto A _ p p ^ C q _ p q ^ B q _ p p ^ q ^ D q . By completeness and substitution, we thushave polynomial-size proofs of (3).Finally, we will make use of the following consequence of the truth conditions: Corollary 3.19.
There are polynomial size eLNDT ` derivations of Γ , Ap p A _ B q Ñ ∆ , A p p A _ B q from hypotheses Γ , A Ñ ∆ , A and Γ , B Ñ ∆ , B , over any positive extension ax-ioms including all extension variables occurring in A and B .Proof. We give the derivation below:Γ , A Ñ ∆ , A Proposition 3.17.(3) A Ñ A p p A _ B q cut Γ , A Ñ ∆ , A p p A _ B q Γ , B Ñ ∆ , B Proposition 3.17.(4) p, B Ñ A p p A _ B q cut Γ , p, B Ñ ∆ , A p p A _ B q p ` - l Γ , Ap p A _ B q Ñ ∆ , A p p A _ B q Note that we have omitted several structural steps, namely w - l, w - r above cut -steps,to match contexts. We will typically continue to omit these, freely using ‘contextsplitting’ and ‘context sharing’ behaviour, under structural rules. (cid:3) p p p p p p ...... ... p n p n p n Figure 4.
An ‘ordered’ branching program (OBDD) forEx nk p p , . . . , p n q , where there are k
0s to the left of the 1, and n ´ k to the right.4. Programs for counting and their basic properties
Let us now consider some of the Boolean counting functions that appeared inour earlier examples more formally. The
Exact functions Ex nk : t , u n Ñ t , u aredefined by: Ex nk p b , . . . , b n q “ ðñ n ÿ i “ b i “ k I.e. Ex nk p b , . . . , b n q “ exactly k of b , . . . , b n are 1.Taking the monotone closures of these functions (cf. Definition 3.6), we obtainthe Threshold functions Th nk : t , u n Ñ t , u by:Th nk p b , . . . , b n q “ ðñ n ÿ i “ b i ě k I.e. Th nk p b , . . . , b n q “ at least k of b , . . . , b n are 1.For consistency with the exposition so far, given a list p “ p , . . . , p n ´ of propo-sitional variables, we construe Ex nk p p q as a Boolean function from assignments toBooleans, writing, say, Ex nk p p qp α q for its (Boolean) output. Similarly for Th nk p p q .4.1. OBDDs for Exact and their representations as eDTs.
It is well-knownthat counting functions like those above are computable by ‘ordered’ branchingprograms, or ‘OBDDs’ (see, e.g., [30]). These are deterministic branching programswhere variables occur in the same relative order on each path. For instance wegive an OBDD for Ex nk p p , . . . , p n q in Figure 4. Thanks to determinism, we canformalise these programs as ( _ -free) eDT formulas as follows: Definition 4.1 (eDTs for Exact) . For each list p of propositional variables, andeach integer k , we introduce an extension variable e p k and write E for the set of all ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 15 extension axioms of the form (i.e. for all choices of p , p and k ),(4) e ε Ø e εk Ø k ‰ e p p k Ø e p k pe p k ´ where we write ε for the empty list.Note that, even though E is an infinite set, we may use it as the underlyingset of extension axioms for proofs, with the understanding that only finitely manywill actually ever be used in a particular proof. We will typically not explicitlycompute this set, but such a consideration will be subsumed by our analysis ofproof complexity.While the extension variables above (and their axioms) do not strictly follow thesubscripting conditions from Definition 2.4, we may understand them to be ‘names’for the appropriate subscripting. It suffices to establish the well-foundedness of theextension axiom set in (4), which is clear by induction on the length of the super-script. We implicitly assume here, and for other well-founded extension axioms,that appropriately small subscripts are assigned to extension variables to satisfythe subscripting condition and to not contribute significantly to proof complexity. Proposition 4.2.
Let p “ p p , . . . , p n q . e p k computes Ex nk p p q , with respect to E .Proof. We show that α ( E e p k ðñ Ex nk p p qp α q “ n of the list p . If n “ k p q attains the value 1 just if k “
0, so the result isimmediate from the first two axioms of (4). For the inductive step we have: α ( E e p p k ðñ α ( E e p k pe p k ´ by (4) and Definition 2.5 ðñ α ( E e p k α p p q “ α ( E e p k ´ α p p q “ ðñ Ex nk p p qp α q “ α p p q “ nk ´ p p qp α q “ α p p q “ ðñ Ex n ` k p p, p qp α q “ (cid:3) Programs for Threshold via positive closure.
Notice that, since the Ex-act programs we considered were OBDDs which are, in particular, read-once, thesemantic characterisation of positive closure by monotone closure from Proposi-tion 3.7 from applies. Looking back to Figure 4, the positive closures we are after(as NBPs) are given in Figure 5. Realising this directly as eNDT formulas andextension axioms we obtain the following:
Definition 4.3 (Positive eNDTs for Threshold) . For each list p of propositionalvariables, and each integer k , we introduce an extension variable t p k and write T for the set of all extension axioms of the form (i.e. for all choices of p , p and k ):(5) t ε Ø t εk Ø k ‰ t p p k Ø t p k p p t p k _ t p k ´ q Again, even though T is an infinite set, we shall typically write eLNDT ` proofswith respect to this set of extension axioms, with the understanding that only p p p p p p ...... ... p n p n p n . . . . . . Figure 5.
The positive closure of the OBDD for Exact from Fig-ure 4, computing Th nk p p , . . . , p n q . Again, there are k
0s to the leftof the 1, and n ´ k to the right.finitely many are ever used in any particular proof. Again, we will typically not ex-plicitly compute this set, but such a consideration will be subsumed by our analysisof proof complexity.Note that the extension variables t p k and extension axioms T above are justthe positive closures of e p k and E earlier, within the eNDT setting. Thus, as aconsequence of Proposition 3.7 we have that, for each non-negative k , t p k computesexactly the threshold function Th nk p p q with respect to T . Corollary 4.4. If k ě , then t p k computes Th nk p p q , with respect to T . Note that, for k negative, we could have alternatively set t εk to be 1. We couldhave also simply set t p to be 1 for arbitrary p . Instead, we have chosen to system-atically take the positive closure of the aforementioned Exact programs, to makeour exposition more uniform.4.3. Small proofs of basic counting properties.
Our main results rely on hav-ing small proofs of characteristic properties of counting formulae, which we dulygive in this section.First we need to establish a basic monotonicity property:
Proposition 4.5 ( t p k is decreasing in k ) . There are polynomial-size eLNDT ` proofsof the following sequents over extension axioms T : (1) Ñ t p (2) t p k ` Ñ t p k (3) t p k Ñ whenever k ą | p | Proof.
We proceed by induction on the length of p . In the base case, when p “ ε ,all three properties follow easily from T initial sequents, weakening and cuts. ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 17
For the inductive steps, we construct polynomial-size proofs as follows:(1) : Ñ t p by the inductive hypothesis Ñ t p p p t p _ t p ´ q by Proposition 3.17.(3) Ñ t p p by extension axioms T (2) : t p p k ` Ñ t p k ` x p t p k ` _ t p k q by extension axioms T Ñ t p k p p t p k _ t p k ´ q by inductive hypotheses and Corollary 3.19 Ñ t p p k by extension axioms T again(3) : t p p k Ñ t p k p p t p k _ t p k ´ q by extension axioms T Ñ t p k , t p k ´ by Proposition 3.17.(2) Ñ t p k ´ by (2) and contraction Ñ by inductive hypothesis (cid:3) The arguments above should be read by obtaining each sequent by the justifica-tion given on the right, possibly with some cuts and structural rules.Note that Corollary 3.19 allows us to apply previously proven implications orequivalences ‘deeply’ within a formula. We will use such reasoning throughout thiswork, but shall typically omit further mentioning such uses of Corollary 3.19 tolighten the exposition.For the complexity bound, note that only polynomially many lines occur: weonly require k ` i ď k . This sort of complexity analysis will usually suffice for laterarguments, in which case we shall suppress them unless further justification isrequired.One of the key points we shall exploit in what follows is the provable symmetry of t p k , in terms of the the ordering of p . We shall establish this through a series ofresults, beginning by showing a form of ‘case analysis’ on a propositional variableoccurring in a list: Lemma 4.6 (Case analysis) . There are polynomial-size eLNDT ` proofs of, t p q q k Ø t q pq k over the extension axioms T .Proof. We proceed by induction on the length of p . The base case, when p isempty, follows immediately by general identity, Proposition 3.16.For the inductive step we construct polynomial-size proofs as follows: t p p q q k Ø t p q q k p p t p q q k _ t p q q k ´ q by T Ø t q pq k p p t q pq k _ t q pq k ´ q by IH and Corollary 3.19 Ø t pq k q p t pq k _ t pq k ´ q p p t pq k q p t pq k _ t pq k ´ q _ t pq k ´ q p t pq k ´ _ t pq k ´ qq by T Ø t pq k p p t pq k _ t pq k ´ q q p t pq k p p t pq k _ t pq k ´ q _ t pq k ´ p p t pq k ´ _ t pq k ´ qq by Example 3.18 Ø t p pq k q p t p pq k _ t p pq k ´ q by T Ø t qp pq k by T (cid:3) Similarly to the proof of Proposition 4.5, the above argument should be readas providing polynomial-size proofs ‘in both directions’, by the justifications givenon the right. Note that, we restrict cedents to singletons when using Ø in thisway, to avoid ambiguity of the comma delimiter. Polynomial proof size is, again,immediate by inspection on the number of lines. Theorem 4.7 (Symmetry) . Let π be a permutation of p . Then there are polynomial-size eLNDT ` proofs over the extension axioms T of: t p k Ø t π p p q k Proof.
Write p “ p ¨ ¨ ¨ p n and write π p p q “ q ¨ ¨ ¨ q n . We construct polynomial-sizeproofs by repeatedly applying Lemma 4.6 as follows: t p k Ø t q n p n k by Lemma 4.6 Ø t q n ´ q n p n ´ ,n k by Lemma 4.6... Ø t q ¨¨¨ q n k by Lemma 4.6 Ø t π p p q k by definition of q where p i,...,n is just p with the elements q i , . . . , q n removed, otherwise preservingrelative order of the propositional variables. (cid:3) Case study: the pigeonhole principle
The pigeonhole principle is usually encoded in propositional logic by a family of -free sequents of the following form: n ` ľ i “ n ł j “ p ij Ñ n ł j “ n ł i “ n ` ł i “ i ` p p ij ^ p i j q Here it is useful to think of the propositional variables p ij as expressing “pigeon i sits in hole j ”. In this way the left-hand side (LHS) above expresses that each pigeon1 , . . . , n ` , . . . , n , and the right-hand side (RHS) expresses thatthere is some hole occupied by two (distinct) pigeons. Note that this encodingallows the mapping from pigeons to holes to be ‘multi-functional’, i.e. the LHS isallows for a pigeon to sit in multiple holes.In the setting of eLNDT ` we may not natively express conjunctions, so we adopta slightly different encoding. Being a sequent, the outermost conjunctions on theLHS above can simply be replaced by commas; the subformulas p ij ^ p i j may beencoded as 0 p ij p _ p i j q .Thus we shall work with the following encoding of the pigeonhole principlethroughout this section: Definition 5.1 (Pigeonhole principle) . PHP n is the following positive sequent: n ł j “ p ij + n ` i “ Ñ n ł j “ n ł i “ n ` ł i “ i ` p ij p _ p i j q We write
LPHP n and RPHP n for the LHS and RHS, respectively, of PHP n .5.1. Summary of proof structure.
The main result of this section is:
Theorem 5.2.
There are polynomial-size eLNDT ` proofs of PHP n . At a high level, we shall employ a traditional proof structure for proving
PHP n ,specialising somewhat to our setting for certain intermediate results. Before sur-veying this, let us introduce some notation. Notation 5.3.
We fix n P N throughout this section and write: ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 19 ‚ p i for the list p i , . . . , p in , and just p for the list p , . . . , p n ` . ‚ p ⊺ j for the list p j , . . . , p n ` j and just p ⊺ for the list p ⊺ , . . . , p ⊺ n .The notation p ⊺ is suggestive since, construing p as an p n ` q ˆ n matrix ofpropositional variables, p ⊺ is just the transpose n ˆ p n ` q matrix.Our approach towards proving PHP n in eLNDT ` (with small proofs) will bebroken up into the three smaller steps, proving the following sequents respectively:(1) LPHP n Ñ t p n ` (2) t p n ` Ñ t p ⊺ n ` (3) t p ⊺ n ` Ñ RPHP n Notice that, since p ⊺ is just a permutation of p , we already have small proofs of(2) from Theorem 4.7. In the next two subsections we shall focus on the other twoimplications, for which the following lemma will be quite useful: Lemma 5.4 (Merging and splitting threshold arguments) . There are polynomial-size eLNDT ` proofs, over extension axioms T of the following sequents: (1) t p k , t q l Ñ t pq k ` l (2) t pq k ` l Ñ t p k ` , t q l Proof.
We proceed by induction on the length of p . In the base case, when p “ ε ,we have two cases for (1): ‚ if k “ t ε , t q l Ñ t q l follows by id and w - l . ‚ if k ‰ t εk Ñ T , whence t εk , t q l Ñ t q k ` l followsby 0, cut and weakenings.For (2), we have polynomial-size proofs of t q k ` l Ñ t q l already from Proposition 4.5,whence we obtain t q k ` l Ñ t εk ` , t q l by w - r .For the inductive step we shall appeal to Corollary 3.19. First, for (1), by theinductive hypothesis we already have a polynomial-size proof of: t p k , t q l Ñ t pq k ` l t p k ´ , t q l Ñ t pq k ` l ´ Thus, by Corollary 3.19 we can derive, t p k p p t p k _ t p k ´ q , t q l Ñ t pq k ` l p p t pq k ` l _ t pq k ` l ´ q whence the required sequent t p p k , t q l Ñ t p pq k ` l follows by the extension axioms T .For (2), by the inductive hypothesis we have a polynomial-size proof of: t pq k ` l Ñ t p k ` , t q l t pq k ` l ´ Ñ t p k , t q l Thus, by Corollary 3.19, we can derive, t pq k ` l p p t pq k ` l _ t pq k ` l ´ q Ñ t p k ` p p t p k ` _ t p k q , t q l whence the required sequent t p pq k ` l Ñ t p p k ` , t q l follows by the extension axioms T . (cid:3) From
LPHP n to p n ` q -threshold. In this subsection we will give smallproofs of the sequent (1) from Section 5.1.
Lemma 5.5.
Let q “ q , . . . , q k ´ . For all j ă k , there are polynomial-size eLNDT ` proofs over extension axioms T of: q j Ñ t q
10 PROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS
Proof.
First we derive,(6) q j Ø t q j as follows: q j Ø q j p _ q by Proposition 3.17, axioms and cut Ø t ε q j p t ε _ t ε q by extension axioms T and Corollary 3.19 Ø t q j by extension axioms T By repeatedly applying Lemma 5.4.(1) we obtain polynomial-size proofs of, t q , . . . , t q j ´ , t q j , t q j ` , . . . , t q k ´ Ñ t q However, we also have small proofs of Ñ t q j by Proposition 4.5.(1), and so applying k ´ t q j Ñ t q The required sequent now follows by simply cutting Eq. (6) against Eq. (7). (cid:3)
Proposition 5.6.
There are polynomial-size eLNDT ` proofs of (1), i.e., LPHP n Ñ t p n ` over extension axioms T .Proof. Let i P t , . . . , n ` u . By Lemma 5.5 above, we have small proofs of, p ij Ñ t p i for each j “ , . . . , n . By applying n ´ _ - l steps we derive:(8) ł p i Ñ t p i Now, applying Lemma 5.4.(1) n times (and using cuts), we obtain small proofs of:(9) t p , . . . , t p n ` Ñ t p n ` Finally, by instantiating Eq. (8) for each i “ , . . . , n ` n ` cut steps against Eq. (9) we derive the required sequent: ł p , . . . , ł p n ` Ñ t p n ` (cid:3) From p n ` q -threshold to RPHP n . Before deriving the final sequent (3) forour proof of
PHP n , we will need some lemmas. Lemma 5.7.
Let q “ q , . . . , q k ´ . There are polynomial-size eLNDT ` proofs of, q, t q Ñ t q p _ q i qu i ă k over extension axioms T .Proof. For each i ă k , by Proposition 3.17.(4) we have a (constant-size) proof of, q, q i Ñ q p _ q i q and so by cutting against appropriate instances of Eq. (6) we obtain: q, t q i Ñ q p _ q i q Instantiating the above for each i ă k and applying several w - r and _ - l steps weobtain:(10) q, ł i ă k t q i Ñ t q p _ q i qu i ă k ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 21
Now, by repeatedly applying Lemma 5.4.(2) (under cuts) and _ - r steps we obtainpolynomial-size proofs of:(11) t q Ñ ł i ă k t q i Finally, we conclude by cutting Eq. (11) above against Eq. (10). (cid:3)
Lemma 5.8.
Let p “ p , . . . , p k ´ . There are polynomial-size eLNDT ` proofs of, t p Ñ t q i p _ q i qu i ă i ă k over extension axioms T .Proof. We proceed by induction on the length k of q . In the base case, when q “ ε ,we have an axiom t ε Ñ T , whence we conclude by a cut against the 0-axiomand w - r .For the inductive step, we obtain by two applications of Lemma 5.4.(2) thefollowing sequents: t q q Ñ t q , t q t q q Ñ t q , t q Now we already have small proofs of t q Ñ q from Eq. (6) and of t q Ñ from Propo-sition 4.5.(3), and so cutting against the respective sequents above we obtain: t q q Ñ q, t q (12) t q q Ñ t q (13)Finally, we combine these sequents using cuts as follows:Eq. (12) t q q Ñ q, t q Eq. (13) t q q Ñ t q Lemma 5.7 q, t q Ñ t q p _ q i qu i ă k cut t q q Ñ t q p _ q i qu i ă k IH t q Ñ t q i p _ q i qu i ă i ă k cut t q q Ñ t q i p _ q i qu i ă i ă k where the proof marked IH is obtained from the inductive hypothesis. (cid:3) Proposition 5.9.
There are polynomial-size eLNDT ` proofs over T of (3), i.e. of: t p ⊺ n ` Ñ RPHP n Proof.
Recall that p ⊺ “ p ⊺ , . . . , p ⊺ n , so by n ´ t p ⊺ n ` Ñ t p ⊺ , . . . , t p ⊺ n Now, instantiating Lemma 5.8 with q “ p ⊺ j we also have small proofs of,(15) t p ⊺ j Ñ t p ij p _ p i j qu ď i ă i ď n for each j “ , . . . , n . Finally, we may apply n cut steps on Eq. (14) against eachinstance of Eq. (15) (for j “ , . . . , n ) and apply _ - r steps to obtain the requiredsequent. (cid:3) Putting it all together.
We are now ready to assemble our proofs for
PHP n . Proof of Theorem 5.2.
We simply cut together the proofs of (1), (2) and (3) thatwe have so far constructed:Proposition 5.6
LPHP n Ñ t p n ` Theorem 4.7 t p n ` Ñ t p ⊺ n ` Proposition 5.9 t p ⊺ n ` Ñ RPHP n cut LPHP n Ñ RPHP n (cid:3) Positive simulation of non-positive proofs
In the previous section we showcased the capacity of the system eLNDT ` to for-malise basic counting arguments by giving polynomial-size proofs of the pigeonholeprinciple. In this section we go further and give a general polynomial simulation of eLNDT , over positive sequents, by adapting a method from [3]. Theorem 6.1. eLNDT ` polynomially simulates eLNDT over positive sequents. While the high-level structure of the argument is similar to that of [3], we mustmake several specialisations to the current setting due to the peculiarities of eNDTformulas and extension axioms.6.1.
Summary of proof structure.
Before giving the low-level details, let ussurvey our approach towards proving Theorem 6.1, in particular comparing it tothe analogous methodology from [3]. Our strategy is divided into three main parts,which mimic the analogous proof structure from [3].In the first part, Section 6.2, we deal with the non-positive formulas occurringin an eLNDT proof. The intuition is similar to what is done in [3] where theyfirst reduced all negations to the variables using De Morgan duality. In our settingformulas are no longer closed under duality but, nonetheless, we are able to devisefor each formula A an appropriate ‘positive normal form’ A ´ . A ´ may containnegative literals (in particular as decision variables), but all decisions themselvesare positive. We duly consider an extension eLNDT `´ of eLNDT ` which admitsnegative literals p and has two extra axioms: p, p Ñ and Ñ p, p . The main result ofthe first part is that eLNDT `´ polynomially simulates eLNDT over positive sequents(Corollary 6.5), by essentially replacing each formula occurrence A by A ´ andlocally repairing the proof (Theorem 6.3).In the second part the aim is to ‘replace’ negative literals in a eLNDT `´ proof bycertain threshold formulas from Definition 4.3. The is the same idea as in [3], but inour setting we must deal with certain technicalities encountered when substitutingextended formulas in eLNDT `´ and eLNDT ` . In particular, if a literal occurs asa decision variable, then we cannot directly substitute it for an extension variable(e.g. a threshold formula t p k ), since the syntax of eLNDT (and its fragments) doesnot allow for this. To handle this issue appropriately, we introduce in Section 6.3a refinement of our previous threshold extension variables and axioms, definedmutually inductively with eNDT formulas themselves, that accounts for all suchsubstitution situations (Definition 6.6).For the remainder of the argument, in Section 6.4 we fix a eLNDT `´ proof P ofΓ Ñ ∆ over extension axioms A and propositional variables p “ p , . . . , p m ´ . Wedefine, for k ě
0, systems eLNDT ` k p P q that each have polynomial-size proofs P k of ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 23 Γ Ñ ∆ (Lemma 6.12). Morally speaking, this simulation is by ‘substituting’ thresh-olds for negative literals, and the consequent new axioms required in eLNDT ` k p P q are parametrised by the threshold k . We point out that eLNDT ` k p P q itself is tai-lored to the specific set of extension axioms A and propositional variables p tofacilitate the choice of threshold formulas and required extension variables/axioms.The final part, Section 6.5, essentially stitches together proofs obtained in each eLNDT ` k p P q for 0 ď k ď m `
1. More precisely, using basic properties of thresholdformulas, we show that each eLNDT ` k p P q proof of a positive sequent Γ Ñ ∆ can bepolynomially transformed into a eLNDT ` proof of t p k , Γ Ñ ∆ , t p k ` , over appropriateextension axioms (Lemma 6.14). We conclude the argument for our main resultTheorem 6.1 by simply cutting these together and appealing to Proposition 4.5.6.2. Positive normal form of eLNDT proofs.
We shall temporarily work witha presentation of eLNDT within eLNDT ` by allowing negative literals, in order tofacilitate our later translations. For this reason, let us introduce, for each proposi-tional variable p , a distinguished variable p , which we shall also refer to as ‘negativeliterals’.The system eLNDT `´ is defined just like eLNDT ` but also allows negative literals p to appear in (positive) decision steps. All syntactic positivity constraints remain.Furthermore, eLNDT `´ has two additional initial sequents: - l p, p Ñ - r Ñ p, p The system eLNDT `´ admits essentially a ‘normal form’ of eLNDT proofs: Definition 6.2.
We define a (polynomial-time) translation from an eLNDT formula A to a eLNDT `´ formula A ´ as follows:0 ´ : “ ´ : “ p ´ : “ pp ´ : “ p e ´ i : “ e i p A _ B q ´ : “ A ´ _ B ´ p ApB q ´ : “ p p _ A ´ q _ p p _ B ´ q For a multiset of formulas Γ “ A , . . . , A n we write Γ ´ : “ A ´ , . . . , A ´ n . For a setof extension axioms A “ t e i Ø A i u i ă n , we write A ´ for t e i Ø A ´ i u i ă n . Theorem 6.3.
Let P be an eLNDT proof of Γ Ñ ∆ over extension axioms A .There is a eLNDT `´ proof P ´ of Γ ´ Ñ ∆ ´ over A ´ of size polynomial in | P | . Notice that, since ¨ ´ commutes with all connectives except for decisions, to provethe above result it suffices to just derive the translation of decision steps. For thisit will be useful to have another ‘truth’ lemma: Lemma 6.4 (Truth for ¨ ´ -translation) . Let A “ t e i Ø A i u i ă n be a set of posi-tive extension axioms and let A and B be formulas over e , . . . , e n ´ . There arepolynomial-size eLNDT `´ proofs of the following sequents over A ´ : (1) p ApB q ´ Ñ A ´ , p (2) p ApB q ´ , p Ñ B ´ (3) A ´ Ñ p ApB q ´ , p (4) p, B ´ Ñ p ApB q ´ Proof.
We give the proofs explicitly below: Ñ w - r Ñ A ´ , p id A ´ Ñ A ´ w - l, w - r p, A ´ Ñ A ´ , p p ` - l p p _ A ´ q Ñ A ´ , p Ñ w - r Ñ A ´ , p id p Ñ p w - l, w - r p, B ´ Ñ A ´ , p p ` - l p p _ B ´ q Ñ A ´ , p _ - l p ApB q ´ Ñ A ´ , p Ñ w - l, w - r , p Ñ B ´ - l p, p Ñ w - l, w - r p, A ´ , p Ñ B ´ p ` - l p p _ A ´ q , p Ñ B ´ Ñ w - l, w - r , p Ñ B ´ id B ´ Ñ B ´ w - l p, B ´ , p Ñ B ´ p ` - l p p _ B ´ q , p Ñ B _ - l p ApB q ´ , p Ñ B ´ - r Ñ p, p w - l, w - r A ´ Ñ p, p id A ´ Ñ A ´ w - r A ´ Ñ , A ´ , p p ` - r A ´ Ñ p p _ A ´ q , p w - r, _ - r A ´ Ñ p ApB q ´ , p id p Ñ p w - l, w - r p, B ´ Ñ , p id B ´ Ñ B ´ w - l, w - r p, B ´ Ñ , B ´ p ` - r p, B ´ Ñ p p _ B ´ q w - r, _ - r p, B ´ Ñ p ApB q ´ (cid:3) We can now prove our polynomial-size interpretation of eLNDT within eLNDT `´ : Proof of Theorem 6.3.
We proceed by a straightforward induction on the length of P . The critical cases are when P ends with decision steps, which we translate asfollows. A left decision step, Γ , A Ñ ∆ , p Γ , p, B Ñ ∆ p - l Γ , ApB Ñ ∆is simulated by the following derivation:Lemma 6.4.(1) p ApB q ´ Ñ A ´ , p Γ , A ´ Ñ ∆ , p cut Γ , p ApB q ´ Ñ ∆ , p Lemma 6.4.(2) p ApB q ´ , p Ñ B ´ Γ , p, B ´ Ñ ∆ cut Γ , p ApB q ´ , p Ñ ∆ cut Γ , p ApB q ´ Ñ ∆A right decision step, Γ Ñ ∆ , A, p Γ , p Ñ ∆ , B p - r Γ Ñ ∆ , ApB is simulated by the following derivation:Γ Ñ ∆ , A ´ , p Lemma 6.4.(3) A ´ Ñ p ApB q ´ , p cut Γ Ñ ∆ , p ApB q ´ , p Γ , p Ñ ∆ , B ´ Lemma 6.4.(4) p, B ´ Ñ p ApB q ´ cut Γ , x Ñ ∆ , p ApB q ´ cut Γ Ñ ∆ , p ApB q ´ (cid:3) Finally we note that the translation above gives rise to a bona fide polynomialsimulation of eLNDT by eLNDT `´ over positive sequents: ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 25
Corollary 6.5. eLNDT `´ polynomially simulates eLNDT , over positive sequents.Proof. From Theorem 6.3 above, it suffices to derive Γ Ñ ∆ from Γ ´ Ñ ∆ ´ . Forthis we shall give short proofs of,(16) A Ø A ´ when A is positive and free of extension variables, whence Γ Ñ ∆ follows fromΓ ´ Ñ ∆ ´ by several cuts. We proceed by structural induction on A , for which thecritical case is when A is a decision formula. We prove the two directions separately.First, note that we have polynomial-size proofs of the following sequents: Ap p A _ B q Ñ A, p by Proposition 3.17.(1)(17) Ap p A _ B q Ñ A _ B by Proposition 3.17.(2) and _ - r (18) A Ñ p Ap p A _ B qq ´ , p by Lemma 6.4.(3) and IH (19) p, A _ B Ñ p Ap p A _ B qq ´ by Lemma 6.4.(4) and IH (20)We arrange these into a proof of the left-right direction as follows:Eq. (17) Eq. (19) A - cut Ap p A _ B q Ñ p Ap p A _ B qq ´ , p Eq. (18) Eq. (20) A _ B - cut Ap p A _ B q , p Ñ p Ap p A _ B qq ´ p - cut Ap p A _ B q Ñ p Ap p A _ B qq ´ Next, note that we have small proofs of the following sequents: A Ñ Ap p A _ B q by Proposition 3.17.(3)(21) p, B Ñ Ap p A _ B q by Proposition 3.17.(4)(22) p Ap p A _ B qq ´ Ñ A, p by Lemma 6.4.(1) and IH (23) p Ap p A _ B qq ´ Ñ A _ B by Lemma 6.4.(2) and IH (24)We arrange these into a proof of the left-right direction as follows:Eq. (23) Eq. (21) A - cut Ap p A _ B q ´ Ñ Ap p A _ B q , p Eq. (24) Eq. (21) A _ B - cut Ap p A _ B q ´ , p Ñ Ap p A _ B q p - cut Ap p A _ B q ´ Ñ Ap p A _ B q (cid:3) Generalised counting formulas.
The argument of [3] relies heavily on sub-stitution of formulas for variables in proofs of LK . Being based on usual Booleanformulae, this is entirely unproblematic in that setting, whereas in our setting wedeal with extension variables that represent NBPs via extension axioms, and sohandling substitutions is much more subtle and notationally heavy.We avoid giving a uniform treatment of this, instead specialising to countingformulas, but we must nonetheless carefully give an appropriate mutually recursiveformulation of formulas and extension variables. Definition 6.6 (Threshold decisions) . We introduce extension variables r A t p k p A _ B q s for each list p of propositional variables, integer k , and formulas A, B .We extend T to include all extension axioms of the following form, with p , k, A, B ranging as just described:(25) r A t ε p A _ B q s Ø A _ B r A t εk p A _ B q s Ø A k ‰ r A t p p k p A _ B q s Ø r A t p k p A _ B q s p pr A t p k p A _ B q s _ r A t p k ´ p A _ B q sq Note that, despite the presentation, r A t p k p A _ B q s is, formally speaking, a singleextension variable, not a decision on the extension variable t p k which, recall, we donot permit. This is why we use the square brackets to distinguish it from otherformulas, though we shall justify this notation shortly.One should view the extension variables above and the notion of an eLNDT ` formula as being mutually defined, so as to avoid foundational issues. For instance,we allow an extension variable r C t p k p C _ D q s , where C or D may themselves containextension variables of the form r A t q k p A _ B q s . By building up formulas and extensionvariables by mutual induction we ensure that such constructions are well-founded.Let us briefly make this formal in the following remark: Remark 6.7 (Well-foundedness of T ) . We may define the extension variables r A t p k p A _ B q s and eLNDT ` formulas ‘in stages’ as follows: ‚ Write Φ for the set of all eLNDT ` formulas over some base set of extensionvariables E “ t e , e , . . . u . ‚ Write T n for the set of all extension variables t p k and of the form r A t p k p A _ B q s with A, B P Φ n . ‚ Write Φ n ` for the set of all formulas built from propositional variables,disjunctions, positive decisions and extension variables from T n , E n and afresh set of new extension variables E n ` “ t e p n ` q , e p n ` q , . . . u .Within each T n we (partially well-)order extension variables by the length of thesuperscript p , and within each E n we (well-)order the e ni s by the subscript i . Fi-nally, we set E n ă T n ă E n ` (i.e. if e P E n , t P T n and e P E n ` then e ă t ă e ).In this way the extension axioms from Eq. (25) (and Eq. (5)) indeed satisfy therequired well-foundedness criterion. We may also admit further extension axiomsallowing elements of E n to abbreviate formulas in Φ n (satisfying the indexing con-dition internal to E n ), preserving well-foundedness. We shall indeed do this later.Note, however, that the required order type on indices of these extension vari-ables, a priori, exceeds the ordinal ω . This causes no issue for us since, in any finiteproof, we will only use finitely many extension variables, and so may construe eachindex as a (relatively small) natural number while preserving the aforementionedorder. We shall gloss over this issue in what follows.As previously mentioned, our notation r A t p k p A _ B q s is designed to be suggestive,justified by the following counterpart of the truth conditions from Proposition 3.17: Proposition 6.8 (Truth conditions for threshold decisions) . There are polynomialsize eLNDT ` proofs over T of: (1) r A t p k p A _ B q s Ñ A, t p k (2) r A t p k p A _ B q s Ñ A, B (3) A Ñ r A t p k p A _ B q s (4) t p k , B Ñ r A t p k p A _ B q s . ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 27
Proof.
We proceed by induction on the length of p . For the base case when p “ ε ,we have the following proofs: Ñ T , cut Ñ t ε w - l, w - r r A t ε p A _ B q s Ñ A, t ε T r A t ε p A _ B q s Ñ A _ B id , _ , cut r A t ε p A _ B q s Ñ A, B id A Ñ A w - r, _ - r A Ñ A _ B T , cut A Ñ r A t ε p A _ B q s id B Ñ B w - l, w - r, _ - r t ε , B Ñ A _ B T , cut t ε , B Ñ r A t ε p A _ B q s For the inductive step for (1) we derive the following sequents: r A t p k p A _ B q s Ñ A, t p k by IH r A t p k ´ p A _ B q s Ñ A, t p k ´ by IH r A t p k p A _ B q s p pr A t p k p A _ B q s _ r A t p k ´ p A _ B q sq Ñ A, t p k p p t p k _ t p k ´ q by Corollary 3.19 r A t p p k p A _ B q s Ñ A, t p p k by T and 2 cut For the inductive step for (2) we derive the following sequents: r A t p k p A _ B q s Ñ A, B by IH p, r A t p k ´ p A _ B q s Ñ A, B by IH and w - l r A t p k p A _ B q s p pr A t p k p A _ B q s _ r A t p k ´ p A _ B q sq Ñ A, B by p ` - l r A t p p k p A _ B q s Ñ A, B by T and cut For the inductive step for (3) we derive the following sequents: A Ñ r A t p k p A _ B q s , p by IH and w - rA Ñ r A t p k p A _ B q s , r A t p k ´ p A _ B q s by IH and w - rA Ñ r A t p k p A _ B q s p pr A t p k p A _ B q s _ r A t p k ´ p A _ B q sq by p ` - rA Ñ r A t p p k p A _ B q s by T and cut For the inductive step for (4) we derive the following sequents: t p k , B Ñ r A t p k p A _ B q s by IH t p k ´ , B Ñ r A t p k ´ p A _ B q s by IH t p k p p t p k _ t p k ´ q , B Ñ r A t p k p A _ B q s p pr A t p k p A _ B q s _ r A t p k ´ p A _ B q sq by Corollary 3.19 t p p k , B Ñ r A t p p k p A _ B q s by T and 2 cut (cid:3) ‘Substituting’ thresholds for negative literals. For the remainder of thissection, let us work with a fixed eLNDT `´ proof P , over extension axioms A “t e i Ø A i p e , . . . , e i ´ qu i ă n , of a positive sequent Γ Ñ ∆ containing propositionalvariables among p “ p , . . . , p m ´ and extension variables among e “ e , . . . , e n ´ .Recall that, since we are eventually trying to give a polynomial simulation of eLNDT by eLNDT ` over positive sequents, our consideration of eLNDT `´ here suf-fices, by Corollary 6.5. We shall also work with the extension axioms T from theprevious subsection, and will soon explain its interaction with A from P .Throughout this section, we shall write p i for p , . . . , p i ´ , p i ` , . . . , p m ´ , i.e. p with the variable p i removed. We shall define yet another intermediary system eLNDT ` k p P q , or rather a family of such systems, one for each k ě
0. Before that,we need to introduce the following translation of formulas.
Definition 6.9 (‘Substituting’ thresholds) . We define a (polynomial-time) trans-lation from an eLNDT `´ formula A (over p , p and e ) to a eLNDT ` formula A k (over p , some extension variables e k and extension variables from T ) as follows: ‚ k : “ ‚ k : “ ‚ p ki : “ p i ‚ p ki : “ t p i k ‚ e ki is a fresh extension variable. ‚ p A _ B q k : “ A k _ B k ‚ p Ap i p A _ B qq k : “ A k p i p A k _ B k q‚ p Ap i p A _ B qq k : “ r A k t p r { p i s k p A k _ B k q s We also define A k : “ t e ki Ø A ki p e k , . . . , e ki ´ qu i ă n , and if Γ “ B , . . . , B l we writeΓ k for B k , . . . , B kl .In what follows, we shall work with the set of extension axioms T Y A k , so let ustake a moment to justify that this set of extension axioms is indeed well-founded. Remark 6.10 (Well-foundedness, again) . Following on from Remark 6.7, well-foundedness of T Y A k follows from a suitable indexing of the extension variablestherein. To this end we assign ‘stages’ to each formula A k by A -induction on A ,using the notation of Remark 6.7: ‚ p ki , k , k P Φ . ‚ p ki P T . ‚ e ki P E m Ď Φ m if A ki p e k , . . . , e ki ´ q P Φ m , with index i (i.e., e ki is e mi ). ‚ p A _ B q k P Φ m if A k , B k P Φ m . ‚ p Ap i p A _ B qq k P Φ m if A k , B k P Φ m . ‚ p Ap i p A _ B qq k P T m Ď Φ m ` if A k , B k P Φ m .Note in particular that stages can grow even for formulas free of e ki since we mayhave nested decisions on negated variables p i .Once again, in terms of proof complexity, we will gloss over this subtlety andsimply count the number of propositional and extension variable occurrences in aproof, assuming that each variable can be equipped with a ‘small’ index.We are now ready to define our intermediary systems. Definition 6.11.
The system eLNDT ` k p P q is defined just like eLNDT ` , but in-cludes additional initial sequents,(26) t - l p i , t p i k Ñ t - r Ñ p i , t p i k and may only use the extension axioms T Y A k . Lemma 6.12.
There is an eLNDT ` k p P q proof of Γ Ñ ∆ of size polynomial in | P | .Proof. We construct the required proof P k by replacing every formula occurrence A in P by A k . Note that all structural steps, identities, cuts remain correct. Anextension axiom for e i from A is just translated to the corresponding extensionaxiom for e ki from A k , and the initial sequents - l and - r from eLNDT `´ aretranslated to the two new initial sequents t - l and t - r , respectively, from Eq. (26)above. It remains to simulate the logical steps. ROOF COMPLEXITY OF POSITIVE BRANCHING PROGRAMS 29
The simulation of _ steps is immediate, since the ¨ k -translation commutes with _ . Similarly for positive decisions on p i . A left positive decision step on p i ,Γ , A Ñ ∆ Γ , p i , B Ñ ∆ p ` i - l Γ , Ap i p A _ B q Ñ ∆is translated to the following derivation: Proposition 6.8.(1) p Ap i p A _ B qq k Ñ A k , p ki Proposition 6.8.(2) p Ap i p A _ B qq k Ñ A k , B k Γ k , p ki , B k Ñ ∆ cut Γ k , p Ap i p A _ B qq k , p ki Ñ ∆ k , A k cut Γ k , p Ap i p A _ B qq k Ñ ∆ k , A k Γ k , A k Ñ ∆ k cut Γ k , p Ap i p A _ B qq k Ñ ∆ k A right positive decision rule on p i ,Γ Ñ ∆ , A, p i Γ Ñ ∆ , A, B p ` i - r Γ Ñ ∆ , Ap i p A _ B q is translated to the following derivation: Γ k Ñ ∆ k , A k , p ki Γ k Ñ ∆ k , A k , B k Proposition 6.8.(4) p ki , B k Ñ Ap i p A _ B q k cut Γ k , p ki Ñ ∆ k , A k , Ap i p A _ B q k cut Γ k Ñ ∆ k , A k , Ap i p A _ B q k Proposition 6.8.(3) A k Ñ Ap i p A _ B q k cut Γ k Ñ ∆ k , Ap i p A _ B q k (cid:3) Putting it all together.
We are now ready to assemble the proof our mainsimulation result, Theorem 6.1. Recall that we are still working with the fixed eLNDT `´ proof P of a positive sequent Γ Ñ ∆ from Section 6.4, over extension ax-ioms A “ t e i Ø A i u i ă n and propositional variables p “ p , . . . , p m ´ . We continueto write p i for p , . . . , p i ´ , p i ` , . . . , p m ´ , i.e. p with p i removed. Proposition 6.13.
For k ě , there are polynomial size eLNDT ` proofs of, p i , t p i k Ñ t p k ` (27) t p k Ñ p i , t p i k (28) over extension axioms T .Proof. We derive Eq. (27) as follows: t p i , t p i k ` Ñ t p i p i k ` by Lemma 5.4.(1) p i , t p i k Ñ t p i p i k ` by Eq. (6) and cut Ñ t p k ` by Lemma 4.6 and cut We derive Eq. (28) as follows: t p k Ñ t p i p i k by Lemma 4.6 Ñ t p i , t p i k by Lemma 5.4.(2) and cut Ñ p i , t p i k by Eq. (6) and cut (cid:3) Lemma 6.14.
For k ě , there are polynomial size eLNDT ` proofs of, t p k , Γ Ñ ∆ , t p k ` over extension axioms T Y A k . Proof.
By Lemma 6.12, we already have a polynomial-size proof eLNDT ` k p P q proof P k of Γ Ñ ∆. By definition of eLNDT ` k p P q , we construe P k as an eLNDT ` deriva-tion of Γ Ñ ∆ over extension axioms T Y A k from hypotheses: p i , t p i k Ñ (29) Ñ p i , t p i k (30)We obtain the required proof by adding t p k to the LHS of each sequent and t p k ` tothe RHS of each sequent in P k . Each local inference step remains correct, exceptthat some weakenings may be required to repair initial steps. Finally we replaceoccurrences of the hypotheses Eq. (29) and Eq. (30) above by the proofs of Eq. (27)and Eq. (28) respectively from Proposition 6.13. (cid:3) We are now ready to prove our main result, that eLNDT ` polynomially simulates eLNDT over positive sequents: Proof of Theorem 6.1 .
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