Going from the huge to the small: Efficient succinct representation of proofs in Minimal implicational logic
aa r X i v : . [ c s . CC ] J a n Going from the huge to the small: Efficientsuccint representation of proofs in Minimalimplicational logic
Edward Hermann Haeusler † Departmento de InformáticaPUC-RioRio de Janeiro, Brasil † Email:[email protected] 26, 2021
Abstract
A previous article shows that any linear height bounded normal proof ofa tautology in the Natural Deduction for Minimal implicational logic ( M ⊃ )is as huge as it is redundant. More precisely, any proof in a family of super-polynomially sized and linearly height bounded proofs have a sub-derivationthat occurs super-polynomially many times in it. In this article, we showthat by collapsing all the repeated sub-derivations we obtain a smaller struc-ture, a rooted Directed Acyclic Graph (r-DAG), that is polynomially upper-bounded on the size of α and it is a certificate that α is a tautology that canbe verified in polynomial time. In other words, for every huge proof of atautology in M ⊃ , we obtain a succinct certificate for its validity. Moreover,we show an algorithm able to check this validity in polynomial time on thecertificate’s size. Comments on how the results in this article are related toa proof of the conjecture N P = CoN P appears in conclusion.
In [13] and [12] we discuss the correlation between the size of proofs and howredundant they can be. A proof or logical derivation is redundant whenever it has1ub-proofs that are repeated many times inside it. The articles [13] and [12] focuson Natural Deduction (ND) proofs in the purely implicational minimal logic M ⊃ .This logic, here called M ⊃ , is PSPACE-complete and simulates polynomially anyproof in Intuitionistic Logic and full minimal logic, being hence an adequate rep-resentative to study questions regarding computational complexity. The fact that M ⊃ has straightforward syntax and ND system is worthy of notice. Moreover,compressing proofs in M ⊃ can provide very good glues to compress proofs in anyone of these mentioned systems, even for the Classical Propositional Logic. In [8]and [7], we prove that for every M ⊃ tautology α there is a two-fold certificate forthe validity of α in M ⊃ . The certificate is polynomially sized on the length of α and verifiable in polynomial time on this length too. This is the general approachof our proof that N P = P SP ACE . It is well-known that
N P = P SP ACE implies
CoN P = N P , so we can conclude also that
CoN P = N P . This arti-cle, together with the articles [13], [12], and one of the appendixes of [13], aimto show an alternative and more intuitive proof that
N P = CoN P . Recently,we deposited in arxiv a note, see [9], that explains how to have a third alternativeproof of
N P = CoN P , this time with a double certificate on linear height normalproofs, without to use Hudelmaier result. This paper aims to show how to usethe inherent redundancy in huge proofs to have polynomial and polytime certifi-cates by removing this redundancy from the proofs. This is made by collapsingall the redundant sub-proofs into only one occurrence in the proof, by type. Westart with tree-like Natural Deduction proofs and end up with a labeled r-DAG(rooted Directed Acyclic Graph). In the sequel we explain and show what is saidin the last two phrases, previously stated. Currently, the use of the redundancytheorem and corollary shown in [12], that is the essence of this proof’s approach,is not easily adaptable to a proof of
N P = P SP ACE . In [8] the linearly heightupper-bounded proofs of the tautologies in M ⊃ are do not have have to be normal.In [13], we identify sets of huge proofs with sets of proofs that, when viewed asstrings, have their length lower-bounded by some exponential function. Moreover,we can consider, without loss of generality, proof/deductions, which are linearlyheight bounded, as stated in [7]. We prove that the exponentially lower-bounded M ⊃ proofs are redundant, in the sense that there is at least one sub-proof foreach proof that occurs exponentially many times in it. In [12], we show that thisresult extends to super-polynomial proofs, i.e., proofs that are lower-bounded byany polynomial. We consider huge proofs/derivations as super-polynomial sizedproofs. Thus, we prove that, in any set of super-polynomially lower-boundedproofs in M ⊃ , of tautologies, all proof are redundant. Redundancy means thatthere is a sub-proof that occurs super-polynomially many times in it for almost2very proof in this set of huge proofs.In section 2, we present a brief presentation and explanation on the proof-theoretical terminology used here and the results from our previous articles. Fora more detailed and comprehensive reference, see [13], [12], [8] and [7]. Sec-tion 3 is the section where we show how to remove redundancies using a recursivemethod adequately. Section 5 shows that the certificate obtained in section 3 is ofpolynomial size and can be checked in polynomial time too. We conclude thisarticle in section 9. Finally, due to have a self-contained article, and the back-ground and previous results that this paper uses, we advise the reader that there isan essential superposition of this article with [12] and [13]. One reason to study redundancy in proofs is to obtain a compressing method basedon redundancy removals. According to the redundancy theorem (theorem 12, page17 in [12]) and its corollary (corollary 13, page 18) proofs belonging to a familyof super-polynomial proofs are super-polynomial redundant. The reader can findall the content of this section in [12] in more detail. This section contains excerptsof [12].The Natural Deduction system, defined in [5]), is taken as a set of inferencerules that settle the concept of a logical deduction. Natural Deduction does nothave axioms. It implements in the level of the logical calculus the (meta)theoremof deduction, Γ , A ⊢ A ⊃ B , via the discharging mechanism. The ⊃ -introductionrule, for example, uses this discharging mechanism in the logic calculus level.[ A ] Π B ⊃ -Intro A ⊃ B We embrace formulas occurrence A in the derivation Π of B from A with apair of [] to indicate the discharge of them. An embraced formula occurrence[ A ] means that from the ⊃ -Intro rule discharging it down to the conclusion of thederivation, the inferred formulas do not depend anymore on this occurrence A .The choice of formulas to be discharged in an ⊃ -Intro rule is arbitrary and liberal.The range of this choice goes from every occurrence of A , the discharged formulauntil none of them. 3he following derivations show two different ways of deriving A ⊃ ( A ⊃ A ) .Observe that in both deductions or derivations, we use numbers to indicate theapplication of the ⊃ -Intro that discharged the marked formula occurrence. For ex-ample, in the right derivation, the upper application discharges the marked occur-rences of A , while in the left derivation, it is the lowest application that dischargesthe formula occurrences A . There is a third derivation that both applications donote discharge any A , and the conclusion A ⊃ ( A ⊃ A ) keep depending on A .This third alternative appears in figure 1. Natural Deduction systems can providelogical calculi without any need to use axioms. In this article, we focus on thesystem formed only by the ⊃ -Intro rule and the ⊃ -Elim rule, as shown below,also known by modus ponens. The logic behind this logical calculus is the purelyminimal implicational logic, M ⊃ . A A ⊃ B ⊃ -Elim B Without loss of generality, we substitute the liberal discharging mechanismby a greedy one that discharges every possible formula occurrence whenever the ⊃ -Intro is applied. As stated and proved in [12], if there is an N.D. proof of α in M ⊃ , then there is a proof of α in M ⊃ that has all applications of ⊃ -Intro as greedyones. [ A ] A ⊃ A A ⊃ ( A ⊃ A ) [ A ] A ⊃ AA ⊃ ( A ⊃ A ) AA ⊃ AA ⊃ ( A ⊃ A ) Figure 1: Two vacuous ⊃ -Intro applicationsIn our previous articles, we consider Natural Deduction as trees with the sakeof having simpler proofs of our results. There is a binary tree with nodes labelledby the formulas and edges linking premises to the conclusion for any ND deriva-tion. The tree’s root is the conclusion of the derivation, and the leaves are itsassumptions. Figure 2 has the tree in figure 3 representing it. In a proof-tree, the4et of formulas that the label of u depends on v labels the edge from v to u . Thisset of formulas is called the dependency set of u from v . The greedy version ofthe ⊃ -intro removes the discharged formula from the corresponding dependencysets, as shown in figure 3. We need one more extra edge and the root node. Thedependency set of the conclusion labels this edge. That is why the edge links theconclusion to the dot in figure 3. [ A ] A ⊃ BB B ⊃ CC A ⊃ C Figure 2: A derivation in M ⊃ . A ⊃ CCBA A ⊃ BB ⊃ C { A ⊃ B, B ⊃ C }{ A, A ⊃ B, B ⊃ C }{ B ⊃ C }{ A, A ⊃ B } { A ⊃ B }{ A } Figure 3: The tree representing the derivation in figure 2Finally, we use bitstrings induced by an arbitrary linear ordering of formulasto have a more compact representation of the dependency sets. Considering thatonly subformulas of the conclusion can be in any dependency set, we only needbitstrings of the size of the conclusion of the proof. Figure 4 shows this final formof tree representing the derivation in figure 2 and 3, when the linear order ≺ is A ≺ B ≺ C ≺ A ⊃ B ≺ B ⊃ C ≺ A ⊃ C . This explanation is an excerpt from[12]. 5 A ⊃ CCBA A ⊃ BB ⊃ C Figure 4: Tree with bitstrings representing the derivation in figure 2Without loss of generality, we consider the additional hypothesis on the linearbound on height of the proof of M ⊃ tautologies. In [7], we show that any tautologyin M ⊃ has a Natural Deduction normal proof of height bound by the size of thistautology. However, proof of the tautology does not need to be normal. On theother hand, if we consider the complexity class CoN P (see the appendix in [13])we are naturally limited to linearly height-bounded proofs. The proofs, in M ⊃ , ofthe non-hamiltonianicity of graphs, are linearly height bounded.We consider the usual definition of the syntax tree for M ⊃ -formulas. Givena formula φ ⊃ φ in M ⊃ , we call φ its right-child and φ its left-child. Theseformulas label the respective right and left child vertexes. A right-ancestral of avertex v in a syntax-tree T α of a formula α is any vertex u , such that, either v isthe right-child of u , or, there is a vertex w , such that v is the right-child of w and u is right-ancestral of w .The left premise of a ⊃ -Elim rule is called a minor premise, and the rightpremise is called the major premise. We should note that the conclusion of thisrule and its minor premise, are sub-formulas of its major premise. A derivation is atree-like structure built using ⊃ -Intro and ⊃ -Elim rules. We have some examplesdepicted in the last section. The derivation conclusion is the root of this tree-likestructure, and the leaves are what we call top-formulas. A proof is a derivationthat has every top-formula discharged by a ⊃ -Intro application in it. The top-formulas are also called assumptions. An assumption that it is not discharged byany ⊃ -Intro rule in a derivation is called an open assumption. If Π is a derivationwith conclusion α and δ , . . . , δ n as all of its open assumptions then we say that Π is a derivation of α from δ , . . . , δ n . 6 efinition 1 (Branch) . A branch in a derivation or proof Π is any sequence β , . . . , β k of formula occurrences in Π , such that: • δ is a top-formula, and; • For every i = 1 , k − , either β i is a ⊃ -Elim major premise of β i +1 or β i isa ⊃ -Intro premise of β i +1 , and; • δ k either is the conclusion of the derivation or the minor premise of a ⊃ -Elim. A normal derivation/proof in M ⊃ is any derivation that does not have any for-mula occurrence simultaneously a major premise of a ⊃ -Elim and the conclusionof a ⊃ -Intro. A formula occurrence that is a conclusion of a ⊃ -Intro and a majorpremise of ⊃ -Elim is called a maximal formula. In [18], Dag Prawitz proves thefollowing theorem for the Natural Deduction system for the full propositionalfragment of minimal logic. Theorem 1 (Normalization) . Let Π be a derivation of α from ∆ = { δ , . . . , δ n } .There is a normal proof Π ′ of α from ∆ ′ ⊆ ∆ . In any normal derivation/proof, a branch’s format provides worth informationon why huge proofs are redundant, as we will see in the next sections. Since noformula occurrence can be a major premise of ⊃ -Elim and conclusion of a ⊃ -Intro rule in a branch we have that the conclusion of a ⊃ -Intro can only be theminor premise of a ⊃ -Elim or it is not a premise of any rule application at all inthe same branch. In this last case, it is the derivation’s conclusion or the minorpremise of a ⊃ -Elim rule. In any case, it is the last formula in the branch. Thus,for any branch, every conclusion of a ⊃ -Intro has to be a premise of a ⊃ -Intro.Hence, any branch in a normal derivation splits into two parts (possibly empty).The E-part starts it with the top-formula, and, every formula occurrence in it isthe major premise of a ⊃ -Elim. We may have then a formula occurrence that isthe conclusion of a ⊃ -Elim and premise of a ⊃ -Intro rule that is called minimalformula of the branch. The minimal formula starts the I-part of the branch, whereevery formula is the premise of a ⊃ -Intro, excepted the last formula of the branch.From the branches’ format, we can conclude that the sub-formula principle holdsfor normal proofs in Natural Deduction for M ⊃ , in fact, for many extensions. Abranch in Π is said to be a principal branch if its last formula is the conclusion of The full propositional fragment is {∨ , ∧ , ⊃ , ¬ , ⊥} . A secondary branch is a branch that is not principal. The primary branch iscalled a 0-branch. Branches, where the last formula is the minor premise of a rulein the E-part of a n -branch, is a n + 1 -branch. Corollary 2 (Sub-formula principle) . Let Π be a normal derivation of α from ∆ = { δ , . . . , δ m } . It is the case that for every formula occurrence β in Π , β is asub-formula of either α or of some of δ i . To facilitate the presentation, we only handle normal proofs in expanded form.
Definition 2.
A normal proof/derivation is in expanded form, if and only if, all ofits minimal formulas are atomic.
Without loss of generality, we can consider that formula in M ⊃ is a tautologyif and only if there is a normal proof in expanded form that proves it. Of course,if it is a tautology, it has proof, and so it has normal proof by normalization. Weuse the following fact to obtain the expanded form from normal proof. In [12]we prove that all tautologies have normal proofs in expanded form. See the firstappendix of [12]In [12], we observed correspondence between each minimal formula of abranch, in a normal and expanded proof Π , employing a one-to-one correspon-dence to the respective top-formula occurrence of its branch. Figure 5 illustratesthe mapping from the proof into the syntax-tree of the proved formula accord-ing to this correspondence. Note that the two positions of the atomic formula q in the syntax tree uniquely indicates the top-formula in the E-part of the Natu-ral Deduction proof/derivation to which it belongs. We can consider that the two q ’s are in fact, different. The top-formula of each q is the biggest in the inversepath (upwards) following the reverse of the right child edge. Definition 5, in thesequel, has the purpose of setting this correspondence in the representation ofproofs formally. With the sake of a more precise presentation, we provide belowthe definition of a formula’s syntax tree. Definition 3 (Syntax tree of a formula) . Let α be a M ⊃ -formula. The syntax treeof α is the triple h V, E left , E right , L i where V is a set, of vertexes, E s ⊆ V × V , s = lef t, right , the corresponding left and right edges, such that h V, E left , E right i is an ordered full binary tree, and, L is a bijective function from V onto the sub-formulas of α , such that: • L ( r ) = α , where r ∈ V is the root of the tree h V, E left , E right i , and; D ] C B [ A ] [ A ⊃ ( B ⊃ ( C ⊃ q ))] B ⊃ ( C ⊃ q ) C ⊃ qqA ⊃ q [( A ⊃ q ) ⊃ ( D ⊃ q )] D ⊃ qqD ⊃ q (( A ⊃ q ) ⊃ ( D ⊃ q )) ⊃ ( D ⊃ q )( A ⊃ ( B ⊃ ( C ⊃ q ))) ⊃ ((( A ⊃ q ) ⊃ ( D ⊃ q )) ⊃ ( D ⊃ q )) αA ⊃ ( B ⊃ ( C ⊃ q )) A ( B ⊃ ( C ⊃ q )) B C ⊃ qC q (( A ⊃ q ) ⊃ ( D ⊃ q )) ⊃ ( D ⊃ q )( A ⊃ q ) ⊃ ( D ⊃ q ) A ⊃ qA q D ⊃ qD q D ⊃ qD q Figure 5: A mapped N.D. proof9
For every formula ϕ ⊃ ϕ ∈ Sub ( α ) , if L ( v ) = ϕ ⊃ ϕ , h v, v i ∈ E left and h v, v i ∈ E right then L ( v ) = ϕ and L ( v ) = ϕ . Definition 4 (Partially mapped ND-proofs) . Let α be a M ⊃ -formula and T α = h V, E left , E right , L i its syntax tree. Let Π be a M ⊃ -ND normal derivation of α .A partially mapped ND-proof of alpha is a structure h Π , T α , l i , where l is a par-tial function from the formula occurrences in Π to V , such that, the followingconditions hold. • If γ is the minimal formula of a branch −→ b in Π then if ℓ ( γ ) is defined then L ( ℓ ( γ )) = γ ; • If γ is the minimal formula of a branch −→ b = h b , . . . , b j = γ, . . . , b k i and ℓ ( γ ) is defined then either ℓ ( b j − ) or ℓ ( b j +1 ) are defined, and; • If ϕ is the conclusion of a ⊃ -Elim rule in Π , that has premises ϕ ⊃ ϕ and ϕ , and ℓ ( ϕ ) = v then there are v and v , such that h v, v i ∈ E right , h v, v i ∈ E left , ℓ ( ϕ ) = v and ℓ ( ϕ ⊃ ϕ ) = v ; • If ϕ ⊃ ϕ is the conclusion of a ⊃ -Intro rule in Π , that has premise ϕ and ℓ ( ϕ ⊃ ϕ ) = v then there is v ′ ∈ V , h v, v ′ i ∈ E right and ℓ ( v ′ ) = ϕ . Definition 5 (E-mapped Natural Deduction Normal Expanded proofs) . Let α bea M ⊃ -formula, T α = h V, E left , E right , L i be the syntax tree of α and Π a normaland expanded proof of α . The triple h Π , T α , l i is an E-mapped Natural Deductionproof, if and only if, l is defined on all formula occurrences that take part in the E-parts of branches in Π , including the minimal formulas. Moreover the followingcondition must hold: • For every branch −→ b , if q is the minimal formula of −→ b , ℓ ( q ) = v ∈ V and β is the top-formula (occurrence) of −→ b then ℓ ( β ) = u , where u is theright-ancestral of v that is left-child of some w ∈ V . In [12] we show that the above definition of E-mapped Natural DeductionNormal Expanded proof,
EmND for short, is well-defined. Moreover, we have thefollowing proposition. We consider a branch as a sequence of formula occurrencesnumbered from top-formula down to the branch’s last formula. The proof of thisproposition is in [12] 10 roposition 3.
Let h Π , T α , l i be a EmND of α . We have that to each branch −→ b = h β , . . . , β k , in Π that has the minimal formula occurrence q = β j , suchthat ℓ ( β j ) = u ∈ V , there exists one and only one path p = h u , . . . , u j i in T α ,with u j = u , and u such that, ℓ ( β i ) = u i , i = 0 , . . . , j . We point out that in proposition 3, above, h β , . . . , β j = q i is the E-part of −→ b . This proposition 3 states that any given E-part h β , . . . , β j i of a branch inan EmND is an instance of at most one path p = h u , . . . , u j i in T α , such that L ( u i ) = β i , i = 0 , . . . , j . Moreover, this path p is as stated in the definition of EmND in the its only item. Given a
EmND Π , for each E-part, in an EmND ,exists a path of the form stated in definition of
EmND , in the syntax tree of theconclusion of the
EmND . The number of such paths in the syntax tree is upper-bounded by its size, then the number of different E-parts types in any
EmND is atmost of the size of the conclusion of this
EmND . We have the following lemma:
Lemma 4 (Linear upper-bound on types of E-parts) . Let Π be an EmND provingthe M ⊃ -formula α . The number of different types of E-parts occurring in this EmND is at most the size of the T α . We remark, as also observed in [12], that we can label the nodes of a NaturalDeduction proof-tree with the nodes (not the labels) of the syntax tree of the con-clusion of the proof-tree. In doing that we will have the same effect on countingdifferent types of E-parts that is stated by lemma 4. M ⊃ mapped derivations Due to the linear speedup theorem, see [19] page 63-64, Theorem 3.10, we canconsider, w.l.o.g a linear height bounded proof of α is a proof which height isupper-bounded by the length of α . In fact, in this article, because of countingdetails, we consider that the upper-bounded is the size of the syntax tree | T α | .Since | T α | = | α | , the definition is equivalent. From [12], we have the followinglemmas. In [12] the reader can fund both proofs. Lemma 5 (Spreading Branchs Repetitions) . Let h Π , T α , l i be a linearly heightbounded EmND proof of α , < p ∈ N and m = | α | . If there is a branch −→ b thathas more than m p instances occurring in Π then there is a level µ , such that, atleast m p − instances of −→ b have the minimal formula q −→ b of −→ b occurring in level µ . emma 6 (Branchs and sub-derivations) . Let Π be a proof of α , and −→ b a branchin Π under the same conditions of the lemma 5 above. Then there is a (sub)derivation Π −→ b of Π , such that, Π −→ b has at least m p − instances occurring in Π . Definition 6 (Linearly height-bounded EmND proofs) . Let Λ be the set of mappedlinearly height-bounded ND M ⊃ proofs. We use the notation c (Π) to denote theformula that is the conclusion of Π . Note that we can consider Λ as a predicate Λ( x ) that is true if and only if x is assigned to a mapped linearly height-boundedND proof. S Λ = { Π ∈ Λ : ∀ p ∈ N , p > , ∃ n , ∀ n > n , (cid:12)(cid:12) T c (Π) (cid:12)(cid:12) = n and | Π | > n p ) } As explained in [12], S Λ contains all huge or hard linearly height upper-bounded proofs in M ⊃ . Of particular interest is the following set. Let T aut M ⊃ bethe set of all ND mapped proofs of M ⊃ tautologies. The following set: Definition 7.
Let Θ be the following set: Θ M ⊃ = { Π ∈ T aut M ⊃ : ∀ p ∈ N , p > , ∃ n , ∀ n > n , (cid:12)(cid:12) T c (Π) (cid:12)(cid:12) = n and | Π | > n p } Θ M ⊃ is the set of super-polynomially sized M ⊃ ND mapped proofs.In [12], we show that every Π ∈ S Λ is redundant. This means that there is atleast one sub-proof Π s of Π that repeats as many times as it is the size of Π . Wehave the following theorem 7, proved in [12]. We emphasise that the proofs in S Λ are linearly heigh-bounded. Theorem 7.
For all p ∈ N , p > , and for all Π ∈ S Λ , such that, |T ( c (Π)) | = m and | Π | > m p , then there is a sub-derivation Π s of Π and a level µ in Π , suchthat, Π s has at least m p − instances occurring in the level µ in Π . From theorem 7 we can roughly state the corollary 8.
Corollary 8.
All, but finitely many, proofs belonging to an arbitrary family ofsuper-polynomial and linearly height upper-bounded proofs, are super polyno-mial redundant. M ⊃ linearly heightbounded proofs Corollary 8 says that any proof in an unbounded set of super-polynomial proofs,linearly bounded on the height, is almost as redundant as it is huge. We can show12hat there are level µ and a derivation Π that occurs as many times in µ as it is thesize of the proof. This section shows a polynomial sized certificate of validity forany huge tautology that belongs to this set of super-polynomial proofs. Our ar-gumentation to prove this is to remove all redundancies in the original derivation,preserving logical consequence.Given a non-empty finite set S , card ( S ) = n , and a total order O S = { s , . . . , s n } on S , the set B ( O S ) is { b . . . b n : b i = 0 or b i = 1 , i = 1 . . . n } . There is a bijec-tion F from B ( O S ) onto the powerset of S given by F ( b . . . b n ) = { s i : b i = 1 } . B ( O S ) is also called the set of bitstrings over O S .Given a M ⊃ formula α , sub ( α ) is the set of all sub-formulas of α . Definition 8 (r-DagProof) . A pre r-DagProof for a M ⊃ formula α is a structure C = h V, E d , E A , r, ℓ, L, ρ, δ, O α i V is a non-empty set of nodes;2. E d ⊆ V × V , deduction edges;3. E A ⊆ V × V , ancestrality edges;4. O α is a total order on sub ( α ) ;5. r ∈ V is the root of the C ;6. ℓ : V → sub ( α ) , for v ∈ V , ℓ ( v ) is the (formula) label of v , sub ( α ) is theset of all sub-formulas of α ;7. L : E d → B ( O ) is a function, such that, for every h u, v i ∈ E d , L ( h u, v i ) ∈ B ( O S ) ;8. δ : E A → N , a total function;9. ρ : E d → N , a partial function;Subjected to the following conditions: (Global) h V, E d , r i is a connected DAG with unique root r and for each node v ∈ V , if h r = u , . . . , u m = v i and h r = v , . . . , v n = v i are two inversepaths from r to v then m = n , i.e., for any v , any longest path from r to v has the same length of a shortest path from r to v . Moreover, for each v ∈ V , if v = r then there is a path from v to r and for every u , h u, u i 6∈ E d . E d − ( l / L ) consistency) For every h u, v i , h v, w i ∈ E d , if there is no h u ′ , v i ∈ E d , such that, u ′ = u and ρ ( h v, w i ) = ↑ , then ℓ ( u ) = ϕ , ℓ ( v ) = ϕ ⊃ ϕ , L ( h v, w i ) = L ( h u, v i ) − ~b ϕ ; ( E d − ( l / L ) consistency) For every h u , v i , h u , v i , h v, w i ∈ E d , there is no h u ′ , v i ∈ E d , such that, u ′ = u i , i = 1 , , and ρ ( h v, w i ) = ↑ , then ℓ ( u ) = ϕ , ℓ ( u ) = ϕ ⊃ ϕ , ℓ ( v ) = ϕ , L ( h v, w i ) = L ( h u , v i ) oplusL ( h u , v i ; ( E A -target consistency) If h u, v i ∈ E A then either there is no h w, v i ∈ E d orthere is h v, w i ∈ E A , w = u ; ( E A -source consistency) If h u, v i ∈ E A then there is h w, u i ∈ E d , w = v and δ ( h u, v i ) = ρ ( h w, u i ) ( E A -ancestrality irreflexivity) For every u , h u, u i 6∈ E A . A pre r-DagProof, satisfying definition 8 above, is a r-DagProof, or a certifi-cate for ℓ ( r ) , if it is sound, or equivalently, if algorithm 3 answers “Correct” whenexecutes on it. In section 5 we define soundness of a r-DagProof and we define thealgorithm 3. In the meanwhile we deal with pre r-DagProofs. Specifically, in thissection, this difference between pre r-DagProofs and r-DagProofs is not relevant.We provide a bit of terminology in this part of the article. Regarding theo-rem 7, we denote by matrix the sub-derivation of the ambient derivation, i.e. thehuge one, that has many instances repeated. Definition 11 define the set of nodesin an r-DagProof that is formed by top-formulas or their representatives after col-lapsing. We need the definition of top-formulas and representative top-formulas. Definition 9 (Top-formula of a r-DagProof) . A node v ∈ V , of a r-DagProof C = h V, E d , E A , r, l, L, P, O α , E δ i , is a top-formula, if and only if, there is no w ∈ V , such that h w, v i ∈ E d or h w, v i ∈ E A . Definition 10 (Representative top-formula of a r-DagProof) . Given a r-DagProof C = h V, E d , E A , r, l, L, ρ, δ, O α i , v ∈ V is a representative top-formula, if and onlyif, there is no w ∈ V , such that h w, v i ∈ E A . Moreover, there is a sequence v = w , . . . , w n , w i ∈ V , i = 1 , n , such that, for every i = 1 , n − , h w i , w i +1 i ∈ E A ,and there is no w ∈ V , such that, h w, w n i ∈ E d . Definition 11.
Given a r-DagProof C = h V, E d , E A , r, ℓ, L, ρ, δ, O α i , its set of ini-tials, I ( C ) is the set of representative top-formulas together with its top-formulas. C ,defined by its root k , from D and link its position to the matrix C accordingly. Inthe following definitions, if C is a r-DagProof h V, E d , E A , r, l, L, ρ, δ, O α i , we usethe notations E A ( C ) , E d ( C ) , V ( C ) , r ( C ) , ℓ ( C ) , etc, to denote E A , E d , V , r , l , etc,respectively. Given a set of nodes V ′ ⊆ V , such that r V ′ , the restriction of C to the set of nodes V ′ is C | V ′ and it is defined below: h V ′ , E d | V ′ , E A | V ′ , r, l | V ′ ) , L | ( E d | V ′ ) , ρ | ( E d | V ′ ) , δ | ( E A | V ′ ) , O α i ,where if E ⊆ V × V is a set of edges and V ′ ⊆ V then E | V ′ = {h v, u i : v ∈ V ′ and u ∈ V ′ } . Definition 12 (Difference of r-DagProofs) . Let C and D be two r-DagProofs, thegraph difference D − C is D| ( V ( D ) − V ( C )) . The difference of r-DagProofs is not a r-DagProof itself. In the definitionbelow we use the notation
D ↑ ( k ) , where D is a r-DagProof and k ∈ V ( D ) , todenote the biggest sub r-DagProof of D that has k as root. In the particular casethat D is a tree, D↑ ( k ) is the sub-tree of D that has root k . Definition 13 (Detach and link sub r-DagProofs function) . Let D be a r-DagProofof α and k a node of D that it is the root of an instance of the r-DagProof C thatis its matrix, as given by theorem 7. Let i ∈ N , a label, and C ′ = D↑ ( k ) . Define D ′ = ( D − C ′ ) ∪ C . We define DetachLink ( D , k, C , i ) as following: h V ( D ′ ) ,E d ( D ′ ) ∪ {h r ( C ) , v i : h k, v i ∈ E d ( D ) } ,E A ( D ′ ) ∪ {h v, w i : h k, v i ∈ E d ( D ) and w ∈ I ( C ) } ,r ( D ′ ) ,ℓ ( D ′ ) ,L ( D ′ ) ∪ {h r ( C ) , v i 7→ L ( h k, v i ) : h k, v i ∈ E d ( D ) } ,ρ ( V ( D ′ )) ∪ {h v, w i 7→ i : h k, v i ∈ E d ( D ) and w ∈ I ( C ) } ,δ ( V ( D ′ )) ∪ {h r ( C ) , v i 7→ i : h k, v i ∈ E d ( D ) } , O α i Figures 6 and 7 illustrate what happens when we apply three DetachLink oper-ations in an ambient r-DagProof. When we repeat this operation to every instanceof a matrix C occurring in a fixed level µ , we say that we performed a collapse ofthe instances of C in D in level µ . This operation is described by algorithm 1 andis denoted by Collapse ( C , D , µ ) . 15onsider a EmND proof that has more than one matrix C with instances oc-curring super-polynomially many times in it. Thus, Lemma 9 shows that we canuse theorem 7 to obtain the list of all matrices having a super-polynomially manyinstances occurring in a fixed level µ of this EmND proof. We remember that amatrix derivation/proof, in our terminology, is nothing but a sub-derivation/sub-proof that has at least one instance in other proof. This lemma obtains a set ofMatrices having all instances occurring in the lowest level in a specific EmNDproof. Moreover, no matrix instance is a proper sub-derivation of any other ma-trix instance in the set. We call this an independent set of matrices from Π to thisset. Definition 14 (Independent set of matrices in a proof/derivation) . Let Π be anEmND proof/derivation. A set of matrices S = { Π ν : Π ν is a matrix in Π which only has instance in a level ν } is an independent set of matrices, if and only if, there is no ν and ν levels in Π , ν = ν , and instances Π and Π of Π ν and Π ν , respectively, Π ν i ∈ S , , ,such that, Π is a sub-derivation of Π , or Π is a sub-derivation of Π . Given a set S of independent matrices of a proof/derivation, we can prove thatif Π ν ∈ S then no instance of Π ν is sub-derivation of Π µ ∈ S , unless Π µ = Π ν .Thus, if Π ν ∈ S then there is no level µ < ν with instances that are super-derivations of instances of Π ν . In a certain sense, ν is a local lowest level. Lemma 9 (List of super-polynomially repeated matrices) . For all p ∈ N , p > ,and for all Π ∈ S Λ , such that, |T ( c (Π)) | = m and | Π | > m p , then there is aset M of independent matrices for sub-derivations instances of Π , such that, forevery Π s ∈ M , Π s has at least m p − instances occurring in some level ξ in Π .Proof. of lemma 9. Using the conditions on lemma, Theorem 7 provides at leastone level µ and a matrix Π s that has at least m p − instances occurring in µ . Thus,the sets S ν = { Π s : Π s is a matrix having at least m p − instances in ν in Π } where ν is a level in Π , form a family. The family ( S ν ) ν ∈ Lev (Π) has at least thenon-empty set S µ . Moreover, if ν < ν and S ν = ∅ then S ν contains all the The set S Λ is defined in defintion 6 S ν that occurs in level ν − ν in the elements (trees) in S ν . If T is a subtree of T ′ then we can say that T ′ is a super-tree of T . The sameapplies to sub-derivations and super-derivations. We define the set L of levels ξ ,such that S ξ has no instance in Π in level ξ that is a super-derivation of an instanceof Π s ∈ S µ , µ = ξ . The set { Π ξ : ξ ∈ L } is an independent set of matrices. It isthe biggest one, indeed.Lemma 9 is used to provide the initial list of instances to collapse in the al-gorithm 2. This list can be alternatively defined as, given p > , the lowest sub-derivations of a proof Π , of size bigger than | Π | p , that occurs at least | Π | p − timesin their respective lowest level µ . Definition 16 introduces the notation LRI (Π) to denote this set. rl l l l k l l l l l l l l l l l l l l . . . v v v v k v v v v v v v v v v v v v v . . . u u u u k u u u u u u u u u u u u u u Figure 6: Some instances of the matrix C in the ambient r-DagProof D l l l l k l l l l l l l l l l l l l l . . . v v v v k v v v v v v v v v v v v v v . . . u u u u k u u u u u u u u u u u u u u Polynomial level Polynomial level b b b b b b b b b b b b Figure 7: Three DetachLink were applied in the ambient r-DagProof D of fig. 6We remember that r C is the root of the r-DagProof C and Starts ( C ) is the setof initials of C as stated by definition 11We observe that the resulting (pre) r-DagProof yielded from the collapse inlevel µ can be bigger than m p yet. With the collapse of m p − sub-proofs/derivations,the size of the resulting r-DagProof is at least |D||C|× m p − . If the mentioned size ofthe resulting r-DagProof is bigger than m p , then there must be two main reasons:(1) The collapsed sub-proof/derivation is bigger then m p by itself, or; (2) Theremust be more matrices in level µ that we consider. The second alternative dealtproceeds by collapsing all instances of all matrices occurring at the lowest level.This is addressed in lines 3 to 11 of algorithm 2. For the first alternative, we onlyhave to recursively find more redundant parts in the sub-proof/derivation that mustexist by theorem 7 in the matrix that had all instances collapsed in the ambient r-DagProof. Since theorem 7 works on EmND proofs, as opposed to derivationswith open assumptions, we need lemma 10 below.
Lemma 10.
Let p ∈ N , p > , Π be an EmND proof of a M ⊃ tautology α , suchthat, |T ( c (Π)) | = m and | Π | > m p . According theorem 7 let Π s be a matrixthat has at least m p − instances occurring in the level µ in Π . If | Π s | > m p thenthere is a matrix Π ′ s that is a sub-derivation of Π s , such that it has at least m p − instances in Π s in some, fixed, level ν > µ .Proof. of 10. The main feature of theorem 7 is the fact that it works for EmNDproofs and Π s does not need to be a proof, it is instead a derivation with open18ssumptions. We can discharge in the correct order, dictated by the syntax tree of α and the mapping l that defines the EmND Π s . To do this, we restrict l to Π s .This restriction does not yield an EmND yet, and we have to consider the syntaxtree of the conclusion of Π s , i.e. the syntax tree T c (Π s ) . However, as we havealready said, we do not have a tautology as the conclusion of Π s . We discharge allopen assumptions in Π s , obtaining a conclusion β that is a tautology. The result isthen an EmND that proves β . The introduction part of the main branch does notdisrupt the condition o definition 5. Finally, since β is smaller than α we choose anew propositional variable q new and define a new formula “ q new ⊃ q new ⊃ . . . β ”with as many q new repetitions as it is enough to have a formula of the same size as α . The EmND adjusted to this new formula is a proof of a tautology, and we canapply Theorem 7 to obtain a matrix Π ′ s that has m p − instances in Π s . Note that (cid:12)(cid:12) T c (Π s ) (cid:12)(cid:12) = | T a | . Since there is only one main branch, the many instances of Π ′ s occur above the original conclusion of Π s . Finally, the I-part of the main branch,from c (Π s ) down to “ q new ⊃ q new ⊃ . . . β ” can be eliminated by merely deletingthe introduction rules applied to draw “ q new ⊃ q new ⊃ . . . β ”, including the rulesused to prove β . In this way we obtain the original matrix Π s and the desired m p − instances of Π ′ s that occurs in it. Finally, as Π ′ s is sub-derivation of Π s thenthe level ν of its root is strictly above µ .By the proof of Lemma 10, we can see that even in the case that Π s is not an EmND proof, if | Π s | > m p , with m = c (Π s ) , then there is a sub-derivation of Π s that it is repeated at least m p − many times in Π s . Corollary 11.
Let p ∈ N , p > , Π be an EmND derivation of α , such that, |T ( c (Π)) | = m and Π s a sub-derivation of Π that occurs in Π in level µ . If | Π s | > m p then there is a sub-derivation Π ′ s of Π s that has at least m p − instancesoccurring in a level ν > µ . The above corollary 11 is used to ensure the termination and correctness of al-gorithm 2 in the section 4. Algorithm 1, below, defines the operation of collapsinga list Y of instances of the r-DagProof matrix C occurring in level µ Definition 15 (Collapse operation inside pre r-DagProofs) . Let D = h V, E d , E A , r, l, L, ρ, δ, O α i We remember that a matrix that occurs in a level µ in an r-DagProof D is any sub-r-DagProofof D that has root in level µ and may have some other instances in level µ too e a pre r-DagProof for a M ⊃ formula α and C be a pre sub r-DagProof of D and Y the set of roots of the instances of C , occuring in level µ that will becollapsed. Algorithm 1 defines and computes the result of collapsing all instancesof C in only one in D . Algorithm 1
Precondition: D , C , r-DagProofs, C ≺ D , and the list Y containing the roots of the instances of C Ensure: a r-DagProof D ′ having all instances of C collapsed in a unique r-DagProof Function
Collapse ( D , Y , C ) D ′ ← D j ← l C ( root ( C )); i ← ; Y ← rest ( Y ) for k ∈ Y do D ′ ← DetachLink ( C , k, D ′ , j @ i ); i ← i + 1 end for Return D Examining algorithm 1, we have the Lemma 12 that provides an upper-boundedfor the resulting r-DagProof after the collapses of all instances of the matrix in theambient r-DagProof D . Lemma 12. |D ′ | ≤ |D| length ( Y ) ×|C| M ⊃ tautologies In this section, we show how to use the
Collapse operation defined in previ-ous section 3. Algorithm 2 defines the operation of compression that collapsesall redundancies that occur in any huge
EmND proof. In line 3 the function
Lemma T )) returns the set of independent matrices in T that exists by Lemma 9.Since it is an independent set, by collapsing all of its instances in only one instanceto each matrix, we do not need to collapse the upper levels that are not local lowestlevels, the roots of the matrices’ elements in the independent set.20 lgorithm 2 Compress an
EmND proof T using corol. Precondition:
Uses the global variable m with value | c ( T ) | Precondition: < p ∈ N , T is a EmND , h ( T ) ≤ | c ( T ) | Ensure: a r-DagProof D proving c ( T ) of size smaller than | c ( T ) | p − Function
Compress( T , p ) if ( m p < |T | ) then LocalLowestLevels ← MinLevel ( Lemma T )) D ← T for lev ∈ LocalLowestLevels upwards h ( T ) do L ← SuperP olySubP roofs ( T , lev ) for hY , C Y i ∈ L do D Y ← Compress ( C Y , p ) D ←
Collapse ( D , Y , D Y ) end for end for Return D else Return T end if Below we have some lemmas that are proved easily by inspecting the code ofalgorithms 1, 2 and definition 13.
Lemma 13.
In algorithm 2 above, the number of recursive calls after an initialinvocation of
Compress ( T , p ) , with m = | c ( T ) | , is at most |T | .Proof. Since m p is constant during all the recursive calls, and the size of the firstargument of Compress is strictly smaller than each previous recursive call. Thus,there must be a call such that |T | < m p . When T is of this size, it is the recursivebasis case, anyway. Taking the (worst) case into account, the number of recursivecalls is upper-bounded by the size of the EmND T itself. If any recursive call, onthe first argument T , is such that |T | > m p , then it obtains the sub-derivationsthat occurs at least m p − , collapsing Compress ( T ) into a unique compressedversion of T . This is done with lesser than |T | recursive calls. In fact, there isat least m p − collapses, so the total amount of recursive calls is at most |T ||T |× m p .Lastly, the compression of T is obtained in a unique call to Compress .Below we find a useful Lemma used to prove that after the compression anylinear height bounded
EmND super-polynomially bounded proof becomes poly-nomial in size.
Definition 16.
Given a
EmND linearly hight bounded proof T , such that, |T | > | α | p , p ≥ . We denote by LRI ( T ) the list of all lowest instances that occur atleast m p − times, for each of the matrices, as provided by Lemma 9. LRI is an acronym to
LowestInstancesRedundant efinition 17 (EmND proofs difference operation) . Let T and T EmND deriva-tions, such that, T is sub-derivation of T occuring in level ν . The differencederivation T ′ = T − T obtains by removing all nodes of T from T , but c ( T ) .Morevore, l T ′ is the restriction of l T to this T ′ new derivation. Definition 18 (EmND proof difference with removal of conclusion) . Let T and T EmND derivations, such that, T is sub-derivation of T occuring in level ν .The greedy difference T ⊖ T is as in defintion 17 extended with the removal ofthe conclusion of T from the yielded sub-derivation. The difference of T and the set of derivations S is the iterating the differenceof T to each member of S .The following Lemma 16 shows that for every proof of size bigger than m p ,when removing its redundant part, that exists in virtue of Theorem 7, what remainsis a rDagProof of size less than m p .We remember that derivations formally are labeled trees, and the later aregraphs. The differnce between graphs is well-defined as defined by definition 17We need a preparatory lemma that is, in a certain sense, a generalization ofLemma 10.We use γ ... γ n ⊃ α to denote the formula γ ⊃ ( γ ⊃ ( . . . ⊃ α ) . . . ) . Lemma 14 (Redundancy Lemma for Normal derivations) . Let Π be a normal andexpanded derivation in N.D. for M ⊃ . Let Γ = { γ , . . . , γ n } be the set of all openassumptions in Π and α the conclusion of Π , m = | α | + P γ ∈ Γ | γ | . If | Π | > m p ,for some p > , then there is a sub-derivation Π s of Π and a level µ from Π , suchthat, there are at least m p − instances of Π s occurring in level µ .Proof. From Π we obtain an expanded normal proof Π ′ of γ ... γ n ⊃ α byapplying a series ⊃ -I rules in the main branch of Π , discharging all open assump-tions in Π . From the parsing tree of γ ... γ n ⊃ α and an adequate partial mapfrom formula occurrences in Π ′ to this parsing tree we obtain an EmND proof of22 γ ... γ n ⊃ α . Note that any adequate mapping produces an EmND proof of thelater formula. We apply Lemma 7 to Π ′ concluding that there is a level µ from Π ′ and a sub-derivation Π s of Π ′ , such that, there are at least m p − instances of Π s occurring in level µ in Π ′ . Finally, we rebuild the orginal derivation Π byremoving all ⊃ -I applications used to construct Π ′ from Π . We reach the desireconclusion, i.e., there is a level µ from Π and a sub-derivation Π s of Π , such that,there are at least m p − instances of Π s occurring in level µ in Π Other very useful lemma is the following. It is inspired in the huge proofs,based on Fibonacci numbers, shown in [ ? ] in its appendix. Lemma 15.
Let β be a M ⊃ formula. There is a family (Π fibn ) n ∈ N ) of M ⊃ N.D.derivations of β from a set of formulas ∆ n , of linear size on n and | β | . If m = | β | + P δ ∈ ∆ n | δ | then, for each n ∈ N , Π fibn is a Normal and expanded derivationof height O (4 × n × | β | ) and size lower-bounded by φ | β |× n √ , where φ ≈ . isthe golden ratio.Proof. In [13], section 4, we show a family of Natural Deduction M ⊃ derivationsof p ⊃ p n , from ∆ n = { p , p ⊃ p } ∪ { p i ⊃ ( p i +1 ⊃ p i +2 ) : i ≤ n − } ofheight n and size φ n √ . We modify this family to have p n = β , getting the statementof the lemma. Lemma 16.
For any EmND proof T of a tautology α and p > , if |T | ≥ | α | p and hT , . . . , T n i = LRI ( T ) , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊖ [ i =1 ...n T i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < m p Proof.
We can consider T as shown in figure 8, where ℓ × m p − ≤ n , and ℓ isthe number of different levels that have sub-derivations occurring at least m p − in them. We remember that the formulas β i , i = 1 . . . n may occur in differentlevels. For each j = 1 . . . ℓ there are at least m p − occurrences of instances ofa sub-derivation T j in level j in T . To facilitate the understanding we re-indexthe instances with the level j , such that, T jj i is the i -th instance of the matrix M j occurring in level j in T . We remember that there are ℓ matrices indicating ℓ redundant parts in T . 23 µ β µ δ µ Π µ β µ ⊃ γ µ γ µ . . . T µ ℓ β µ ℓ δ µ ℓ Π µ ℓ β µ ℓ ⊃ γ µ ℓ γ µ ℓ Π q ... α Figure 8: The proof T in Lemma 16We prove by induction on the number of different lowest levels, i.e, on ℓ that |T | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊖ [ i =1 ...n T i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < m p Induction on ℓ :Basis There is only one level µ and one matrix T µ , hence ℓ = 1 , that is repeated atleast m p − times in level µ in T . Let T = T ⊖ S i =1 ...n T i , where T µ i i are allthe instances of T µ , for all i = 1 . . . n . We notice that T is not a derivationanymore, it is only a tree, depicted in figure 9. δ µ Π µ β µ ⊃ γ µ γ µ . . . δ µ n Π µ n β µ n ⊃ γ µ n γ µ n Π q ... α Figure 9: The tree T Let us suppose that: |T | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ⊖ [ i =1 ...n T i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ m p
24e observe that by Lemma 14, for each i , | Π µ i i | < m p and | Π | < m p .If any of these derivations were bigger than m p there would be a matrixoccurring at least m p − in some level of them. The only instances that oc-cur at least m p − in T are T i , i = 1 , . . . , n by hypotheses. They occurin level µ , contradicting the above assumptions on the size of the deriva-tions Π µ i i and Π . We note that |T | = | Π | + P i =1 ,n | Π µ i i | ≥ m p and, byprevious observation on the size of each of the summands, there must bea sub-sequence (Π µ r i ) i =1 ,r of the sequence (Π µ i ) i =1 ,n of derivations, suchthat, | Π | + P i =1 ,r | Π µ r i | ≥ m p . We have to consider two cases: Proper
We have that r < m p − . So, we build the derivation T ++0 in fig-ure 10, by re-introducing in T the respective minor premiss β µ r i foreach Π µ r i derivation of the major premiss β µ r i ⊃ γ µ r i . Moreover, theother derivations Π µ i not in the sub-sequence (Π µ r i ) i =1 ,r are erasedfrom T . We re-inforce the observation that all β µ i are the same for-mula.Moreover, we delete the I − part of the main branch of T thatoccurs in T ++0 . This is the final derivation T ++0 show in the figure 10. q is the minimal formula of the main branch. γ µ . . . β µ r δ µ r Π µ r β µ r ⊃ γ µ r γ µ r . . . γ µ i . . . β µ r r δ µ r r Π µ r r β µ r r ⊃ γ µ r r γ µ r r . . . γ µ n Π q Figure 10: The derivation T ++0 We must observe that T ++0 is a valid derivation bigger than m p , thusby Lemma 14 there is a level ν , such that, there are at least m p − instances of the matrix M . We have a contradiction.• If ν > µ then there is at least m p − instances of a derivation ina level that exists in all the sub-derivations (Π µ r i ) i =1 ,r . Replacingback all the instances T i , i = 1 , n , we obtain T and a proof thatit has another level, besides µ , that has at least m p − repeatedinstances of the same matrix. We have a contradiction.25 If ν = µ , since β = β µ i , for each i , occurs less than m p − in level µ , then there must be other instances than T i occurring at least m p − in T in µ . We have a contradiction, when we replace backthe derivations T i , for there will be more than one matrix with atleast m p − repeated occurrences in level µ = ν .• If ν < µ we have a contradiction, by a reasoning analogous to theprevious item. Improper
We have that m p − ≤ r . So, we build the derivation T +++0 infigure 11, by re-introducing in T the respective minor premiss β asin T ++0 , but only from r to r m p − . Moreover, for each i = m p − +1 , . . . , r we remove Π µr i from T and replace β µi − m p − , the derivationof the minor premiss of the ⊃ -Elim rule by the smallest derivation ofthe form stated by Lemma 15 that is bigger than (cid:12)(cid:12) Π µr i (cid:12)(cid:12) . We denoteby Σ Π µri this derivation. Note that the size of Σ Π µri compensates theremoval of Π µ i , for every i = m p − + 1 to i = r . It is importantto note that the height of Σ µr i is linear on m , so it is T +++0 . From i = r + 1 to i = n there is only a simply removal of Π µ r i , without anycompensation. At the end the size of T +++0 is bigger than m p and thereis less than m p − occurrences of β in the level µ = µ . Thus, by ananalysis similar to the above case, with the application of Lemma 14,we obtain a contradiction. We observe that we introduced a linearnumber of assumptions in the Lemma 15. We notice that we reasonin terms of the big-O notation, thus we have that T +++0 is bigger than m p , in terms of m . γ µ . . . Σ µ r β µ r δ µ r Π µ r β µ r ⊃ γ µ r γ µ r . . . γ µ i . . . β µ r r δ µ r r Π µ r r β µ r r ⊃ γ µ r r γ µ r r . . . γ µ n Π q Figure 11: The derivation T +++0 Inductive Step In this case we have ℓ > levels with repetitions of the respective sub-derivations at least m p − . We analyse the highest level of repetitions in a26imilar way that was done in the basis step, with the hypothesis that theremoval of all repetitions in the lowest ℓ − levels result in a tree of sizelesser than m p . With this inductive hypothesis and the reasonning on therepetitions removal in the highest level we reach the desired conclusion. Lemma 17.
For any EmND proof T of a tautology α and p > , if |T | ≥ | α | p then | Compress ( T , p ) | < | α | p Proof.
We prove by induction on the number of recursive calls that | Compress ( T , p ) | < | α | p , for in lemma 13 we have already proven that the algorithm stops for any validinput pair hT , p i .• Basis
No recursive call: In this case we already have | Compress ( T , p ) | < | α | p . The “ else ” of the “if” in line 2 of algorithm 2 is used.• I.H.
Suppose |T | ≥ | α | p holds. Thus, a call to Lemma T ) , in line 3return the list of all occurrences of sub-derivations, instances of the inde-pendent set of matrices given by Lemma 9, that occurs more than m p − inthe lowest levels of T . By inductive hypothesis, there is less one recursivecall, Compress ( T i ) of each instance T i of the list returns a r-Dag of sizeless than | α | p . By Lemma 16 the part of T that it is not collapsed is lessthan | α | p and by Lemma 12 we obtain that | Compress ( T , p ) | < | α | p .The above upper-bound is not tight. A tighter one obtains by counting thenumber of matrices for each level, according to Theorem 7 and corollary 11. How-ever, in this article, we do not need a tighter upper-bound than what we state inlemma 13. The following definitions are central in proving that r-DagProofs are certificatesfor M ⊃ formulas validity. Let C = h V, E d , E A , r, l, L, ρ, δ, O α i be a pre r-DagProofof α from M ⊃ . We associate to each v ∈ V an entailment relation. The entail-ment represents the logical consequency relation carried within C from the DAG’sleaves downwards until v . Due to the collapse operation, and the many down-ward detours that the collapses introduce, we use an environment function that27eeps track of the entailment relation related to each detour. We use the notation Emt ( α ) to denote the set { ∆ ⊢ β : ∆ ⊆ Sub ( α ) and β ∈ Sub ( α ) } of all possibleentailments between sets of (sub)formulas of α , ∆ , and (sub)formulas of α . Definition 19.
Let ∆ ⊢ δ ∈ Emt ( α ) , and O be a total order on the subformulasof alpha . We define Ant O (∆ ⊢ δ ) = b O (∆) . In what follows, consider a pre r-DagProof C = h V, E d , E A , r, ℓ, L, ρ, δ, O α i of α . A node v ∈ V is called a deductive leaf, iff, it has no incoming deductiveedge, otherwise we call it as deductive (internal) node. The nodes of C that havemode than one different Deductive Edges outcoming from it are called divergentnode. Definition 20 (Local Entailment) . Given a pre r-DagProof C = h V, E d , E A , r, ℓ, L, ρ, δ, O α i of α , we define the mapping M C⊢ : V × N −→ Emt ( α ) ∪ { ′ ⊢ ′ } , for each v ∈ V recursively as follows: Deductive Leaf, no Ancestrality If v ∈ V and, there is no u ∈ V , such that h u, v i ∈ E d and, there is no w ∈ V , such that h w, v i ∈ E A then M C⊢ ( v,
0) = { ℓ ( v ) } ⊢ ℓ ( v ) , and M C⊢ ( v, j ) = ′ ⊢ ′ , for j ∈ N , j = 0 , and; Deductive Leaf with Ancestrality If v ∈ V and, there is no u ∈ V , such that h u, v i ∈ E d and, there is w ∈ V , such that h w, v i ∈ E A then M C⊢ ( v, i ) = { ℓ ( v ) } ⊢ ℓ ( v ) , for each i , such that there is h w, v i ∈ E A with δ ( h w, v i ) = i ,and M C⊢ ( v, j ) = ′ ⊢ ′ , for every j ∈ N , such that, there is no h w, v i ∈ E A with δ ( h w, v i ) = j , and; Deductive Internal Node, no Ancestrality If v ∈ V and, there is u ∈ V , suchthat h u, v i ∈ E d and, there is no w ∈ V , such that h w, v i ∈ E A then wehave two cases:1. There are only two u , u ∈ V , such that, h u i , v i ∈ E d , for i = 1 , , ℓ ( u ) = δ , ℓ ( u ) = δ ⊃ δ and, ℓ ( v ) = δ . Moreover, let I i = { j : M C⊢ ( u i , j ) = ′ ⊢ ′ } . Thus, we have that: If I = I then for every j ∈ I = I , we have that: M C⊢ ( v, j ) = (cid:26) ∆ ∪ ∆ ⊢ δ if M C⊢ ( u i , j ) = ∆ i ⊢ ℓ ( u i ) ′ ⊢ ′ otherwise, and every j ∈ N − ( I ∪ I ) , M C⊢ ( v, j ) = ′ ⊢ ′ . Moreover, if L ( h u i , v i ) ↓ , i = 1 , , then L ( h u i , v i ) = Ant O α ( M C⊢ ( u i )) , and; f I = I then for every j ∈ N , M C⊢ ( v, j ) = ′ ⊢ ′
2. There is only one u ∈ V , such that, h u, v i ∈ E d , ℓ ( u ) = δ and ℓ ( v ) = δ ⊃ δ . Moreover, let I = { j : M C⊢ ( u, j ) = ′ ⊢ ′ } . If I = ∅ , andhence, for every j ∈ I , we have that: M C⊢ ( v, j ) = (cid:26) ∆ − { δ } ⊢ δ ⊃ δ if M C⊢ ( u, j ) = ∆ ⊢ ℓ ( u ) and ℓ ( u ) = δ ′ ⊢ ′ otherwise, and every j ∈ N − I , M C⊢ ( v, j ) = ′ ⊢ ′ , and; For every j ∈ N , M C⊢ ( v, j ) = ′ ⊢ ′ . Moreover, if L ( h u, v i ) ↓ then L ( h u, v i ) = Ant CO α ( M ⊢ ( u )) .If I = ∅ then M C⊢ ( v, j ) = ′ ⊢ ′ , for every j ∈ N . Deductive Internal Node with Ancestrality If v ∈ V and, there is u ∈ V , suchthat h u, v i ∈ E d and, there is w ∈ V , such that h w, v i ∈ E A then we havetwo cases:1. There are only two u , u ∈ V , such that, h u i , v i ∈ E d , for i = 1 , , ℓ ( u ) = δ , ℓ ( u ) = δ ⊃ δ and, ℓ ( v ) = δ . Moreover, let I i = { j : M C⊢ ( u i , j ) = ′ ⊢ ′ } . Thus, we have that: If I = I then for every j ∈ I = I , we have that: M C⊢ ( v, j ) = (cid:26) ∆ ∪ ∆ ⊢ δ if M C⊢ ( u i , j ) = ∆ i ⊢ ℓ ( u i ) ′ ⊢ ′ otherwise, and every j ∈ N − ( I ∪ I ) , M C⊢ ( v, j ) = ′ ⊢ ′ , and; If I = I then for every j ∈ N , M C⊢ ( v, j ) = ′ ⊢ ′
2. There is only one u ∈ V , such that, h u, v i ∈ E d , ℓ ( u ) = δ and ℓ ( v ) = δ ⊃ δ . Moreover, let I = { j : M C⊢ ( u, j ) = ′ ⊢ ′ } . If I = ∅ , andhence, for every j ∈ I , we have that: M C⊢ ( v, j ) = (cid:26) ∆ − { δ } ⊢ δ ⊃ δ if M C⊢ ( u, j ) = ∆ ⊢ ℓ ( u ) and ℓ ( u ) = δ ′ ⊢ ′ otherwise, and every j ∈ N − I , M C⊢ ( v, j ) = ′ ⊢ ′ , and; For every j ∈ N , M C⊢ ( v, j ) = ′ ⊢ ′ . If I = ∅ then M C⊢ ( v, j ) = ′ ⊢ ′ , for every j ∈ N . Divergent Deductive Internal Node
In this case v should not be the target of anancestrality edge, otherwise C is not a valid pre rDagProof and M C⊢ ( v, j ) = ′ ⊢ ′ ,for every j ∈ N . Moreover, the set S = {h w : h v, w i ∈ E d } has at least twonodes . We have two cases to consider: This is just the case for divergent deductive nodes . There are only two u , u ∈ V , such that, h u i , v i ∈ E d , for i = 1 , , ℓ ( u ) = δ , ℓ ( u ) = δ ⊃ δ and, ℓ ( v ) = δ . Moreover, let I i = { j : M C⊢ ( u i , j ) = ′ ⊢ ′ } and T = { ρ ( h v, w i ) : w ∈ S } . Thus, we have that: If I = I = I then for every j ∈ I = I = T , we have that: M C⊢ ( v, j ) = (cid:26) ∆ ∪ ∆ ⊢ δ if M C⊢ ( u i , j ) = ∆ i ⊢ ℓ ( u i ) ′ ⊢ ′ otherwise, and every j ∈ N − ( I ∪ I ) , M C⊢ ( v, j ) = ′ ⊢ ′ , and; If I = I or I i = T , i = 1 or i = 2 then for every j ∈ N , M C⊢ ( v, j ) = ′ ⊢ ′
2. There is only one u ∈ V , such that, h u, v i ∈ E d , ℓ ( u ) = δ and ℓ ( v ) = δ ⊃ δ . Moreover, let I = { j : M C⊢ ( u, j ) = ′ ⊢ ′ } , T = { ρ ( h v, w i ) : w ∈ S } . If T = I = ∅ then for every j ∈ I , we have that: M C⊢ ( v, j ) = (cid:26) ∆ − { δ } ⊢ δ ⊃ δ if M C⊢ ( u, j ) = ∆ ⊢ ℓ ( u ) and ℓ ( u ) = δ ′ ⊢ ′ otherwise, and every j ∈ N − I , M C⊢ ( v, j ) = ′ ⊢ ′ , and; For every j ∈ N , M C⊢ ( v, j ) = ′ ⊢ ′ . Moreover, if L ( h u, v i ) ↓ then L ( h u, v i ) = Ant O α ( M ⊢ ( u )) .If I = ∅ then M C⊢ ( v, j ) = ′ ⊢ ′ , for every j ∈ N . Target of Divergent Deductive Internal Node
In this case v is such that thereis a divergent deductive node u with h u, v i ∈ E d and ρ ( h u, v i ) is defined.Thus, we have two cases: v is not target of an ancestrality edge M C⊢ ( v,
0) = M C⊢ ( u, ρ (( h u, v i ) and M C⊢ ( v, j ) = ′ ⊢ ′ , for every j = 0 ; v is target of an ancestrality edge There is h w, v i ∈ E A . Hence we set M C⊢ ( v, δ ( h w, v i )) = M C⊢ ( u, ρ ( h u, v i ) and M C⊢ ( v, j ) = ′ ⊢ ′ , for every j = δ ( h w, v i ) ; Obs:
In what follows, sometimes we use the notation M ⊢ ( v, i ) instead of M C⊢ ( v, i ) , whenever C can be easily infered from the context.A full subgraph of a graph A = h V A , E A i is any graph B = h V A , E i , with E ⊆ E A . The labelled version of full subgraph keeps all labels that label theelements of V A and E with the same value they have in A .30 efinition 21 (Underlying-deductive-structure of an rDagProof) . Given a pre rDagProof C = h V, E d , E A , r, ℓ, L, ρ, δ, O α i . The full sub-graph of C when weconsider all and only all of the edges in E d is denoted by C| E d . It is called theunderlying deductive strutucture of C . Definition 22 (maximal-path) . Given a pre rDagProof C = h V, E d , E A , r, ℓ, L, ρ, δ, O α i and v k , . . . , v , v i ∈ V , such that, h v i +1 , v i i ∈ E d , i = 1 , . . . , k − . We say that v , . . . , v k is a maximal path in C| E d , if and only if, v k is a top-formula. We saythat the maximal-path starts in v . The lenght of the sequence of nodes v k , . . . , v , is k . Definition 23 (reverse-deductive height) . Given pre r-DagProof C = h V, E d , E A , r, ℓ, L, ρ, δ, O α i of α . Let C| E d the sub-graph of C restricted to deductive edges only ( E d ). The re-verse deductive height of a node v ∈ V in C| E d , named rdh ( v ) is defined as: rdh ( v ) = max { k : v , . . . , v k is a maximal-path with v = v } The above definition 20 of M C⊢ is recursive. Given a pre rDagProof C , wecan assign to each node v ∈ V C the value of rhd ( v ) . By the recursion theorem,from set theory, we have that the function M C⊢ is well-defined for every node v andnatural number i in any rDagProof C . Acoording to this assignment of values, wehave that the value assigned to the root of C| E d is h ( C ) , the value of all of its leavesis 0 (zero) and the value of the children of any node is smaller than the value oftheir respective parent. Thus, M C⊢ is well-defined and unique for any C .We note the following well-known facts, regarding usual Kripke semantics for M ⊃ [21], denoted by | = M ⊃ . Fact 1 (Soundness of ND M ⊃ rules) . Consider ∆ and ∆ two sets of M ⊃ formulas,and, δ and δ two M ⊃ formulas. We have that:1. If ∆ | = M ⊃ δ and ∆ | = M ⊃ δ ⊃ δ then ∆ ∪ ∆ | = M ⊃ δ , and;2. If ∆ | = M ⊃ δ then ∆ − { δ } | = M ⊃ δ ⊃ δ Note that the above items are just the ⊃ -Intro and ⊃ -Elim rules. ConcerningItem 2, we have both cases δ ∈ ∆ and δ ∆ , as it is the case with the ⊃ -Introrule. In what follows we omit the symbol M ⊃ in the notation | = M ⊃ whenever itsmeaning as the minimal entailment is made clear.31 efinition 24 (rDagProof correctness) . Let C = h V, E d , E A , r, l, L, ρ, δ, O α i be a pre r-DagProof. We say that C is correct iff M ⊢ ( r, = ′ ⊢ ′ and, for each i = 0 , i ∈ N , M ⊢ ( r, i ) = ′ ⊢ ′ . When a pre rDagProof C is correct we simply call it rDagProof. Given acorrect rDagProof C , such that, M C⊢ ( r C ,
0) = ∆ ⊢ β , we have that C is a certificatethat β as logical consequence of ∆ in M ⊃ . This is what we state in Lemma 18below. Lemma 18 (Local entailment sounds) . Let C = h V, E d , E A , r, l, L, ρ, δ, O α i bea correct rDagProof of α . Thus, for every v ∈ V , for every i ∈ N , such that M C⊢ ( v, i ) = ⊢ , then, if M C⊢ ( v, i ) = ∆ ⊢ ℓ ( v ) , ℓ ( v ) = β , then ∆ | = M ⊃ β .Proof. By induction on the definition of M ⊢ definition and using Lemma 1. Corollary 19.
If in the stating of Lemma 18 above, we consider that: • There is a formula δ , subformula of α , such that, it is top-formula in C and, there is no deductive path from this top-formula to the root r of C thatapplies an ⊃ introduction rule having δ as the antecedent of the formulathat it is the conclusion of this application.Then M C⊢ ( v, i ) = ∆ ⊢ ℓ ( v ) , ℓ ( v ) = β , with δ ∈ ∆ .Proof. This corollary is a consequence of the proof of Lemma 18. Its proof is anextension of the induction proof of the lemma, by the inclusion the condition inthe statement and verify that the formula δ is not removed during the evaluation of M C⊢ ( v, i ) . Since it is a top formula, δ is included in the local entailment antecedent,in the basic step of the induction and, we do not remove it anymore. Theorem 20 (Completeness of rDagProofs) . For any M ⊃ formula α and set ofsubformulas ∆ of α and subformula β of α , we have that if ∆ | = ⊢ β holds thenthere is a correct rDagProof C , such that, M C⊢ ( r C ,
0) = ∆ ′ ⊢ β , with ∆ ′ ⊆ ∆ .Proof. The system of Natural Deduction for M ⊃ is sound and complete regardedthe usual Kripke semantics for M ⊃ [21]. Since N D M ⊃ proofs are particular casesof rDagProofs we have completeness of rDagProofs. Thus, if ∆ | = β then thereis a derivation Π having β as conclusion and a set ∆ ′ of open assumptions, with ∆ ′ ⊆ ∆ . Taking Π as a rDagProof and by Corollary 19 we have that M C⊢ ( r C ,
0) =∆ ′ ⊢ β . 32 heorem 21 (Soundness of rDagProofs) . If C is a correct rDagProof and M C⊢ ( r,
0) =∆ ⊢ β then ∆ | = ⊢ β .Proof. This theorem is an immediate consequence of Lemma 18
Corollary 22. If C is a correct rDagProof of α then α is a M ⊃ tautology We have the following lemmata that help us to show that a correct r-DagProof,compressed by the technique that algorithm 2 implements, is sound. Moreover,in the next section, we show an algorithm that checks whether a pre rDagProofis correct or not. This verification is efficient (linear) on the size of the pre rDag-Proof.In the next section, we show that for any Natural Deduction Π that has itsheight linearly bounded by the size of its conclusion, Compress (Π) is a correctrDagProof of α . From this result, we can conclude that any M ⊃ tautology has asuccinct (polynomial) and correct rDagProof. In conclusion, we describe how touse this result to show that N P = CoN P . This section contains some lemmata that help us to prove that the correctness ofrDagProofs are preserved by the compression of rDagProofs.
Definition 25 (A-consistent sub-rDagProof) . Let D be a pre rDagProof of α and C a sub-rDagProof of D . We say that C is an A-consistent sub-rDagProof of D , ifand only if, D ↑ is graph-isomorphic to C and for every h u, v i ∈ E A , we have that v ∈ V C and u V C , if and only if, v ∈ V D↑ k and u V D↑ k . Lemma 23 (Local Entailment preservation under DetachLink) . Let D be a pre rDagProof and k a node of D that is the root of an instance of a A-consistent sub-rDagProof, cf. definition 25, C of D . Let i ∈ N be such that i does not label anyof the E d edges going out of k . Moreover, consider D ′ = DetachLink ( D , k, C , i ) as defined in Definition 13. We have that for every v ∈ V D ′ and j ∈ N , j = i , thefollowing conditions hold: • If v V ( D ) ↑ k then M D⊢ ( v, j ) = M D ′ ⊢ ( v, j ) , and; • If v ∈ V ( D ) ↑ k then M C⊢ ( h − ( v ) , j ) = M D ′ ⊢ ( v, j ) , and; If v ∈ V ( D ) ↑ k then M C⊢ ( h − ( v ) ,
0) = M D ′ ⊢ ( v, i ) ;Where h is the (full) labeled graph-isomorphism from C into D ↑ k .Proof. By inspecting Definition 13, we observe that D ′ = ( D − C ′ ) ∪ C , where D ↑ k = C ′ = h ( C ) . We note that M D ′ ⊢ definition is then either on the complementof D ↑ k , or on C . The former agrees with M D⊢ and the later agrees with M D↑ k ⊢ = M h ( C ) ⊢ . Finaly, the third item in the statement of the lemma is related to the newedge that links the root of C to the former target of the E d edge that linked k withthis target. The top-formulas that are related to the basis steps of M ⊢ recursion areaccordingly accordingly associated to each respective part of D ′ , the same can besaid about the recursive steps. So we have the desired result. Corollary 24 (Soundness of DetachLink) . Consider the conditions of Lemma 23,above. We have that if C is correct and D is also correct then DetachLink ( D , k, C , i ) is correct. Moreover, the local entailment is preserved, modulo the the isomor-phism between D ↑ k and C . An immediate consequence of Corollary 24 is that it the DetackLink opera-tion can be repeated many times, without disturbing the soundness of the yieldedrDagProof. Due to this, we have the following lemma.
Lemma 25 (Soundness of Collapse) . Let D and C be r-DagProofs. Let C be A-consistent subgraph of D . Let Y be a list containing the roots of the instances of C in a fixed level µ . We have that if D and C are correct then Collapse ( D , Y , C ) is correct too. Moreover the local entailment is preserved as stated in Lemma 23. The above Lemma 25 is the correctness proof of algorithm 1.The following theorem proves the correctness of algorithm 2
Theorem 26 (Soundness of Compressed rDagProofs) . Let D = h V, E d , E A , r, l, L, ρ, δ, O α i be a pre rDagProof for a M ⊃ formula α . If D is correct then for every < pCompress ( D , p ) is correct. Moreover, the local entailment is preserved, i.e.,Proof. This proof proceeds by induction on the number of, recursive, calls to
Compress . In the proof of the termination of algorithm 2, we have seen that ithalts for every D and p , < p . The basis step is trivial, and we use the inductivehypothesis together with Lemma 25 to prove the inductive (recursive) step.34 On the complexity of verifying that a pre rDag-Proof is correct or not
The following algorithm performs a top-down sweeping in any pre rDagProof tocheck whether it is correct or not. It is an iterative implementation of M C⊢ thatprints can check whether the pre rDagProof is correct, and certifies a tautology ornot. In the case, it is not a tautology it prints “DERIVATION”. Finally, it prints“INCORRECT” if the rDagProof is not correct. The definition of M ⊢ points outthe correctness of the algorithm. We analyse its computational complexity inthe sequel. We have to note that the update − and − check , inside the itera-tion structures in lines 17 and 35 is responsible by updating the local entailmentdata-structure (the Reg indexed structure) with ′ ⊢ ′ to indicate that the checkingalgorithm detected an incorrect pre rDagProof.35 lgorithm 3 Verifies whether a r-DagProof is valid Function
Check-rDagProof( C ) for k = height ( C ) downto do L ← T opF ormulas ( k ) for top ∈ L do Reg ( top ) ← ∅ for edge ∈ AncestorEdges ( top ) do Reg ( top ) ← Reg ( top ) ∪ { δ ( edge ) ℓ ( top ) ⊢ ℓ ( top ) } end for Reg ( top ) ← Reg ( top ) ∪ { ℓ ( top ) ⊢ ℓ ( top ) } end for I ← InternalNodes ( k ) for v ∈ I do q ← P remisses ( v ) if Divergent ( v ) then Lv ← {h v, w i : h v, w i ∈ E A } if { i : Defined ( Reg ( q, i } = { ρ ( e ) : e ∈ Lv } then for i ∈ Reg ( q ) do Reg ( v, i ) ← update − and − check ( v, q, i ) end for else Reg ( v, ← ′ ⊢ ′ end if else if T argetDivergent ( v ) then u ← ι { u : h u, v i ∈ E d } if Defined ( ρ ( h u, v i ) ∧ ¬∃h w, v i ∈ E A then Reg ( v, ← Reg ( u, ρ ( h u, v i )) Reg ( v, j ) ← ′ ⊢ ′ ∀ j = 0 end if if Defined ( ρ ( h u, v i ) ∧ ∃h w, v i ∈ E A then Reg ( v, δ ( h w, v i )) ← Reg ( u, ρ ( h u, v i )) Reg ( v, j ) ← ′ ⊢ ′ ∀ j = δ ( h w, v i )) end if else for i ∈ Reg ( q ) do Reg ( v, i ) ← update − and − check ( v, q, i ) end for end if end if end for end for if Reg ( r ) = ′ ⊢ ′ then CORRECT if Reg ( r ) == ⊢ ℓ ( r ) then T AUT OLOGY else
DERIV AT ION end if else
INCORRECT end if
Let C = h V, E d , E A , r, l, L, ρ, δ, O α i be a pre rDagProof. We set n v = | V | , n A = | E A | , m = |O α | = | T α | ≤ len ( α ) and h = height ( C ) = height ( h V, E d i ) .36n the sequel, all the line references are in algorithm 3. We proceed to a worstcase analysis to find an upper-bounded for the number of steps to check whether C is correct or not. The loop that starts in line 2 consumes h steps, for each ofthese steps, we have at most n v possible nodes that are top-formulas, this is whatline 4 sweeps using the “for” statement. Inside this “for” there is other nestediteration on the set of ancestor edges that target the top-formulas, and a consequentupdating in the list of local entailments stored in the Reg data-structure. Line 9 isresponsible by the update of the main local-entailment (indexed by 0). After that,the loop that starts in line 12 takes care of the internal nodes, and for each internalnode, we have two possible cases, either it is a target of a divergent node and theneeded udpate on the local indexed by the index 0 is made, or, the update of allindexes, including the 0, of the local entailment structure is made. The choice ismade by the “if” statement in line 24. The cost of with steps that takes care ofthe top-formulas is h × n V × n A . To this we have to add the cost of processingthe internal nodes that is h × n v × n A too, due to the cost of updating the indexesrelated to each Ancestor edge in E A . However, as top-formulas and internal nodesare disjoint then we have that algorithm 3 when applied on C performs the numberof steps upper-bounded by disequality 1: Steps ( C ) ≤ h × n v × n A (1) Steps ( C ) ≤ n v × n v = n v (2)Observing that n A ≤ n v and h ≤ n v we have the upper-bound in disequality 2Since we are counting steps, we can say that the time complexity to check whethera pre rDagProof C is correct or not is polynomial, 4th power indeed, on the sizeof C . C oN P = N P
We have already discussed in the introduction of this text that when consideringthe complexity class
CoN P , we are naturally limited to linearly height-boundedproofs. The proofs, in M ⊃ , of the non-hamiltonianicity of graphs, are linearlyheight bounded. See the appendix in [13] or [9] for a detailed explanation on this.If N P = CoN P then the set of non-hamiltonian graphs there is no polynomiallysized and verifiable in polynomial time certificate for each of its elements. Thus,the set S of all formulas that have Normal Natural Deduction proofs linear height-bounded contains the valid for the non-hamiltonian graphs. Hence, by assuming37hat N P = CoN P , we have to conclude that S is a family of normal super-polynomial proofs with linear height. If we consider any proof in S , either it ispolynomially sized, and we have nothing to prove, or it is bigger than m p , for some p > , where m is the size of the proof’s conclusion. We observe that the case p ≤ is subsumed by p > , anyway. Thus, we can apply Theorem 7 to show thatthis big proof is redundant, so we can apply the compression algorithm 2 to obtaina correct rDagProof of size smaller than m p , according to Lemma 17. Finally,the algorithm 3 can check the correctness of this polynomially sized rDagProof intime upper-bounded by m p .The last paragraph provided a precise argumentation showing polynomial cer-tificates for each non-hamiltonicity of each non-hamiltonian graph. We can checkeach of them is a (correct) certificate in polynomial time too. We can conclude that CoN P ⊆ N P , since non-hamiltonicity of graphs is a
CoN P -complete problem.Having proved that
CoN P ⊆ N P we have proof that
N P = CoN P , as the fol-lowing reasoning shows, where L is the set-theoretical complement of L . We haveused the logically simplest definition of the class CoN P class as { L : L ∈ N P } . ⇒ CoN P ⊂ N P ⇓ L ∈ N P , iff, L ∈ CoN P , CoN P ⊆ N P so L ∈ N P , iff, L ∈ CoN P ⇓ CoN P = N P
More details and mathematical precision on this argument contain an alterna-tive proof to the conjecture
CoN P = N P and are a matter for a further article.
This article shows that for any huge proof of a tautology in M ⊃ we obtain a suc-cinct certificate for its validity. Moreover, we offer an algorithm able to checkthis validity in polynomial time on the certificate’s size. We can use this resultto provide a compression method to propositional proofs. Moreover, we can ef-ficiently check the compressed proof without uncompressing it. Thus, we havemany advantages over traditional compression methods based on strings. Thecompression ratio of techniques based on collapsing redundancies seems to bebigger, as shown in [22] that reports some experiments with a variation of the38orizontal Compression method compared with Huffman compression. The sec-ond and more important advantage is the possibility to check for the validity ofthe compressed proof without having to uncompress it. In general, the originalproof is huge, super-polynomial and hard check computationally.Another application of the results in this article is to provide an alternativeproof of N P = CoN P . In [8] we have a proof that
N P = N P SP ACE . Animmediate consequence of this equality is that
N P = CoN P . The approach thatarises from the results we have shown here does not need Hudelmaier [15] linearlybounded sequent calculus for M ⊃ logic. The proof reported in [8], on the otherhand, needs Hudelmaier Sequent Calculus and a translation to Natural Deductionproofs that preserves the linear upper-bound. However, the resulted translationis not normal, and it is well-known that normalization does not preserve upper-bounds in general. Thus, we cannot apply our approach to the whole class of M ⊃ tautologies to prove that N P SP ACE ⊆ N P , for the use of normal proofsis essential to obtain the redundancy lemma, i.e., Lemma 7. However, the com-pression method reported in this article, due to the redundancy lemma, providesknowledge to prove M ⊃ short tautologies automatically. It seems easier than theuse of the double certificate approach in [8].
10 Acknowledgement
We would like very much to thank professor Lew Gordeev for the work we havedone together and the inspiration to follow this alternative approach. Thank Pro-fessor Luiz Carlos Pereira for his support, lessons and ideas on Proof Theory sincethe first course I have taken with him as a student. Thank the proof-theory groupat Tuebingen-University, led by prof. Peter Schroeder-Heister. Many thanks toprofs Gilles Dowek (INRIA) and Jean-Baptiste Joinet (univ. Lyon) for the intenseinteraction during this work’s elaboration. Finally, we want to thank all students,former students, and colleagues who discussed with us in many stages during thiswork. We must have forgotten to mention someone, and we hope we can mendthis memory failure in a nearer future.
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