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On the Provability of Consistency of PV Ebrahim A Larijani Submitted to the University of Wales in fulfilment for the Degree of Master of Research in Logic and Computation Department of Computer Science Swansea University October 2008 Declaration This work has not been previously accepted in substance for any degree and is not being concurrently submitted in candidature for any degree. Signed ............................................................ Date ............................................................ (candidate) Statement 1 This thesis is the result of my own investigations, except where otherwise stated. Other sources are acknowledged by footnotes giving explicit references. A bibliography is appended. Signed ............................................................ Date ............................................................ (candidate) Statement 2 I hereby give my consent for my thesis, if accepted, to be available for photocopying and for inter-library loan, and for the title and summary to be made available to outside organisations. Signed ............................................................ Date ............................................................ (candidate) Abstract This thesis is a contribution to the separation problem of Bounded Arithmetic. It is shown the consistency of induction free fragment of the equational theory PV of polynomially verifiable equations is not provable within PV. The main body of the thesis is a workout and improvement of the corresponding result by Buss and Ignjatovic [6]. The improvement has done by dropping the axiom concerning squaring function from PV. Contents 1 Introduction 1.1 Bounded Arithmetic(BA) . . . . . . . 1.2 BA hierarchy . . . . . . . . . . . . . . 1.3 Polynomial Hierarchy . . . . . . . . . 1.3.1 PH (machine dependent ) . . 1.3.2 PH (machine independent) . 1.4 Separation of BA hierarchy and PH . 1.5 Separation problem and consistency 1.6 Main results and structure of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 5 6 6 6 9 10 12 2 Proof System PV 2.1 Language of PV . . . . . . . . . . . . . . . . 2.2 Axioms of PV . . . . . . . . . . . . . . . . . . 2.3 Proofs in PV . . . . . . . . . . . . . . . . . . 2.3.1 Structure of Proofs in PV . . . . . . 2.3.2 E − , P V − , P V . . . . . . . . . . . . . . 2.3.3 Interpretation of formulae as terms . 2.4 Arithmetisation of PV . . . . . . . . . . . . . 2.4.1 Gödelisation of Lp . . . . . . . . . . . 2.4.2 Gödelisation of Proofs . . . . . . . . 2.4.3 Provability Predicate for PV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 15 17 19 19 22 23 25 26 27 28 3 Examples of upper bounds for the length of proofs in P V − 29 4 Speed-up of induction for P V − 4.1 Speed-up of induction with quantifiers . . . . . . . . . . . . . . 4.2 Speed-up of induction for quantifier free equational theories . 4.3 Provability of consistency of P V − . . . . . . . . . . . . . . . . . 38 38 40 54 1 . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 Appendix 58 Bibliography 59 2 Chapter 1 Introduction Logical complexity theory consists of logical frameworks to study problems in computational complexity. Such logical approaches give us better understanding of computational complexity classes like P, NP, co-NP, etc. which are first defined by S.Cook and L.Levin precisely [9] their investigations into the complexity theory raised famous open problem whether P equals NP. One framework to tackle latter question regarding proof complexity is equational theory PV that has been introduced by Cook [10] and deals with feasibly constructive proofs for reasoning about equations between polynomial time computable functions. In another aspect Buss showed [2] theories of Bounded Arithmetic (subsytems of Peano Arithmetic)are strongly related to the P V . It has been shown [12, 11] that separation problem for complexity classes (P, N P, co−N P, ...)are strongly related to the separation problem of Bounded Arithmetic theories which means whether the Bounded Arithmetic hierarchy is strict? A possible way of studying separation problem is to examine Gödel type consistency statements for Bounded Arithmetic theories. In this chapter, first we introduce Bounded Arithmetic and polynomial hierarchy and their relation. This will be followed by an overview of main results in separation problem. 3 1.1 Bounded Arithmetic(BA) Definition 1. We define language of BA, Lb as follows: 1 Lb = {0, S, +, ⋅, ⌊ a⌋, ♯, ≤, ∣a∣} 2 where 0, S, +, ⋅, ⌊ 21 a⌋ and ≤ are constant zero, successor function, multiplication, addition, shift right function and less-than-or-equal-to relation. ∣a∣ denotes the length of binary representation of a and a ♯ b = 2∣a∣⋅∣b∣ . In BA all quantifiers are bounded : ∀x ≤ t, ∃x ≤ t or sharply bounded: ∀x ≤ ∣t∣, ∃x ≤ ∣t∣. Now for defining Bounded Arithmetic hierarchy we need some definitions: Definition 2. For quantifier free formulae ϕ in the language of BA, the class of Σbi -fromulae(Πbi -formulae) is defined as follows: Σb1 ∶ ∃x1 ≤ s1 ∀y ≤ ∣t∣ϕ(x1 , y) Σb2 ∶ ∃x1 ≤ s1 ∀x2 ≤ s2 ∃y ≤ ∣t∣ϕ(x1 , x2 , y) ⋮ By replacing ∃and∀ in the above Σbi -formulae we obtain: Πb1 , Πb2 , .... The difference between Bounded Arithmetic and Peano Arithmetic is that induction in BA is restricted only to the bounded formulae Σbi or Πbi . In the following we define axiom scheme of BA. Definition 3. We define BA induction axioms as follows: Let Ψ be either the class of Σbi -formulae or Πbi -formulae and let A(x) ∈ Ψ then 1. Ψ − IN D ∶ A(0) ∧ (∀x)(A(x) ⊃ A(Sx)) ⊃ (∀x)A(x) 2. Ψ − P IN D ∶ A(0) ∧ (∀x)(A⌊ 12 x⌋) ⊃ A(x)) ⊃ (∀x)A(x) 3. Ψ − LIN D ∶ A(0) ∧ (∀x)(A(x) ⊃ A(Sx)) ⊃ (∀x)A(∣x∣) Now we can define theories of Bounded Arithmetic by using BASIC axioms (c.f. appendix A) and above induction schemes: Definition 4. S2i is defined as a first order theory over the language Lb and it has following sets of axioms: 4 • BASIC • Σbi − P IN D T2i is defined as a first order theory over the language Lb and it has following sets of axioms: • BASIC • Σbi − IN D Definition 5. S2 = ⋃ S2i i T2 = ⋃ T2i i 1.2 BA hierarchy With the above definition of S2i and T2i we can form a hierarchy of BA theories. In this line Buss stated following theorem to illustrate relationships between bounded arithmetic theories [2] : Theorem 1. For i ≥ 1, S2i and T2i in the Definition 4 we have T2i ⊢ S2i S2i ⊢ T2i−1 In particular we have following sequence of BA theories S21 ⊆ T21 ⊆ S22 ⊆ T22 ⊆ . . . It has been shown in [2] that S21 and equational theory P V are strongly related. Furthermore Buss proved some conservation results between P V and S21 . In BA hierarchy we are interested to know which functions can be defined at each level, in particular, Σbi -definable functions in S2i . Definition 6. Let A(x, y) be a Σbi (Πbi ) formula. A function f ∶ N → N is called Σbi -definable in S2i (T2i ) if 1. ∀x, A(x, f (x)) must be a valid formula . 2. S2i ⊢ (∀x)(∃y)A(x, y) 3. S2i ⊢ (∀x)(∀y)(∀z)(A(x, y) ∧ A(x, z) ⊃ y = z) 5 1.3 Polynomial Hierarchy Polynomial hierarchy (PH) first introduced by L.Stockmeyer [15] to generalise computational complexity classes P, NP, co-NP and beyond.There are two equivalent approaches to define PH, first approach is machine dependent definition in terms of Turing machine and the second is machine independent definition by using limited iteration and bounded quantification to determine complexity classes. 1.3.1 PH (machine dependent ) In this section we define PH with the aid of Turing machines. First we define classes for language or decision problems: [14] Definition 7. For every i ≥ 1 , Let L be a binary language (decision problem) we say L ∈ Σpi if there exist a polynomial time Turing machine M and a polynomial q such that : x ∈ L ⇔ ∃u1 ∈ {0, 1}q(∣x∣) ∀u2 ∈ {0, 1}q(∣x∣) . . . Qi ui ∈ {0, 1}q(∣x∣) M (x, u1 , . . . , ui ) = 1 Where Qi denotes ∃ or ∀ depending on whether i is even or odd. similarly by changing universal and existential quantifiers we obtain class of languages or decision problems, Πpi . Now we define PH: P H = ⋃ Σpi i p F P Σi denotes the class of functions which are computable by a polynomial time Turing machine with an oracle in Σpi . 1.3.2 PH (machine independent) Polynomial Hierarchy can be obtained by defining certain kinds of function classes. For this in this section first we define an algebra on a set of initial functions by operations “limited iteration” and composition then we introduce the set of “polynomial time growth rate” functions and finally we define PH in this setting.[2] Definition 8. Set of basic functions B: • 0 (the constant 0) 6 • x ↦ Sx (the successor function) • x ↦ ⌊ 12 x⌋ (the shift right function) • x ↦ 2 ⋅ x (shift left function) ⎧ ⎪ ⎪1 • (x, y) ↦ x ≤ y = ⎨ ⎪ ⎪ ⎩0 x ≤ y, x>0 ⎧ ⎪ ⎪y • (x, y, z) ↦ Choice(x, y, z) = ⎨ ⎪ ⎪ ⎩z x > 0, x=0 Definition 9. Let k ≥ 0, let g ∶ Nk → N and h ∶ Nk+2 → N be arbitrary functions let p and q be positive polynomials.We say f ∶ Nk → N is defined from g, h and time bound p and space bound q by “limited iteration” if the following conditions hold. For τ ∶ Nk+1 → N τ (x1 , . . . , xk , 0) = g(x1 , . . . , xk ) τ (x1 , . . . , xk , n + 1) = h(x1 , . . . , xk , n, τ (x1 , . . . , xk , n)) → → → (∀n ≤ p(∣Ð x ∣)(∣τ (Ð x , n)∣ ≤ q(∣Ð x ∣) → → → f (Ð x ) = τ (Ð x , p(∣Ð x ∣)) Definition of limited iteration is similar to the notion of limited recursion on notation which we will use it in the next chapter in order to define the language of PV. Definition 10. A function f ∶ Nk → N has polynomial growth rate iff there is a positive polynomial p(a polynomial with only nonnegative integer coef→ → → ficients.) such that for all Ð x , we have ∣f (Ð x )∣ < p(∣Ð x ∣). Let C be a set of functions of polynomial growth rate. The Polynomial-time closure of C, PTC(C), is the smallest class of functions which (1) contains C and B and (2) is closed under composition and limited iteration. Definition 11. PRED(C): is the set of predicates (boolean functions ) in the set a of functions C. 7 In particular, if Q and R are functions then (∀y ≤ Q(x1 , . . . , xn ))R(x1 , . . . , xn , y) is predicate (boolean function) which has the value 1 iff for all y ≤ Q(x1 , . . . , xn ) the value of R(x1 , . . . , xn , y) is nonzero. Accordingly we could define bounded quantification for existential quantifiers. Now we can explain “polynomially bounded quantification” and “logarithmically bounded quantification” by using different bounds for quantifiers: Definition 12. Let C be a set of functions closed under composition and Let ∣x∣ denote the length of binary representation of x . Then P B∃(C) is the set of predicates Q such that: 1. Q ∶ Ni → N for some i ∈ N 2. There is an R ∈ P RED(C) and a positive polynomial p such that for → all Ð x: Q(x1 , . . . , xk ) = ∃y ≤ 2p(∣x1 ∣,...,∣xk ∣) R(x1 , . . . , xk , y) P B∀(C) can be defined in a similar way by replacing existential quantifier with universal quantifier. Definition 13. Let C be a set of functions closed under composition. Then LB∃(C) is the set of predicates Q such that 1. Q ∶ Ni → N for some i ∈ N 2. There is an R ∈ P RED(C) and a positive polynomial p such that for → all Ð x: Q(x1 , . . . , xk ) = ∃y ≤ p(∣x1 ∣, . . . , ∣xk ∣)R(x1 , . . . , xk , y) LB∀(C) can be defined in a similar way by replacing existential quantifier with universal quantifier. Now we are ready to define Polynomial Hierarchy according to [2]: Definition 14. Polynomial Hierarchy(PH) is defined by induction on k as follows 1. ◻p0 is the smallest set of functions containing B and closed under composition, LB∃ and LB∀. 2. ∆p0 = Σp0 = Πp0 = P RED(◻p0 ). 8 3. ◻pk+1 = P T C(Σpk ) 4. ∆pk+1 = P RED(◻pk+1 ) 5. Σpk+1 = P B∃(∆pk+1 ) 6. Πpk+1 = P B∀(∆pk+1 ) 7. P H = ⋃k Σpk p In particular in the above definition we have ◻pk+1 = F P Σi (see last subsection). Furthermore in [15] it is shown that these two definition of PH (machine dependent and machine independent) are equivalent.(see following figure from [4] ) 1.4 Separation of BA hierarchy and PH One major open problem in Bounded Arithmetic is if the inclusion of theories as described in 1.2, is proper? or the hierarchy of Bounded Arithmetic could 9 be separated? It is shown by Buss et al. that hierarchy of BA and PH are strongly related in the following we present some results which show this relation. Theorem 2. Class of Σbi definable functions in S2i is equal to ◻pi (F P Σi−1 ) b This implies for example, in the case i = 1, the class of Σb1 definable functions in S21 is equivalent to the class of functions ◻p1 which are computed by a polynomial time Turing machine(F P ). For another example, the class of Σb2 definable functions in S22 is equivalent to the class of functions in ◻p2 which can be computed by a polynomial time Turing machine with an oracle in N P . We conclude this section by presenting following important theorems concerning the relation of separation problem in BA hierarchy and PH. Theorem 3. [12, 11] If the levels of the Polynomial time Hierarchy (PH) are separated, then the levels of bounded arithmetic theories (BA) are separated as well. In particular, if Σpi+2 ≠ Πpi+2 then S2i ≠ S2i+1 . Theorem 4. [17, 3] BA collapses iff PH collapses provably in BA 1.5 Separation problem and consistency As mentioned before in order to approach separation problem in BA hierarchy one way is to study Gödel type consistency statements of BA theories. We know from Gödel incompleteness theorem that, for subsystems of PA, namely IΣ1 where induction is restricted to the Σ1 formulae (of the form ∃xφ(x) where φ is bounded formula) we have: IΣ1 ⊬ Con(IΣ1 ) Where Con(IΣ1 ) denotes consistency of IΣ1 . Also for system IΣ2 where induction is applied on Σ2 formulae (of the form ∃x∀yφ(x, y)) we know that: IΣ2 ⊢ Con(IΣ1 ) Buss [2] showed the general notion of consistency is too strong since he proved: S2 ⊬ Con(S2−1 ) 10 where S2−1 is induction-free fragment of S21 . This fact lead Buss to introduce the notion of bounded consistency (BDCon) which is concerned with proofs containing only bounded formulae (formula which all quantifiers have to be bounded). Buss has shown in [2] that for at most one i: S2i+1 ⊢ BDCon(S2i ) This has been improved by Pudlak in [13]: S2 ⊬ BDCon(S21 ) In this line also Takeuti showed S2i can not prove the consistency of S2−1 where only Σbi+5 formulae appear.[16] On the other hand for equational theory P V Cook proved :[10] P V ⊬ Con(P V ) By similar argument to the Gödel incompleteness theorem. Furthermore Buss and Ignjatovic showed in [6] that even for induction-free fragment of P V , P V − , consistency is not provable in P V : P V ⊬ Con(P V − )(∗) The above negative results for provability of consistency led Takeuti to raised following conjecture [7]: “Let S2−∞ be the equational theory involving equations s = t, when s and t are closed terms in the language of S2 with natural rules based on the recursive definition of function symbols then S2 ⊬ Con(S2−∞ ).” The consequence of this conjecture would be impossibility of positive answer to the separation problem of BA by using consistency statements. Despite that in [1] it is shown that Takeuti conjecture is not true and consistency of S2−∞ is provable in S21 . In fact in [1] it is shown that S21 proves more general claim for consistency of “pure” equational theory S2−∞ . In particular, consistency of “pure” equational theory in the language of P V which is based solely on the recursive definition of function symbols is provable in S21 . This seems to contradict (∗) since S21 and P V are quiet similar (conservativity of S2 (P V ) over P V [2]). A deeper look reveals that these two results are sligthy different: 11 1. In [1] consistency statements consider proofs in pure equational theory .i.e. lines in proofs are equations and rules are from equational logic. 2. In (∗) [6] proofs considered by consistency statements have more liberal form such that each line consists of sequent and each sequent has formulae with boolean conjections. Also rules come from propositional logic and equational logic. Conclusion: The above comparison shows that consistency statements of the kind Con(P V − ) may be a candidate for separation of BA theories. 1.6 Main results and structure of thesis This thesis is a work out of results in [6] section (1,4) , regarding “unprovability of consistency of P V − ”. The contribution of the thesis is threefold: 1. Precision of arguments in [6] by providing full backgrounds which are necessary to get results. 2. New observation from proofs in [6] that theory P Vkr which can be obtained by restricting induction axioms to the “restricted k-induction” and has following schematic form ϕ(b) → ϕ(b + 1) ϕ(0) → ϕ(∣t∣k ) where ∣t∣k denotes the k-th binary length of term t, has the same expressive power to the P V (c.f. Theorem 12). 3. Improving results in [6] by weakening theory P V − . For this we dropped function symbol sq denoting squaring function, and its corresponding axiom. Furthermore we argued that the consistency of resulting theory which we also denote by Con(P V − ) is still unprovable in P V . So the structure of thesis as follows: In Chapter 2 we define theory P V and P V − precisely by fixing the language, axiom scheme, deduction scheme and Gödelisation of proof system P V . In Chapter 3 we show some examples of feasible proofs in P V − which we need them in Chapter 4. In particular Theorem 9 is proved in a new form different 12 than original proof (c.f. Theorem 8) [6]. Chapter 4 will discuss speed of induction in equational theories. This is based on speed-up technique in [6] but in a new form by introducing “restricted k-induction”. This is followed by a brief discussion of formalisation of proofs in P V − in a metalevel which enables us to get statements about consistency of P V − 13 Chapter 2 Proof System PV The proof system PV has been introduced by Cook [10] to capture the notion of feasibility of computation. The first motivation of Cook for developing PV came from the main question in complexity theory that whether P equals NP. His approach is to show that they are not equal, by trying to prove that the set of tautologies is not in NP or equivalently there is no propositional proof system with short proofs for all tautologies. The second motivation came from constructive mathematics; A constructive proof of a statement ∀xA provides efficient methods of finding a proof of A for each value of x, but it does not say anything about growth of proof’s length in terms of x and this led Cook to the notion of feasibly constructive proofs: If equation f (x) = g(x) has a proof Π in PV then there is a polynomial PΠ such that Π provides a uniform method of verifying f (x0 ) = g(x0 ), for a given natural number x0 , in PΠ (∣x0 ∣) steps. In this case we say f (x) = g(x) is Polynomially Verifiable . Like Church’s thesis for recursive functions, Cook proposed Verifiability thesis for equations in PV[10]: “An equation t = u of PV is provable in PV if and only if it is Polynomially Verifiable.” Cook used pure equational theory for the definition of PV. Obviously working with that proof system is very cumbersome therefore he developed proof system PV1 which allows logical connectives such as conjuction, negation and etc. Also in[10] it is shown that every equation is provable in PV1 if and only if it is provable in PV. We will develop PV1 further, and add new 14 symbols to the language as well as new set of axioms to obtain proofs in a more liberal way. 2.1 Language of PV In order to specify the syntax of PV we will add new symbols to the language of Bounded Arithmetic Lb and symbols for all polynomial computable functions. For this we shall introduce a mechanism for recursion called Limited Recursion on Notation (LRN) as well as Cobham’s theorem which ensures that all of polynomial computable functions could be defined in this framework. Definition 15. Limited Recursion on Notation (LRN): Function f (x, y1 , . . . , yn ) is obtained by LRN from functions g(y1 , . . . , yn ) , h(x, z, y1 , . . . , yn ) and k(x, y1 , . . . , yn ) if : 1. f (0, y1 , . . . , yn ) = g(y1 , . . . , yn ) . 2. f (x, y1 , . . . , yn ) = h(x, f (⌊ 12 x⌋, y1 , . . . , yn ), y1 , . . . , yn ) for x > 0 3. f (x, y1 , . . . , yn ) ≤ k(x, y1 , . . . , yn ) There is another characterisation of Limited Recursion on Notation by defining function Min which we normally use it in this thesis. Let M in(a, b) ∶= a (a b) Then we could rewrite LRN as follows: 1. f (0, y) = M in(g(y), k(0, y)) 2. f (x, y1 , . . . , yl ) = M in(h(x, f (⌊ 12 x⌋, y1 , . . . , yl ), y1 , . . . , yl ), k(x, y1 , . . . , yl )) Limited Recursion on Notation defines an algebra for all polynomial computable functions. This means that every polynomial computable functions can be defined by LRN and composition on initial functions. Later we use this fact to define the language of PV. The algebra of polynomial computable functions first studied by Cobham.[8] We state Cobham’s theorem in a sligthtly different way [6] rather than original one by Cobham and Cook [10] . 15 Definition 16. Cobham’s Class is the set of functions f ∶ Nl → N , such that for some Turing machine T and some polynomial p , T computes f (x1 , . . . , xl ) within p(∣x1 ∣ + . . . + ∣xl ∣) steps. Theorem 5. (Cobham’s theorem) Cobham’s Class is the least class of functions which include initial functions of Le and closed under operations LRN and Composition. Definition 17. Le is an extension of the language of Lb (see Definition 1) by the following new symbols: {Exp(a, b), , sq(a), ⟨a, b⟩, (a)1 , (a)2 } where Exp(a, b) = 2min{a,∣b∣} , is limited subtraction , sq(a) = a⋅a and ⟨a, b⟩, (a)1 , (a)2 are pairing function and its corresponding projections. Definition 18. Let ar(f ) denote arity of function symbol f and be defined in usual way. Then the language Lp can inductively be defined. For this the base clause is as follows: 1. Pil , which is the projection function (i.e. Pil (x1 , . . . , xl ) = xi ) and ar(Pil ) = l ) . 2. Zero function 0l , 0l (x1 , . . . , xl ) = 0 3. Le ⊆ Lp . And the inductive clause is : 1. Composition: If h, g1 . . . gk ∈ Lp , ar(h) = k and ar(g1 ) = . . . = ar(gk ) = l then : f ≡ Compk,l (h, g1 . . . , gk ) ∈ Lp and ar(Compk,l (h, g1 . . . , gk )) = l 2. LRN : If g, h, k ∈ Lp ,ar(g) = l , ar(h) = l + 2 then : f ≡ LRN l (g, h, k) ∈ Lp , ar(LRN l (g, h, k)) = l + 1 16 Notation:“≡” in the Definition 18 denotes defining function symbols in the meta language. ⎧ ⎪ ⎪y x = 0 Example:If Cond(x, y, z) = ⎨ then ⎪z x > 0 ⎪ ⎩ Cond ≡ LRN 3 (g, h, k) where g(y, z) = y (P12 ), h(x, w, y, z) = z(P44 ) and k(x, y, z) = y + z with this example we can easily defined following functions which we need them later: sg(x) = Cond(x, 0, s(0)) sg(x) = Cond(x, s(0), 0) Definition 19. By inductive definition we define terms in PV as following: 1. 0 is a term. 2. Every variable x and function symbol of arity 0 are terms. 3. If t1 , . . . , tk be terms then for function symbols f of arity k > 0, f (t1 , . . . , tk ) is a term. Definition 20. By inductive definition we define formulae in PV as following: 1. For terms t and u, t = u and t ≤ u are formulae. 2. If A and B are formulae then ¬A, A ⊃ B, A ∨ B, A ∧ B are formulae. 2.2 Axioms of PV Bounded Arithmetic is axiomatized by the finite set BASIC which is defined in [2][p 32]. Furthermore in this subsection we define theories corresponding to the languages in previous subsection . We will add new axioms to the BASIC to obtain the axiom scheme of PV which is defined by Lp . Earlier version of PV was a pure equational theory and had equality axioms only together with induction and substitution. Our PV also uses equality axioms as a part of its axiom scheme as well as new axioms which are introduced in the following definitions: 17 Definition 21. Equality axioms 1. t = t (Reflexivity) 2. s = t ⊃ t = s(Symmetry) 3. (s = t ∧ t = u) ⊃ s = u(Transitivity) 4. f ∈ Lp , ar(f ) = k (⋀ki=1 si = ti ) ⊃ f (s1 , . . . , sk ) = f (t1 , . . . , tk ) (Function compatibility) 5. in particular for the predicate symbol ≤ we have: s1 = t1 , s2 = t2 , t1 < t2 ⊃ s1 < s2 Definition 22. BASICe : The set of axioms BASICe can be obtained from the set of axiom BASIC by adding following new axioms: 1. ∣a∣ ≤ a 2. ∣a ⋅ b∣ ≤ ∣a∣ + ∣b∣ 3. Exp(a, 0) = 1 4. Exp(0, c) = 1 5. a + b ≤ ∣c∣ ⊃ Exp(a + b, c) = Exp(a, c) ⋅ Exp(b, c) 6. c ≠ 0 ⊃ (Exp(1, c) = 2 ∧ Exp(a, c) < 2 ⋅ c) 7. a ≤ b ↔ a b = 0 8. a b = 0 ↔ (b a) + a = b 9. sq(a) = a ⋅ a 10. ⟨(a, b⟩)1 = a 11. ⟨(a, b⟩)2 = b 12. ⟨(a)1 , (a)2 ⟩ = a 13. ∣⟨a, b⟩∣ ≤ 2 ⋅ (1 + ∣a∣ + ∣b∣) 18 14. ⟨a, b⟩ = ⌊ 12 ((a2 + b2 + 2a ⋅ b + a + 1) b))⌋ 15. t ≤ Exp(∣t∣, t) 16. ∣t∣ ≤ u ⊃ Exp(u, t) = Exp(∣t∣, t) Definition 23. BASICp is inductively defined as follows, the base clause is: BASICe ⊆ BASICp and the inductive clause is: 1. If f ∈ Lp and f ≡ Compk,l (h, g1 . . . , gk ) then: f (x, y1 , . . . , yl ) = h(g1 (y1 , . . . , yl ), . . . , gk (y1 , . . . , yl )) is in BASICp 2. If f ∈ Lp and f ≡ LRN l (g, h, k) then: f (0, y) = M in(g(y), k(0, y)) and x > 0 ⊃ f (x, y1 , . . . , yl ) = M in(h(x, f (⌊ 12 x⌋, y1 , . . . , yl ), y1 , . . . , yl ), k(x, y1 , . . . , yl )) are in BASICp 2.3 Proofs in PV In this section we describe structure of proofs in PV then we define P V − , and PV precisely. 2.3.1 Structure of Proofs in PV We use Gentzen sequent calculus PK[5] to define proofs in PV and its fragments. Each line of proofs in this system has expresions which are called “Sequents”. Eeach sequent consists of P V -formulae, A1 , . . . Ak and B1 , . . . Bl and sequent arrow “→”: A i , . . . , A k →B 1 , . . . B l The meaning of such sequent can be interpreted by the following formula: k l i=1 j=1 ⋀ Ai ⊃ ⋁ Bj Every proof in this system is a tree which every node represents a sequent. Every leaf in this tree has a sequent of the form →A where A is an axiom in 19 BASICp or logical axiom of the form A→A where A is a arbitrary atomic formula. Sequents can be derived by applying following inference rules to their antecedent sequents. Finally the root in the proof’s tree represents the endsequent of proof. Rules of inference in the sequent calculus PK are classified to the three groups: Weak Structural Rules, Cut Rule and Propositional Rules and illustrated by sequents as follows:(Γ, ∆ are sequences of formulae and A, B are PV-formulae) • Weak Structural Rules: (1) Exchange:left (2) Exchange:right 20 Γ, A, B, Π→∆ Γ, B, A, Π→∆ Γ→∆, A, B, Λ Γ→∆, B, A, Λ (3) Contraction:left (4) Contraction:right (5) Weakening:left A, A, Γ→∆ A, Γ→∆ Γ→∆, A, A Γ→∆, A Γ→∆ A, Γ→∆ (6) Weakening:right Γ→∆ Γ→∆, A • (7) The Cut Rule Γ→∆, A A, Γ→∆ Γ→∆ • The Propositional Rules: (8) ¬:left Γ→∆, A ¬A, Γ→∆ (9) ¬:right A, Γ→∆ Γ→∆, ¬A (10) ∧ ∶ left A, B, Γ→∆ A ∧ B, Γ→∆ (11) ∧ ∶ right (12) ∨ ∶ left Γ→∆, A Γ→∆, B Γ→∆, A ∧ B A, Γ→∆ B, Γ→∆ A ∨ B, Γ→∆ (13) ∨ ∶ right (14) ⊃∶ left Γ→∆, A, B Γ→∆, A ∨ B Γ→∆, A B, Γ→∆ A ⊃ B, Γ→∆ (15) ⊃∶ right 21 A, Γ→∆, B Γ→∆, A ⊃ B (16) Substitution (17) Induction Γ(a)→∆(a) Γ(t)→∆(t) Γ, A(b)→A(b + 1), ∆ → Γ, A(0)→A(t(Ð a )), ∆ In latter two cases t is an arbitrary term and b is an eigenvariable, that is a variable which must not appear in the lower sequent. Definition 24. Length of proof is the total number of occurrence of sequents in the proof. Definition 25. Size of proof P is the total number of symbols in P and is showed by ∣P ∣. 2.3.2 E −, P V −, P V In this subsection we define equational theories P V , P V − and E − . Every theory has a collection of language, set of axioms and set of inference rules. Proofs in these theories can be obtained by similar method in the subsection 2.3.1 w.r.t the suitable language, set of axioms and rules of inferences. Definition 26. E − is an equational theory which is determined by the language Le , set of axioms BASICe and equality axioms and rules of inference 1-16 (without induction rule) in the subsection 2.3.1 . Definition 27. P V − is an equational theory which is determined by the language Lp , set of axioms BASICp and equality axioms and rules of inference 1-16 (without induction rule) in the subsection 2.3.1 . Definition 28. P V is an equational theory which is determined by the language Lp , set of axioms BASICp and equality axioms and rules of inference 1-17 in the subsection 2.3.1 . Let B be a Lp -formula , we write P V ⊢ B if 22 →B is provable in P V . 2.3.3 Interpretation of formulae as terms In this section we explain how to interpret formulae as terms in P V − . This enables us to define functions with several conditions in P V by using P V formulae. We will use this kind of functions in Chapter 4 to speed-up induction in P V − . To this end first we define following P V functions:(for definition of sg(x) and sg(x) see example in section 2.1) • N OT (x) = 1 x • AN D(x, y) = M in(x, y) • M ax(a, b) = b + (a b) • OR(x, y) = M ax(x, y) ⎧ ⎪ ⎪1 • sg(x) = ⎨ ⎪ ⎪ ⎩0 ⎧ ⎪ ⎪1 • sg(x) = ⎨ ⎪ ⎪ ⎩0 x ≠ 0, x=0 x = 0, x≠0 Definition 29. We inductively define for each P V − f ormula, φ it’s interpretation φI : 1. if φ ≡ a ≤ b, then φI ≡ sg(a b). 2. if φ ≡ a = b, then φI ≡ AN D((a ≤ b)I , (b ≤ a)I ). 3. if φ ≡ ¬ψ, then φI ≡ N OT (ψ I ). 4. if φ ≡ ψ0 ∧ ψ1 , then φI ≡ AN D(ψ0I , ψ1I ). 5. if φ ≡ ψ0 ∨ ψ1 , then φI ≡ OR(ψ0I , ψ1I ) It is easy to check that if Lemma 1. Let φ be a P V − f ormula then: P V − ⊢ φ ↔ φI = 1 23 Proof. This lemma can be proved by induction on the structure of formula φ. For instance we prove the lemma for atomic formula a ≤ b and for the conjunction of two formulae. For atomic case φ ≡ a ≤ b we have from Axiom 7 of BASICe that: PV − ⊢ a ≤ b ↔ a b = 0 so from definition of sg we have: P V − ⊢ a ≤ b ↔ sg(a b) = 1 thus we proved that P V − ⊢ φ ↔ φI = 1. If φ ≡ ψ0 ∧ ψ1 then induction hypothesis implies: P V − ⊢ ψ0 ↔ ψ0I = 1 P V − ⊢ ψ1 ↔ ψ1I = 1 So we get the proof of P V − ⊢ ψ0 , ψ1 →M in(ψ0I , ψ1I ) = 1 P V − ⊢ ψ0 ∧ ψ1 →AN D(ψ0I , ψ1I ) = 1 This finishes the proof for conjunction of formulae. Above discussion helps us to define P V function with several conditions w.r.t P V formulae: Example: Let A be a PV-formula and let: Choice(A, x, y) ≡ x.sg(AI ) + y.sg(AI ) In particular from the Lemma 1 we have P V − ⊢ A→AI = 1 so: P V − ⊢ A→sg(AI ) = 1 P V − ⊢ A→sg(AI ) = 0 with the same argument for ¬A we obtain: P V − ⊢ ¬A→sg(AI ) = 0 P V − ⊢ ¬A→sg(AI ) = 1 Consequently we get the proofs of following sequents: 24 1. P V − ⊢ A→Choice(A, x, y) = x 2. P V − ⊢ ¬A→Choice(A, x, y) = y For example for f (x) such that : ⎧ ⎪ ⎪g(x) if A, f (x) = ⎨ ⎪ ⎪ ⎩h(x) if ¬A f is defined by f (x) = Choice(A, g(x), h(x))andf (x) ∈ Lp . Specifically we have following sequents P V − ⊢ A→f (x) = g(x) P V − ⊢ ¬A→f (x) = h(x) For another example if functions g1 (x), g2 (x), g3 (x) ∈ Lp and formulae A, B ∈ P V then for τ (x) such that: ⎧ g1 (x) if A, ⎪ ⎪ ⎪ ⎪ τ (x) = ⎨g2 (x) if ¬A ∧ B, ⎪ ⎪ ⎪ ⎪ ⎩g3 (x) if ¬A ∧ ¬B we have τ (x) ∈ Lp such that τ (x) = Choice(A, g1 (x), Choice(B, g2 (x), g3 (x))) . Also we get P V − ⊢ A→τ (x) = g1 (x) P V − ⊢ ¬A ∧ B →τ (x) = g2 (x) P V − ⊢ ¬A ∧ ¬B →τ (x) = g3 (x) 2.4 Arithmetisation of PV In order to study consistency of P V and P V − we need to arithmetize metamathematical concepts like proofs and formulas. In this section we describe a coding method for the syntax of PV by assigning Gödel number to symbols in Lp . We explain how to code terms, formulae and proofs in P V to Gödel numbers. Finally we will define provability predicate for P V . 25 2.4.1 Gödelisation of Lp Definition 30. (coding of sequence)To assign Gödel numbers to a sequence of natural numbers we use following method [2]: For the sequence a1 , . . . , an we reverse the order of binary representation of a1 , . . . , an which consists of 0,1 and commas. Then we replace each 0 by “10” and each 1 by “11” and each comma by “00”. Latter string of ones and zeros is binary representation of Gödel number of ⟨a1 . . . , an ⟩. The Gödel number of empty sequence ⟨⟩ = 0 First we assign to each symbol of PV a Gödel number as follows: • Logical Symbols: ¬−0 ⊃−1 ∧−2 ∨−3 (−4 )−5 ,−6 →−7 • Function Symbols: 0−9 S−13 1 ⋅−17 ⌊ 2 a⌋−19 ≤−23 Exp(a, b)−25 sq(a)−29 ⟨a, b⟩−31 (a)2 −35 +−15 ♯−21 −27 (a)1 −33 • Polynomial Computable Functions: As defined before, the language of P V , Lp , has symbols for every polynomial time computable functions. For assigning Gödel number to them we distinguish following cases: If f ≡ Compk,l (h, g1 . . . , gk ) then Gödel number of f is: ⌜f ⌝ = ⟨⟨0, k, l⟩, ⌜h⌝, ⌜g1 ⌝, . . . ⌜gk ⌝⟩ × 2 If f ≡ LRN l (g, h, k) then Gödel number of f is: ⌜f ⌝ = ⟨⟨1, l⟩, ⌜g⌝, ⌜h⌝, ⌜k⌝⟩ × 2 Free Variables: x1 - 37 x2 - 41 x3 - 45 ⋮ 26 Thus for every sequent “Γ→∆” of the form Ai , . . . , Ak →B1 , . . . , Bl we have: ⌜Γ→∆⌝ = ⌜⟨Ai , . . . , Ak →B1 , . . . , Bl ⟩⌝ Also for formula A, l(A) denotes ∣⌜A⌝∣ and for term t, l(t) denotes ∣⌜t⌝∣. 2.4.2 Gödelisation of Proofs As explained in 2.3.1, every proof in P V is a tree where each node corresponds to a sequent like Γ→∆. In order to arithmetize P V − proof s, first we assign to each node Γ→∆ a pair ⟨x, w⟩ where w is the Gödel number of that sequent which is computed according to the last subsection. x shows the number of inference rule which is applied to the immediate upper sequents of Γ→∆. For leaves of proof’s tree where it is a logical axiom of the form A→A or axioms from BASICp or equality axioms of the form →A, x is equal to zero. So far, we have assigned to every node in proof tree a number that codes inference details as well as sequent, now we can easily map every tree to a string by depth first order [2] with the aid of two symbols “[”, “]”, which denote moving down or up in a tree respectively. This can be shown by the following example: a b g c h e The above tree can be map into the following sequence: a[b[gh]c[e]] By assigning Gödel numbers to “[” and “]”, the Gödel number of above tree is equal to the code of following sequence: ⟨⌜a⌝, ⌜[⌝, ⌜b⌝, ⌜[⌝, ⌜g⌝, ⌜h⌝, ⌜]⌝, ⌜c⌝, ⌜[⌝, ⌜e⌝, ⌜]⌝, ⌜]⌝⟩ . Note: For xn = 16 and xn = 17, at every occurrence of substitution rule or induction rule we have to check that whether the substitution or induction are applied to the right variables or not? This can be done by introducing two polynomial computable PV-functions which get the number of sequent Γ→∆ as an input and gives all of occurring variables and terms in the sequent: 27 • V ar(⌜Γ→∆⌝) = {a1 , . . . , an } • T erm(⌜Γ→∆⌝) = {t1 , . . . tn } By above functions and Gödelisation method, verifying that substitution of variables or induction on variables is leading to the lower sequent with expected terms, can be done in polynomial time and that guarantees that substitution and induction are done in a proper way. Notation: For proof P , l(P ) denotes Gödel number of P : l(P ) = ∣⌜P ⌝∣. 2.4.3 Provability Predicate for PV Generally for a proof system, T, provability predicate can be defined as following: Definition 31. The provability predicate for a theory T can be defined as follows: P rovT (x) ≡ ∃yP rfT (y, x) which P rfT (y, x) is proof predicate and it is true when y is the Gödel number of a T − proof of formula with the Gödel number x. Definition 32. For a proof system T , Con(T ), consistency predicate is defined as follows: Con(T ) ≡ ¬P rovT (⌜0 = 1⌝) So by above definition of P rovT (x) we have : Con(P V ) ≡ ∀y¬P rfT (y, ⌜0 = 1⌝) Because our definition of PV is quantifier free, thus we drop ∀ from above formula and define consistency predicate for PV as follows: Con(P V ) ≡ ¬P rfP V (y, ⌜0 = 1⌝) 28 Chapter 3 Examples of upper bounds for the length of proofs in P V − In this chapter we study how to obtain upper bounds for terms in P V − such that the length of proving those upper bounds in P V − be polynomially bounded. We give proof of results in [2] concerning the length of P V − -proofs in detail and we will dicuss about the complexity of such proofs. Definition 33. Terms sq i (x) can be defined inductively as follows: 1. sq 0 (x) = x 2. sq k+1 (x) = sq(sq k (x)) Regarding the above definition we have following theorem: Lemma 2. BASICe ⊢ x ≤ y →sq i (x) ≤ sq i (y) for 0 ≤ i with a proof of length linear in i. Proof. From the set of axioms BASICe we can easily show that: x ≤ y →sq(x) ≤ sq(y) Applying i times substitution in (3.1) yields proof of Lemma 2. Theorem 6. E − ⊢ x ≤ sq m (y), y ≤ sq k (z)→x ≤ sq k+m (z) with a proof of length linear in m. 29 (3.1) Proof. By Lemma 2 we know a ≤ b→sq m (a) ≤ sq m (b) , applying substitution rules two times such that: a ↦ y and b ↦ sq k (z) gives: y ≤ sq k (z)→sq m (y) ≤ sq k+m (z). (3.2) Using the Axiom 8 of BASIC gives: x ≤ sq m (y), sq m (y) ≤ sq k+m (z)→x ≤ sq k+m (z) (3.3) finally applying Cut to the (3.2) and (3.3) gives the proof of: x ≤ sq m (y), y ≤ sq k (z)→x ≤ sq k+m (z) The length of proof in Theorem 6 is linear in m since we only used Lemma 2 and constant number of substitutions and cuts. Theorem 7. Let t(a1 , . . . , ak ) be an arbitrary Lp − term and c be a variable. Then P V − proves: ⋀(∣ai ∣ ≤ c), (1 < c)→∣t( a)∣ ≤ sq l(t) (c) → i≤k with a proof of length quadratic in l(t) We prove this theorem by induction on the complexity of terms in P V − . In the Lemma 4 we will prove above theorem for function symbols belong to the P V − . Later the general case for arbitrary terms will be discussed. Lemma 3. There is a constant 0 < B such that for every 0 < r, P V − proves: →c ≤ sqr (c) (3.4) With a proof of length bounded in B ⋅ l(r) Proof of Lemma 3. Proof is by induction on k, from axioms we have P V − ⊢ →c ≤ sq(c) by substitution sq r (c) to c in that we obtain P V − ⊢ →sq r (c) ≤ sq r+1 (c) 30 (3.5) From Axiom (8) of BASIC we have c ≤ sq r (c), sq r (c) ≤ sq r+1 (c)→c ≤ sq r+1 (c) (3.6) Also induction hypothesis implies : P V − ⊢ →c ≤ sq r (c) (3.7) with a proof of length bounded in B ⋅ l(r). So by applying cut (also weakening) to the (3.6) and (3.5), (3.7), respectively, we obtain: P V − ⊢ →c ≤ sq k+1 (c) (3.8) with a proof of length bounded in B ⋅ l(r). Lemma 4. There is a constant 0 < A such that for every function symbols f ∈ Lp , P V − proves: ⋀(∣ai ∣ ≤ c), (1 < c)→∣f ( a)∣ ≤ sq l(f ) (c) → i≤k with a proof of length bounded in A ⋅ l(f )2 . Proof of Lemma 4. We prove Lemma 4 by induction on the complexity of the definition of f . For this we have following cases: 1. If f is defined by LRN . 2. If f is defined by composition. → For the Case 1, f (x, Ð y ) is defined by limited recursion on notation (LRN) → → → from g(Ð y ) and h(x, z, Ð y ) and k(x, Ð y ). To prove this case, by induction hypothesis we have: P V − ⊢ ⋀(∣ai ∣ ≤ c), (1 < c)→∣g( a)∣ ≤ sq l(g) (c) → (3.9) i≤k P V − ⊢ ⋀ (∣ai ∣ ≤ c), (1 < c)→∣h( a)∣ ≤ sq l(h) (c) → (3.10) i≤k+2 P V − ⊢ ⋀ (∣ai ∣ ≤ c), (1 < c)→∣k( a)∣ ≤ sq l(k) (c) → i≤k+1 31 (3.11) With proofs of length bounded in A⋅l(g)2 , A⋅l(h)2 and A⋅l(k)2 , respectively. By properties of M in (as described in Chapter 2) and axioms of P V concerning LRN and equality axioms we have: → → P V − ⊢ →∣f (Ð a )∣ ≤ ∣k(Ð a )∣with a “constant” length proof. (3.12) By Lemma 3 we have : P V − ⊢ →c ≤ sq l(f )−l(k) (c) with a proof of length bounded in B ⋅(l(f )−l(k)). Applying one substitution c ↦ sq l(k) (c) gives P V − ⊢ →sq l(k) (c) ≤ sq l(f ) (c) (3.13) with a proof length bounded in B ⋅ (l(f ) − l(k))(our coding guarantees that l(k) ≤ l(f ). By using Axiom (8)of BASIC we have following derivations : → → → → P V − ⊢ ∣f (Ð a )∣ ≤ ∣k(Ð a )∣, ∣k(Ð a )∣ ≤ sq l(k) (c)→∣f (Ð a )∣ ≤ sq l(k) (c) (3.14) → → P V − ⊢ ∣f (Ð a )∣ ≤ sq l(k) (c), sq l(k) (c) ≤ sq l(f ) (c)→∣f (Ð a )∣ ≤ sq l(f ) (c) (3.15) Now by applying cuts and weakening as before, to the (3.15) and (3.14), (3.13), induction hypothesis (3.11) also (3.12), respectively, we get the final derivation in the Lemma 4. Complexity Analysis: In order to measure the size of above proof first we count the number of sequents in the derivation: by using induction hypothesis (3.11) the number of sequents will be quadratic in l(k) namely A ⋅ l(k)2 derivation (3.13) increases the length of proof B ⋅ (l(f ) − l(k)) and for other derivations from axioms as well as using Cut rule and Weakenings we increase the length of proof constantly by C. Thus, since we have l(k) ≤ l(f ) and above analysis, the constants B and C are independent of l(f ) so we have: A ⋅ l(k)2 + B ⋅ (l(f ) − l(k)) + C ≤ A ⋅ l(f )2 This shows that the length of proof is quadratic in l(f ).◻ → → → → For the Case 2, f (x, Ð y ) is defined by “Composition” from h(Ð z ) and gi (Ð y ), . . . , gk (Ð y ). From induction hypothesis we have : P V − ⊢ ⋀(∣ai ∣ ≤ c), (1 < c)→∣g1 ( a)∣ ≤ sq l(g1 ) (c) → i≤k 32 ⋮ P V − ⊢ ⋀(∣ai ∣ ≤ c), (1 < c)→∣gk ( a)∣ ≤ sq l(gk ) (c) → i≤k With proofs of length bounded in A ⋅ l(g)2 , . . . , A ⋅ l(gk )2 , respectively. → P V − ⊢ ⋀(∣bi ∣ ≤ c), (1 < c)→∣h( b )∣ ≤ sq l(h) (c) (3.16) i≤k with a proof of length bounded in A ⋅ l(h)2 . → From induction hypothesis (3.16), applying k substitutions bi ↦ gi ( a) and the substitution c ↦ sq m (c) where m = max {l(gi ) ∣i ≤ k} yields: P V − ⊢ ⋀(∣gi ( a ∣) ≤ sq m (c)), (1 < sq m (c))→∣h(g1 ( a), . . . , gk ( a)∣ ≤ sq l(h)+m (c) → → → i≤k (3.17) Lemma 3 implies following derivations: P V − ⊢ →1 ≤ sq m (c) (3.18) And proves for each i we have: P V − ⊢ c ≤ sq m−l(gi ) (c) so we get P V − ⊢ →sq l(gi ) (c) ≤ sq m (c) (3.19) Using this and induction hypothesis for gi and Axiom (8) of BASIC gives: P V − ⊢ ⋀(∣ai ∣ ≤ c), (1 < c)→∣g1 ( a)∣ ≤ sq m (c) → i≤k ⋮ P V − ⊢ ⋀(∣ai ∣ ≤ c), (1 < c)→∣gk ( a)∣ ≤ sq m (c) → (3.20) i≤k Now cuts of (3.17) to every derivation in (3.20) and (3.18) yields: P V − ⊢ ⋀(∣ai ∣ ≤ c), (1 < c)→∣h(gi ( a))∣ ≤ sq l(h)+m (c) → i≤k From axioms for composition of function we have → → → f (Ð a ) = h(g1 ( a), . . . , gk ( a) 33 (3.21) using this, (3.22)and Compatibility axiom concerning ∣x∣ and ≤ gives: P V − ⊢ ⋀(∣ai ∣ ≤ c), (1 < c)→∣f ( a)∣ ≤ sq l(h)+m (c) → (3.22) i≤k Again by using Lemma 3 and considering l(f ) > l(h) + m according to our coding in the case of composition, we will derive: P V − ⊢ →sq l(h)+m (c) ≤ sq l(f ) (c) (3.23) With similar technique as before by using Axiom (8)of BASIC and applying cuts, finally we obtain final derivation in the Lemma 4. Complexity Analysis: At (3.17) the length of proof is A ⋅ l(h)2 (by induction hypothesis and using substitutions) for occurrence of cuts, weak structural rules and derivations from axioms we consider constant C. Each derivation in (3.20) has the complexity: A ⋅ l(gi )2 + B.(m − l(gi )) so that part of derivation increase the length by A ⋅ m2 (because of applying k cuts). Using (3.23) for the final derivation increase the length of proof by B ⋅ (l(f ) − (m + l(h))). Thus for the whole proof we have: A⋅l(h)2 +A⋅m2 +B ⋅(l(f )−(m+l(h)))+C. This shows the constants “C” and “B” can be chosen independent of l(f ) consequently the whole derivation has the length A ⋅ l(f )2 . Proof of Theorem 7. For arbitrary term t which is defined by composition of function f and sub terms t1 , . . . , tk : t = f (t1 , . . . , tk ) Since our coding implies l(f ) + l(t1 ) + . . . + l(tk ) ≤ l(t), by similar way to the Lemma 4, in the case of composition, we get the proof of Theorem 7. So far it is shown that Theorem 7 has a proof of length quadratic in l(t), namely A ⋅ l(t)2 . We observe by inspecting the proof of Theorem 7 that each sequent appears in the proof has the size linear in l(t) e.g. A ⋅ l(t). Consequently the size of proof of Theorem 7 is bounded by A ⋅ l(t)3 (cubic in l(t)). In the following we discuss proof of upper bounds for triple length of terms. [6] proves Theorem 8 by using Lemma 5 which is involved with squaring function. Because we do not follow this way and we will prove Theorem 8 instead of using squaring function, we only explain proof of Theorem 8 in [6] brief. 34 Lemma 5. [6] Let ∣∣x∣∣ denotes double length of x (binary length of binary length of x), for all n ∈ N, P V − can prove: ∣∣sq n (x)∣∣ ≤ n + ∣∣x∣∣ with a proof of length linear in l(t). Theorem 8. P V − ⊢ ⋀(∣ai ∣ ≤ c), (1 < c)→∣∣∣t( a)∣∣∣ ≤ l(t) + ∣∣c∣∣ → i≤k with a proof of length quadratic in l(t). Proof. : From previous lemma we have in P V − : →∣∣sql(t)(c) ≤ l(t) + ∣∣c∣∣ (3.24) and its proof has the length quadratic in l(t). Also from axiom (12) of BASIC we have following derivations in P V − : ∣t( a)∣ ≤ sq l(t) (c)→∣∣t( a)∣∣ ≤ ∣sq l(t) (c)∣ → → ∣∣t( a)∣∣ ≤ ∣sq l(t) (c)∣→∣∣∣t( a)∣∣∣ ≤ ∣∣sq l(t) (c)∣∣ → → (3.25) So together with (3.25), (3.26) we get in P V − the proof of: ∣t( a)∣ ≤ sq l(t) (c)→∣∣∣t( a)∣∣∣ ≤ l(t) + ∣∣c∣∣ → → (3.26) Applying cut to (3.27) and end sequent in the Theorem 7 gives the proof of Theorem 8. Complexity Analysis: The proof of theorem 8 consists of two parts one from theorem 7 another from Lemma 5 and each parts has the length quadratic in l(t) so the proof of theorem 8 is of the length quadratic in l(t). In Theorem 9 we will prove upper bounds for triple length of terms different than Theorem 8. Our proof involves successor function which is recursively defined in the following definition. Definition 34. We define k-th successor of x, S k (x) and k-th length of term t, ∣t∣k by recursion on k: 35 ⎧ ⎪ ⎪S 0 (x) = x, • ⎨ k+1 k ⎪ ⎪ ⎩S (x) = S(S (x)) • ∣t∣0 = t • ∣t∣k+1 = ∣∣t∣k ∣ → Theorem 9. Let t( a) be a P V term and 0 < A be a constant then: P V − ⊢ ⋀(∣ai ∣3 ≤ c), (1 < c)→∣t( a)∣3 ≤ S l(t) (c) → i≤k with a proof of length bounded by A ⋅ (l(t))2 Proof. Like Theorem 7 the proof of this theorem is by induction on the complexity of terms. First we prove it for function symbols. If f is defined from g, h and k by LRN, from definition of LRN and Compatibility axiom for ∣t∣3 and ≤ we have following derivations:(we omit variables of function for the sake of simplicity) P V − ⊢ →∣f ∣ ≤ ∣k∣ P V − ⊢ →∣f ∣3 ≤ ∣k∣3 (3.27) Induction hypothesis in this case implies: P V − ⊢ →∣k∣3 ≤ S l(k) (c) This and (3.27) show that: P V − ⊢ →∣f ∣3 ≤ S l(k) (c) (3.28) Similar to the Lemma 3, for constant 0 < A, P V − proves →c ≤ S l(f )−l(k)(c) with a proof of length bounded by A ⋅ (l(f ) − l(k)) Applying one substitution gives: P V − ⊢ →S l(k) (c) ≤ S l(f ) (c), where our coding shows l(k) ≤ l(f ) From above derivations we can easily get the proof of P V − ⊢ →∣f ∣3 ≤ S l(f ) (c) 36 (3.29) The complexity analysis is similar to the Theorem 7 in particular derivation (3.29) has a proof length bounded by A⋅(l(f )−l(k)). Also from induction hypothesis, (3.28) has a proof of length bounded by A ⋅ (l(k)2 ). Thus the length of proof of Theorem 9 is bounded by A ⋅ (l(f )2 ) If f is defined by composition from h and g1 , . . . , gk then we have following derivations: → P V − ⊢ ⋀ ∣gi ∣3 ≤ c→∣h(gi )∣3 ≤ S l(h)+m (c) i≤k applying cuts to this and induction hypothesis for every gi gives the proof of: P V − ⊢ ⋀ ∣ai ∣3 ≤ c→∣h(gi , . . . , gk )∣3 ≤ S l(h)+m (c) (3.30) i≤k With axiom for composition and also the fact that l(h) + m ≤ l(f ) as well as (3.30) we get the proof of: P V − ⊢ ⋀ ∣ai ∣3 →∣f ( a)∣3 ≤ S l(f ) (c) → i≤k The estimation of proof’ length is similar to the Theorem 7. We conclude that Theorem 9 has a proof of length bounded by A ⋅ (l(t))2 . 37 Chapter 4 Speed-up of induction for P V − As defined in chapter 2 induction is given schematically in P V by : (17) Induction Γ, A0 (b)→A0 (b + 1), ∆ → Γ, A0 (0)→A0 (t(Ð a )), ∆ where A0 (x) is an open formula and t is an arbitrary term with eigenvariable b. Thus in order to do full induction we need following implications: ((A0 (0) ⊃ A0 (1))∧(A0 (1) ⊃ A0 (2))∧. . .∧(A0 (t−1) ⊃ A0 (t))) ⊃ (A0 (0) ⊃ A0 (t)) (with applying repeatedly Modes Pones). But in speed-up technique which will be explained later, we introduce a formula A∗0 (z) and shorten (c.f. Theorem 10) the number of steps up to the length of t (∣t∣): ((A∗0 (0) ⊃ A∗0 (1))∧(A∗0 (1) ⊃ A∗0 (2))∧. . .∧(A∗0 (∣t∣−1) ⊃ A∗0 (∣t∣))) ⊃ (A0 (0) ⊃ A0 (t)) In P V − we do not have Induction rule. In order to simulate Induction feasibly, we use above technique which is called cut shortening [6] by the aid of polynomial time computable functions and applying bounded number of cuts. This is discussed in Section (4.2). 4.1 Speed-up of induction with quantifiers In this section we state results from [6] concerning the first order theory over P V − which is denoted by P V1− . 38 Definition 35. For an open P V − f ormula, A0 and P V − term, t we define A∗0 as follows: A∗0 (z) ≡ (∀y ≤ t)(∀y ′ ≤ t)(y ′ ≤ y ∧ (y ≤ y ′ + Exp(z, t)) ∧ A0 (y ′ ) ⊃ A0 (y) Theorem 10. For above definition of A∗ we have: 1. P V1− ⊢ (∀x < t)(A0 (x) ⊃ A0 (x + 1)) ⊃ A∗0 (0) 2. P V1− ⊢ A∗0 (z) ⊃ A∗0 (z + 1) 3. P V1− ⊢ A∗0 (∣t∣) ⊃ (A0 (0) ⊃ A0 (t)) Proof. To prove Theorem 10, (1) we argue informally in P V1− . By definition and axiom (4) of BASICe , A∗0 (0) is equivalent to (∀y ≤ t)(∀y ′ ≤ t)(y ′ ≤ y ∧ (y ≤ y ′ + 1) ∧ A0 (y ′ ) ⊃ A0 (y) For the critical case y = y ′ , (1) is trivial. For the critical case y = y ′ + 1, by using premise of implication in Theorem 10, (1) and axioms we finish the proof. We informally prove Theorem 10, (2) in P V1− by distinguishing following cases 39 For Case 1, By definition premise of implication in Theorem 10, (2) is equivalent to: (∀y ≤ t)(∀y ′ ≤ t)(y ′ ≤ y ∧ (y ≤ y ′ + Exp(z, t)) ∧ A0 (y ′ ) ⊃ A0 (y) Also axioms imply that Exp(z, t) ≤ Exp(z + 1, t). This and above formula gives the result. For Case 2, first we apply premise of implication in Theorem 10, (2) to the values y ′ and y ′ + Exp(z, t) this gives us: (y ′ ≤ y ′ +Exp(z, t)∧(y ′ +Exp(z, t) ≤ y ′ +Exp(z, t))∧A0 (y ′ ) ⊃ A0 (y ′ +Exp(z, t)) Secondly we apply the premise of implication in to the y ′ + Exp(z, t) and y ′ to get: (y ′ + Exp(z, t) ≤ y ∧ (y ≤ y ′ + Exp(z + 1, t)) ∧ A0 (y ′ + Exp(z, t)) ⊃ A0 (y) By combining above results we finish the proof of Theorem 10, (2). Instantiating premise of Theorem 10, (3) by 0, t for y ′ , y and using the axiom t ≤ Exp(∣t∣, t)(axiom (15) of BASICe ) gives: (0 ≤ t ∧ (t ≤ 0 + Exp(∣t∣, t)) ∧ A0 (0) ⊃ A0 (t) which implies (3). 4.2 Speed-up of induction for quantifier free equational theories Since we do not have quantifiers in P V − we have to develop a speed-up induction technique which is different from the one in the previous subsection. Theorem 11, Lemma 6 and Lemma 8 gives a method to speed-up induction in quantifier free equational theories, namely P V − by using polynomial time computable functions called Skolem functions. Theorem 11. Let a = ⟨y ′ , y⟩, A0 and t be a PV formula and a PV term also A′0 be defined as: A′0 (z, ⟨y ′ , y⟩) ≡ (y ≤ t) ∧ (y ′ ≤ y) ∧ (y ≤ y ′ + Exp(z, t)) ∧ A0 (y ′ ) ⊃ A0 (y) 40 then there is a polynomial-time computable function F (z, a) in P V − such that, for A′ as above, P V − proves: A′0 (z, F (z, a)) ⊃ A′0 (z + 1, F (z + 1, a)) (4.1) A′0 (∣t∣, F (∣t∣, a)) ←→ A′0 (∣t∣, a) (4.2) And Let Â(z, a) = A′ (z, F (z, a)) then we have: P V − ⊢ Â(z, a) ⊃ Â(z + 1, a) (4.3) P V − ⊢ (A0 ((F (0, a))1 ) ⊃ A0 ((F (0, a))1 + 1)) ⊃ Â(0, a) (4.4) P V − ⊢ (∣t∣ ≤ u) ∧ Â(u, ⟨0, t⟩) ⊃ (A0 (0) ⊃ A0 (t)) (4.5) It should be mentioned that the proof of Theorem 11 does not depend on the structure of formula A0 and we use it only schematically. Now we define a Skolem function τ (z, a) for the pair a = ⟨y ′ , y⟩ which enables us to eliminate quantifiers in the induction step. For this we prove Lemma 6 as follows: Lemma 6. Let A0 and t be a PV formula and a PV term A′0 be the formula as described in Theorem 11 and a = ⟨y ′ , y⟩, then there is a P V term τ such that: P V − ⊢ A′0 (z, τ (z, a)) ⊃ A′0 (z + 1, a) (4.6) Proof. We prove this lemma by direct construction of τ from A0 : ⎧ ⟨y ′ , y⟩ y ≤ y ′ + Exp(z, t), ⎪ ⎪ ⎪ ⎪ τ (z, a) = ⎨⟨y ′ + Exp(z, t), y⟩ y > y ′ + Exp(z, t), A0 (y ′ + Exp(z, t)) ⎪ ⎪ ⎪ ′ ′ ′ ⎪ ⎩⟨y, y + Exp(z + 1, t)⟩ y > y + Exp(z, t), ¬A0 (y + Exp(z, t)), Above construction shows that τ is 1. In P V because ⟨, ⟩ ∈ Lp and Exp ∈ Lp (see examples in section 2.3.3). 2. uniformly defined from A0 such that l(τ ) and the length of proof of lemma are linearly bounded by l(A0 ). We prove (4.6) by informal argument within P V . For this we distinguish three cases: 41 For the first case τ (z, ⟨y ′ , y⟩) = ⟨y ′ , y⟩ we must prove: A′0 (z, ⟨y ′ , y⟩)→A′0 (z + 1, ⟨y ′ , y⟩) And this is supported by axioms of BASICe since we have : y ≤ y ′ + Exp(z, t)→y ≤ y ′ + Exp(z + 1, t) For the case 2 we prove assertion (4.6) by contradiction. That is to show in PV − ¬A′0 (z + 1, a) ⊃ ¬A′0 (z, τ (z, a))(∗) In this case we have τ (z, a) = ⟨y ′ + Exp(z, t), y⟩. For proving (∗) we need to show A0 (y ′ ), ¬A0 (y)→A0 (y ′ + Exp(z, ∣t∣)) ∧ ¬A0 (y) which is true since by definition of τ , A0 (y ′ + Exp(z, ∣t∣)) is true. Similarly in the case 3 we prove (4.6) by contradiction. From definition of τ we know τ (z, a) = ⟨y ′ , y ′ + Exp(z, ∣t∣)⟩. So for proving (∗) we need to show A0 (y ′ ), ¬A0 (y)→A0 (y ′ ) ∧ ¬A0 (y ′ + Exp(z, t)) 42 which is true since from definition of τ in Case 3, ¬A0 (y ′ + Exp(z, t)) is true. Definition 36. Modified limited recursion on notation(MLRN) Let t be a PV term then f is defined by MLRN from g, h, k ∈ P V if: 1. f (0, y) = g(y) 2. f (u + 1, y) = h(u, f (u, y)) where u < ∣t∣ 3. f (u, y) = f (∣t∣, y) where ∣t∣ ≤ u 4. f (u, y) ≤ k(u, y) Lemma 7. Let t be a PV term and g, h, k ∈ P V and let f be defined by MLRN from g, h, k, then f is definable in P V such that the conditions of MLRN are provable in P V − Proof. To prove this lemma we define f from another function which is defined by LRN and is in Lp We define h∗ , g∗ , k∗ ∈ Lp as follows: g∗ (y) = g(y) u h∗ (u, z, y) = h(∣⌊ ⌋∣, z, y) 2 k∗ (u, y) = k(∣u∣, y) Let f∗ ∈ Lp be the function which is defined by LRN from h∗ , g∗ , k∗ . Then we define f (u, y): f∗ (Exp(u, t) 1, y) So f which is defined by MLRN can be defined by LRN. Also above definition of f shows that conditions 1-4 of MLRN is provable in P V − . For the first part of theorem we explicitly construct F (z, a). Let t∗ be a P V -term such that ⟨∣t∣, ∣t∣⟩ ≤ ∣t∗ ∣ now we define F∗ (z, a) by modified limited recursion on notation (MLRN) as following: 1. F∗ (0, ⟨y ′ , y⟩) = ⟨y ′ , y⟩ 2. F∗ (u, ⟨y ′ , y⟩) = τ (∣t∣ u, F∗ (u 1, ⟨y ′ , y⟩)) for 1 ≤ u ≤ ∣t∣ 3. F∗ (u, ⟨y ′ , y⟩) = F∗ (∣t∣, ⟨y ′ , y⟩) for ∣t∣ ≤ u 4. ∣F∗ (u, ⟨y ′ , y⟩)∣ ≤ ∣t∗ ∣ 43 Definition 37. F (z, a) ≡ F∗ (∣t∣ z, a) It is directly followed from Axiom 7 of BASIC that: F (∣t∣, a) = F∗ (0, a) = a Lemma 8. P V − ⊢ z < ∣t∣ ⊃ F (z, a) = τ (z, F (z + 1, a)) Proof. : We argue informally in P V − . First we observe from BASICe we have following properties of for b ≤ a: • (a b) 1 = a (b + 1) • a (a b) = b The definition of F (z, a) and above facts from BASICe show that for z ≤ ∣t∣ we have: F (z, a) = F∗ (∣t∣ z, a) = τ (∣t∣ (∣t∣ z), F∗ ((∣t∣ z) 1, a)) = = τ (z, F∗ (∣t∣ (z + 1), a)) = τ (z, F (z + 1, a)) . Proof of theorem 11. By substituting a in lemma 6 with F (z + 1, a) we have P V − ⊢ A′0 (z, τ (z, F (z + 1, a))) ⊃ A′0 (z + 1, F (z + 1, a)). This and Lemma 8 yield (4.1). (4.2) is immediately followed from the fact that F (∣t∣, a) = a. (4.3) is implied by rewriting of (4.1) with formula Â. In (4.4) we have Â(0, a) ≡ A′0 (0, F (0, a)). Let m′ = (F (0, a))1 and m = (F (0, a))2 then A′ (0, m′ , m) is equivalent to m ≤ t, m′ ≤ m ∧ m ≤ m′ + 1 ∧ A(m′ ) ⊃ A(m) Critical case m = m′ is trivial. Critical case m = m′ + 1 is implied by axioms and premise of (4.4). Finally for (4.5), definition of Â and axiom (16) of BASICe imply that: P V − ⊢ (∣t∣ ≤ u) ∧ Â(u, ⟨x′ , x⟩) ⊃ Â(∣t∣, ⟨x′ , x⟩) 44 On the other hand Â(∣t∣, ⟨0, t⟩) is defined as A′ (∣t∣, F (∣t∣, ⟨0, t⟩) which is equivalent to A′ (∣t∣, 0, t) By definition the latter formula means: (t ≤ t) ∧ (0 ≤ t) ∧ (t ≤ Exp(∣t∣, t)) ∧ A0 (0) ⊃ A0 (t) Thus we proved Â(∣t∣, ⟨0, t⟩) ⊃ (A0 (0) ⊃ A0 (t)) Note: The proofs of (4.3)-(4.5) only use A schematically with constant number of cuts and substitutions so the length of proofs are linearly bounded by l(A0 ). In [6] authors repeated arguments in Theorem 11 three times for ∣t∣, ∣∣t∣∣ and ∣∣∣t∣∣∣ in order to speed up induction. In the following we introduce new kind of induction, k-Induction which is more general and allows to argue about ∣t∣k . This method simplifies the proofs in [6]. Definition 38. For an open formula B(b) we define: k-Induction Γ, B(b)→B(b + 1), ∆ Γ, B(0)→B(∣s∣k ), ∆ P Vkr is a theory with the language Lp , sets of axioms BASICp and equality axioms, inference rules 1-16 and k-induction . Lemma 9. r P Vkr = P Vk+1 r That is a sequent is provable in P Vkr if and only if it is provable in P Vk+1 . Furthermore the length of proof only increases linearly. r r Proof. To prove Lemma 9 we show P Vkr ⊇ P Vk+1 and P Vkr ⊆ P Vk+1 . For the r first case we have to show that every P Vk+1 -proof can be transformed to a P Vkr -proof. We prove this by induction on the height of proofs considering various rules for the last inference. The only nontrivial case happens when r -proof we do (k +1)-Induction in the last inference. Assume in this case P Vk+1 has the following form. 45 P1 Γ, B(b) → B(b + 1), ∆ (k + 1)-Induction Γ, B(0) → B(∣s∣k+1 ), ∆ Induction hypothesis implies: P Vkr ⊢ Γ, B(b) → B(b + 1), ∆ Applying one k-induction using the fact that ∣s∣k+1 = ∣∣s∣∣k gives: P Vkr ⊢ Γ, B(0) → B(∣s∣k+1 ), ∆ r This finishes the proof of first case. In the case P Vkr ⊆ P Vk+1 we have to r r show that every P Vk -proof can be transformed into a P Vk+1 -proof, this can be done by induction on the height of proofs. Like previous case we consider different rules for the last inference. If the last inference be substitution then the P Vkr -proof is of the form P1 Γ(b) → ∆(b) Substitution Γ(t) → ∆(t) induction hypothesis implies : r P Vk+1 ⊢ Γ(b) → ∆(b) applying one substitution gives: r P Vk+1 ⊢ Γ(t) → ∆(t) Other strong inferences except k-induction are similar to this case. For the case of induction the structure of P Vkr − proof is P1 Γ, B(b)→B(b + 1), ∆ k-Induction Γ, B(0)→B(∣s∣k ), ∆ From induction hypothesis we have: r P Vk+1 ⊢ Γ, B(b)→B(b + 1), ∆ Let B̂ and F (z, a) be defined as Theorem 11 for formula B. Substitution of b with F (z, a)1 and inference rule for implication gives the proof of: r P Vk+1 ⊢ Γ→B(F (z, a)1 ) ⊃ B(F (z, a)1 + 1), ∆ 46 On the other hand from Theorem 11, (4.4) we have P V − ⊢ (B((F (0, a))1 ) ⊃ B((F (0, a))1 + 1)) ⊃ B̂(0, a) By applying cut to this and above sequent we get r P Vk+1 ⊢ Γ→B̂(0, a), ∆ (4.7) Theorem 11, (4.3) implies that: r P Vk+1 ⊢ Γ, B̂(z, a)→B̂(z + 1, a), ∆ So by definition of k-induction we have the proof of: r P Vk+1 ⊢ Γ, B̂(0, a)→B̂(∣s∣k+1 , a), ∆ Applying cut to this and derivation in (4.7) yields: r P Vk+1 ⊢ Γ→B̂(∣∣s∣k ∣, a), ∆ By another cut to this and derivation in Theorem 11, (4.5)we get the proof of : r P Vk+1 ⊢ Γ, B(0)→B(∣s∣k ), ∆ The above proof is based on the Theorem 11 which used input formula B only schematically. In addition we used constant number of substitutions and cuts. This shows that the length of proof only increases linearly. Theorem 12. For a fixed k ≥ 0 there is a polynomial q such that for every P V -proof, P , of sequent Γ→∆ there is a P Vkr -proof, P ∗ , of Γ→∆ and ∣P ∗ ∣ ≤ q(∣P ∣) Proof. The proof is by induction on k. Base case is trivial: If k = 0 then P V0r = P V , by definition. Induction hypothesis says that there is a polynomial q such that for every P V -proof, P there is P Vkr -proof, P ∗ which proves the same end sequent as r . The size of P and also ∣P ∗ ∣ ≤ q(∣P ∣). By Lemma 9 we have P Vkr = P Vk+1 proof in the Lemma 9 is polynomial in ∣P ∣, this and induction hypothesis for q shows the whole proof has polynomially bounded size. Definition 39. A proof P is in “Variable Normal Form” (VNF) if for each cut occurring in P of the form: 47 Γ→∆, A A, Γ→∆ Γ→∆ we have: V ar(A) ⊆ V ar(Γ, ∆) . Lemma 10. For each P Vkr -proof, P , of sequent Γ→∆, there is a P Vkr -proof, P ∗ , in VNF of Γ→∆ such that ∣P ∗ ∣ is polynomial in ∣P ∣. Proof. The proof is by induction on the height of proofs. The only nontrivial case occurs when the last inference is an application of cut rule. In this case P has the form: P1 P2 Γ → ∆, A A, Γ → ∆ Cut Γ→∆ where V ar(A) ⊈ V ar(Γ, ∆). By induction hypothesis we have P Vkr -proofs P1∗ and P2∗ in VNF. Now we substitute each variable in V ar(A)∖V ar(Γ, ∆) by 0 to reach formula A′ which has the property V ar(A′ ) ⊆ V ar(Γ, ∆). Applying one cut to this gives us the final proof. This derivation is as follows: P1∗ P2∗ Γ → ∆, A A, Γ → ∆ Substitution ⋮ ⋮ ′ ′ Γ → ∆, A A ,Γ → ∆ Cut Γ→∆ Since at most we need ∣P ∣ substitutions, the size of ∣P ∗ ∣ is polynomially bounded by ∣P ∣. Definition 40. Γ→∆ has “numerically restricted variables ” if for every free variable aj , in Γ and ∆ there exists a corresponding formula of the form ∣aj ∣3 ≤ S nj (0), nj ∈ N occurring in Γ. Definition 41. For a proof P , si(P ) denotes the number of inferences different than W:l. 48 Theorem 13. There is a polynomial p such that every P V -proof , P , of sequent Γ→∆ with numerically restricted variables can be transformed to a P V − -proof, P ∗ , of Γ→∆ and ∣P ∗ ∣ ≤ p(∣P ∣) . Proof of Theorem 13. We prove Theorem 13 in the following steps: 1. Transforming every P V -proof, P into a P V2r -proof by Theorem 12. 2. Obtaining P V2r -proof of P in VNF by Lemma 10 3. Proving that every P V2r -proof, P in VNF of sequent Γ → ∆ with NRV can be transformed into a P V − -proof, P ∗ of Γ → ∆ such that si(P ∗ ) ≤ p(si(P )). We prove (3) by induction on si(P ). We distinguish cases according to the last inferences. We disuss weakening to the left later and we will finish the proof by complexity analysis. First We prove (3) if the last inference in the proof P of Γ → ∆ is : 1. Cut 2. Substitution 3. Induction Other cases are similar to these cases. Cut: The structure of proof in this case has the following form: P2 P1 Γ → ∆, A A, Γ → ∆ Cut Γ→∆ Induction hypothesis implies that for P V2r -proofs , P1 and P2 , there are P V − proofs, P1∗ and P2∗ with the same end sequents. Applying one cut gives us P V − -proof, P ∗ of Γ → ∆. Substitution: The structure of proof in the case of substitution is: (b is an eigenvariable) P1 Substitution Γ(b)→∆(b) Γ(t)→∆(t) 49 We define m = max {nj ∣j ≤ k − 1} (variables are numerically restricted)and s = l(t) + m for P V -term, t. By applying one W:l to P1 we obtain P V2r -proof of ∣b∣3 ≤ S s (0), Γ(b)→∆(b) We observe that induction hypothesis is applicable because W:l does not increase the si(P1 ) and also the last sequent has NRV. So induction hypothesis implies that there is a P V − -proof of P V − ⊢ ∣b∣3 ≤ S s (0), Γ(b)→∆(b) applying substitution to the above sequent gives: P V − ⊢ ∣t∣3 ≤ S s (0), Γ(t)→∆(t) Finally by doing cut to this and the result from Theorem 9, Chapter 3 which says: P V − ⊢ ⋀ ∣aj ∣3 ≤ S nj (0)→∣t∣3 ≤ S s (0) j≤k we get the proof of: P V − ⊢ Γ(t)→∆(t) Induction: If the last inference in the P V2r -proof of P is induction (restricted 2-induction, c.f.Definition 38) then the structure of proof will be: P1 Γ, A(b)→A(b + 1), ∆ 2-Induction Γ, A(0)→A(∣∣t∣∣), ∆ Let m = max {nj ∣j ≤ k − 1} (variables are numerically restricted)and s = l(t)+ m for P V -term, t. By applying one W:l to P1 we get P V2r ⊢ ∣b∣3 ≤ S s (0), Γ, A(b)→A(b + 1), ∆ (4.8) Now we can apply induction hypothesis since W:l does not change the si(P1 ) and the last end sequent has NRV. Thus we have P V − ⊢ ∣b∣3 ≤ S s (0), Γ, , b ≤ t, A(b)→A(b + 1)(∗∗) Theorem 9 in Chapter 3 implies P V − ⊢ Γ, ⋀ ∣aj ∣3 ≤ S nj (0)→∣t∣3 ≤ S s (0)(∗ ∗ ∗) j≤k 50 also from BASIC we have P V − ⊢ b ≤ t→∣b∣3 ≤ ∣t∣3 by using axiom (8) of BASIC and applying cuts to these sequents we get P V − ⊢ b ≤ t→∣b∣3 ≤ S s (0) Applying cut to this and derivation (∗∗) gives us P V − ⊢ Γ, b ≤ t, A(b)→A(b + 1), ∆ (4.9) From equality axioms we have b ≤ t→b ≤ t applying implication rule (left(14))gives: b ≤ t ⊃ A(b), Γ, b ≤ t→A(b + 1) with using cut to this and sequent b + 1 ≤ t→b ≤ t we get the proof of: b + 1 ≤ t, b ≤ t ⊃ A(b), Γ→A(b + 1), ∆ Using implication rule (right (15)) yields: Γ, b ≤ t ⊃ A(b)→b + 1 ≤ t ⊃ A(b + 1), ∆ Now we define A0 as follows: A0 (b) ≡ b ≤ t ⊃ A(b) thus by rewriting above sequent we have the proof of: P V − ⊢ Γ→∆, A0 (b) ⊃ A0 (b + 1) (4.10) Let u = ∣∣t∣∣ we define A′ , Â, F (z, a) w.r.t formula A0 like Theorem 11. Let a = ⟨0, u⟩, then substituting b in (4.10) with F (0, a)1 and applying cut to the (4.4) gives: P V − ⊢ Γ→Â(0, ⟨0, u⟩), ∆ (4.11) From (4.3) we obtain P V − ⊢ Â(z, ⟨0, u⟩)→Â(S(z), ⟨0, u⟩) 51 applying cut to this and (4.11), S s (0) many times gives us : Γ→Â(s, ⟨0, u⟩), ∆ (4.12) From Theorem 11 we have : P V − ⊢ (∣u∣ ≤ S s (0)) ∧ Â(S s (0), ⟨0, u⟩)→(A0 (0) ⊃ A0 (u)) By applying a cut to this and (4.12) and another cut to the (∗ ∗ ∗) we get the proof of: P V − ⊢ Γ→A0 (0) ⊃ A0 (∣∣t∣∣), ∆ . From definition of A0 and applying some inference rules we can replace formula A0 with A we get : P V − ⊢ Γ, A(0)→A(∣∣t∣∣), ∆ Weakening:left If the last inference in the P V2r -proofs of P be a weak structural rule then the si(P ) does not change, consequently we could not apply induction hypothesis. To solve this problem we should find the last strong inferences and apply induction hypothesis at that point. In the following we discuss a case which the last strong inference be the rule (12)∨ ∶ lef t of the form: A, Γ→∆ B, Γ→∆ A ∨ B, Γ→∆ And the last inference be a weakening to the left. Thus the structure of proof is : P1 P2 A, Γ → ∆ B, Γ → ∆ ∨ ∶ lef t A ∨ B, Γ → ∆ Weakening : left ⋮ Ð → C , A ∨ B, Γ → ∆ In order to apply induction hypothesis we need to obtain sequents in NRV thus first we apply weakenings to the end sequents of P1 and P2 so we have following subderivations: 52 P1 A, Γ → ∆ Weakening : left ⋮ Ð → C , A, Γ → ∆ P2 B, Γ → ∆ Weakening : left ⋮ Ð → C , B, Γ → ∆ By applying induction hypothesis to the above sub derivations we get their P V − − proof s, P1∗ and P2∗ using (12)∨ ∶ lef t and some exchange rule gives the proof of P ∗ , this derivation is as follows: P1∗ P2∗ Ð → C , A, Γ → ∆ Exchange : left Ð → A, C , Γ → ∆ ∨ ∶ lef t Ð → C , B, Γ → ∆ Exchange : left Ð → B, C , Γ → ∆ Ð → A ∨ B, C , Γ → ∆ Exchange : left Ð → C , A ∨ B, Γ → ∆ Complexity Analysis: For the second part of Theorem 13 concerning size of proofs, first we estimate the length of P V − -proof, P ∗ . We showed in the step 3 of proof of Theorem 13 by induction on si(P ) that there is a polynomial p such that every P V2r proof, P in VNF with NRV, can be transformed to a P V − -proof, P ∗ which proves the same end sequent such that si(P ∗ ) ≤ p(si(P )). As it is shown in the proof of Theorem 13 in the case of using cut, substitution and induction for the last inference, the length of proof only increases linearly and we did certain number of W:l for each occurrence of cut,substitution and induction which is bounded by C ⋅ ∣P ∣ for a constant C. So the whole proof consists of two parts one comes from P and another part consists of weakenings which are necessary to get NRV. Therefore the overall estimation of proof’s length will be length of P ∗ ≤ si(P ∗ ) ⋅ C ⋅ ∣P ∣ 53 Let bi and ti be eigenvariables and terms which are involved in every occurrance of substitution or induction. According to the premiss of Theorem 13 every parameter variable ai must satify the condition of NRV so they have the form ∣aj ∣3 ≤ S nj (0) Let m be maximum of nj . To ensure that the condition of NRV holds when we apply substitution or cut we used weakenings namely with fromulae of the form ∣bi ∣3 ≤ S ni (0) Where ni = m + l(ti ). Hence ni is bounded by sum of all l(ti ) and we have ni ≤ m + ∑ l(ti ) ≤ ∣P ∣ This shows the size of each sequent is bounded by C.∣P ∣2 for a constant C. Finally in contrast of above discussion for length of proof we conclude that the size of P ∗ is polynomial in ∣P ∣. 4.3 Provability of consistency of P V − In this section we want to show that the consistency of P V − is not provable in P V . For this first we formalise Theorem 13 in S21 (P V ) then by using conservativity of S21 (P V ) over P V we show in the Theorem 15 that P V ⊬ Con(P V − )The complete workout of theorem 14 and 15 is beyond the scope of this MRes thesis so we only sketch their proofs. Theorem 14. Theorem 13 can be formalised in S21 (P V ) in the following way: Let t(p) be a P V -term and N RV (s) be a predicate which is true when a sequent with the number s has numerical restricted variables. (this predicate is definable in P V ) Then S21 (P V ) proves ∀p∀s∃q(P rfP V (p, s) ∧ N RV (s) ⊃ q ≤ t(p) ∧ P rfP V − (q, s)) Proof. Proof of Theorem 13 is built by Theorem 11, Lemma 9 and Lemma 10. The proof of Lemma 11 does not involve induction and uses input formula only schematically. Formalisation of this theorem in S21 (P V ) is straightforward. In Lemma 9, induction is on the height of proofs. In order to formalize this lemma we follow these steps: 54 1. Let P be a P Vkr -proof and l be length of P , we form sequence of subproofs p1 . . . pl which are sorted according to the number of inferences which they use. r 2. we show by induction on j that there are P Vk+1 − proof s, qj , with size polynomial in the size of pj , and proves the same end sequent as pj . 3. The above construction is an application of Σb1 − LIN D which is applicable in S21 (P V ). Formalisation of Lemma 10 is similar to the Lemma 9. In the proof of Theorem 13 we observe that induction hypothesis is applied to modified subproofs(to ensure the condition of NRV is hold ) in other words we applied induction on h in the following formula: ∀p∀s(si(p) ≤ h ∧ P rfP V2r (p, s) ∧ V N F (p) ∧ N RV (s) ⊃ ∃qP rfP V − (q, s)) This induction is an application of Πb2 − LIN D which is not applicable in S21 (P V ) unless BA hierarchy collapses. One way to solve this problem is to construct a sequence of suitable P V2r -proofs (with VNF and NRV) such that the elements of that sequence can be transformed into P V − -proofs successively this can be done in the following steps: 1. Fix a P V2r -proof, P which has NRV and VNF. Let b1 . . . bl be the eigenvariables occurring in P and a1 . . . ak be parameter variables. Without loss of generality we assume b1 . . . bl have following property: “ If the inference which eliminates bi occurs earlier than inference which eliminates bj then i < j” Let ti be the term which bounds bi in its eliminating inference, e.g. Γ, A(bi ) → ∆, A(bi + 1) Γ, A(0) → ∆, A(ti ) thus we have a sequence of terms t1 . . . tl . 2. We fix numerical restrictions for b1 . . . bl in similar way to the proof of theorem 13. The end sequent of P contains formulae of the form ∣ai ∣3 ≤ S nj (0), which occurred in the anticedent of endsequent of P . Let m = max{n1 . . . nk }. Now for each bi we define numerical restriction as following formula : ∣bi ∣3 ≤ S m+l(ti )+⋯+l(tl ) (0) 55 3. Construct the list L1 of all subproofs of P which are not the premise of weakening left (W:l), including P . 4. For each p′ ∈ L1 we add necessary weakening to the left (W:l) with numerical restriction from (2.) in a canonical way such that the resulting proofs has NRV. Let L2 be the list of such proofs. 5. Let p1 , p2 , . . . be the list of proofs in L2 sorted according to the order of number of inferences different than (W:l) 6. By induction on the j it can be shown that there are P V − -proof, qj of size polynomial in the size of pj which has the same end sequent as pj . Apparently this is an application of Σb1 − LIN D which is available in S21 (P V ). In section 2.4 we introduced Con(P V ) as the formula ∀y¬P rfT (y, ⌜0 = 1⌝) The appropriate notion for P V is developed by Cook [10] as follows: ⎧ 1 If m is the number of an equation t = u, ⎪ ⎪ ⎪ ⎪ P roofP V (m, n)=⎨ and n is the number of a proof in P V of t = u ⎪ ⎪ ⎪ ⎪ ⎩0 Otherwise These two definition for proof predicate are equivalent in S21 (P V ). Theorem 15. P V ⊬ Con(P V − ) Proof. For the sake of contradiction suppose P V ⊢ Con(P V − ) that means: P V ⊢ P roofP V − (p, ⌜0 = 1⌝) = 0 S21 (P V ) ⊢ P rfP V − (p, ⌜0 = 1⌝) = 0 S21 (P V ) ⊢ ∀p¬P rfP V − (p, ⌜0 = 1⌝) Applying Theorem 14 we gives : S21 (P V ) ⊢ ∀p¬P rfP V (p, ⌜0 = 1⌝) = 0 56 and from conservativity of S21 (P V ) over P V [2] we get: P V ⊢ P roofP V (p, ⌜0 = 1⌝) = 0 which contradicts the result in [10] by Cook that P V ⊬ Con(P V ) 4.4 Future works In this thesis we have worked out the result P V ⊬ Con(P V − ) in detail. Our proof for this result does not involve with additional function symbol, Squaring function and its corresponding axiom. But still there are some other axioms which distinct P V − from pure equational theory, in particular in contrast of S2−∞ [1]. One way to approach this problem is to weaken P V − by dropping further axioms and still maintaining provability of consistency. This will include also carrying out speed-up techniques in pure equational theories. Another direction for future work is to strengthen techniques in [1] to show provability of consistency of pure equational theories which are only based on recursive definition of function symbols as axioms. 57 Chapter 5 Appendix BASIC 1. y ≤ x ⊃ y ≤ Sx 2. x ≠ Sx 3. 0 ≤ x 4. x ≤ y ∧ x ≠ y ↔ Sx ≤ y 5. x ≠ 0 ⊃ 2 ⋅ x ≠ 0 6. y ≤ x ∨ x ≤ y 7. x ≤ y ∧ y ≤ x ⊃ x = y 8. x ≤ y ∧ y ≤ z ⊃ x ≤ z 9. ∣0∣ = 0 10. x ≠ 0 ⊃ ∣2 ⋅ x∣ = S(∣x∣) ∧ ∣S(2 ⋅ x)∣ = S(∣x∣) 11. ∣S0∣ = 0 12. x ≤ y ⊃ ∣x∣ ≤ ∣y∣ 13. ∣x ♯ y∣ = S(∣x∣ ⋅ ∣y∣) 14. 0 ♯ y = S0 58 15. x ≠ 0 ⊃ 1 ♯(2 ⋅ x) = 2(1) ∧ 1 ♯(S(2 ⋅ x)) = 2(1) 16. x = y 17. ∣x∣ = ∣y∣ ⊃ x = y 18. ∣x∣ = ∣u∣ + ∣v∣ ⊃ x = (u ♯ y) ⋅ (v ♯ y) 19. x ≤ x + y 20. x ≤ y ∧ x ≠ y ⊃ S(2 ⋅ x) ≤ 2 ⋅ y ∧ S(2 ⋅ x) ≠ 2 ⋅ y 21. x + y = y + x 22. x + 0 = x 23. x + Sy = S(x + y) 24. (x + y) + z = x + (y + z) 25. x + y ≤ x + z ↔ y ≤ z 26. x ⋅ 0 = 0 27. x ⋅ (Sy) = (x.y) + x 28. x ⋅ y = y ⋅ x 29. x ⋅ (y + z) = (x ⋅ y) + (x ⋅ z) 30. x ≥ S0 ⊃ (x ⋅ y ≤ x ⋅ z ↔ y ≤ z) 31. x ≠ 0 ⊃ ∣x∣ = S(∣⌊ 12 x⌋∣) 32. x = ⌊ 12 y⌋ ↔ (2 ⋅ x = y ∨ S(2 ⋅ x) = y) 59 Bibliography [1] Arnold Beckmann. Proving consistency of equational theories in bounded arithmetic. J. Symbolic Logic, 67(1):279–296, 2002. [2] Samuel R. Buss. Bounded arithmetic, volume 3 of Studies in Proof Theory. Lecture Notes. Bibliopolis, Naples, 1986. [3] Samuel R. Buss. Relating the bounded arithmetic and polynomial time hierarchies. Ann. Pure Appl. Logic, 75(1-2):67–77, 1995. Proof theory, provability logic, and computation (Berne, 1994). [4] Samuel R. Buss. Bounded arithmetic and propositional proof complexity. In Logic of computation (Marktoberdorf, 1995), volume 157 of NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., pages 67–121. Springer, Berlin, 1997. [5] Samuel R. Buss, editor. Handbook of proof theory, volume 137 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1998. [6] Samuel R. Buss and Aleksandar Ignjatovic. Unprovability of consistency statements in fragments of bounded arithmetic. Ann. Pure Appl. Logic, 74(3):221–244, 1995. [7] Peter Clote and Jan Krajı́ček, editors. Arithmetic, proof theory, and computational complexity, volume 23 of Oxford Logic Guides. The Clarendon Press Oxford University Press, New York, 1993. Papers from the conference held in Prague, July 2–5, 1991, Oxford Science Publications. 60 [8] Alan Cobham. The intrinsic computational difficulty of functions. In Logic, Methodology and Philos. Sci. (Proc. 1964 Internat. Congr.), pages 24–30. North-Holland, Amsterdam, 1965. [9] Stephen A. Cook. The complexity of theorem-proving procedures. In STOC ’71: Proceedings of the third annual ACM symposium on Theory of computing, pages 151–158, New York, NY, USA, 1971. ACM. [10] Stephen A. Cook. Feasibly constructive proofs and the propositional calculus (preliminary version). In Seventh Annual ACM Symposium on Theory of Computing (Albuquerque, N.M., 1975), pages 83–97. Assoc. Comput. Mach., New York, 1975. [11] Jan Krajı́ček. Fragments of bounded arithmetic and bounded query classes. Trans. Amer. Math. Soc., 338(2):587–598, 1993. [12] Jan Krajı́ček, Pavel Pudlák, and Gaisi Takeuti. Bounded arithmetic and the polynomial hierarchy. Ann. Pure Appl. Logic, 52(1-2):143–153, 1991. International Symposium on Mathematical Logic and its Applications (Nagoya, 1988). [13] P. Pudlák. A note on bounded arithmetic. Fund. Math., 136(2):85–89, 1990. [14] Boaz Barak Sanjeev Arora. Complexity Theory: A Modern Approach. Cambridge University Press, 2009. [15] Larry J. Stockmeyer. The polynomial-time hierarchy. Theoret. Comput. Sci., 3(1):1–22 (1977), 1976. [16] Gaisi Takeuti. Some relations among systems for bounded arithmetic. In Mathematical logic, pages 139–154. Plenum, New York, 1990. [17] Domenico Zambella. Notes on polynomially bounded arithmetic. J. Symbolic Logic, 61(3):942–966, 1996. 61