Characterising Complexity Classes by Inductive Definitions in Bounded Arithmetic
aa r X i v : . [ m a t h . L O ] J a n Characterising Complexity Classes by InductiveDefinitions in Bounded Arithmetic ⋆ Naohi Eguchi ⋆⋆ Institute of Computer Science, University of Innsbruck, Austria [email protected]
Abstract.
Famous descriptive characterisations of P and PSPACE arerestated in terms of the Cook-Nguyen style second order bounded arith-metic. We introduce an axiom of inductive definitions over second orderbounded arithmetic. We show that P can be captured by the axiom ofinflationary inductive definitions whereas PSPACE can be captured bythe axiom of non-inflationary inductive definitions.
The notion of inductive definitions is widely accepted in logic and mathematics.Although inductive definitions usually deal with infinite sets, we can also dis-cuss finitary inductive definitions. Let S be a finite set and Φ : P ( S ) → P ( S )an operator , a mapping over the power set P ( S ) of S . For a natural m , define asubset P mΦ of S inductively by P Φ = ∅ and P m +1 Φ = Φ ( P mΦ ). If the operator Φ is inflationary , i.e., if X ⊆ Φ ( X ) holds for any X ⊆ S , then there exists a natural k ≤ | S | such that P k +1 Φ = P kΦ , where | S | denotes the number of elements of S ,and hence the operator Φ has a fixed point. On the side of finite model theory, afamous descriptive characterisation of the class of P of polytime predicates wasgiven by N. Immerman [6] and M. Y. Vardi [11]. It is shown that the class Pcan be captured by the first order predicate logic with fixed point predicates offirst order definable inflationary operators. In case that the operator Φ is notinflationary, it is not in general possible to find a fixed point of Φ . One canhowever find two naturals k, l ≤ | S | such that l = 0 and P k + lΦ = P kΦ . Basedon this observation, it is shown that the class PSPACE of polyspace predicatescan be captured by the first order predicate logic with fixed point predicates offirst order definable (non-inflationary) operators, cf. [4]. On the side of boundedarithmetic, it was shown by S. Buss that P can be captured by a first order sys-tem S whereas PSPACE can be captured by a second order extension U of S ,cf. [2]. An alternative way to characterise P was invented by D. Zambella [12].As well as Buss’ characterisation by S , P can be captured by a certain form ofcomprehension axiom over a weak second order system of bounded arithmetic.A modern formalisation of Zambella’s idea including further discussions can be ⋆ This work started as part of
Philosophical frontiers in Reverse Mathematics spon-sored by the John Templeton Foundation. ⋆⋆ The author is supported by JSPS posdoctoral fellowships for young scientists. ound in the book [3] by S. Cook and P. Nguyen. More recently, A. Skelleyin [8] extended this idea to a third order formulation of bounded arithmetic,capturing PSPACE as well as Buss’ characterisation by U . On the other side,as discussed by K. Tanaka [9,10] and others, cf. [7], inductive definitions overinfinite sets of naturals can be axiomatised over second order arithmetic themost elegantly. All these motivate us to introduce an axiom of inductive defi-nitions over second order bounded arithmetic. Let us recall that for each i ≥ B i of formulas is defined in the same way as the class Σ i of secondorder formulas, but only bounded quantifiers are taken into account. We showthat, over a suitable base, system the class P can be captured by the axiom ofinductive definitions under Σ B0 -definable inflationary operators (Corollary 5.2)whereas PSPACE can be captured by the axiom of inductive definitions underΣ B0 -definable (non-inflationary) operators (Corollary 7.2). There is likely no di-rect connection, but this work is also partially motivated by the axiom AID ofAlogtime inductive definitions introduced by T. Arai in [1].After the preliminary section, in Section 3 we introduce a system Σ B0 -IIDof inductive definitions under Σ B0 -definable inflationary operators and a systemΣ B0 -ID of inductive definitions under Σ B0 -definable (non-inflationary) operators.In Section 4 we show that every polytime function can be defined in Σ B0 -IID. InSection 5 we show that conversely the system Σ B0 -IID can only define polytimefunctions by reducing Σ B0 -IID to Zambella’s system V . In Section 6 we show thatevery polyspace function can be defined in Σ B0 -ID. In Section 7 we show thatconversely the system Σ B0 -ID can only define polyspace functions by reducingΣ B0 -ID to Skelley’s system W . The two-sorted first order vocabulary L A consists of 0, 1, +, · , | | , = , = , ≤ and ∈ . At the risk of confusion, we also call L A the second order vocabularyof bounded arithmetic . Note that = and = respectively denote the first orderand the second order equality, and t = s or U = V will be simply written as t = s or U = V . First order elements x, y, z, . . . denote natural numbers whereasseconder order elements X, Y, Z, . . . denote binary strings. The formula of theform t ∈ X is abbreviated as X ( t ). Under a standard interpretation, | X | denotesthe length of the string X , and X ( i ) holds if and only if the i th bit of X is 1. Let L be a vocabulary such that L A ⊆ L . We follow a convention that for an L -term t , a string variable X and a formula ϕ , ( ∃ X ≤ t ) ϕ stands for ∃ X ( | X | ≤ t ∧ ϕ )and ( ∀ X ≤ t ) ϕ stands for ∀ X ( | X | ≤ t → ϕ ). Furthermore ( ∃ x ≤ t ) ϕ stands for( ∃ x ≤ t ) · · · ( ∃ x k ≤ t k ) ϕ if x = x , . . . , x k and t = t , . . . , t k . We follow similarconventions for ( ∀ x ≤ t ) ϕ , ( ∃ X ≤ t ) ϕ and ( ∀ X ≤ t ) ϕ . A quantifier of the form( Qx ≤ t ) or ( QX ≤ t ) is called a bounded quantifier . Specific classes Σ B i ( L ) andΠ B i ( L ) (0 ≤ i ) are defined by the following clauses.1. Σ B0 ( L ) = Π B0 ( L ) is the set of L -formulas whose quantifiers are boundednumber ones only. 2. Σ B i +1 ( L ) (Π B i +1 ( L ) resp.) is the set of formulas of the form ( ∃ X ≤ t ) ϕ ( X )(( ∀ X ≤ t ) ϕ ( X ) resp.), where ϕ is a Π B i ( L )-formula (a Σ B i ( L )-formula resp.)and t is a sequence of L -terms not involving any variables from X .Finally the class ∆ B i ( L ) is defined in the most natural way for each i ≥
0. Wesimply write Σ B i (Π B i resp.) to denote Σ B i ( L A ) (Π B i ( L A ) resp.) if no confusionlikely arises. Let us recall that for each i ≥ i is axiomatised over L A by the defining axioms for numerical and string function symbols in L A (B1–B12, L1, L2 and SE, see [3, p. 96]) and the axiom (Σ B i -COMP) of comprehensionfor Σ B i formulas: ∀ x ( ∃ Y ≤ x )( ∀ i < x )[ Y ( i ) ↔ ϕ ( i )] , (Σ B i -COMP)where ϕ ∈ Σ B i . We will use the following fact frequently. Proposition 2.1 (Zambella [12]). (Cf. [3, p. 98, Corollary V.1.8])
The axiom (Σ B i - IND) of induction for Σ B i formulas holds in V i . Let L A ⊆ L . For a string function f , a class Φ of L -formulas and a system T over L , we say f is Φ -definable in T if there exists an L -formula ϕ ( X , Y ) ∈ Φ such that – ϕ does not involve free variables other than X nor Y , – the graph f ( X ) = Y of f is expressed by ϕ ( X , Y ) under a standard inter-pretation as mentioned at the beginning of this section, and – the sentence ∀ X ∃ ! Y ϕ ( X , Y ) is provable in T .Note that every function over natural numbers can be regarded as a string oneby representing naturals in their binary expansion. Proposition 2.2 (Zambella [12]). (Cf. [3, p. 135, Theorem VI.2.2])
A func-tion is polytime computable if and only if it is Σ B1 -definable in V . In this section we introduce an axiom of inductive definitions. We work over aconservative extension of V . For the sake of readers’ convenience, from Cook-Nguyen [3], we recall several string functions, all of which have Σ B0 -definablebit-graphs. Let h x, y i = ( x + y )( x + y + 1) + 2 y be a standard numerical paringfunction. Clearly the paring function is definable in L A .( String encoding [3, p. 114 Definition V.4.26]) The x th component Z [ x ] of astring Z is defined by the axiom Z [ x ] ( i ) ↔ i < | Z | ∧ Z ( h x, i i ).( Encoding of bounded number sequences [3, p. 115 Definition V.4.31]) The x th element ( Z ) x of the sequence encoded by Z is defined by the axiom( Z ) x = y ↔ [ y < | Z | ∧ Z ( h x, y i ) ∧ ( ∀ z < y ) ¬ Z ( h x, z i )] ∨ [ y = | Z | ∧ ( ∀ z < y ) ¬ Z ( h x, z i )] . String paring [3, p. 243, Definition VIII.7.2]) The string function h X, Y i isdefined by the axiom h X , X i ( i ) ↔ ( ∃ j ≤ i )[( i = h , j i ∧ X ( j )) ∨ ( i = h , j i ∧ X ( j ))] . Correspondingly, a pair of strings can be unpaired as h Z , Z i [ i ] = Z i ( i = 0 , String constant, string successor, string addition [3, p. 112, Example V.4.17])The string constant ∅ is defined by the axiom ∅ ( i ) ↔ i <
0. The string successor S ( X ) is defined by the axiom S ( X )( i ) ↔ i ≤ | X | ∧ [ X ( i ) ∧ ( ∃ j < i ) ¬ X ( j )] ∨ [ ¬ X ( i ) ∧ ( ∀ j < i ) X ( j )] . The string addition X + Y is defined by the axiom( X + Y )( i ) ↔ ( i < | X | + | Y | ∧ ( X ( i ) ⊕ Y ( i ) ⊕ Carry ( i, X, Y ))) , where ⊕ denotes “exclusive or”, i.e., p ⊕ q ≡ ( p ∧ ¬ q ) ∨ ( ¬ p ∧ q ), and Carry ( i, X, Y ) ↔ ( ∃ k < i )[ X ( k ) ∧ Y ( k ) ∧ ( ∀ j < i )( k < j → X ( j ) ∨ Y ( j )] . ( String ordering [3, p. 219, Definition VIII.3.5]) The string relation
X < Y is defined by the axiom
X < Y ↔ | X | ≤ | Y | ∧ ( ∃ i ≤ | Y | )[( ∀ j ≤ | Y | )( i < j ∧ X ( j ) → Y ( j )) ∧ Y ( i ) ∧ ¬ X ( i )] . We write X ≤ Y to denote X = Y ∨ X < Y . In addition, we write x ˙– y to denotethe limited subtraction : x ˙– y = max { , x − y } , and | x | to denote the devision of x by 2: | x | = ⌊ x/ ⌋ . We will write x − y = z if x ˙– y = z and y ≤ x . We expand thenotion of “ Φ -definable in T ” (presented on page 3) to those functions involvingthe numerical sort in addition to the string sort in an obvious way. Then itcan be shown that both x ˙– y and | x | are Σ B0 -definable in V , cf. [3, p. 60].Furthermore, though much harder to show, it can be also shown that a limitedform of exponential, Exp ( x, y ) = min { x , y } , is Σ B0 -definable in V , cf. [3, p. 64].It is known that if V is augmented by adding a collection of Σ B0 -defining axiomsfor numerical and string functions, then the resulting system is a conservativeextension of V , cf. [3, p. 110, Corolalry V.4.14]. Hence we identify V with thesystem resulting by augmenting V by adding the Σ B0 -defining axioms for thosenumerical and string functions and relations defined above.Furthermore we work over a slight extension of the vocabulary L A . For aformula ϕ ( i, X ) let P ϕ ( i, x, X ) denote a fresh predicate symbol, where ϕ maycontain free variables other than i and X . We write L to denote the vocabularyexpanded with the new predicate P ϕ for each ϕ . Definition 3.1 (Extension by fixed point predicates).
For a system T overa vocabulary L such that L A ⊆ L , T ( L ) denotes the conservative extension of T obtained by augmenting T with the following defining axioms for P ϕ .1. ( ∀ i < x )[ P ϕ ( i, x, ∅ ) ↔ i < ∀ X ≤ x +1)( ∀ i < x ) (cid:2) P ϕ ( i, x, S ( X )) ↔ ϕ ( i, P Xϕ,x ) (cid:3) , where ϕ ( i, P Xϕ,x ) denotesthe result of replacing every occurrence of Y ( j ) in ϕ ( i, Y ) with P ϕ ( j, x, X ) ∧ j < x .Now we introduce an axiom of inductive definitions . Definition 3.2 (Axiom of Inductive Definitions).
Let Φ be a class of formu-las. Then the axiom schema ( Φ -ID) of inductive definitions denotes (the universalclosure of) the following formula, where ϕ ∈ Φ .( ∃ U, V ≤ x + 1) [ V = ∅ ∧ ( ∀ i < x ) ( P ϕ ( i, x, U + V ) ↔ P ϕ ( i, x, U ))] ( Φ -ID)We write ( Φ -IID) for ( Φ -ID) if additionally the formula ϕ ∈ Φ is inflationary ,i.e., if ( ∀ Y ≤ x )( ∀ i < x )[ Y ( i ) → ϕ ( i, Y )] holds.For notational convention, we write P Xϕ,x = Y to denote ( ∀ i < x )[ P ϕ ( i, x, X ) ↔ Y ( i )]. By definition, P Xϕ,x denotes the string consisting of the first x bits of thestring obtained by X -fold iteration of the operator defined by the formula ϕ (starting with the empty string). Definition 3.3.
Let Φ be a class of L A -formulas.1. Φ -ID := V ( L ) + ( Φ -ID).2. Φ -IID := V ( L ) + ( Φ -IID).By definition, the inclusion Φ -IID ⊆ Φ -ID holds for any class Φ of L A -formulas. It is important to note that (Σ B i ( L )-COMP) is not allowed in V i ( L )for any i ≥
0, and hence ∀ x ∀ X ( ∃ Y ≤ x ) P Xϕ,x = Y does not hold in V ( L ).The main theorem in this paper is stated as follows. Theorem 3.1.
1. A function is polytime computable if and only if it is Σ B1 ( L ) -definable in Σ B0 - IID .2. A function is polyspace computable if and only if it is Σ B1 ( L ) -definable in Σ B0 - ID . Theorem 4.1.
Every polytime function is Σ B1 ( L ) -definable in Σ B0 - IID .Proof.
Suppose that a function f is polytime computable. Assuming withoutloss of generality that f is a unary function such that f ( X ) can be computedby a single-tape Turing machine M in a step bounded by a polynomial p ( | X | )in the binary length | X | of an input X .We can assume that each configuration of M on input X is encoded into abinary string whose length is exactly q ( | X | ) for some polynomial q . The polyno-mial q can be found from information on the polynomial p since | f ( X ) | ≤ p ( | X | )holds. Let the predicate Init M denote the initial configuration of M and Next M the next configuration of M . More precisely,5 Init M ( i, X ) is true if and only if the i th bit of the binary string that encodesthe initial configuration of M on input X is 1, and – Next M ( i, X, Y ) is true if and only if Y encodes a configuration of M oninput X and the i th bit of the binary string that encodes the successorconfiguration of Y is 1. Note that Next M ( i, X, Y ) never holds if Y does notencode a configuration of M , or if Y encodes the final configuration of M .Careful readers will see that both Init and
Next can be expressed by Σ B0 -formulas.We define MSP ( j, Y ), the last j bits of a string Y , which is also known as the most significant part of Y , by MSP ( j, Y )( i ) ↔ i < j ∧ Y ( | Y | ˙– j + i ) . Let ϕ ( i, X, Y ) denote the formula i < | Y | + q ( | X | ) ∧ [ Y ( i ) ∨ Init M ( i, X ) ∨ Next M ( i ˙– | Y | , X, MSP ( q ( | X | ) , Y ))] . Clearly ϕ is a Σ B -formula.Now reason in Σ B0 -IID. It is not difficult to see that ϕ ( i, X, Y ) is infla-tionary with respect to Y . Hence, by the axiom (Σ B -IID) of Σ B0 inflationaryinductive definitions, we can find two strings U and V such that | U | , | V | ≤ q ( | X | ) · ( p ( | X | ) + 1), V = ∅ and P U + Vϕ,q ( | X | ) · ( p ( | X | )+1) = P Uϕ,q ( | X | ) · ( p ( | X | )+1) . Hencethe following Σ B1 ( L ) formula ψ f ( X, Y ) holds.( ∃ U, V ≤ q ( | X | ) · ( p ( | X | ) + 1)) [ V = ∅ ∧ P U + Vϕ,q ( | X | ) · ( p ( | X | )+1) = P Uϕ,q ( | X | ) · ( p ( | X | )+1) ∧ Y = Value ( MSP ( q ( | X | ) , P Uϕ,q ( | X | ) · ( p ( | X | )+1) ))] , where Value ( Z ) denotes the function Σ B0 -definable in V (depending on the un-derlying encoding) which extracts the value of the output from Z if Z encodes thefinal configuration of M . By the definition of ϕ , MSP ( q ( | X | ) , P Uϕ,q ( | X | ) · ( p ( | X | )+1) )encodes the final configuration of M , since in any terminating computation thesame configuration does not occur more than once. Hence ψ f ( X, Y ) defines thegraph f ( X ) = Y of f . It is easy to see that ∀ X ∃ Y ψ f ( X, Y ) also holds. Theuniqueness of Y such that ψ f ( X, Y ) can be shown accordingly, allowing us toconclude. ⊓⊔ In this section we show that every function Σ B1 ( L )-definable in the systemΣ B0 -IID of Σ B0 inflationary inductive definitions is polytime computable by re-ducing Σ B0 -IID to the system V . 6 efinition 5.1. A function val ( x, X ) , which denotes the numerical value of thestring consisting of the last x bits of a string X , is defined by val ( x, ∅ ) = 0 , or otherwise, val (0 , X ) = 0 , val ( x + 1 , X ) = val ( x, X ) if | X | ≤ x, · val ( x, X ) if x < | X | & ¬ X (( | X | ˙ –
1) ˙ – x ) , · val ( x, X ) + 1 if x < | X | & X (( | X | ˙ –
1) ˙ – x ) . Lemma 5.1.
The function ( x, X ) val ( x, X ) is ∆ B1 -definable in V if x ≤ | y | for some y . More precisely, the relation val ( x, X ) = z can be expressed by a ∆ B1 formula ψ val ( x, y, z, X ) if x ≤ | y | , and the sentence ∀ y ( ∀ x ≤ | y | ) ∀ X ∃ ! zψ val ( x, y, z, X ) is provable in V .Proof. Let ψ ( x, z, X, Y ) denote the formula expressing that z = 0 if | X | = 0, orotherwise ( Y ) = 0, ( Y ) x = z , and for all j < x , – | X | ≤ j → ( Y ) j +1 = ( Y ) j , – j < | X | ∧ ¬ X ( | X | ˙– j ˙– 1) → ( Y ) j +1 = 2( Y ) j , and – j < | X | ∧ X ( | X | ˙– j ˙– 1) → ( Y ) j +1 = 2( Y ) j + 1.Define ψ val ( x, y, z, X, Y ) to be ( ∃ Y ≤ h x, y +1 i +1) ψ ( x, z, X, Y ). Clearly ψ val is aΣ B1 formula expressing the relation val ( x, X ) = z in case x ≤ | y | . Note that 2 | y | ≤ y +1 for all y . Hence if x ≤ | y | , then val ( x, X ) ≤ x ≤ | y | ≤ y +1. Reason in V .One can show that if x ≤ | y | , then ( ∃ z ≤ y +1)( ∃ Y ≤ h x, y +1 i +1) ψ ( x, z, X, Y )holds by induction on x . Accordingly the uniqueness of those z and Y above canbe also shown. From the uniqueness of z and Y , val ( x, X ) = z is equivalent toa Π B1 formula ( ∀ u ≤ y + 1)( ∃ Y ≤ h x, y + 1 i + 1)[ ψ ( x, y, u, X, Y ) → u = z ].Hence ψ val is a ∆ B1 formula. ⊓⊔ Lemma 5.2.
Let ϕ ( x, X ) be a Σ B0 formula. Then the relation ( x, X, Y ) P Xϕ,x = Y can be expressed by a ∆ B1 formula ψ P ϕ ( x, y, X, Y ) if | X | ≤ | y | . Moreprecisely, corresponding to Definition 3.1.1 and 3.1.2, ψ P ϕ enjoys the following.1. ψ P ϕ ( x, y, ∅ , ∅ ) .2. ( ∀ X ≤ x + 1)( | X | ≤ | y | → ∀ Y, Z [ ψ P ϕ ( x, y, X, Y ) ∧ ψ P ϕ ( x, y, S ( X ) , Z ) → ( ∀ i < x )( Z ( i ) ↔ ϕ ( i, Y ))]) .Furthermore, the sentence ∀ x, y ( ∀ X ≤ | y | )( ∃ ! Y ≤ x ) ψ P ϕ ( x, y, X, Y ) is provablein V .Proof. Let ψ ( x, X, Y, Z ) denote a formula which expresses that – ( ∀ j ≤ val ( | y | , X )) | ( Z ) j | ≤ x , – Z [0] = ∅ , Z [ val ( | y | ,X )] = Y , and – ( ∀ j < val ( | y | , X ))( ∀ i < x )[ Z [ j +1] ( i ) ↔ ϕ ( i, Z [ j ] )].7efine ψ P ϕ ( x, y, X, Y ) to be ( ∃ Z ≤ h val ( | y | , X ) , x i + 1) ψ ( x, X, Y, Z ). Then, since ϕ is a Σ B0 formula, ψ P ϕ is a Σ B1 formula expressing the relation P Xϕ,x = Y if | X | ≤ | y | . Reason in V . One can show | X | ≤ | y | → ( ∃ Y ≤ x ) ψ P ϕ ( x, y, X, Y, Z )by induction on val ( | y | , X ). The uniqueness of such strings Y and Z can be alsoshown. Hence, as in the previous proof, thanks to the uniqueness of Y and Z , ψ P ϕ is a ∆ B1 formula. ⊓⊔ Definition 5.2.
1. A string function
Ones ( y ) , which denotes the string con-sisting only of of length y , is defined by the axiom Ones ( y )( i ) ↔ i < y .2. The string predecessor P ( X ) is by the axiom P ( X )( i ) ↔ i < | X | ∧ [( X ( i ) ∧ ( ∃ j < i ) X ( j )) ∨ ( ¬ X ( i ) ∧ ( ∀ j < i ) ¬ X ( j ))] . Lemma 5.3.
1. In V , if < | X | , then S ( P ( X )) = X holds.2. In V , if x < | y | , then the following holds. val ( | y | , S ( Ones ( x ))) = val ( | y | , Ones ( x )) + 1 . (1)
3. In V , if < | X | ≤ | y | , then val ( | y | , P ( X )) + 1 = val ( | y | , X ) holds.Proof.
1. We reason in V . Suppose 0 < | X | . Then X ( i ) holds for some i < | X | .Since the axiom (Σ B i -MIN) of minimisation for Σ B i formulas holds in V i , cf. [3,p. 98, Corollary V.1.8], there exists an element i < | X | such that X ( i ) and( ∀ j < i ) ¬ X ( j ) hold. Define a string Y with use of (Σ B0 -COMP) by | Y | ≤ | X | and ( ∀ i < | X | )[ Y ( i ) ↔ ( i < i ∧ X ( i )) ∨ i < i ] . (2)We show (i) S ( Y ) = X and (ii) P ( X ) = Y . It is not difficult to see | S ( Y ) | = | X | and | P ( X ) | = | Y | . For (i) suppose i < | S ( X ) | and S ( X )( i ). If Y ( i ) and( ∃ j < i ) ¬ Y ( j ) hold, then i < i and X ( i ) hold by the definition of Y . If ¬ Y ( i )and ( ∀ j < i ) Y ( j ) hold, then i = i holds. By the choice of i , X ( i ) and ( ∀ j
Let us recall a numerical function numones ( x, X ) which denotes the num-ber of elements of X , or equivalently the number of 1 occurring in the string X ,not exceeding x (See [3, p. 149]). It can be shown that numones is Σ B1 -definablein V (See [3, p. 149]). As we observed in the proof of Lemma 5.1 or Lemma 5.2, numones is even ∆ B1 -definable in V .Let ϕ ∈ Σ B0 . Reason in V . Suppose that ϕ is inflationary, i.e., ( ∀ Y ≤ x )( ∀ j < x )[ Y ( i ) → ϕ ( i, Y )] holds. By contradiction we show the existenceof a string U such that U ≤ Ones ( | x | ) and the condition (3) holds. Since | S ( Ones ( | x | )) | = | x | + 1 = | x | , by Lemma 5.2 ( ∃ ! Y ≤ x ) P Xϕ,x = Y holds for any X ≤ S ( Ones ( | x | )). Hence it suffices to find a string U such that U ≤ Ones ( | x | )and P S ( U ) ϕ,x = P Uϕ,x . Assume that such a string U does not exist. Then for any X ≤ Ones ( | x | ) there exists i < x such that P S ( X ) ϕ,x ( i ) but ¬ P Xϕ,x ( i ). This meansthat numones ( x, P Xϕ,x ) < numones ( x, P S ( X ) ϕ,x ) holds for any X ≤ Ones ( | x | ). Claim. If X ≤ S ( Ones ( | x | )), then val ( | x | + 1 , X ) ≤ numones ( x, P Xϕ,x ) holds.We show the claim by induction on val ( | x | +1 , X ). The base case that val ( | x | +1 , X ) = 0 is clear. For the induction step, consider the case val ( | x | + 1 , X ) > < | X | , and hence by Lemma 5.3.3 val ( | x | + 1 , P ( X )) + 1 = val ( | x | +1 , X ) holds. Hence by IH val ( | x | +1 , P ( X )) ≤ numones ( x, P P ( X ) ϕ,x ) holds.By Lemma 5.3.1, S ( P ( X )) = X holds. This together with IH yields val ( | x | +1 , X ) = val ( | x | + 1 , P ( X )) + 1 ≤ numones ( x, P Xϕ,x ) since numones ( x, P P ( X ) ϕ,x ) Suppose ≤ i . If Σ B i ( L ) formula ψ is provable in Σ B0 - IID ,then there exists a Σ B i formula ψ ′ provable in V and provably equivalent to ψ in V ( L ) .Proof. The theorem can be shown by an induction argument on the length ofa formal Σ B0 -IID-proof resulting in ψ . We only discuss the axiom (Σ B0 -IID) ofΣ B0 inflationary inductive definitions and kindly refer details to readers. Let ϕ aΣ B0 formula. We reason in V . Fix a natural x arbitrarily. Then, since S ( X ) = X + S ( ∅ ), Theorem 5.1 yields two strings U and V such that | U | , | V | ≤ | x | ≤ x +1, V = ∅ , and the following hold. ∀ Y, Z [ ψ P ϕ ( x, x, U + V, Y ) ∧ ψ P ϕ ( x, x, U, Z ) → ( ∀ i < x )( Y ( i ) ↔ Z ( i ))] . | U | , | U + V | ≤ | x | , Lemma 5.2 yields unique two strings Y and Z suchthat | Y | , | Z | ≤ x + 1, ψ P ϕ ( x, x, U + V, Y ) and ψ P ϕ ( x, x, U, Z ) hold. Hence,by Lemma 5.2.2, Z ( i ) ↔ ϕ ( i, Y ) holds for any i < x . This together withLemma 5.2.1 allows us to conclude that the statement ( i < x ) ( Z ( i ) ↔ ϕ ( i, Y ))is provably equivalent to ( i < x ) ( P ϕ ( i, x, U + V ) ↔ P ϕ ( i, x, U )) in V ( L ). ⊓⊔ Corollary 5.1. Every function Σ B1 ( L ) -definable in Σ B0 - IID is polytime com-putable.Proof. Suppose that a Σ B1 ( L ) sentence ψ is provable in Σ B0 -ID. Then by The-orem 5.2 we can find a Σ B1 sentence ψ ′ provable in V and provably equivalentto ψ in V ( L ). In particular ψ and ψ ′ are equivalent under the underlyingstandard interpretation. Hence every function Σ B1 ( L )-definable in Σ B0 -IID isΣ B1 -definable in V . Now employing Proposition 2.2 enables us to conclude. ⊓⊔ Corollary 5.2. A predicate belongs to P if and only if it is ∆ B1 ( L ) -definablein Σ B0 - IID . Theorem 6.1. Every polyspace computable function is Σ B ( L ) -definable in Σ B - ID .Proof. The theorem can be shown in a similar manner as Theorem 4.1. Sup-pose that a function f is polyspace computable. As in the proof of Theorem4.1 we can assume that f is a unary function such that f ( X ) can be com-puted by a single-tape Turing machine M using a number of cells bounded bya polynomial p ( | X | ) in | X | . Assuming a standard encoding of configurationsof M into binary strings, the binary length of every configuration is exactly q ( | X | ) for some polynomial q . Let Init M denote the predicate defined on page6. A new predicate Next ′ M ( i, X, Y ) denotes the successor configuration of Y ,but in contrast to Next M , Next ′ M ( i, X, Y ) does not change if Y encodes thefinal configuration. More precisely, if Y encodes the final configuration, then( ∀ i < q ( | X | ))( Next ′ M ( i, X, Y ) ↔ Y ( i )) holds. In contrast to the definition of ϕ on page 6, let ϕ ( i, X, Y ) denote the formula i < q ( | X | ) ∧ [ Init M ( i, X ) ∨ Next ′ ( i, X, Y )] . It is not difficult to convince ourselves that ϕ is a Σ B0 formula. Hence, reasoning inΣ B0 -ID, by the axiom (Σ B -ID) of Σ B0 inductive definitions, we can find two strings U and V such that | U | , | V | ≤ q ( | X | ) + 1, V = ∅ and P U + Vϕ,q ( | X | )+1 = P Uϕ,q ( | X | )+1 hold. Hence the following Σ B1 ( L ) formula ψ f ( X, Y ) holds.( ∃ U, V ≤ q ( | X | ) + 1) [ V = ∅ ∧ P U + Vϕ,q ( | X | )+1 = P Uϕ,q ( | X | )+1 ∧ Y = Value ( P Uϕ,q ( | X | )+1 )] , Value ( Z ) denotes the extraction function Σ B0 -definable in V as in theproof of Theorem 4.1. As we observed, P Uϕ,q ( | X | )+1 encodes the final configurationof M . Hence ψ f ( X, Y ) defines the graph f ( X ) = Y of f . Now it is clear that ∀ X ∃ Y ψ f ( X, Y ) holds. The uniqueness of Y follows accordingly, allowing us toconclude. ⊓⊔ In this section we show that every function Σ B1 ( L )-definable in the systemΣ B0 -ID of Σ B0 inductive definitions is polyspace computable by reducing Σ B0 -IDto a third order system W of bounded arithmetic which was introduced by A.Skelley in [8]. The third order vocabulary L A is defined augmenting the sec-ond order vocabulary L A with the third order membership relation ∈ . As inthe case of the second order membership, the formula of the form Y ∈ X is abbreviated as X ( Y ). Third order elements X , Y , Z , . . . would denote hyper strings, i.e., X ( Y ) holds if and only if the Y th bit of X is 1. Classes Σ B i , Π B i and ∆ B i (0 ≤ i ) are defined in the same manner as Σ B i , Π B i and ∆ B i but thirdorder quantifiers are taken into account instead of second order ones. For in-stance, Σ B = S ≤ i Σ B i ( L A ), and a Σ B formula is of the form ∃X ψ ( X ), where nothird order quantifier appears in ψ . For a class Φ of L A -formulas, the axiom of(Φ-3COMP) is defined by ∀ x ∃Z ( ∀ Y ≤ x )[ Z ( Y ) ↔ ϕ ( Y )] , (Φ-3COMP)where ϕ ∈ Φ. The system W consists of the basic axioms of second orderbounded arithmetic (B1–B12, L1, L2 and SE, [3, p. 96]), (Σ B -IND), (Σ B -COMP)and Σ B -3COMP. Proposition 7.1 (Skelley [8]). A function is polyspace computable if and onlyif it is Σ B -definable in W .Remark 7.1. In the original definition of W presented in [8], the axiom (IND)of induction is allowed only for a class ∀ Σ B of formulas, which is slightly morerestrictive than Σ B . However it can be shown that every Σ B formula is provablyequivalent to a ∀ Σ B formula in W (See [8, Theorem 2 and Cororally 3]).We show that a stronger form of Σ B0 inductive definitions holds in W . Definition 7.1 (Axiom of Relativised Inductive Definitions). We assumea new predicate symbol P ϕ ( i, x, X, Y ) instead of P ϕ ( i, x, X ) for each ϕ . Wereplace Definition 3.1.1 and 3.1.2 respectively with the following defining axioms.1. ( ∀ i < x )[ P ϕ ( i, x, ∅ , Y ) ↔ Y ( i )].2. ( ∀ X ≤ x +1)( ∀ i < x ) (cid:2) P ϕ ( i, x, S ( X ) , Y ) ↔ ϕ ( i, P Xϕ,x [ Y ]) (cid:3) , where ϕ ( i, P Xϕ,x [ Y ])denotes the result of replacing every occurrence of X ( j ) in ϕ ( i, X ) with P ϕ ( j, x, X, Y ) ∧ j < x . 11hen a relativised form of the axiom of inductive definitions denotes the followingstatement, where ϕ ∈ Φ .( ∀ Y ≤ x )( ∃ U, V ≤ x + 1) [ V = 0 ∧ ( ∀ i < x ) ( P ϕ ( i, x, U + V, Y ) ↔ P ϕ ( i, x, U, Y ))]As in the case of the predicate P ϕ ( i, x, X ), we write P Xϕ,x [ Y ] = Z instead of( ∀ i < x )( P ϕ ( i, x, X, Y ) ↔ Z ( i )). Apparently the axiom of relativised inductivedefinitions implies the original axiom of inductive definitions. Definition 7.2. 1. The complementary string Y C x of a string Y of length x isdefined by the axiom Y C x ( i ) ↔ i < x ∧ ¬ Y ( i ) .2. The string subtraction X ˙ – Y is defined by the axiom ( X ˙ – Y )( i ) ↔ ( X ≤ Y ∧ i < ∨ ( Y < X ∧ i < | X | ∧ ( X + S ( Y C | X | ))( i )) . It can be shown that in V , if | Y | ≤ x , then Y + Y C x = Ones ( x ), and hence Y + S ( Y C x ) = S ( Ones ( x )) holds. Thus one can show that | ( X + Y ) ˙– Y | = | X | and, for any i < | X | , [( X + Y ) ˙– Y ]( i ) ↔ [ X + S ( Ones ( | X + Y | ))]( i ) ↔ X ( i ),concluding ( X + Y ) ˙– Y = X . Lemma 7.1. Let ϕ ( x, X ) be a Σ B0 formula. Then the relation ( x, y, X, Y, Z ) P Xϕ,x [ Y ] = Z can be expressed by a ∆ B formula ψ P ϕ ( x, X, Y, Z ) if | X | , | Y | ≤ y inthe same sense as in Lemma 5.2. Furthermore the sentence ∀ x, y ( ∀ X ≤ y )( ∀ Y ≤ x )( ∃ ! Z ≤ x ) ψ P ϕ ( x, y, X, Y, Z ) is provable in W .Notation. We define a string function ( Z ) X , which denotes the X th componentof a hyper string Z , by the axiom ( Z ) X = Y ↔ Z ( h X, Y i ). For a hyper string Z we write ∃ ! Z ≤ x to refer to the uniqueness up to elements of length notexceeding x , i.e., ( ∃ ! Z ≤ x ) ψ ( Z ) denotes ∃Z ψ ( Z ) and additionally, ∀Z , Z [ ψ ( Z ) ∧ ψ ( Z ) → ( ∀ Y ≤ x )( Z ( Y ) ↔ Z ( Y ))] . (4) Proof. Let ψ ( x, y, X, Y, Z, Z ) denote the Σ B formula expressing – ( ∀ U ≤ y )( U ≤ X → | ( Z ) U | ≤ x ), – ( Z ) ∅ = Y , ( Z ) X = Z , and – ( ∀ U ≤ y )( U < X → ( ∀ i < x )[( Z ) S ( U ) ( i ) ↔ ϕ ( i, ( Z ) U )]).By the definition of ψ , the relation P Xϕ,x [ Y ] = Z is expressed by the Σ B for-mula ∃Z ψ ( x, y, X, Y, Z, Z ) if | X | ≤ y . It suffices to show that ( ∀ Y ≤ x )( ∃ ! Z ≤ x )( ∃ ! Z ≤ h| X | , x i ) ψ ( x, X, Y, Z, Z ) hols in W .Reason in W . We only show the existence of such a string Z and a hyperstring Z . The uniqueness in the sense of (4) can be shown accordingly. By in-duction on | X | we derive the Σ B formula ( ∀ Y ≤ x )( ∃ Z ≤ x ) ∃Z ψ ( x, X, Y, Z, Z ).The argument is based on a standard “divide-and-conquer method”. In the basecase, | X | = 0, i.e., X = ∅ , and hence the assertion is clear. The case that | X | = 1,i.e., X = S ( ∅ ), is also clear. Suppose that | X | > 1. Then we can find two strings X and X such that | X | = | X | = | X | − X = X + X . Fix a string Y so that | Y | ≤ x . Then by IH we can find a string Z and a hyper string Z such12hat | Z | ≤ x and ψ ( x, X , Y, Z , Z ) hold. Since | Z | ≤ x , another applicationof IH yields Z and Z such that | Z | ≤ x and ψ ( x, X , Z , Z , Z ) hold. Definea hyper string Z with use of (Σ B -3COMP) by( ∀ U ≤ h| X | , x i )[ Z ( U ) ↔ ( U [0] ≤ X ∧ ( Z ) U [0] = U [1] ) ∨ ( X < U [0] ∧ ( Z ) U [0] ˙– X = U [1] )] . (5)Intuitively Z denotes the concatenation Z a Z , the hyper string Z followedby Z . Then by definition ψ ( x, X, Y, Z , Z ) holds. Due to the uniqueness ofthe string Z and the hyper string Z , the Σ B formula ∃Z ψ ( x, y, X, Y, Z, Z ) isequivalent to the Π B formula ( ∀ V ≤ x )( ∀Z ≤ h| X | , x i )( ψ ( x, X, Y, V, Z ) → V = Z ), and hence is also a ∆ B formula. ⊓⊔ Lemma 7.2. The following holds in W . ∀ x, y ( ∀ X ≤ y )( ∀ Y ≤ y )( ∀ Z ≤ x )( | Y + X | ≤ y → P Xϕ,x [ P Yϕ,x [ Z ]] = P Y + Xϕ,x [ Z ]) . Proof. By the previous lemma the relation P Xϕ,x [ P Yϕ,x [ Z ]] = P Y + Xϕ,x [ Z ] can beexpressed by a ∆ B formula if | X | , | Y | ≤ y . Reason in W . We show that | X | ≤ y → ( ∀ Y ≤ y )( ∀ Z ≤ x )( | Y + X | ≤ y → P Xϕ,x [ P Yϕ,x [ Z ]] = P Y + Xϕ,x [ Z ])holds by induction on | X | . The base case that | X | = 0 or | X | = 1 is clear. Suppose | X | > 0. Then we can find two strings X and X such that | X | = | X | = | X |− X + X = X . Fix a string Z so that | Z | ≤ x . Since | X | = | X | < | X | ≤ y and | X + X | = | X | ≤ y , IH yields P X ϕ,x [ P X ϕ,x [ Z ]] = P X + X ϕ,x [ Z ]. Hence P Yϕ,x [ P Xϕ,x [ Z ]] = P Yϕ,x [ P X ϕ,x [ P X ϕ,x [ Z ]]] . (6)On the other hand, since | X | ≤ y , | Y + X | ≤ | Y + X | ≤ y and | P X ϕ,x [ Z ] | ≤ x ,another application of IH yields P Yϕ,x [ P X ϕ,x [ P X ϕ,x [ Z ]]] = P Y + X ϕ,x [ P X ϕ,x [ Z ]] . (7)Farther, since | Y + X | ≤ y and | X | ≤ | X | ≤ x , the final application of IH yields P Y + X ϕ,x [ P X ϕ,x [ Z ]] = P Y + X + X ϕ,x [ Z ] = P Y + Xϕ,x [ Z ] . (8)Combining equation (6), (7) and (8) allows us to conclude. ⊓⊔ Definition 7.3. A string function numones [ Y ]( X, X ) , which counts the numberof elements of X (starting with Y ) such that ≤ X , is defined by numones [ Y ]( ∅ , X ) = Y, numones [ Y ]( S ( X ) , X ) = ( S ( numones [ Y ]( X, X )) if X ( X ) holds, numones [ Y ]( X, X ) if ¬X ( X ) holds. Lemma 7.3. The function numones is ∆ B -definable in W . roof. Let ψ numones ( X, Y, Z, X , Y ) denote the Σ B formula expressing – Z ≤ Y + X , – ( Y ) ∅ = Y , – ( Y ) X = Z , – ( ∀ U ≤ | X | )[ U < X ∧ X ( U ) → ( Y ) S ( U ) = S (( Y ) U )], and – ( ∀ U ≤ | X | )[ U < X ∧ ¬X ( U ) → ( Y ) S ( U ) = ( Y ) U ].Then by definition the Σ B formula ∃Y ψ numones ( X, Y, Z, X , Y ) defines the graph numones [ Y ]( X, X ) = Z of numones . We show that if | X | ≤ x , then( ∀ Y ≤ x )[ | Y + X | ≤ x → ( ∃ ! Z ≤ x )( ∃ ! Y ≤ h| X | , x i ) ψ numones ( X, Y, Z, X , Y )]holds in W . Reason in W . Given x , we only show the existence of such a string Z and a hyper string Y by induction on | X | . The uniqueness can be shown ina similar manner. Fix x and Y so that | Y | ≤ x and | Y + X | ≤ x . In case that | X | = 0, i.e., X = ∅ , define Y by( ∀ U ≤ h , x i )[ Y ( U ) ↔ ( U = h∅ , Y i )] . Then | Y | ≤ x , Y ≤ Y + ∅ and ψ numones ( ∅ , Y, Y, X , Y ) hold. In the case that | X | = 1, i.e., X = S ( ∅ ), define Y by( ∀ U ≤ h , x i )[ Y ( U ) ↔ U = h∅ , Z i∨ ( X ( ∅ ) ∧ U = h S ( ∅ ) , S ( Y ) i ) ∧ ( ¬X ( ∅ ) ∧ U = h S ( ∅ ) , Y i )] . Clearly | ( Y ) S ( ∅ ) | ≤ | S ( Y ) | = | Y + S ( ∅ ) | , ( Y ) S ( ∅ ) ≤ S ( Y ) = Y + S ( ∅ ) and ψ numones ( S ( ∅ ) , ( Y ) S ( ∅ ) , Y, X , Y ) hold. For the induction step, suppose | X | > X and X such that | X | = | X | = | X | − X + X = X . By assumption | Y + X | ≤ | Y + X | ≤ x . Hence IH yields a string Z and a hyper string Y such that | Z | ≤ x and ψ numones ( X , Y , Z , X , Y )hold. In particular Z ≤ Y + X holds. This implies | Z + X | ≤ | Y + X + X | = | Y + X | ≤ x . Thus another application of IH yields Z and Y such that | Z | ≤ x and ψ numones ( X , Z , Z , X , Y ) hold. Define Y in the same way as(5) in the proof of Lemma 7.1, i.e., Y = Y a Y . It is not difficult to see that ψ numones ( X, Y, Z , X , Y ) holds. Thanks to the uniqueness of Z and Y , one cansee that ∃Y ψ numones ( X, Y, Z, X , Y ) is a ∆ B formula. ⊓⊔ Lemma 7.4. The axiom (Σ B - of third order comprehension for Σ B formulas holds in W . Readers might recall that the V can be axiomatised by (Σ B0 -COMP) and (Σ B1 -IND)instead of (Σ B1 -COMP), cf. [3, p. 149, Lemma VI.4.8]. Lemma 7.4 can be shownwith the same idea as the proof of this fact. For the sake of completeness, wegive a proof in the appendix. Theorem 7.1. The axiom (Σ B - ID) of Σ B0 inductive definitions holds in W inthe same sense as in Theorem 5.2. roof. Instead of showing that the axiom (Σ B0 -ID) holds in W , we show thateven the axiom of relativised Σ B0 inductive definitions holds in W . Let ϕ ∈ Σ B0 .We reason in W . Fix x arbitrarily. Given X and Y , we define a hyper string P X [ Y ] with use of (Σ B -3COMP) by( ∀ Z ≤ x )[ P X [ Y ]( Z ) ↔ ( ∃ U ≤ | X | )[ U < X ∧ P Uϕ,x [ Y ] = Z ]] . Claim. For a string W , if x < | W | , then the following holds.( ∀ Y ≤ x ) [ numones ( W, P X [ Y ]) ≤ X → ( ∃ U, V ≤ | X | )( U < V ≤ X ∧ P Vϕ,x [ Y ] = P Uϕ,x [ Y ])] . (9)Assume the claim. Since numones ( S ( Ones ( x )) , P S ( Ones ( x )) [ Y ]) ≤ S ( Ones ( x )) bythe definition of numones and x < x + 1 = | S ( Ones ( x )) | , (9) then implies theinstance of (Σ B0 -ID) in case of ϕ .The rest of the proof is devoted to prove the claim. Let us observe that (9)is a Σ B statement. We show the claim by induction on | X | . In the base case, X = ∅ and hence (9) trivially holds. The case that X = S ( ∅ ) is also trivial. Forthe induction step, suppose | X | > 1. Then there exist strings X and X suchthat | X | = | X | = | X | − X + X = X . Fix a string Y so that | Y | ≤ x and suppose numones ( W, P X [ Y ]) ≤ X . By the definition of the hyper string P X [ Y ] and Lemma 7.2, for any Z , if | Z | ≤ x , then P X [ Y ]( Z ) ↔ P X [ Y ]( Z ) ∨P X [ P X ϕ,x [ Y ]]( Z ) holds, i.e., P X [ Y ] = P X [ Y ] ∪P X [ P X ϕ,x [ Y ]]. On the other handwe can assume that ( ∀ U < X )( ∀ V < X ) P Uϕ,x [ Y ] = P Vϕ,x [ P X ϕ,x [ Y ]] holds, i.e., P X [ Y ] ∩ P X [ P X ϕ,x [ Y ]] = ∅ . This yields numones ( W, P X [ Y ])= numones ( W, P X [ Y ]) + numones ( W, P X [ P X ϕ,x [ Y ]]) . (10) Case. numones ( W, P X [ Y ]) ≤ X : In this case IH yields two strings U and V such that | U | , | V | ≤ | X | , U < V ≤ X and P V ϕ,x [ Z ] = P U ϕ,x [ Z ]. Since | X | ≤ | X | and ≤ X ≤ X , we can define U and V by U = U and V = V . Case. X < numones ( W, P X [ Y ]): In this case, numones ( W, P X [ P X [ Y ]]) ≤ X by the equality (10). Since | P X [ Y ] | ≤ x by definition, another applicationof IH yields two strings U and V such that | U | , | V | ≤ | X | , U < V ≤ X and P V ϕ,x [ P X ϕ,x [ Y ]] = P U ϕ,x [ P X ϕ,x [ Y ]] hold. Define strings U and V by U = X + U and V = X + V . Since P Vϕ,x [ Y ] = P V ϕ,x [ P X ϕ,x [ Y ]] and P Uϕ,x [ Y ] = P U ϕ,x [ P X ϕ,x [ Y ]]by Lemma 7.2, now it is easy to check that the assertion (9) holds. ⊓⊔ Corollary 7.1. Every function Σ B1 ( L ) -definable in Σ B0 - ID is polyspace com-putable.Proof. Suppose that a Σ B1 ( L ) formula ψ is provable in Σ B0 -ID. Then, as inthe proof of Theorem 5.2, from Lemma 7.1 and Theorem 7.1 one can find aΣ B formula ψ ′ provable in W and provably equivalent to ψ in W ( L ). Inparticular ψ and ψ ′ are equivalent under the underlying interpretation. Henceevery string function Σ B1 ( L )-definable in Σ B0 -ID is Σ B -definable in W . Thusemploying Proposition 7.1 enables us to conclude. ⊓⊔ orollary 7.2. A predicate belongs to PSPACE if and only if it is ∆ B1 ( L ) -definable in Σ B0 - ID . In this paper we introduced a novel axiom of finitary inductive definitions overthe Cook-Nguyen style second order bounded arithmetic. We have shown thatover a conservative extension V ( L ) of V by fixed point predicates, P can becaptured by the axiom of inductive definitions under Σ B0 -definable inflationaryoperators whereas PSPACE can be captured by the axiom of inductive definitionsunder (non-inflationary) Σ B0 -definable operators. It seems also possible for each i ≥ i th level of the polynomial hierarchy by the axiom ofinductive definitions under Σ B i -definable inflationary operator, e.g., a predicatebelongs to NP if and only if it is ∆ B2 ( L )-definable in Σ B2 -IID. As shown by Y.Gurevich and S. Shelah in [5], over finite structures the fixed point of a first orderdefinable inflationary operator can be reduced the least fixed point of a first orderdefinable monotone operator. In accordance with this fact, it is natural to askwhether the axiom Σ B0 -IID of inflationary inductive definitions for Σ B0 -definableoperators can be reduced a suitable axiom of monotone inductive definitions forΣ B0 -definable operators. Acknowledgment The author thanks Shohei Izawa for the initial discussions about this subjectwith him. He also acknowledges valuable discussions with Toshiyasu Arai duringhis visit at Chiba University. He pointed out that the numerical function val andthe string one P Xϕ,x ( | X | ≤ | y | ) are not only Σ B1 -definable but even ∆ B1 -definablein V (Lemma 5.1, 5.2) and P Xϕ,x and numones are even ∆ B -definable in W aswell (Lemma 7.1, 7.3). This observation made later arguments easier. Finally,he would like to thank Arnold Beckmann for his comments about this subject.The current formulation of inductive definitions stems from his suggestion . References 1. T. Arai. A Bounded Arithmetic AID for Frege Systems. Annals of Pure AppliedLogic , 103(1-3):155–199, 2000.2. S. Buss. Bounded Arithmetic . Bibliopolis, 1986.3. S. Cook and P. Nguyen. Logical Foundations of Complexity . Cambridge UniversityPress, 2010.4. H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Second edition . Perspectivesin Mathematical Logic. Springer, 1999.5. Y. Gurevich and S. Shelah. Fixed-point Extensions of First-order Logic. Annalsof Pure and Applied Logic , 32(3):265–280, 1986. The previous formulation can be found at http://arxiv.org/abs/1306.5559v3 . . N. Immerman. Relational Queries Computable in Polynomial Time (ExtendedAbstract). In Proceedings of the 14th Annual ACM Symposium on Theory of Com-puting , pages 147–152, 1982.7. W. Pohlers. Subsystems of Set Theory and Second Order Number Theory. In S. R.Buss, editor, Handbook of Proof Theory , pages 210–335. North Holland, 1998.8. A. Skelley. A Third-order Bounded Arithmetic Theory for PSPACE. In Proceedingsof the 18th EACSL Annual Conference on Computer Science Logic (CSL 2004),Lecture Notes in Computer Science , volume 3210, pages 340–354, 2004.9. K. Tanaka. The Galvin-Prikry Theorem and Set Existence Axioms. Annals ofPure Applied Logic , 42(1):81–104, 1989.10. K. Tanaka. Weak Axioms of Determinacy and Subsystems of Analysis II (Σ Games). Annals of Pure Applied Logic , 52(1-2):181–193, 1991.11. M. Y. Vardi. The Complexity of Relational Query Languages (Extended Abstract).In Proceedings of the 14th Annual ACM Symposium on Theory of Computing , pages137–146, 1982.12. D. Zambella. Notes on Polynomially Bounded Arithmetic. Journal of SymbolicLogic , 61(3):942–966, 1996. A Proving (Σ B -3COMP) in W In the appendix we show Lemma 7.4 which states that the axiom (Σ B -3COMP)of third order comprehension for Σ B formulas (presented on page 11) holds inW . We start with showing a couple of auxiliary lemmas. Lemma A.1. In W for any number x , string X and hyper string Z , if | X | ≤ x and ∅ < numones ( X, Z ) , then the following holds. ( ∃ Y ≤ x )( Y < X ∧ S ( numones ( Y, Z )) = numones ( X, Z )) . Proof. Reason in W . Fix x and Z . We show the following stronger assertionholds by induction on | X | ≤ x .( ∀ U ≤ x ) | U + X | ≤ x ∧ numones ( U, Z ) < numones ( U + X, Z ) → ( ∃ Y ≤ x )( Y < X ∧ S ( numones ( U + Y, Z )) = numones ( U + X, Z )) . If | X | = 0, i.e., X = ∅ , then numones ( U, Z ) = numones ( U + X, Z ), andhence the assertion trivially holds. In the case | X | = 1 , i.e., X = S ( ∅ ), if numones ( U, Z ) < numones ( U + S ( ∅ ) , Z ), then the assertion is witnessed by Y = ∅ . For the induction step, suppose | X | > 1. Then there exist two strings X and X such that | X | = | X | = | X |− X + X = X . Fix a string U so that | U | ≤ x and suppose that | U + X | ≤ x and numones ( U, Z ) < numones ( U + X, Z ) hold. Then | U + X | ≤ x . Case. numones ( U, Z ) = numones ( U + X , Z ): By IH there exists a string Y < X < X such that | Y | ≤ x and S ( numones ( U + Y, Z )) = numones ( U + X , Z ) = numones ( U + X + X , Z ) = numones ( U + X, Z ). Case. numones ( U, Z ) < numones ( U + X , Z ): In this case by IH there existsa string Y < X such that | Y | ≤ x and S ( numones ( U + Y , Z )) = numones ( U +17 , Z ) holds. If numones ( U + X , Z ) = numones ( U + X, Z ), then the witnessingstring Y can be defined to be Y . Consider the case numones ( U + X , Z ) < numones ( U + X, Z ). Then another application of IH yields a string Y < X suchthat | Y | ≤ x and S ( numones (( U + X )+ Y , Z )) = numones (( U + X )+ X , Z )hold. Define a string Y by X + Y . Then | Y | ≤ | X | ≤ x , Y = X + Y Reason in W . Fix x and Z . We show the following stronger assertionholds by induction on | Z | .( ∀ X ≤ x )( ∀ U ≤ x ) {| U + Z | ≤ x ∧ ∅ < Z ≤ numones ( X, Z ) → ( ∃ Y ≤ x )( Y < X ∧ numones ( Y, Z ) + U + Z = numones ( X, Z ) + U ) } . If | Z | = 0, i.e., Z = ∅ , then the assertion trivially holds. In the case | Z | = 1,i.e., Z = S ( ∅ ), since numones ( Y, Z ) + U + S ( ∅ ) = S ( numones ( Y, Z )) + U , theassertion follows from Lemma A.1. For the induction step, suppose | Z | > Z and Z such that | Z | = | Z | = | Z | − Z + Z = Z . Fix two strings X and U so that | X | , | U | ≤ x and | U + Z | ≤ x and suppose that ∅ < Z ≤ numones ( X, Z ). Then, since | U + Z | ≤ | U + Z | ≤ x and ∅ < Z < Z ≤ numones ( X, Z ), IH yields a string Y < X such that | Y | ≤ x and numones ( Y , Z ) + U + Z = numones ( X, Z ) + U hold. Since | Y | ≤ | X | ≤ x and | U + Z | ≤ | U + Z | ≤ x , another application ofIH yields a string Y < Y < X such that | Y | ≤ x and numones ( Y , Z ) + ( U + Z ) + Z = numones ( Y , Z ) + U + Z = numones ( X, Z ) + U holds. Thus thewitnessing string Y can be defined to be Y . ⊓⊔ Notation. In contrast to the empty string ∅ , we write ∅ to denote the emptyhyper string defined by the axiom ∅ ( X ) ↔ | X | < Proof (of Lemma 7.4). Suppose a Σ B formula ϕ ( Z ). We have to show the ex-istence of a hyper string Y such that ( ∀ Z ≤ x )( Y ( Z ) ↔ ϕ ( Z )) holds. Let ψ ( x, U, X, Y ) denote the following formula.( ∀ Z ≤ x )( Y ( Z ) → ϕ ( Z )) ∧ X = U + numones ( S ( Ones ( x )) , Y ) . By Lemma 7.3, ψ is a Σ B formula, and hence so is ∃Y ψ ( x, U, X, Y ). Reason inW . The argument splits into two (main) cases. Case. ( ∃ X ≤ x + 1)[ X < S ( Ones ( x )) ∧ ∃Y ψ ( x, ∅ , X, Y ) ∧ ( ∀ Y ≤ x + 1)( Y ≤ S ( Ones ( x )) ∧ X < Y → ¬∃Y ψ ( x, ∅ , Y, Y ))]: Suppose that a string X witnessesthis case. Let ψ ( x, ∅ , X , Y ). Then clearly ( ∀ Z ≤ x )( Y ( Z ) → ϕ ( Z )) holds. Weshow the converse inclusion by contradiction. Assume that there exists a string Z such that | Z | ≤ x , ϕ ( Z ) but ¬Y ( Z ). Define a hyper string Y ′ by( ∀ Z ≤ x )[ Y ′ ( Z ) ↔ ( Z = Z ∨ Y ( Z ))] . ∀ Z ≤ x )( Y ′ ( Z ) → ϕ ( Z )) by definition, and also numones ( S ( Ones ( x )) , Y )= X < S ( X ) = numones ( S ( Ones ( x )) , Y ′ ) ≤ S ( Ones ( x )). But this contradictsthe assumption of this case. Case. The previous case fails: Namely, ( ∀ X ≤ x + 1)[ X < S ( Ones ( x )) ∧∃Y ψ ( x, ∅ , X, Y ) → ( ∃ Y ≤ x + 1)( Y ≤ S ( Ones ( x )) ∧ X < Y ∧ ∃Y ψ ( x, ∅ , Y, Y ))]holds. We derive the following Σ B formula by induction on | X | .( ∀ U ≤ x + 1)[ U + X ≤ S ( Ones ( x )) → ( ∃ Y ≤ x + 1)( ∃Y ψ ( x, U, Y, Y ) ∧ U + X ≤ Y ≤ S ( Ones ( x )))] . (11)Assume the formula (11) holds. Let U = ∅ and X = S ( Ones ( x )). Then by(11) we can find a string Y and a hyper string Y such that | Y | ≤ x + 1 and numones ( S ( Ones ( x )) , Y ) = Y = S ( Ones ( x )). This means that ( ∀ Z ≤ x ) Y ( Z )holds, and hence in particular ( ∀ Z ≤ x )[ ϕ ( Z ) → Y ( Z )] holds.In the base case, if | X | = 0, i.e., X = ∅ , then ψ ( x, U, U, ∅ ) holds. This implies ψ ( x, ∅ , ∅ , ∅ ). Hence by the assumption of this case, we can find a string Y and ahyper string Y such that | Y | ≤ x + 1, Y ≤ S ( Ones ( x )), ∅ < Y and ψ ( x, ∅ , Y, Y ).These imply the case | X | = 1, i.e., ψ ( x, U, U + Y, Y ) and U + S ( ∅ ) ≤ U + Y .For the induction step, suppose | X | > 1. Then there exist two strings X and X such that | X | = | X | = | X | − X + X = X . Fix a string U so that | U + X | ≤ x + 1. Then by IH we can find a string Y and a hyper string Y suchthat | Y | ≤ x + 1, ψ ( x, U, Y , Y ) and U + X ≤ Y . Subcase. U + X = Y : In this subcase, another application of IH yieldsa string Y and a hyper string Y such that | Y | ≤ x + 1, ψ ( x, Y , Y , Y ) and Y + X ≤ Y . Since U + X = U + X = Y + X ≤ Y , it can be observed that ψ ( x, U, Y , Y a Y ) holds. Subcase. U + X < Y : In this subcase we can assume that Y < U + X holds. Hence by Lemma A.2, we can find a string V < S ( Ones ( x )) such that numones ( V, Y ) = U + X holds. Define a hyper string Y ↾ V by( ∀ Z ≤ x )[( Y ↾ V )( Z ) ↔ Z < V ∧ Y ( Z )] . Then numones ( S ( Ones ( x )) , Y ↾ V ) = U + X holds by definition. Now we canproceed in the same way as the previous subcase but we define the witnessinghyper string Y by Y = ( Y ↾ V ) a Y . This completes the proof of Lemma 7.4. ⊓⊔⊓⊔