Ergodic Theorems for PSPACE functions and their converses
aa r X i v : . [ c s . CC ] F e b Ergodic Theorems for PSPACE functions and their converses
Satyadev Nandakumar and Subin Pulari Department of Computer Science and Engineering, Indian Institute of TechnologyKanpur, Kanpur, Uttar Pradesh, India. { satyadev,subinp } @cse.iitk.ac.inFebruary 16, 2021 Abstract
We initiate the study of effective pointwise ergodic theorems in resource-bounded settings.Classically, the convergence of the ergodic averages for integrable functions can be arbitrar-ily slow [15]. In contrast, we show that for a class of PSPACE L functions, and a class ofPSPACE computable measure-preserving ergodic transformations, the ergodic average existsfor all PSPACE randoms and is equal to the space average on every EXP random. We establisha partial converse that PSPACE non-randomness can be characterized as non-convergence ofergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse -a point x is SUBEXP-space random if and only if the corresponding effective ergodic theoremholds for x . In Kolmogorov’s program to found information theory on the theory of algorithms, we investigatewhether individual “random” objects obey probabilistic laws, i.e. , properties which hold in samplespaces with probability 1. Indeed, a vast and growing literature establishes that every
Martin-L¨ofrandom sequence (see for example, [4] or [22]) obeys the Strong Law of Large Numbers [28], theLaw of Iterated Logarithm [29], and surprisingly, the Birkhoff Ergodic Theorem [30], [20], [10], [1]and the Shannon-McMillan-Breiman theorem [8], [9], [24]. In effective settings, the theorem forMartin-L¨of random points implies the classical theorem since the set of Martin-L¨of randoms hasLebesgue measure 1, and hence is stronger.In this work, we initiate the study of ergodic theorems in resource-bounded settings. This isa difficult problem, since classically, the convergence speed in ergodic theorems is known to bearbitrarily slow ( e.g. see Bishop [3], Krengel [15], and V’yugin [30]). However, we establish ergodictheorems in resource-bounded settings which hold on every resource-bounded random object ofa particular class. The main technical hurdle we face is the lack of sharp tail bounds. The onlygeneral tail bound in ergodic settings is the maximal ergodic inequality, which yields only an inverselinear bound in the number of sample points, in contrast to the inverse exponential bounds in theChernoff and the Azuma-Hoeffding inequalities.Rapid L convergence of subsequences of ergodic averages suffices to establish that the ergodicaverage at all PSPACE randoms exist, and is equal to the space average on all EXP randoms.A non-trivial connection with the theory of uniform distribution of sequences modulo 1 [16], [19],123], [21] enables us to show that the canonical example of the Bernoulli measure and the leftshift satisfies our convergence assumption. In general, such assumptions are unavoidable since anadaptation of V’yugin’s counterexample [30] shows that there are PSPACE computable ergodicMarkov systems where the convergence rate to the ergodic average is not even computable.It is known that at the level of Martin-L¨of randomness, we can use ergodicity to characterizerandomness. Franklin and Towsner [5] show that for every non-Martin-L¨of random x , there is aneffective ergodic system where the ergodic average at x does not converge to the space average.We first show that our PSPACE effective ergodic theorem admits a partial converse of this form.PSPACE non-randoms can be characterized as points where the PSPACE ergodic theorem fails.Since the theorem holds on the smaller set of EXP randoms, it is important to know whether thereis a class of resource-bounded randoms on which an effective ergodic theorem holds with an exactconverse. We show that the class of SUBEXP-space randoms is one such - on every SUBEXP-spacerandoms, the SUBEXP-space ergodic theorem holds, and on every SUBEXP-space non-random, itfails. We summarize our results in Table 1.The proofs of these results are adapted from the techniques of Rute [24], Ko [14], Galatolo,Hoyrup & Rojas [7], [11], and Huang & Stull [12]. Our proofs involve several new quantitativeestimates, which may of general interest.Class of functions Convergence of ergodic averages ∀ f ( A fn → R f dµ ) ∃ f ( A fn R f dµ )PSPACE L EXP randoms(Theorem 5.5) PSPACE non-randoms(Theorem 6.1)SUBEXP-space L SUBEXP-spacerandoms (Theorem7.16) SUBEXP-spacenon-randoms (Theorem7.18)Table 1: Summary of the results involving PSPACE/SUBEXP-space systems.
Let Σ = { , } be the binary alphabet. Denote the set of all finite binary strings by Σ ∗ and theset of infinite binary strings by Σ ∞ . For σ ∈ Σ ∗ and y ∈ Σ ∗ ∪ Σ ∞ , we write σ ⊑ y if σ is a prefixof y . For any infinite string y and any finite string σ , σ [ n ] and y [ n ] denotes the character at the n th position in y and σ respectively. For any infinite string y and any finite string σ , σ [ n, m ] and y [ n, m ] represents the strings σ [ n ] σ [ n + 1] . . . σ [ m ] and y [ n ] y [ n + 1] . . . y [ m ] respectively. For any x ∈ Σ ∞ and n ∈ N , x ↿ n denotes the string x [1 , n ]. We denote finite strings using small Greekletters like σ , α etc. The length of a finite binary string σ is denoted by | σ | .For any finite string σ , the cylinder [ σ ] is the set of all infinite sequences with σ as a prefix. χ σ denotes the characteristic function of [ σ ]. For any set of strings S ⊆ Σ ∗ , [ S ] is the union of [ σ ]over all σ ∈ S . Extending the notation, χ S denotes the characteristic function of [ S ]. The Borel σ -algebra generated by the set of all cylinders is denoted by B (Σ ∞ ). There are alternative approaches to the proof in Martin-L¨of settings, like that of V’yugin [30]. However, the toolhe uses for establishing the result is a lower semicomputable test defined on infinite sequences - this is difficult toadapt to resource-bounded settings requiring the output value within bounded time or space. Moreover, the functionsin V’yugin’s approach are continuous. We consider the larger class of L functions, which can be discontinuous ingeneral. n ∈ N is represented in the binary alphabet. As is typical inresource-bounded settings, some integer parameters are represented in unary. The set of unarystrings is represented as 1 ∗ , and the representation of n ∈ N in unary is 1 n , a string consisting of n ones. For any n , n ∈ N , [ n , n ] represents the set { n ∈ N : n ≤ n ≤ n } .Throughout the paper we take into account the number of cells used in the output tape and theworking tape when calculating the space complexity of functions. We assume a finite representationfor the set of rational numbers Q satisfying the following: there exists a c ∈ N such that if r ∈ Q has a representation of length l then r ≤ l c . Following the works of Hoyrup, and Rojas [11], weintroduce the notion of a PSPACE-computable probability space on the Cantor space by endowingit with a PSPACE-computable probability measure. Definition 2.1.
Consider the probability space (Σ ∞ , B (Σ ∞ )). A Borel probability measure µ : B (Σ ∞ ) → [0 , -probability measure if there is a PSPACE machine M : Σ ∗ × ∗ → Q such that for every σ ∈ Σ ∗ , and n ∈ N , we have that | M ( σ, n ) − µ ([ σ ]) | ≤ − n .A PSPACE -probability Cantor space is a pair (Σ ∞ , µ ) where Σ ∞ is the Cantor space, and µ isa PSPACE probability measure.In order to define PSPACE (EXP) randomness using PSPACE (EXP) tests we require thefollowing method for approximating sequences of open sets in Σ ∞ in polynomial space (exponentialtime). Definition 2.2 (PSPACE/EXP sequence of open sets [12]) . A sequence of open sets h U n i ∞ n =1 is aPSPACE sequence of open sets if there exists a sequence of sets (cid:10) S kn (cid:11) k,n ∈ N , where S kn ⊆ Σ ∗ suchthat1. U n = ∪ ∞ k =1 [ S kn ], where for any m > µ (cid:0) U n − ∪ mk =1 [ S kn ] (cid:1) ≤ m .2. There exists a controlling polynomial p such that max {| σ | : σ ∈ ∪ mk =1 S kn ) } ≤ p ( n + m ).3. The function g : Σ ∗ × ∗ × ∗ → { , } such that g ( σ, n , m ) = 1 if σ ∈ S mn , and 0 otherwise,is decidable by a PSPACE machine.The definition of EXP sequence of open sets is similar but the bound in condition 2 is replacedwith 2 p ( n + m ) and the machine in condition 3 is an EXP-time machine.Henceforth, we study the notion of resource bounded randomness on (Σ ∞ , µ ). Definition 2.3 (PSPACE/EXP randomness [26]) . A sequence of open sets h U n i ∞ n =1 is a PSPACE test if it is a PSPACE sequence of open sets and for all n ∈ N , µ ( U n ) ≤ n .A set A ⊆ Σ ∞ is PSPACE null if there is a PSPACE test h U n i ∞ n =1 such that A ⊆ ∩ ∞ n =1 U n . Aset A ⊆ Σ ∞ is PSPACE random if A is not PSPACE null.The EXP analogues of the above concepts are defined similarly except that h U n i ∞ n =1 is an EXPsequence of open sets.By considering the sequence (cid:10) ∪ ki =1 S in (cid:11) k,n ∈ N instead of (cid:10) S kn (cid:11) k,n ∈ N , without loss of generality, wecan assume that for each n , (cid:10) S kn (cid:11) ∞ k =1 is an increasing sequence of sets.In order to establish our ergodic theorem, it is convenient to define a PSPACE version of Solovaytests, where the relaxation is that the measures of the sets U n can be any sufficiently fast convergentsequence. We later show that this captures the same set of randoms as PSPACE tests.3 efinition 2.4 (PSPACE Solovay test) . A sequence of open sets h U n i ∞ n =1 is a PSPACE Solovaytest if it is a PSPACE sequence of open sets and there is a polynomial p such that ∀ m ≥ P ∞ n = p ( m )+1 µ ( U n ) ≤ m . A set A ⊆ Σ ∞ is PSPACE Solovay null if there exists a PSPACESolovay test h U n i ∞ n =1 such that A ⊆ ∩ ∞ i =1 ∪ ∞ n = i U n . A ⊆ Σ ∞ is PSPACE Solovay random if A is notPSPACE Solovay null. Theorem 2.5.
A set A ⊆ Σ ∞ is PSPACE null if and only if A is PSPACE
Solovay null.Proof.
It is easy to see that if A is PSPACE null then A is PSPACE Solovay null. Conversely,let A be PSPACE Solovay null and let h U n i ∞ n =1 be any Solovay test which witnesses this fact. Let V n = ∪ ∞ i = p ( n )+1 U n . We show that h V n i ∞ n =1 is a PSPACE test. Let (cid:10) S kn (cid:11) n,k ∈ N be any sequence of setsapproximating h U n i ∞ n =1 as in definition 2.2 such that (cid:10) S kn (cid:11) ∞ k =1 is increasing for each n . We define asequence of sets (cid:10) T kn (cid:11) n,k ∈ N approximating V n as follows.Let r ( n, k ) = max { p ( n ) + 1 , p ( k + 1) } . Define T kn = r ( n,k ) [ i = p ( n )+1 S r ( n,k ) − p ( n )+ k +1 i . We can easily verify the first three conditions in definition 2.2. Using the machine M andcontrolling polynomial p witnessing that h U n i ∞ n =1 is a PSPACE sequence of open sets, we canconstruct the corresponding machines for h V n i ∞ n =1 in the following way. Machine N on input( σ, n , k ) does the following:1. For each i ∈ [ p ( n ) + 1 , r ( n, k )] do the following:(a) Output 1 if M ( σ, i , r ( n,k ) − p ( n )+ k +1 ) = 1.2. Output 0 if none of the above computations results in 1.It is straightforward to verify that N is a PSPACE machine.The set of PSPACE Solovay randoms and PSPACE randoms are equal, hence to prove PSPACErandomness results, it suffices to form Solovay tests. PSPACE L computability The resource-bounded ergodic theorems in our work hold for PSPACE- L functions, the PSPACEanalogue of integrable functions. In this section, we briefly recall standard definitions for PSPACEcomputable L functions and measure-preserving transformations. The justifications and proofs ofequivalences of various notions are present in Stull’s thesis [25] and [26]. We initially define PSPACEsequence of simple functions, and define PSPACE integrable functions based on approximationsusing these functions. Definition 3.1 (PSPACE sequence of simple functions [26]) . A sequence of simple functions h f n i ∞ n =1 where each f n : Σ ∞ → Q is a PSPACE sequence of simple functions if1. There is a controlling polynomial p such that for each n , there exist k ( n ) ∈ N , { d , d , . . . , d k ( n ) } ⊆ Q and { σ , σ , . . . , σ k ( n ) } ⊆ Σ p ( n ) satisfying f n = P k ( n ) i =1 d i χ σ i . This implies that P ∞ n =1 µ ( U n ) < ∞
4. There is a PSPACE machine M such that for each n ∈ N , σ ∈ Σ ∗ M (1 n , σ ) = ( f n ( σ ∞ ) if | σ | ≥ p ( n )? otherwise.Note that since M is a PSPACE machine, { d , d . . . d k ( n ) } is a set of PSPACE representablenumbers. Now, we define PSPACE L -computable functions in terms of limits of convergentPSPACE sequence of simple functions. Definition 3.2 (PSPACE L -computable functions [26]) . A function f ∈ L (Σ ∞ , µ ) is PSPACE L -computable if there exists a PSPACE sequence of simple functions h f n i ∞ n =1 such that for every n ∈ N , k f − f n k ≤ − n . The sequence h f n i ∞ n =1 is called a PSPACE L -approximation of f .A sequence of L functions h f n i ∞ n =1 converging to f in the L -norm need not have pointwiselimits. Hence the following concept ([24]) is important in studying the pointwise ergodic theoremin the setting of L -computability Definition 3.3 ( e f for PSPACE L -computable f ) . Let f ∈ L (Σ ∞ , µ ) be PSPACE L -computableand let h f n i ∞ n =1 be any PSPACE sequence of simple functions in L (Σ ∞ , µ ) approximating f (as inDefinition 3.2). Define e f : Σ ∞ → R ∪ { undefined } by e f ( x ) = lim n →∞ f n ( x ) if this limit exists, andis undefined otherwise. To define ergodic averages, we restrict ourselves to the following class of transformations.
Definition 3.4 (PSPACE simple transformation) . A measurable function T : (Σ ∞ , µ ) → (Σ ∞ , µ )is a PSPACE simple transformation if there is a controlling constant c and a PSPACE machine M such that such that for any σ ∈ Σ ∗ T − ([ σ ]) = k ( σ ) [ i =1 [ σ i ]such that:1. { σ i } k ( σ ) i =1 is a prefix free set and for all 1 ≤ i ≤ k ( σ ), | σ i | ≤ | σ | + c
2. For each σ, α ∈ Σ ∗ , M ( σ, α ) = | α | ≥ | σ | + c and α ∞ ∈ T − ([ σ ])0 if | α | ≥ | σ | + c and α ∞ T − ([ σ ])? otherwiseIt is easy to verify that if T is a PSPACE simple transformation then for any n ≥ T n is alsoa PSPACE simple transformation. We need the following stronger assertion in the proof of theergodic theorem. Lemma 3.5.
Let T : (Σ ∞ , µ ) → (Σ ∞ , µ ) be a PSPACE simple transformation with controllingconstant c . There exists a PSPACE machine N such that for each n ∈ N and σ, α ∈ Σ ∗ , N (1 n , σ, α ) = if | α | ≥ | σ | + cn and α ∞ ∈ T − n ([ σ ])0 if | α | ≥ | σ | + cn and α ∞ T − n ([ σ ])? otherwise The definition of e f is dependent on the choice of the approximating sequence h f n i ∞ n =1 . However, due to Lemma5.1, we use e f in a sequence independent manner. roof. Let M be the machine witnessing the fact that T is a PSPACE simple transformationwith the polynomial space complexity bound p ( n ). Let the machine N do the following on input(1 n , σ, α ):1. If α < | σ | + cn , then output ?.2. If n = 1 then, run M ( σ, α ) and output the result of this simulation.3. Else:(a) For all strings α ′ of length | σ | + c ( n −
1) do the following:i. If N (1 n − , σ, α ′ ) = 1 then, output 1 if M ( α ′ , α ) = 1.4. If no output is produced in the above steps, output 0.When n = 1, N uses at most p ( | σ | + | α | + cn ) + O (1) space. Inductively, assume that for n = k , N uses at most (2 k − p ( | σ | + | α | + cn ) + O (1) space. For n = k + 1, the storage of α ′ and the twosimulations inside step 3a can be done in 2 p ( | σ | + | α | + cn ) + (2 k − p ( | σ | + | α | + cn ) + O (1) =(2( k + 1) − p ( | σ | + | α | + cn ) + O (1) space. Hence, N is a PSPACE machine.PSPACE computability as defined above, relates naturally to convergence of L norms. Butthe pointwise ergodic theorem deals with almost everywhere convergence, and its resource-boundedversions deal with convergence on every random point. We introduce the modes of convergence wedeal with in the present work. Definition 3.6 (PSPACE-rapid limit point) . A real number a is a PSPACE -rapid limit point of the real number sequence h a n i ∞ n =1 if there exists a polynomial p such that for all m ∈ N , ∃ k ≤ p ( m ) such that | a k − a | ≤ m .Note that this requires rapid convergence only on a subsequence. We remark that the aboveis equivalent to the existence of a PSPACE machine computing the speed of convergence on input1 m . The following definition is the L version of the above. Definition 3.7 (PSPACE-rapid L -limit point) . A function f ∈ L (Σ ∞ , µ ) is a PSPACE -rapid L -limit point of a sequence h f n i ∞ n =1 of functions in L (Σ ∞ , µ ) if 0 is a PSPACE-rapid limit pointof k f n − f k .Now we define PSPACE analogue of almost everywhere convergence ([24]). Definition 3.8 (PSPACE-rapid almost everywhere convergence) . A sequence of measurable func-tions h f n i ∞ n =1 is PSPACE -rapid almost everywhere convergent to a measurable function f if thereexists a polynomial p such that for all m and m , µ ( x : sup n ≥ p ( m m | f n ( x ) − f ( x ) | ≥ m )! ≤ m . Notation.
Let A f,Tn = f + f ◦ T + f ◦ T + ...f ◦ T n − n denote the n th Birkhoff average for any function f and transformation T . We prove the ergodic theorem in measure preserving systems where R f dµ is a PSPACE-rapid L -limit point of A f,Tn . In the rest of the paper we denote A f,Tn simply by A fn .The transformation T involved in the Birkhoff sum is implicit.6 efinition 3.9 (PSPACE ergodic transformations) . A measurable function T : (Σ ∞ , µ ) → (Σ ∞ , µ )is PSPACE ergodic if T is a PSPACE simple measure preserving transformation such that for anyPSPACE L -computable f ∈ L (Σ ∞ , µ ), R f dµ is a PSPACE-rapid L limit point of A fn .V’yugin [30] shows that the speed of a.e. convergence to ergodic averages in computable ergodicsystems is not computable in general. This leads us to consider some assumption on the rapidityof convergence in resource-bounded settings. We show that the requirement on L rapidity ofconvergence of A fn is sufficient to derive our result. Several probabilistic laws like the Law of LargeNumbers, Law of Iterated Logarithm satisfy this criterion, hence the assumption is sufficientlygeneral. Moreover, as we show now, in the canonical example of Bernoulli systems with the left-shift, every PSPACE L function exhibits PSPACE rapidity of A fn , showing that the latter propertyis not artificial. The proof of this theorem is a non-trivial application of techniques from uniformdistribution of sequences modulo 1 [16], [23], [19], [21]. Theorem 3.10.
Let f ∈ L (Σ ∞ , B (Σ ∞ ) , µ ) where µ is the Bernoulli measure µ ( σ ) = | σ | and let T be the left shift transformation. If f is PSPACE L -computable, then there exists a polynomial q satisfying the following: given any m ∈ N , for all n ≥ q ( m ) , k A fn − R f dµ k ≤ − m . An equivalent statement is the following: The left-shift transformation on the Bernoulli proba-bility measure is PSPACE ergodic . Theorem 3.10 gives an explicit bound on the speed of conver-gence in the L ergodic theorem for an interesting class of functions over the Bernoulli space. Suchbounds do not exist in general for the L ergodic theorem as demonstrated by Krengel in [15].The above theorem can be obtained from the following assertion regarding PSPACE-rapidconvergence of characteristic functions of long enough cylinders. Lemma 3.11.
Let T be the left shift transformation T : (Σ ∞ , B (Σ ∞ ) , µ ) → (Σ ∞ , B (Σ ∞ ) , µ ) where µ is the Bernoulli measure µ ( σ ) = 2 −| σ | . There exist polynomials q , q such that for any m ∈ N and σ ∈ Σ ∗ with | σ | ≥ q ( m ) we get k A χ σ n − µ ( σ ) k ≤ − m for all n ≥ | σ | q ( m ) . Now we prove Theorem 3.10 by assuming Lemma 3.11.
Proof of Theorem 3.10.
Let h f n i ∞ n =1 be a PSPACE sequence of simple functions witnessing the factthat f is PSPACE L -computable. Let p be a controlling polynomial and let t be a polynomialupper bound for the space complexity of the machine associated with h f n i ∞ n =1 . Let q , q be thepolynomials from Lemma 3.11. Let c ∈ N be any number such that if a r ∈ Q has a representationof length l then r ≤ l c (see Section 2). Observe that for any m ∈ N , k A fn − Z f dµ k ≤ k A fn − A f q m +3) n k + k A f q m +3) n − Z f q ( m +3) dµ k + k Z f q ( m +3) dµ − Z f dµ k ≤ q ( m +3) + k A f q m +3) n − Z f q ( m +3) dµ k + 12 q ( m +3) . ≤ m +3 + k A f q m +3) n − Z f q ( m +3) dµ k + 12 m +3 . We know that there exists { σ , σ . . . σ k } ⊆ Σ p ( q ( m +3)) such that A f q m +3) n = P k ( q ( m +3)) i =1 d i χ σ i where each d i ≤ t ( q ( m +3)+ p ( q ( m +3))) c . Hence, k A f q m +3) n − Z f q ( m +3) dµ k ≤ t ( q ( m +3)+ p ( q ( m +3))) c k ( q ( m +3)) X i =1 k A χ σi n − µ ( σ i ) k Equivalently, there exists a constant c such that for all n > k A fn − R fdµ k ≤ −⌊ log( n ) c ⌋ . | σ i | ≥ p ( q ( m + 3)) ≥ q ( m + 3), using Lemma 3.11, for n ≥ p ( q ( m + 3)) q ( t ( q ( m +3)+ p ( q ( m +3))) c + p ( q ( m +3))+ m +3) we get that, k A f q m +3) n − Z f q ( m +3) dµ k ≤ t ( q ( m +3)+ p ( q ( m +3))) c + p ( q ( m +3)) t ( q ( m +3)+ p ( q ( m +3))) c + p ( q ( m +3))+ m +3 ≤ m +3 . Hence, for all n ≥ p ( q ( m + 3)) q ( t ( q ( m +3)+ p ( q ( m +3))) c + p ( q ( m +3))+ m +3) we have k A fn − R f dµ k ≤ . − ( m +3) < − m .Now, we give a proof for Lemma 3.11. Proof of Lemma 3.11.
The major difficulty in directly approximating k A χ σ n − µ ( σ ) k is that for any n, m ∈ N , A χ σ n and A χ σ m may not be independent . In order to overcome this, we use constructionssimilar to those used in proving Pillai’s theorem (see [23], [19] for normal numbers, [21] for con-tinued fractions) in order to approximate each A χ σ n with sums of disjoint averages. These disjoint averages turns out to be averages of independent random variables. Hence, elementary results fromprobability theory regarding independent random variables can be used to show that A χ σ n convergesto R f dµ sufficiently fast.Observe that for any x ∈ Σ ∞ A χ σ n ( x ) = |{ i ∈ [0 , n − | T i x ∈ [ σ ] }| n Let k = | σ | . As in the proof of Theorem 3.1 from [21], the following is a decomposition of the aboveterm as disjoint averages, |{ i ∈ [0 , n − | T i x ∈ [ σ ] }| n = g ( n ) + g ( n ) + · · · + g (1+ ⌊ log nk ⌋ ) ( n ) + ( k − .O (log n ) n where, g p ( n ) = n − |{ i | T ki x ∈ [ σ ] , ≤ i ≤ ⌊ n/k ⌋}| , if p = 1 n − P k − j =1 |{ i | T (2 p − ) ki x ∈ [ S j ] , ≤ i ≤ ⌊ n/ p − k ⌋}| , if 1 < p ≤ (1 + ⌊ log ( n/k ) ⌋ )0 , otherwisewhere S j is the finite collection of 2 ( p − k length blocks s.t σ occurs in it at starting position(2 ( p − k − j + 1) th position i.e S j is the set of strings of the form, u a a . . . a k v where u is somestring of length 2 p − k − j , and v is some string of length 2 p − k − k + j .When p = 1, g ( n ) = ⌊ nk ⌋ P i =1 X , i n where, X , i ( x ) = ( x [ ik + 1 , ( i + 1) k ] = σ < p ≤ ⌊ log ( n/k ) ⌋ , g p ( n ) = ⌊ n p − k ⌋ P i =1 k − P j =1 X p,ji n where, X p,ji ( x ) = ( x [2 p − k − j + 1 , p − k − j + k ] = σ A χ σ n ( x ) = ⌊ nk ⌋ P i =1 X , i ( x ) n + ⌊ log ( nk ) ⌋ X p =2 k − X j =1 ⌊ n p − k ⌋ P i =1 X p,ji n + ( k − .O (log n ) n An important observation that we use later in the proof is that for any fixed p and j , { X p,ji } ∞ i =1 isa collection of i.i.d Bernoulli random variables such that µ ( { x : X p,ji ( x ) = 1 } ) = 2 −| σ | . We showthat the conclusion of the lemma holds when q ( m ) = 2( m + 6) and q ( m ) = 5( m + 6). For any m ∈ N , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X p = m +5+2 k − X j =1 n ⌊ n p − k ⌋ X i =1 X p,ji (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ∞ X p = m +5+2 p − ≤ m +5 (1)And for n ≥ | σ | q ( m ) > | σ | m +5) , (cid:13)(cid:13)(cid:13)(cid:13) ( k − O (log( n )) n (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) ( k − O (log( n )) √ n √ n (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:12)(cid:12)(cid:12)(cid:12) k − √ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) k − k m +5 (cid:12)(cid:12)(cid:12)(cid:12) ≤ m +5 (2)Let, D σn,m ( x ) = ⌊ nk ⌋ P i =1 X , i ( x ) n + m +5+2 X p =2 k − X j =1 ⌊ n p − k ⌋ P i =1 X p,ji n From (1) and (2), we get that k A χ σ n − D σn,m k ≤ m +5 . Let, E σn,m ( x ) = ⌊ nk ⌋ P i =1 X , i ( x ) ⌊ nk ⌋ − k ⌊ nk ⌋ n + m +5+2 X p =2 k − X j =1 ⌊ n p − k ⌋ P i =1 X p,ji ⌊ n p − k ⌋ − k ⌊ n p − k ⌋ n D σn,m ( x ) − E σn,m ( x ) = 12 k k + m +5+2 X p =2 k − X j =1 k ⌊ n p − k ⌋ n It follows that, k D σn,m ( x ) − E σn,m k ≤ k + m +5+2 X p =2 k − X j =1 k p − k ≤ k + m +5+2 X p =2 k p − ≤ k + m +5+2 X p =2 k ≤ m + 5 + 22 k Hence, if | σ | = k ≥ q ( m ) = m + 5 + 2 + m + 5 then, k D σn,m ( x ) − E σn,m k ≤ m +5 and, k A χ σ n − µ ( σ ) k ≤ k A χ σ n − D σn,m k + k D σn,m ( x ) − E σn,m k + k E σn,m k + 12 k ≤ m +5 + k E σn,m k + 12 m +12 ≤ m +5 + k E σn,m k . Hence, in order to show that for all n ≥ | σ | q ( m ) , k A χ σ n − µ ( σ ) k ≤ k A χ σ n − µ ( σ ) k ≤ − m , it isenough to show that for all n ≥ | σ | q ( m ) , k E σn,m k ≤ − ( m +5) . Observe that, k E σn,m k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ⌊ nk ⌋ ⌊ nk ⌋ X i =1 X , i ( x ) − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + m +5+2 X p =2 k − X j =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ⌊ n p − k ⌋ ⌊ n p − k ⌋ X i =1 X p,ji − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Let Y , Y , . . . Y n be i.i.d Bernoulli random variables, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n n X i =1 Y i − E ( Y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = vuuut E n n X i =1 Y i − E ( Y ) ! = vuut Var n n X i =1 Y i ! = r n n Var( Y ) ≤ p Var( Y ) √ n ≤ √ n . Hence, if n ≥ | σ | q ( m ) = | σ | m +6) then, j nk k > k m +6) and j n p − k k ≥ k m +6) m +5+1 k > k m +6) . Hence for all n ≥ | σ | q ( m ) = | σ | m +6) , k E σn,m k ≤ k m +6) + ( m + 6) k k m +6) < m +6 + 12 m +6 ≤ m +5 . Hence we obtain the desired conclusion.We remark that since Lemma 3.11 is true with the L -norm replaced by the L -norm, Theorem3.10 is also true in the L setting. i.e, if a function f is PSPACE L -computable (replacing L normswith L norms in definition 3.2) then there exists a polynomial q satisfying the following: givenany m ∈ N , for all n ≥ q ( m ) , k A fn − R f dµ k ≤ − m . Hence, for PSPACE L -computable functionsand the left shift transformation T , we get bounds on the convergence speed in the von-Neumann’sergodic theorem.We now show that PSPACE ergodicity is a stronger version of ln -ergodicity introduced in [6]. Lemma 3.12.
Let T : Σ ∞ → Σ ∞ be any measurable transformation. T is PSPACE ergodic if andonly if for any f ∈ L ∞ (Σ ∞ , µ ) , there exist c > and k ∈ N such that for all n > , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X i =0 Z f ◦ T i .f − Z f dµ Z f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c (ln n ) k . Proof.
We prove the forward implication first. The proof uses techniques from the proof of Theorem4 in [6]. From the hypothesis there exist c > k ∈ N such that for all n > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X i =0 f ◦ T i .f − (cid:18)Z f dµ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < c (ln n ) k By replacing f with f − R f dµ , without loss of generality we assume that R f dµ = 0. First we showthat A fn is PSPACE-rapid almost everywhere convergent to R f dµ . Following the steps in proof ofLemma 6 in [6] we get that, µ (cid:18)(cid:26) x : (cid:12)(cid:12)(cid:12)(cid:12) A fn − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) > m (cid:27)(cid:19) ≤ m k f k ∞ n + c (ln n ) k ! Hence, there exists a polynomial q such that, µ ( x : sup n ≥ q ( m ,m (cid:12)(cid:12)(cid:12)(cid:12) A f √ n − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) > m )! ≤ m m such that √ n ≤ m ≤ √ n + 1 and β n = √ n √ n +1 , k A f √ n − A fm k ∞ ≤ − β n ) k f k ∞ Let l , l be any numbers such that 2 l ≥ k f k ∞ and 2 l ≥ k . Let p ( n ) = 2 n −
2. It is easy to seethat for all n ≥ p ( m + l ) , √ n √ n + 1 = r − n + 1 ≥ − m + l +2 ≥ − m +2 k f k ∞ From the two previous inequalities we get that for n ≥ p ( m + l ) and m such that √ n ≤ m ≤√ n + 1, k A f √ n − A fm k ∞ ≤ m +1 Hence, [ √ n ≤ m ≤√ n +1 (cid:18)(cid:26) x : (cid:12)(cid:12)(cid:12)(cid:12) A fm − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) > m (cid:27)(cid:19) ⊆ (cid:18)(cid:26) x : (cid:12)(cid:12)(cid:12)(cid:12) A f √ n − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) > m +1 (cid:27)(cid:19) Let r ( m , m ) = q ( m + 1 , m ) + p ( m + l ). Now, µ ( x : sup n ≥ r ( m ,m (cid:12)(cid:12)(cid:12)(cid:12) A fn − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) > m )! ≤ µ ( x : sup n ≥ q ( m ,m (cid:12)(cid:12)(cid:12)(cid:12) A f √ n − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) > m )! ≤ m Hence A fn is PSPACE-rapid almost everywhere convergent to R f dµ . Now, given any i , for n ≥ r ( i +1 ,i + l +1) , k A fn − Z f dµ k ≤ i +1 + k f k ∞ i + l +1 ≤ i Hence, R f dµ is a PSPACE-rapid L -limit point of A fn .Now, we prove the backward direction. Observe that, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X i =0 Z f ◦ T i .f − Z f dµ Z f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X i =0 Z f ◦ T i .f − Z (cid:18)Z f dµ (cid:19) f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X i =0 Z (cid:18) f ◦ T i − Z f dµ (cid:19) f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z n n − X i =0 f ◦ T i − Z f dµ ! f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k A fn − Z f dµ k k f k A fn is PSPACE-rapid almost everywhere convergent to R f dµ . Hence, there exists a polynomial p such that for any m and m , µ ( x : sup n ≥ p ( m m (cid:12)(cid:12)(cid:12)(cid:12) A fn ( x ) − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) > m )! ≤ m . Let l be any number such that 2 l ≥ k f k ∞ ≥ k f k . Now, for any given m >
0, for all n ≥ p (2 m +2 l +1 , m +5 l +1) , k A fn − Z f dµ k ≤ m +2 l +1 + k f k ∞ m +5 l +1 ≤ m +2 l . Hence, there exists j > m >
0, for all n ≥ ( m + l ) j k A fn − Z f dµ k ≤ m + l . Now given n >
0, let m be any number such that 2 ( m + l ) j ≤ n ≤ ( m + l +1) j , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X i =0 Z f ◦ T i .f − Z f dµ Z f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k f k m + l ≤ m Since n ≤ ( m + l +1) j , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X i =0 Z f ◦ T i .f − Z f dµ Z f dµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ l +1 (ln n ) j The result follows with c = 2 l +1 . PSPACE -rapid almost everywhere convergence of ergodic aver-ages
In the earlier section, we related the rapidity of L convergence of f n to f , to the L convergencespeed of A fn to R f . Now we present PSPACE versions of Theorem 2 and Proposition 5 from [7],relating the L convergence of A fn to R f to its almost everywhere convergence. The main estimatewhich we require in this section is the maximal ergodic inequality, which we now recall. Lemma 4.1 (Maximal ergodic inequality [2]) . If f ∈ L (Σ ∞ , µ ) and δ > then, µ (cid:18)(cid:26) x : sup n ≥ | A fn ( x ) | > δ (cid:27)(cid:19) ≤ k f k δ . Using this lemma, we now prove the almost everywhere convergence of ergodic averages. Incontrast to [7], we give a direct proof of the theorem for L functions with possibly infinite essentialsupremum using Markov’s inequality. Theorem 4.2.
Let f be any function in L (Σ ∞ , µ ) and let T be a measure preserving transforma-tion. If R f dµ is a PSPACE -rapid L -limit point of A fn then A fn is PSPACE -rapid almost everywhereconvergent to R f dµ . roof. By replacing f with f − R f dµ we can assume without loss of generality that R f dµ = 0.We construct a polynomial q such that for any m and m , µ ( x : sup n ≥ q ( m m | A fn ( x ) | > − m )! ≤ − m . Since, R f dµ = 0 is a PSPACE-rapid L -limit point of A fn there is a polynomial p such that ∃ k ≤ p ( m + m +2) with k A fk k ≤ m m .Applying the maximal ergodic inequality to g = A fk , we get µ (cid:18)(cid:26) x : sup n ≥ | A gn ( x ) | > m +1 (cid:27)(cid:19) ≤ m +1 . (3)Expanding A gn , A gn = A fn + u ◦ T n − unk , where u = ( k − f + ( k − f ◦ T + · · · + f ◦ T k − . Note that k u k ≤ k ( k − k f k . Let M be any upper bound for k f k . And let n = (2 p ( m + m +2) − M m + m +2 . From the above,we get k A gn − A fn k ≤ m m for any n ≥ n . Now from Markov’s inequality it follows that, µ (cid:18)(cid:26) x : sup n ≥ n | A fn ( x ) − A gn ( x ) | > m +1 (cid:27)(cid:19) ≤ k A gn − A fn k m +1 ≤ m +1 (4)Hence, from 3 and 4, we get µ (cid:18)(cid:26) x : sup n ≥ n | A fn ( x ) | > m (cid:27)(cid:19) ≤ m . Since n is upper bounded by a term of the form 2 q ( m + m ) for a polynomial q , the claim follows.If f ∈ L ∞ , the converse of Theorem 4.2 can be easily obtained by expanding k A fn − R f dµ k . PSPACE L functions We now establish the main theorem in our work, namely, that for PSPACE L computable func-tions, the ergodic average exists, and is equal to the space average, on every EXP random. Weutilize the almost everywhere convergence results proved in the previous section, to prove theconvergence on every PSPACE/EXP random. The convergence notions involved in proving thePSPACE/SUBEXP-space ergodic theorems and their interrelationships are summarized in Figure1. The following fact was shown in [12]. However, for our ergodic theorem we require an alternateproof of this fact using techniques from [24]. Lemma 5.1.
Let h f n i ∞ n =1 be a PSPACE sequence of simple functions which converges
PSPACE -rapid almost everywhere to f ∈ L (Σ ∞ , µ ) . Then, . A fn Z f dµ A fn PSPACE −−−−−→ a.e Z f dµ SUBEXP ergodic theorem(7 . Theorem 4 . f ∈ L ∞ f ∈ P S P A C E L f ∈ S U B E X P L Figure 1: Relationships between the major convergence notions involving PSPACE simple measurepreserving transformations. A fn R f dµ denotes that R f dµ is a PSPACE-rapid L -limit point of A fn . PSPACE/SUBEXP-space ergodicity is required only for obtaining the ergodic theorems fromPSPACE a.e convergence. lim n →∞ f n ( x ) exists for all EXP random x .2. Given a PSPACE sequence of simple functions h g n i ∞ n =1 which is PSPACE -rapid almost ev-erywhere convergent to f , lim n →∞ g n ( x ) = lim n →∞ f n ( x ) for all EXP random x .Proof. We initially show 1. For each k ≥
0, since h f n i ∞ n =1 is PSPACE-rapid almost everywhereconvergent to f , we have a polynomial q such that µ ( x : sup n ≥ q ( k ) | f n ( x ) − f ( x ) | ≥ k +2 )! ≤ k +2 . It is easy to verify that µ ( x : sup n ≥ q ( k ) | f n ( x ) − f q ( k ) ( x ) | ≥ k +1 )! ≤ k +1 . Define U k = (cid:26) x : max q ( k ) ≤ n ≤ q ( k +1) | f n ( x ) − f q ( k ) ( x ) | ≥ k +1 (cid:27) . Observe that µ ( U k ) ≤ µ ( x : sup n ≥ q ( k ) | f n ( x ) − f q ( k ) ( x ) | ≥ k +1 )! ≤ k +1 . Let r be the controlling polynomial and let M be the PSPACE machine witnessing the fact that h f n i ∞ n =1 is a PSPACE sequence of simple functions. U k is hence a union of cylinders of length atmost r (2 q ( k +1) ). Let the machine N on input ( σ, k ) do the following:1. If | σ | < r (2 q ( k +1) ) then, output 0. 15. Compute f q ( k ) ( σ ∞ ) by running M (1 q ( k ) , σ ) and store the result.3. For each n ∈ [2 q ( k ) , q ( k +1) ] do the following:(a) Compute f n ( σ ∞ ) by running M (1 n , σ ) and store the result.(b) Check if | f n ( σ ∞ ) − f q ( k ) ( σ ∞ ) | ≥ k +1 . If so, output 1.4. Output 0.Since N rejects any σ with length less than r (2 q ( k +1) ), the simulation of M (1 q ( k ) , σ ) is always apolynomial space operation. Hence, N is an EXP-time machine witnessing the fact that h U k i ∞ k =1 is an EXP sequence of open sets.Define, V k = ∞ [ i = k U i . Since µ ( U k ) ≤ − ( k +1) , machine N above can be easily modified to show that h V k i ∞ k =1 is an EXPtest.If x ∈ Σ ∞ is an EXP random then x is in at most finitely many V k and hence in only finitelymany U k . Hence, for large enough k and for all n ≥ q ( k ) we have | f n ( x ) − f q ( k ) ( x ) | ≤ ∞ X j = k j +1 ≤ k . This shows that f n ( x ) is a Cauchy sequence. This completes the proof of 1.Given h g n i ∞ n =1 which is PSPACE-rapid almost everywhere convergent to f , the interleavedsequence f , g , f , g , f , g . . . can be easily verified to be PSPACE-rapid almost everywhere con-vergent to f . 2 now follows directly from 1.The following immediately follows from the above lemma. Corollary 5.2.
Let f ∈ L (Σ ∞ , µ ) be a PSPACE L -computable function with an L approximating PSPACE sequence of simple functions h f n i ∞ n =1 . Then,1. lim n →∞ f n ( x ) exists for all EXP random x .2. Given a PSPACE sequence of simple functions h g n i ∞ n =1 L approximating f , lim n →∞ g n ( x ) =lim n →∞ f n ( x ) for all EXP random x .Proof. For any m , m ≥ µ (cid:18)(cid:26) x : sup n ≥ m + m +1 | f n ( x ) − f ( x ) | ≥ m (cid:27)(cid:19) ≤ ∞ X n = m + m +1 µ (cid:18)(cid:26) x : | f n ( x ) − f ( x ) | ≥ m (cid:27)(cid:19) ≤ ∞ X n = m + m +1 k f n − f k m ≤ ∞ X i =1 m + i = 2 − m . Hence, h f n i ∞ n =1 is PSPACE-rapid almost everywhere convergent to f . The claim follows due toLemma 5.1. 16he following properties satisfied by PSPACE simple transformations and PSPACE L -computablefunctions are useful in our proof of the PSPACE ergodic theorem. Lemma 5.3.
Let f be a PSPACE L -computable function. Let I f : Σ ∞ → Σ ∞ be the constantfunction taking the value R f dµ over all x ∈ Σ ∞ . Then, I f is PSPACE L -computable and e I f ( x ) = R f dµ for all EXP random x .Proof. Let h f n i ∞ n =1 be a PSPACE sequence of simple functions, M be a PSPACE machine and p be a controlling polynomial witnessing the fact that f is PSPACE L -computable. We constructa PSPACE sequence of simple functions h f ′ n i ∞ n =1 where each f ′ n is the constant function taking thevalue R f n dµ . Since k f ′ n − I f k = | R f n dµ − R f dµ | ≤ k f n − f k ≤ − n , it follows that I f is PSPACE L -computable. Now, from part 2 of Lemma 5.1, we get that e I f ( x ) = lim n →∞ f ′ n ( x ) = lim n →∞ R f n dµ = R f dµ for all EXP random x .On input (1 n , σ ), let machine N do the following:1. Let Sum = 0.2. For each α ∈ Σ p ( n ) do the following:(a) Run M (1 n , α ) and add the result to Sum.3. Output Sum / p ( n ) .Let t be a polynomial upper bound for the space complexity of M . Then, the result of M (1 n , α )is always upper bounded by 2 t ( n + p ( n )) and representable in t ( n + p ( n )) space. The sum of at most2 p ( n ) many such numbers is upper bounded by 2 t ( n + p ( n ))+ p ( n ) and representable in t ( n + p ( n ))+ p ( n )space. Dividing the running sum by 2 p ( n ) can also be done in polynomial space. Hence, N is aPSPACE machine computing the sequence h f ′ n i ∞ n =1 where each f ′ n = R f n dµ . Lemma 5.4.
Let f be a PSPACE L -computable function with an L approximating PSPACE sequence of simple functions h f n i ∞ n =1 . Let T be a PSPACE simple transformation and p be apolynomial. Then, D A f p ( n ) n E ∞ n =1 is a PSPACE sequence of simple functions.Proof.
Let q be a controlling polynomial and M f be a machine witnessing the fact that h f n i ∞ n =1 isa PSPACE sequence of simple functions. Let c T be a controlling constant witnessing the fact that T is a PSPACE simple transformation. For any n ≥
1, we have A f p ( n ) n = f p ( n ) + f p ( n ) ◦ T + f p ( n ) ◦ T + . . . f p ( n ) ◦ T n n . The functions (cid:10) f p ( n ) ◦ T i (cid:11) ni =1 are simple functions defined on cylinders of length at most q ( p ( n )) + c T n . Hence, the polynomial r ( n ) = q ( p ( n )) + c T n is a controlling polynomial for the sequence offunctions D A f p ( n ) n E ∞ n =1 as in condition 1 of Definition 3.1. Now, let us verify condition 2 of Definition3.1. Let M be the machine from Lemma 3.5. We construct a machine N such that for each n ∈ N and α ∈ Σ ∗ , N (1 n , α ) = ( A f p ( n ) n ( α ∞ ) if | α | ≥ r ( n )? otherwise.On input (1 n , α ) if | α | < r ( n ) then N outputs ? else it operates as follows:17. Let Sum = 02. For each i ∈ [1 , n ], do the following:(a) For each string σ of length q ( p ( n )), do the following:i. If M (1 i , σ, α ) = 1, then let Sum = Sum + M f (1 p ( n ) , σ ).3. Output Sum /n .If t is a polynomial upper bound for the space complexity of M and t is a polynomial upperbound for the space complexity of M f then, each f p ( n ) ◦ T i can be computed in O ( t (2 q ( p ( n )) + c T n + n ) + t ( q ( p ( n )) + p ( n ))) space for any i ≤ n . The results of computations of f p ( n ) ◦ T i for i ∈ [1 , n ] can be added up and divided by n in polynomial space. N outputs the result of thiscomputation. Since N is a PSPACE machine, the proof is complete.Now, we prove the ergodic theorem for PSPACE L functions, which is our main result. Theproof involves adaptations of techniques from Rute [24], together with new quantitative boundswhich yield the result within prescribed resource bounds. Theorem 5.5.
Let T : (Σ ∞ , B (Σ ∞ ) , µ ) → (Σ ∞ , B (Σ ∞ ) , µ ) be a PSPACE ergodic measure preserv-ing transformation. Then, for any
PSPACE L -computable f , lim n →∞ f A fn = R f dµ on EXP randoms.Proof.
Let h f m i ∞ m =1 be any PSPACE sequence of simple functions L approximating f . We initiallyapproximate A fn with a PSPACE sequence of simple functions h g n i ∞ n =1 which converges to R f dµ on EXP randoms. Then we show that e A fn has the same limit as g n on PSPACE randoms and henceon EXP randoms.For each n , it is easy to verify that D A f m n E ∞ m =1 is a PSPACE sequence of simple functions L approximating A fn with the same rate of convergence. Using techniques similar to those in Lemma5.1 and Corollary 5.2, we can obtain a polynomial p such that µ ( x : sup m ≥ p ( n + i ) | A f m n ( x ) − A f p ( n + i ) n ( x ) | ≥ n + i +1 )! ≤ n + i +1 . For every n >
0, let g n = A f p ( n ) n . We initially show that h g n i ∞ n =1 converges to R f dµ on EXPrandoms. Let m , m ≥
0. From Theorem 4.2, A fn is PSPACE-rapid almost everywhere convergentto R f dµ . Hence there is a polynomial q such that µ ( x : sup n ≥ q ( m m | A fn ( x ) − Z f dµ | ≥ m +1 )! ≤ m +1 . Let N ( m , m ) = max { m , m , q ( m + m ) } . Then, X n ≥ N ( m ,m ) k +1 = 12 N ( m ,m ) ≤ min (cid:26) m +1 , m +1 (cid:27) . µ ( x : sup n ≥ N ( m ,m ) | g n − Z f dµ | > m )! ≤ X n ≥ N ( m ,m ) µ (cid:18)(cid:26) x : | g n − A fn ( x ) | > m +1 (cid:27)(cid:19) + µ ( x : sup n ≥ q ( m m | A fn ( x ) − Z f dµ | ≥ m +1 )! ≤ X n ≥ N ( m ,m ) n +1 + 12 m +1 ≤ m . Note that N ( m , m ) is bounded by 2 ( m + m ) c for some c ∈ N . Hence, g n is PSPACE-rapid almosteverywhere convergent to R f dµ . From Lemma 5.4 it follows that h g n i ∞ n =1 = D A f p ( n ) n E ∞ n =1 is aPSPACE sequence of simple functions (in parameter n ). Let I f : Σ ∞ → Σ ∞ be the constantfunction taking the value R f dµ over all x ∈ Σ ∞ . From the above observations and Lemma 5.1 weget that lim n →∞ g n ( x ) = e I f ( x ) for any x which is EXP random. Now, from Lemma 5.3, we get thatlim n →∞ g n ( x ) = R f dµ for any x which is EXP random.We now show that lim n →∞ e A fn = lim n →∞ g n on PSPACE randoms. Define U n,i = (cid:26) x : max p ( n + i ) ≤ m ≤ p ( n + i +1) | A f m n ( x ) − A f p ( n + i ) n ( x ) | ≥ n + i +1 (cid:27) . We already know µ ( U n,i ) ≤ n + i +1 . U n,i can be shown to be polynomial space approximable inparameters n and i in the following sense. There exists a sequence of sets of strings h S n,i i i,n ∈ N andpolynomial p satisfying the following conditions:1. U n,i = [ S n,i ].2. There exists a controlling polynomial r such that max {| σ | : σ ∈ S n,i ) } ≤ r ( n + i ).3. The function g : Σ ∗ × ∗ × ∗ → { , } such that g ( σ, n , i ) = ( σ ∈ S n,i N computing the function g above. Let M f bea computing machine and let q be a controlling polynomial for h f n i ∞ n =1 . Let c be a controllingconstant for T . Let M ′ be the machine from Lemma 3.5. Machine N on input ( σ, n , i ) does thefollowing:1. If | σ | > q ( p ( n + i + 1)) + cn , then output 0.2. Compute A f p ( n + i ) n ( σ ∞ ) as in Lemma 5.4 by using M f and M ′ and store the result.3. For each m ∈ [ p ( n + i ) , p ( n + i + 1)] do the following:19a) Compute A f m n ( σ ∞ ) as in Lemma 5.4 by using M f and M ′ and store the result.(b) Check if | A f m n ( σ ∞ ) − A f p ( n + i ) n ( σ ∞ ) | ≥ n + i +1 . If so, output 1.4. Output 0.It can be easily verified that N is a PSPACE machine. r ( n + i ) = q ( p ( n + i + 1)) + cn is thecontrolling polynomial for h U n,i i n,i ∈ N . Now, define V m = [ n,i ≥ n + i = m U n,i . Note that, µ ( V m ) ≤ m m . It can be shown that for any j , X n>j m m = 12 j − + j j . Given any k ≥
0, let p ( k ) = 3( k + 1). Hence, we have ∞ X n = p ( k )+1 m m = 12 k +1) + 3( k + 1)2 k +1) < k +1 + 12 k +1 k + 1)2 k +1) < k +1 = 12 k . The last inequality holds since 3( k + 1) < k +1) for all k ≥
0. Since each V m is a finite unionof sets from h U n,i i n,i ∈ N , the machine computing h U n,i i n,i ∈ N can be easily modified to constructa machine witnessing that h V m i ∞ m =1 is a PSPACE approximable sequence of sets. From theseobservations, it follows that h V m i ∞ m =1 is a PSPACE Solovay test. Now, let x be a PSPACE random. x is in at most finitely many V m and hence in at most finitely many U n,i . Hence for some largeenough N for all n ≥ N , i ≥ m such that p ( n + i ) ≤ m ≤ p ( n + i + 1), we have | A f m n ( x ) − A f p ( n + i ) n ( x ) | < n + i +1 . It follows that for all n ≥ N and for all m ≥ p ( n ) that, | A f m n ( x ) − g n ( x ) | = | A f m n ( x ) − A f p ( n ) n ( x ) | ≤ ∞ X i =0 n + i +1 ≤ − n . Therefore, lim n →∞ e A fn ( x ) = lim n →∞ g n ( x ) on all PSPACE random x and hence on all x which is EXPrandom.Hence, we have shown that lim n →∞ e A fn = R f dµ on EXP randoms which completes the proof ofthe theorem. PSPACE
Ergodic Theorem
In this section we give a partial converse to the PSPACE ergodic theorem (Theorem 5.5). Weshow that for any PSPACE null x , there exists a function f and transformation T satisfying all theconditions in Theorem 5.5 such that f A fn ( x ) does not converge to R f dµ .Let us first observe that due to Corollary 5.2, Theorem 5.5 is equivalent to the following:20 heorem. Let T be a PSPACE ergodic measure preserving transformation such that for any
PSPACE L -computable f , R f dµ is an PSPACE -rapid L -limit point of A fn . Let { g n,i } be anycollection of simple functions such that for each n , h g n,i i ∞ i =1 is a PSPACE L -approximation of f A fn .Then, lim n →∞ lim i →∞ g n,i ( x ) = R f dµ for any EXP random x . Hence, the ideal converse to Theorem 5.5 is the following:
Theorem.
Given any
EXP null x , there exists a PSPACE ergodic measure preserving transforma-tion T and PSPACE L -computable f ∈ L (Σ ∞ , µ ) such that the following conditions are true:1. R f dµ is an PSPACE -rapid limit point of A fn .2. There exists a collection of simple functions { g n,i } such that for each n , h g n,i i ∞ i =1 is a PSPACE L -approximation of A fn but lim n →∞ lim i →∞ g n,i ( x ) = R f dµ . But, we prove the following partial converse to Theorem 5.5.
Theorem 6.1.
Given any
PSPACE null x , there exists a PSPACE L -computable f ∈ L (Σ ∞ , µ ) such that for any PSPACE simple measure preserving transformation, the following conditions aretrue:1. For all n ∈ N , k A fn − R f dµ k = 0 . Hence, R f dµ is an PSPACE -rapid L -limit point of A fn .2. There exists a collection of simple functions { g n,i } such that for each n , h g n,i i ∞ i =1 is a PSPACE L -approximation of A fn but lim n →∞ lim i →∞ g n,i ( x ) = R f dµ . A proof of the above theorem requires the construction in the following lemma.
Lemma 6.2.
Let h U n i ∞ n =1 be a PSPACE test. Then there exists a sequences of sets
D b S n E ∞ n =1 suchthat for each n ∈ N , b S n ⊆ Σ ∗ satisfying the following conditions:1. µ ([ b S n ]) ≤ − n .2. ∩ ∞ m =1 ∪ ∞ n = m [ b S n ] ⊇ ∩ ∞ n =1 U n .3. There exists c ∈ N such that for all n , σ ∈ b S n implies | σ | ≤ n c .4. There exists a PSPACE machine N such that N ( σ, n ) = 1 if σ ∈ b S n and 0 otherwise.Proof of Lemma 6.2. Let (cid:10) S kn (cid:11) n,k be the collection of approximating sets and M be the machinecomputing h U n i ∞ n =1 as in Definition 2.2. Define U = ∪ ∞ n =1 ∪ ∞ k =1 S kn . Now, define T n = n +1 [ i =1 S n +2 i Observe that, [ U ] \ [ T n ] ⊆ n +1 [ i =1 U i − [ S n +2 i ] ! [ ∞ [ i = n +2 U i ! µ ([ U ] \ [ T n ]) ≤ n +1 X i =1 n +2 + X i = n +2 i ≤ n + 12 n +1+ n +1 + 12 n +1 ≤ n . From the definition of T n , it follows that there is a c ∈ N such that the length of strings in T n isupper bounded by n c . Now, if b S n = { σ ∈ T n +1 : ∀ α ⊑ σ ( α T n ) } , we have µ ([ b S n ]) ≤ µ ([ U ] \ [ T n ]) ≤ − n . Conditions 2 and 3 can be readily verified to be true. We now construct a PSPACE machine N satisfying the condition in 4. N on input ( σ, n ) does the following:1. For each i ∈ [1 , ( n + 1) + 1] simulate M ( σ, i , n +1)+2 ). If all simulations result in 0, output0.2. Else, for each m ∈ [1 , n ] do the following:(a) For each α ⊑ σ do the following:i. For each i ∈ [1 , m + 1] simulate M ( α, i , m +2 ). If any of these simulations resultin a 1 then, output 0.3. Output 1. N can be easily verified to be a PSPACE machine. Hence, our constructions satisfy all the desiredconditions.Now, we prove Theorem 6.1. Proof of Theorem 6.1.
Let h V n i ∞ n =1 be any PSPACE test such that x ∈ ∩ ∞ n =1 V n . Now, from Lemma6.2, there exists a collection of sets D b S n E ∞ n =1 such that ∩ ∞ m =1 ∪ ∞ n = m [ b S n ] ⊇ ∩ ∞ n =1 V n . Let, U n = { σ : [ σ ] ∈ b S i for some i such that 2 n + 1 ≤ i ≤ n + 1) + 1 } Now, let f n = nχ U n . Since, µ ( U n ) ≤ n +1)+1 X i =2 n +1 i ≤ n it follows that k f n k ≤ n n + n ≤ n . Now, using the properties of
D b S n E ∞ n =1 , it can be shown that h f n i ∞ n =1 is a PSPACE L -approximationof f = 0. We construct a machine M computing h f n i ∞ n =1 . The other conditions are easily verified.Let N be the machine from Lemma 6.2. On input (1 n , σ ), M does the following:1. If | σ | < (2( n + 1) + 1) c then, output ?. 22. Else, for each i ∈ [2 n + 1 , n + 1) + 1] do the following:(a) For each α ⊆ σ , do the following:i. If N (1 i , α ) = 1 then, output n .3. Output 0. M uses at most polynomial space and computes h f n i ∞ n =1 . Now, define g n,i = f i + f i ◦ T + · · · + f i ◦ T n − n For any fixed n ∈ N , since T is a PSPACE simple transformation, as in Lemma 5.4 it can be shownthat h g n,i i ∞ i =1 is a PSPACE L -approximation of A fn . We know that there exist infinitely many m such that x ∈ [ b S m ]. For any such m , let i be the unique number such that 2 i + 1 ≤ m ≤ i + 1) + 1.For this i , f i ( x ) = i . This shows that there exist infinitely many i such that f i ( x ) = i . Since each f i is a non-negative function, it follows that there are infinitely many i with g n,i ≥ i/n . Hence, iflim i →∞ g n,i ( x ) exists, then it is equal to ∞ . It may be the case that lim i →∞ g n,i ( x ) does not exist. Ineither case, lim n →∞ lim i →∞ g n,i ( x ) cannot be equal to R f dµ = 0. Hence, our construction satisfies all thedesired conditions. SUBEXP -space randoms and its converse
In the previous sections, we demonstrated that for PSPACE L -computable functions and PSPACEsimple transformations, the Birkhoff averages converge to the desired value over EXP randoms.However, the converse holds only over PSPACE non-randoms. The two major reasons for this gap are the following: PSPACE-rapid convergence necessitates exponential length cylinders whileconstructing the randomness tests, and PSPACE L -computable functions are not strong enoughto capture all PSPACE randoms. In this section, we demonstrate that for a different notion of ran-domness - SUBEXP-space randoms and a larger class of L -computable functions (SUBEXP-space L -computable), we can prove the ergodic theorem on the randoms and obtain its converse on thenon-randoms. Analogous to Towsner and Franklin [5], we demonstrate that the ergodic theorem forPSPACE simple transformations and SUBEXP-space L -computable functions satisfying PSPACErapidity, fails for exactly this class of non-random points. We first introduce SUBEXP-space testsand SUBEXP-space randomness. Definition 7.1 (SUBEXP-space sequence of open sets) . A sequence of open sets h U n i ∞ n =1 is aSUBEXP -space sequence of open sets if there exists a sequence of sets (cid:10) S kn (cid:11) k,n ∈ N , where S kn ⊆ Σ ∗ such that1. U n = ∪ ∞ k =1 [ S kn ], where for any m > µ (cid:0) U n − ∪ mk =1 [ S kn ] (cid:1) ≤ m − log( m ) .2. There exists a controlling polynomial p such that max {| σ | : σ ∈ ∪ mk =1 S kn ) } ≤ p (log( n )+log( m )) .3. The function g : Σ ∗ × ∗ × ∗ → { , } such that g ( σ, n , m ) = 1 if σ ∈ S mn , and 0 otherwise,is decidable by a PSPACE machine. Definition 7.2 (SUBEXP-space randomness) . A sequence of open sets h U n i ∞ n =1 is a SUBEXP -space test if it is a SUBEXP-space sequence of open sets and for all n ∈ N , µ ( U n ) ≤ n − log( n ) .23 set A ⊆ Σ ∞ is SUBEXP -space null if there is a SUBEXP-space test h U n i ∞ n =1 such that A ⊆ ∩ ∞ n =1 U n . A set A ⊆ Σ ∞ is SUBEXP -space random if A is not SUBEXP-space null. The slower decay rate of n − log( n ) = 2 − log( n ) enables us to obtain an ergodic theorem and anexact converse in the SUBEXP-space setting .The following results are useful in manipulating sumsinvolving terms of the form 2 − (log( n )) k for k ≥ Lemma 7.3.
For any k, m ∈ N , P ∞ i = m +1 1 n k +1 ≤ m k Proof.
Let k = 1. Then, for any m > ∞ X i = m +1 n < m ( m + 1) + 1( m + 1)( m + 2) + . . . = 1 m − m + 1 + 1 m + 1 − m + 2 + . . . = 1 m For k > ∞ X i = m +1 n k +1 < m k − ∞ X i = m +1 n The proof now follows by applying the result when k = 1 to bound the summation on the rightwith m − . Lemma 7.4.
For any m ∈ N , P ∞ n =2(2 m +1) nn log( n ) ≤ m log( m ) .Proof. ∞ X n =2(2 m +1) nn log( n ) ≤ ∞ X n =2(2 m +1) n log( n ) − ≤ ∞ X n =2 m +1 n log(2 m ) ≤ ∞ X n =2 m +1 n ⌊ log( m ) ⌋ +1 ≤ m ) ⌊ log( m ) ⌋ ) ≤ m log( m ≤ m log( m ) The fourth inequality above follows from Lemma 7.3. It is easy to see that the set of SUBEXP-space randoms is smaller than the set of PSPACE randoms. But, wedo not know if any inclusion holds between SUBEXP-space randoms and EXP-randoms.
24 similar inequality can be trivially seen to be true on replacing n/n log( n ) with 1 /n log( n ) . Now,we introduce the Solovay analogue of SUBEXP-space randomness and prove that these notions areanalogous. Definition 7.5 (SUBEXP-space Solovay test) . A sequence of open sets h U n i ∞ n =1 is a SUBEXP -space Solovay test if it is a SUBEXP-space sequence of open sets and there exists a polynomial p such that ∀ m ≥ P ∞ n = p ( m )+1 µ ( U n ) ≤ m log( m ) . A set A ⊆ Σ ∞ is SUBEXP -space Solovay null if there exists a SUBEXP-space Solovay test h U n i ∞ n =1 such that A ⊆ ∩ ∞ i =1 ∪ ∞ n = i U n . A ⊆ Σ ∞ isSUBEXP -space Solovay random if A is not SUBEXP-space Solovay null. Lemma 7.6.
A set A ⊆ Σ ∞ is SUBEXP -space null if and only if A is SUBEXP -space Solovaynull.Proof.
Using Lemma 7.4, it is easy to see that if A is SUBEXP-space null then A is SUBEXPSolovay null. Conversely, let A be SUBEXP-space Solovay null and let h U n i ∞ n =1 be any Solovay testwhich witnesses this fact. Let V n = ∪ ∞ i = p ( n )+1 U n . We show that h V n i ∞ n =1 is a SUBEXP-space test.Let (cid:10) S kn (cid:11) n,k ∈ N be any sequence of sets approximating h U n i ∞ n =1 as in definition 7.1 such that (cid:10) S kn (cid:11) ∞ k =1 is increasing for each n . We define a sequence of sets (cid:10) T kn (cid:11) n,k ∈ N approximating V n as follows.Let r ( n, k ) = max { k + 1) + 1) + 1) , p ( n ) + 1 } . Define T kn = r ( n,k ) [ i = p ( n )+1 S ( r ( n,k ) − p ( n ))2(2 k +1) i . We can easily verify conditions 1 and 3 in definition 7.1. From the definition of T kn , it can beverified that µ ( V n − [ T kn ]) ≤ r ( n,k ) X i = p ( n )+1 − log( r ( n,k ) − p ( n )) − log(2(2 k +1)) + ∞ X n = r ( n,k )+1 − log( n ) ≤ r ( n,k ) X i = p ( n )+1 − log( r ( n,k ) − p ( n )) − log(2(2 k +1)) + ∞ X n =2((2(2 k +1)+1) +1) − log( n ) ≤ r ( n, k ) − p ( n )2 log( r ( n,k ) − p ( n )) log(2(2 k +1)) + 12 log(2(2 k +1)+1) < log(2(2 k +1)) + 12 log(2(2 k +1)+1) < log( k ) . The third inequality and the last inequality above follows from Lemma 7.4. Using the machine M and controlling polynomial p witnessing that h U n i ∞ n =1 is a PSPACE sequence of open sets, wecan construct the corresponding machines for h V n i ∞ n =1 in the following way. Machine N on input( σ, n , k ) does the following:1. For each i ∈ [ p ( n ) + 1 , r ( n, k )] do the following:(a) Output 1 if M ( σ, i , ( r ( n,k ) − p ( n ))2(2 k +1) ) = 1. This implies that P ∞ n =1 µ ( U n ) < ∞
25. Output 0 if none of the above computations results in 1.It is straightforward to verify that N is a PSPACE machine.Now, we define SUBEXP-space analogues of concepts from Section 3. Definition 7.7 (SUBEXP-space sequence of simple functions) . A sequence of simple functions h f n i ∞ n =1 where each f n : Σ ∞ → Q is a SUBEXP -space sequence of simple functions if1. There is a controlling polynomial p such that for each n , there exist k ( n ) ∈ N , { d , d . . . , d k ( n ) } ⊆ Q and { σ , σ . . . σ k ( n ) } ⊆ Σ p (log( n )) such that f n = P k ( n ) i =1 d i χ σ i , where χ σ i is the characteristicfunction of the cylinder [ σ i ].2. There is a PSPACE machine M such that for each n ∈ N , σ ∈ Σ ∗ M (1 n , σ ) = ( f n ( σ ∞ ) if | σ | ≥ p (log( n )) ? otherwise. Definition 7.8 (SUBEXP-space L -computable functions) . A function f ∈ L (Σ ∞ , µ ) is SUBEXP-space L -computable if there exists a SUBEXP-space sequence of simple functions h f n i ∞ n =1 suchthat for every n ∈ N , k f − f n k ≤ n − log( n ) . The sequence h f n i ∞ n =1 is called a SUBEXP -space L -approximation of f .We require the following equivalent definitions of PSPACE-rapid convergence notions for work-ing in the setting of SUBEXP-space randomness. Lemma 7.9.
A real number a is a PSPACE -rapid limit point of the real number sequence h a n i ∞ n =1 if and only if there exists a polynomial p such that for all m ∈ N , ∃ k ≤ p (log( m )) such that | a k − a | ≤ m − log( m ) .Proof. We prove the forward direction first. If a is a PSPACE-rapid limit point of h a n i ∞ n =1 , thereexists a polynomial q such that for all l ∈ N , ∃ k ≤ q ( l ) such that | a l − a | ≤ − m . Substituting l = ⌈ log( m ) ⌉ , we get that for all m ∈ N , ∃ k ≤ q ( ⌈ log( m ) ⌉ ) ≤ q (log( m )+1) such that | a l − a | ≤ −⌈ log( m ) ⌉ ≤ − log( m ) . Conversely, assume that there exists a polynomial p such that for all m ∈ N , ∃ k ≤ p (log( m )) such that | a k − a | ≤ − log( m ) . Substituting m = 2 ⌈ l ⌉ , we get that for all m ∈ N , ∃ k ≤ q ( ⌈ l ⌉ ) ≤ q ( l ) such that | a l − a | ≤ −⌈ l ⌉ ≤ − l . Hence, a is a PSPACE-rapid limit point of h a n i ∞ n =1 . Lemma 7.10.
A sequence of measurable functions h f n i ∞ n =1 is PSPACE -rapid almost everywhereconvergent to a measurable function f if and only if there exists a polynomial p such that for all m and m , µ ( x : sup n ≥ p (log( m m | f n ( x ) − f ( x ) | ≥ m log( m )1 )! ≤ m log( m )2 . The same technique used in the proof of Lemma 7.9 can be used to prove this claim.Before addressing the main result, let us define SUBEXP-space ergodicity.
Definition 7.11 (SUBEXP-space ergodic transformations) . A measurable function T : (Σ ∞ , µ ) → (Σ ∞ , µ ) is SUBEXP -space ergodic if T is a PSPACE simple transformation such that for anySUBEXP-space L -computable f ∈ L (Σ ∞ , µ ), R f dµ is a PSPACE-rapid L limit point of A f,Tn .26ow, we prove SUBEXP analogues of the auxiliary lemmas from Section 5. Lemma 7.12.
Let h f n i ∞ n =1 be a SUBEXP -space sequence of simple functions which converges
PSPACE -rapid almost everywhere to f ∈ L (Σ ∞ , µ ) . Then,1. lim n →∞ f n ( x ) exists for all SUBEXP -space random x .2. Given a SUBEXP -space sequence of simple functions h g n i ∞ n =1 which is PSPACE -rapid almosteverywhere convergent to f , lim n →∞ g n ( x ) = lim n →∞ f n ( x ) for all SUBEXP -space random x .Proof. We initially show 1. For each k ≥
0, since h f n i ∞ n =1 is PSPACE-rapid almost everywhereconvergent to f , there exists c ∈ N such that µ ( x : sup n ≥ ⌈ log( k ) c ⌉ | f n ( x ) − f ( x ) | ≥ log(2 k ) )! ≤ log(2 k ) . Since (log(2 k )) = (log( k ) + 1) ≥ log( k ) + 1, we get µ ( x : sup n ≥ ⌈ log( k ) c ⌉ | f n ( x ) − f ( x ) | ≥ log( k ) +1 )! ≤ log( k ) +1 . It is easy to verify that µ ( x : sup n ≥ ⌈ log( k ) c ⌉ | f n ( x ) − f ⌈ log( k ) c ⌉ ( x ) | ≥ log( k ) )! ≤ log( k ) . Define U k = (cid:26) x : max ⌈ log( k ) c ⌉ ≤ n ≤ ⌈ log( k +1) c ⌉ | f n ( x ) − f ⌈ log( k ) c ⌉ ( x ) | > ⌊ log( k ) ⌋ (cid:27) . Observe that µ ( U k ) ≤ µ ( x : sup n ≥ ⌈ log( k ) c ⌉ | f n ( x ) − f ⌈ log( k ) c ⌉ ( x ) | ≥ log( k ) )! ≤ log( k ) . Let q be the controlling polynomial and let M be the PSPACE machine witnessing the fact that h f n i ∞ n =1 is a SUBEXP-space sequence of simple functions. U k is hence a union of cylinders of lengthat most 2 q (log(2 ⌈ log( k +1) c ⌉ )) = 2 q ( ⌈ log( k +1) c ⌉ ) which is upper bounded by 2 log( k ) d for some d ∈ N . The fact that functions of the form 2 log( n ) i areclosed under composition enables us to obtain convergence on SUBEXP-space randoms instead ofEXP randoms as in Lemma 5.1. The machine M can be used to construct a machine N that oninput ( σ, k ) outputs 1 if [ σ ] ⊆ U k and outputs 0 otherwise. Let the machine N on input ( σ, k ) dothe following:1. If | σ | < q ( ⌈ log( k +1) c ⌉ ) then, output 0.2. Compute f ⌈ log( k ) c ⌉ ( σ ∞ ) by running M (1 ⌈ log( k ) c ⌉ , σ ) and store the result.27. For each n ∈ [2 ⌈ log( k ) c ⌉ , ⌈ log( k +1) c ⌉ ] do the following:(a) Compute f n ( σ ∞ ) by running M (1 n , σ ) and store the result.(b) Check if | f n ( σ ∞ ) − f ⌈ log( k ) c ⌉ ( σ ∞ ) | ≥ ⌊ log( k )2 ⌋ . If so, output 1.4. Output 0Since N rejects any σ with length less than 2 q ( ⌈ log( k +1) c ⌉ ) , the simulation of M (1 ⌈ log( k ) c ⌉ , σ ) is alwaysa polynomial space operation. Hence, N witnesses the fact that h U k i ∞ k =1 is an EXP sequence ofopen sets.Define V k = ∞ [ i =2(2 k +1) U i . It follows from Lemma 7.4 that µ ( V k ) ≤ − log( k ) and from the above observations it can be verifiedthat h V k i ∞ k =1 is an SUBEXP-space test.If x ∈ Σ ∞ is an SUBEXP-space random then x is in at most finitely many V k and hence in onlyfinitely many U k . Hence, for some k and for all n ≥ ⌈ log( k ) c ⌉ using Lemma 7.3 we have, | f n ( x ) − f ⌈ log( k ) c ⌉ ( x ) | ≤ ∞ X j = k log( j ) ≤ ∞ X j = k j ⌊ log( k ) ⌋ ≤ k − ( ⌊ log( k ) ⌋− . This shows that f n ( x ) is a Cauchy sequence. This completes the proof of 1.Given h g n i ∞ n =1 which is PSPACE-rapid almost everywhere convergent to f , the interleavedsequence f , g , f , g , f , g . . . can be easily verified to be PSPACE-rapid almost everywhere con-vergent to f . 2 now follows directly from 1.The following immediately follows from the above lemma. Corollary 7.13.
Let f ∈ L (Σ ∞ , µ ) be a SUBEXP -space L -computable function with an L approximating SUBEXP -space sequence of simple functions h f n i ∞ n =1 . Then,1. lim n →∞ f n ( x ) exists for all SUBEXP random x .2. Given a SUBEXP -space sequence of simple functions h g n i ∞ n =1 L approximating f , lim n →∞ g n ( x ) =lim n →∞ f n ( x ) for all SUBEXP random x . roof. For any m , m ≥ µ ( x : sup n ≥ m m ) +1) | f n ( x ) − f ( x ) | ≥ log( m ) )! ≤ ∞ X n =2(2( m m ) +1) µ (cid:18)(cid:26) x : | f n ( x ) − f ( x ) | ≥ log( m ) (cid:27)(cid:19) ≤ ∞ X n =2(2( m m ) +1) k f n − f k log( m ) ≤ log( m ) ∞ X n =2(2( m m ) +1) log( n ) ≤ log( m ) log( m m ) ≤ log( m ) log( m ) +log( m ) = 12 log( m ) . Hence, h f n i ∞ n =1 is PSPACE-rapid almost everywhere convergent to f . The claim now follows fromLemma 7.12.The following are SUBEXP-space analogues of Lemma 5.4 and Lemma 5.3. Lemma 7.14.
Let f be a SUBEXP -space L -computable function. Let I f : Σ ∞ → Σ ∞ be the con-stant function taking the value R f dµ over all x ∈ Σ ∞ . Then, I f is SUBEXP -space L -computableand e I f ( x ) = R f dµ for all SUBEXP -space random x .Proof. Let h f n i ∞ n =1 be a SUBEXP-space sequence of simple functions, M be a PSPACE machineand p be a controlling polynomial witnessing the fact that f is SUBEXP-space L -computable.We construct a SUBEXP-space sequence of simple functions h f ′ n i ∞ n =1 where each f ′ n is the constantfunction taking the value R f n dµ . Since k f ′ n − I f k = | R f n dµ − R f dµ | ≤ k f n − f k ≤ n − log( n ) ,it follows that I f is SUBEXP L -computable. Now, from part 2 of Lemma 7.12, we get that e I f ( x ) = lim n →∞ f ′ n ( x ) = lim n →∞ R f n dµ = R f dµ for all SUBEXP-space random x .On input (1 n , σ ), let machine N do the following:1. If | σ | < ⌈ p (log( n )) ⌉ output ?. Else, do the following:2. Let Sum = 0.3. For each α ∈ Σ ⌈ p (log( n )) ⌉ do the following:(a) Run M (1 n , α ) and add the result to Sum.4. Output Sum / ⌈ p (log( n )) ⌉ .Let t be a polynomial upper bound for the space complexity of M . Then, the result of M (1 n , α ) isalways upper bounded by 2 t ( n +2 ⌈ p (log( n )) ⌉ ) and representable in t ( n +2 ⌈ p (log( n )) ⌉ ) space. The sum of atmost 2 ⌈ p (log( n )) ⌉ many such numbers is upper bounded by 2 t ( n +2 ⌈ p (log( n )) ⌉ )+2 ⌈ p (log( n )) ⌉ and representablein t ( n + 2 ⌈ p (log( n )) ⌉ ) + 2 ⌈ p (log( n )) ⌉ space. Since N rejects any σ with | σ | < ⌈ p (log( n )) ⌉ , calculating therunning sum can be done in polynomial space. Dividing the running sum by 2 ⌈ p (log( n )) ⌉ can also bedone in polynomial space due to the same reason. Hence, N is a PSPACE machine computing thesequence h f ′ n i ∞ n =1 where each f ′ n = R f n dµ . 29 emma 7.15. Let f be a SUBEXP -space L -computable function with an L approximating SUBEXP -space sequence of simple functions h f n i ∞ n =1 . Let T be a PSPACE simple transformation and p be apolynomial. Then, D A f p ( n ) n E ∞ n =1 is a SUBEXP -space sequence of simple functions.Proof.
Let q be a controlling polynomial and M f be a machine witnessing the fact that h f n i ∞ n =1 is a SUBEXP-space sequence of simple functions. Let c T be a controlling constant witnessing thefact that T is a PSPACE simple transformation. For any n ≥
1, we have A f p ( n ) n = f p ( n ) + f p ( n ) ◦ T + f p ( n ) ◦ T + . . . f p ( n ) ◦ T n n . The functions (cid:10) f p ( n ) ◦ T i (cid:11) ni =1 are simple functions defined on cylinders of length at most 2 q (log( p ( n ))) + c T n . Hence, r ( n ) = 2 q ( ⌈ log( p ( n )) ⌉ ) + c T n , which can be upper bounded by 2 log( n ) d for some d ∈ N is a controlling function for the sequence of functions D A f p ( n ) n E ∞ n =1 as in condition 1 of Definition7.7. Now, we verify condition 2 of Definition 7.7. Let M be the machine from Lemma 3.5. Weconstruct a machine N such that for each n ∈ N and σ ∈ Σ ∗ , N (1 n , σ ) = ( A f p ( n ) n ( σ ∞ ) if | σ | ≥ r ( n )? otherwise.On input (1 n , α ) if | α | < r ( n ) = then N outputs ? else it operates as follows:1. Let Sum = 02. For each i ∈ [1 , n ], do the following:(a) For each string σ of length 2 q ( ⌈ log( p ( n )) ⌉ ) , do the following:i. If M (1 i , σ, α ) = 1, then let Sum = Sum + M f (1 p ( n ) , σ ).3. Output Sum /n .If t is a polynomial upper bound for the space complexity of M and t is a polynomial upperbound for the space complexity of M f then, each f p ( n ) ◦ T i can be computed in O ( t (2 q ( ⌈ log( p ( n )) ⌉ ) + c T n + n ) + t (2 q ( ⌈ log( p ( n )) ⌉ ) + p ( n ))) space for any i ≤ n . Since every α such that | α | < r ( n ) =2 q ( ⌈ log( p ( n )) ⌉ ) + c T n is rejected by N , the computations of f p ( n ) ◦ T i is done in polynomial space. Theresults of computations of f p ( n ) ◦ T i for i ∈ [1 , n ] can be added up and divided by n in polynomialspace. N outputs the result of this computation. Since N is a PSPACE machine, the proof iscomplete.Now, we proceed onto proving the SUBEXP-space ergodic theorem. Theorem 7.16.
Let T : (Σ ∞ , B (Σ ∞ ) , µ ) → (Σ ∞ , B (Σ ∞ ) , µ ) be a SUBEXP -space ergodic measurepreserving transformation. Then, for any
SUBEXP -space L -computable f , lim n →∞ f A fn = R f dµ on SUBEXP -space randoms.Proof.
Let h f m i ∞ m =1 be any SUBEXP-space sequence of simple functions L approximating f .We initially approximate A fn with a SUBEXP-space sequence of simple functions h g n i ∞ n =1 whichconverges to R f dµ on SUBEXP-space randoms. Then we show that e A fn has the same limit as g n on SUBEXP-space randoms. 30or each n , it is easy to verify that D A f m n E ∞ m =1 is a SUBEXP sequence of simple functions L approximating A fn with the same rate of convergence. Using techniques similar to those in Lemma7.12 and Corollary 7.13, we can obtain a polynomial p such that µ ( x : sup m ≥ p ( n + i ) | A f m n ( x ) − A f p ( n + i ) n ( x ) | ≥ log( n + i ) )! ≤ log( n + i ) . For every n >
0, let g n = A f p ( n ) n . We initially show that h g n i ∞ n =1 converges to R f dµ on SUBEXP-space randoms. Let m , m ≥
0. From Theorem 4.2, A fn is PSPACE-rapid almost everywhereconvergent to R f dµ . Hence there is a d ∈ N such that µ ( x : sup n ≥ (log( m m d | A fn ( x ) − Z f dµ | ≥ log( m ) +1 )! ≤ log( m ) +1 . Let N ( m , m ) = max { m ) + 1) , log( m + m ) d } . Using Lemma 7.4, X n ≥ N ( m ,m ) log( n ) ≤ X n ≥ m ) +1) log( n ) ≤ log(2 m ) Now, we have µ ( x : sup n ≥ N ( m ,m ) | g n − Z f dµ | > m )! ≤ X n ≥ N ( m ,m ) µ (cid:18)(cid:26) x : | g n − A fn ( x ) | > log( m ) +1 (cid:27)(cid:19) + µ ( x : sup n ≥ log( m m d | A fn ( x ) − Z f dµ | ≥ log( m ) +1 )! ≤ X n ≥ N ( m ,m ) log( n ) + 12 log( m ) +1 ≤ log(2 m ) + 12 log( m ) +1 ≤ log( m ) +1 + 12 log( m ) +1 ≤ log( m ) . Hence, g n is PSPACE-rapid almost everywhere convergent to R f dµ . From Lemma 7.15 it followsthat h g n i ∞ n =1 = D A f p ( n ) n E ∞ n =1 is a SUBEXP-space sequence of simple functions (in parameter n ). Let I f : Σ ∞ → Σ ∞ be the constant function taking the value R f dµ over all x ∈ Σ ∞ . From the aboveobservations and Lemma 7.12 we get that lim n →∞ g n ( x ) = e I f ( x ) for any x which is SUBEXP-spacerandom. Now, from Lemma 7.14, we get that lim n →∞ g n ( x ) = R f dµ for any x which is SUBEXP-spacerandom.We now show that lim n →∞ e A fn = lim n →∞ g n on SUBEXP-space randoms. Define U n,i = (cid:26) x : max p ( n + i ) ≤ m ≤ p ( n + i +1) | A f m n ( x ) − A f p ( n + i ) n ( x ) | ≥ log( n + i ) (cid:27) .
31e already know µ ( U n,i ) ≤ − log( n + i ) . U n,i can be shown to be polynomial space approximablein parameters n and i in the following sense. There exists a sequence of sets of strings h S n,i i i,n ∈ N and polynomial p satisfying the following conditions:1. U n,i = [ S n,i ].2. There exists a controlling polynomial r such that max {| σ | : σ ∈ S n,i } ≤ r (log( n )+log( i )) .3. The function g : Σ ∗ × ∗ × ∗ → { , } such that g ( σ, n , i ) = ( σ ∈ S n,i N computing the function g above. Let M f be a computing machine and let q be a controlling polynomial for h f n i ∞ n =1 . Let c be a controllingconstant for T . Let M ′ be the machine from Lemma 3.5. Machine N on input ( σ, n , i ) does thefollowing:1. If | σ | > q ( ⌈ log( p ( n + i +1)) ⌉ ) + cn , then output 0.2. Compute A f p ( n + i ) n ( σ ∞ ) as in Lemma 7.15 by using M f and M ′ and store the result.3. For each m ∈ [ p ( n + i ) , p ( n + i + 1)] do the following:(a) Compute A f m n ( σ ∞ ) as in Lemma 7.15 by using M f and M ′ and store the result.(b) Check if | A f m n ( σ ∞ ) − A f p ( n + i ) n ( σ ∞ ) | ≥ log( n + i )2 . If so, output 1.4. Output 0.It can be easily verified that N is a PSPACE machine. The second condition follows from the factthat | σ | ≤ q ( ⌈ log( p ( n + i +1)) ⌉ ) + cn for any σ ∈ S n,i . Now, define V m = [ n,i ≥ n + i = m U n,i . Now, we show that h V m i ∞ m =1 is a SUBEXP Solovay test. Note that µ ( V m ) ≤ m log( m ) . Since each V m is a finite union of sets from h U n,i i n,i ∈ N , the machine computing h U n,i i n,i ∈ N can beeasily modified to construct a machine witnessing that h V m i ∞ m =1 is a SUBEXP-space approximablesequence of sets. From the above observations and Lemma 7.4, it follows that h V m i ∞ m =1 is aSUBEXP-space Solovay test. Now, let x be a SUBEXP-space random. x is in at most finitelymany V m and hence in at most finitely many U n,i . Hence for some large enough N for all n ≥ N , i ≥ m such that p ( n + i ) ≤ m ≤ p ( n + i +1), we have | A f m n ( x ) − A f p ( n + i ) n ( x ) | < − log( n + i ) .It follows that for all n ≥ N and for all m ≥ p ( n ) that, | A f m n ( x ) − g n ( x ) | = | A f m n ( x ) − A f p ( n ) n ( x ) | ≤ ∞ X i =0 log( n + i ) ≤ ∞ X i =0 n + i ) ≤ π n . n →∞ e A fn ( x ) = lim n →∞ g n ( x ) on all SUBEXP-space random x . Hence, we have shown that lim n →∞ e A fn = R f dµ on SUBEXP-space randoms whichcompletes the proof of the theorem.An important reason for investigating SUBEXP-space randomness is that the SUBEXP-spaceergodic theorem has an exact converse unlike the PSPACE ergodic theorem which only seems tohave a partial converse (Theorem 6.1). Before proving the converse, we show a SUBEXP-spaceanalogue of the construction in Lemma 6.2. Lemma 7.17.
Let h U n i ∞ n =1 be a SUBEXP -space test. Then there exists a sequences of sets
D b S n E ∞ n =1 such that for each n ∈ N , b S n ⊆ Σ ∗ satisfying the following conditions:1. µ ([ b S n ]) ≤ − log( n ) ∩ ∞ m =1 ∪ ∞ n = m [ b S n ] ⊇ ∩ ∞ n =1 U n
3. There exists c ∈ N such that for all n , σ ∈ b S n implies | σ | ≤ log( n ) c
4. There exists a
PSPACE machine N such that N ( σ, n ) = 1 if σ ∈ b S n and 0 otherwise.Proof. Let (cid:10) S kn (cid:11) n,k be the collection of approximating sets for h U n i ∞ n =1 as in Definition 7.1. Define U = ∪ ∞ n =1 ∪ ∞ k =1 S kn . Now, define T n = n +1) [ i =1 S n +1)2 ni Observe that, [ U ] \ [ T n ] ⊆ n +1) [ i =1 U i − [ S n +1)2 ni ] [ ∞ [ i =2(4 n +1)+1 U i Hence, µ ([ U ] \ [ T n ]) ≤ n +1) X i =1 log(2(4 n +1)2 n ) + ∞ X i =2(4 n +1)+1 log( i ) < n +1) X i =1 log(2(4 n +1)2 n ) + ∞ X i =2(4 n +1) log( i ) ≤ n + 1)2 log(2(4 n +1)2 n ) + 12 log(2 n ) < n + 1)2 log(2(4 n +1)) +log(2 n ) + 12 log(2 n ) < log( n ) +1 + 12 log( n ) +1 = 12 log( n ) T n , it follows that there is a c ∈ N such that the length of strings in T n is upper bounded by 2 log( n ) c . Now, if b S n = { σ ∈ T n +1 : ∀ α ⊑ σ ( α T n ) } , we have µ ([ b S n ]) ≤ µ ([ U ] \ [ T n ]) ≤ − log( n ) . Conditions 2 and 3 can be readilyverified to be true. We now construct a PSPACE machine N satisfying the condition in 4. N oninput ( σ, n ) does the following:1. For each i ∈ [1 , n + 1) + 1)] simulate M ( σ, i , n +1) +1)2( n +1) ). If all simulations resultin 0, output 0.2. Else, for each m ∈ [1 , n ] do the following:(a) For each α ⊑ σ do the following:i. For each i ∈ [1 , m + 1)] simulate M ( α, i , m +1)2 m ). If any of these simula-tions result in a 1 then, output 0.3. Output 1. N can be easily verified to be a PSPACE machine. Hence, our constructions satisfy all the desiredconditions. Theorem 7.18.
Given any
SUBEXP -space null x , there exists a SUBEXP -space L -computable f ∈ L (Σ ∞ , µ ) such that for any PSPACE simple measure preserving transformation, the followingconditions are true:1. For all n ∈ N , k A fn − R f dµ k = 0 . Hence, R f dµ is an PSPACE -rapid L -limit point of A fn .2. There exists a collection of simple functions { g n,i } such that for each n , h g n,i i ∞ i =1 is a SUBEXP -space L -approximation of A fn but lim n →∞ lim i →∞ g n,i ( x ) = R f dµ .Proof. Let h V n i ∞ n =1 be any SUBEXP-space test such that x ∈ ∩ ∞ n =1 V n . Now, from Lemma 7.17,there exists a collection of sets D b S n E ∞ n =1 such that ∩ ∞ m =1 ∪ ∞ n = m [ b S n ] ⊇ ∩ ∞ n =1 V n and σ ∈ b S n implies | σ | ≤ log( n ) c for some c . Let, U n = { σ : [ σ ] ∈ b S i for some i such that 2(2( n ) + 1) ≤ i ≤ n + 1) ) + 1) } Now, let f n = nχ U n . Using Lemma 7.4, µ ( U n ) ≤ n +1) ) +1) X i =2(2( n ) +1) log( i ) ≤ log( n ) ≤ log( n ) +log( n ) it follows that k f n k ≤ n log( n ) +log( n ) ≤ log( n ) . Now, using the properties of
D b S n E ∞ n =1 , it can be shown that h f n i ∞ n =1 is a SUBEXP-space L -approximation of f = 0. We construct a machine M computing h f n i ∞ n =1 . The other conditions areeasily verified. Let N be the machine from Lemma 7.17. On input (1 n , σ ), M does the following:1. If | σ | < ⌈ log(2(2(( n +1) ) +1)) ⌉ c then, output ?.34. Else, for each i ∈ [2(2( n ) + 1) , n + 1) ) + 1)] do the following:(a) For each α ⊆ σ , do the following:i. If N (1 i , α ) = 1 then, output n .3. Output 0. M uses at most polynomial space and computes h f n i ∞ n =1 . Now, define, g n,i = f i + f i ◦ T + · · · + f i ◦ T n − n For any fixed n ∈ N , since T is a PSPACE simple transformation, as in Lemma 7.15 it canbe shown that h g n,i i ∞ i =1 is a SUBEXP-space L -approximation of A fn . We know that there existinfinitely many m such that x ∈ b S m . For any such m , let i be the unique number such that2(2( i ) + 1) ≤ m ≤ n + 1) ) + 1). For this i , f i ( x ) = i . This shows that there exist infinitelymany i such that f i ( x ) = i . Since each f i is a non-negative function, it follows that for infinitelymany i with g n,i ≥ i/n . Hence, if lim i →∞ g n,i ( x ) exists, then it is equal to ∞ . It may be the case thatlim i →∞ g n,i ( x ) does not exist. In either case, lim n →∞ lim i →∞ g n,i ( x ) cannot be equal to R f dµ = 0. Hence,our construction satisfies all the desired conditions. PSPACE randomness
The study of resource bounded randomness was initiated by Lutz([17],[18]) using resource boundedmartingales. Huang and Stull introduced weak PSPACE randomness (definition 2.3) in [13] as aresource bounded analogue of randomness defined in terms of c.e tests. In this section we give acharacterization of PSPACE randoms in terms of PSPACE martingales, demarking on the differencebetween Lutz’s notion of PSPACE randomness and our notion of PSPACE randomness. Our resultand its proof are significantly different from those given in [27]. We demonstrate this in thesetting of (Σ ∞ , B (Σ ∞ ) , µ ) where µ is the Bernoulli measure µ ([ σ ]) = | σ | . Now we define PSPACEcomputable martingales. Definition 8.1 (PSPACE computable martingales) . A function D : Σ ∗ → [0 , ∞ ) is a PSPACEcomputable martingale if for each σ ∈ Σ ∗ , D ( σ ) = D ( σ
0) + D ( σ M such that for each σ ∈ Σ ∗ and n ∈ N , M ( σ, n ) ∈ Q suchthat, | D ( σ ) − M ( σ, n ) | ≤ n Now, we give a characterization of PSPACE randomness in terms of PSPACE computablemartingales.
Theorem 8.2. x ∈ Σ ∞ is PSPACE null if and only if there exists a
PSPACE computable martin-gale D and k ∈ N such that there exist infinite many n ∈ N satisfying D ( x ↿ n ) ≥ ⌊ n k ⌋ / (5)35 ∈ Σ ∞ is strong PSPACE null (Lutz [17][18]) if there exists a PSPACE computable martingalesuch that lim sup n →∞ D ( x ↿ n ) = ∞ . Hence, if x is PSPACE null then x is strong PSPACE null .Stull [25] proved the existence of a strong
PSPACE null which is PSPACE random. The proof belowis a careful adaptation of Theorem 7.3.3 from [22] into the PSPACE setting using the constructionin Lemma 6.2.
Proof of Theorem 8.2.
We show the forward implication first. Let h U n i ∞ n =1 be a PSPACE testwitnessing the fact that x is PSPACE null. Now, from Lemma 6.2, there exists a collection ofsets D b S n E ∞ n =1 such that ∩ ∞ m =1 ∪ ∞ n = m [ b S n ] ⊇ ∩ ∞ n =1 U n and σ ∈ b S n implies | σ | < n k for some k . Let U = S ∞ n =1 [ b S n ]. Hence, X σ ∈U| σ |≥ n k | σ | ≤ X n ′ >n X σ ∈ b S n ′ | σ | ≤ n . Hence, X σ ∈U| σ |≥ n | σ | ≤ ⌊ n k ⌋ = 12 ⌊ n k ⌋ / Let f ( n ) = ⌊ n k ⌋ /
2. Hence, X σ ∈U| σ |≥ n | σ | ≤ f ( n ) For every r ∈ N , define s r = X σ ∈U f ( | σ | ) ≥ r f ( | σ | ) −| σ | It can be easily verified that s r ≤ − r .Now, we define a martingale D such that D wins on x in the sense of 5. For each i ∈ N , let G i = { σ ∈ U : f ( | σ | ) ≥ i } . It can be readily verified that µ ( G i ) ≤ − i for all i >
0. For each i ∈ N define E i ( σ ) = X α ∈ G i α ⊏ σ f ( | α | ) F i ( σ ) = 2 | σ | X α ∈ G i α ⊒ σ f ( | α | ) −| α | As in the proof of [22] Lemma 7.3.4, it can be verified that D i = E i + F i is a martingale. Nowdefine D = P ∞ i =1 D i . It can be easily seen that D is a martingale.36e now show that D is PSPACE computable. Let σ be a string on which D ( σ ) needs to beapproximated with error at most 2 − m . Observe that, E i ( σ ) = X α ∈ G i α ⊏ σ | α | f ( | α | ) −| α | ≤ | σ | X α ∈ G i α ⊏ σ f ( | α | ) −| α | Now, D i ( σ ) = E i ( σ ) + F i ( σ ) ≤ | σ | X α ∈ G i f ( | α | ) −| α | ≤ | σ | X α ∈U f ( | α | ) ≥ i f ( | α | ) −| α | ≤ | σ | s i ≤ | σ |− i Hence, if we can approximate P | σ | + m +1 i =1 D i with error at most 2 − ( m +1) , we can get a 2 − m errorapproximation for D .Let N be a machine computing D b S n E ∞ n =1 as in Lemma 6.2. For the given σ , we can sum 2 f ( | α | ) for all α ⊏ σ and α ∈ G i in PSPACE. For each i , E i can be computed in the following way:1. Let Sum = 0.2. For each α ⊏ σ do the following:(a) Check if there exists 1 ≤ m ≤ | α | such that N (1 m , α ) = 1. If yes, check if f ( | α | ) ≥ i . Ifthis is true, then add f ( | α | ) to Sum.3. Output Sum. α b S n for n ≥ | α | + 1 since for any such n , µ ( b S n ) < −| α | . Hence, the procedure above exactlycomputes E i . The procedure can be easily seen to be a polynomial space operation. Hence, P | σ | + m +1 i =1 E i can be computed without error in polynomial space.Observe that for all m ∈ N ,2 | σ | X α : f ( | α | ) ≥ m + | σ | +1 f ( | α | ) −| α | ≤ | σ | ∞ X r = m + σ +1 s r ≤ − m . So, we can ignore all but finitely many terms in the summation defining F i as in the case of D i . Forany α , if f ( | α | ) = ⌊| α | k ⌋ / < m + | σ | + 1 then its length | α | can be upper bounded by a polynomialin m + | σ | , say ( m + | σ | ) d . Hence, the terms that are left in the summation defining F i can becomputed as in the computation of E i by going over all α ⊒ σ with | α | ≤ ( m + | σ | ) d in polynomialspace. It follows that D = F + E is a PSPACE computable martingale.Now, we show that condition 5 is satisfied by D for infinitely many n ∈ N . We know that forinfinitely many n ∈ N , x ↿ n ∈ U . For any such n , we have x ↿ n ∈ G f ( n ) . Observe that, D ( x ↿ n ) ≥ D f ( n ) ( x ↿ n ) ≥ F f ( n ) ( x ↿ n ) ≥ n . f ( n ) − n = 2 f ( n ) n ∈ N as desired.Now we prove the converse. Let U i = { σ ∈ Σ ∗ : M ( σ, ≥ ⌊| σ | k ⌋ / − ≥ i +1 } . Since M ( σ, ≥ i +1 implies d ( σ ) > i , using the Kolmogorov’s inequality (see [22] Proposition 7.1.9)we get that µ ( U i ) ≤ − i . For any m ∈ N , | σ | ≥ (2( m + 2)) k implies 2 ⌊| σ | k ⌋ / ≥ m +2 . Hence, forany σ ∈ U i , | σ | ≥ (2( m + 2)) k implies that d ( σ ) ≥ m +2 − > m . Now, using the Kolmogorov’sinequality again we get that µ (cid:0)(cid:8) σ ∈ U i : | σ | ≥ (2( m + 2)) k (cid:9)(cid:1) ≤ − m . From the two previousobservations, it follows that h U i i ∞ i =1 is a PSPACE sequence of open sets. Since, there are infinitelymany n ∈ N such that D ( x ↿ n ) ≥ ⌊ n k ⌋ / , it follows that there are infinitely many n such that M ( x ↿ n, ≥ ⌊ n k ⌋ / −
1. Since, h U i i ∞ i =1 is a PSPACE test, it follows that x is PSPACE null. PSPACE random which is
EXP non-random
It was shown in Stull’s thesis [25] that weak polynomial time randomness is strictly weaker thanpolynomial time randomness. Using a similar approach, we give below an explicit construction ofa PSPACE random which is EXP non-random. Let x be any Martin-L¨of random. We define theinfinite sequence y as follows, y [ n ] = ( , if n = 2 m for some m ∈ N x [ n ] , otherwise.It is easy to construct a PSPACE computable martingale D : Σ ∗ → [0 , ∞ ) such that lim sup n →∞ D ( y ↿ n ) = ∞ . Define, D ( σ ) = ( ⌊ log( | σ | ) ⌋ , if σ [2 i ] = 0 for all 1 ≤ i ≤ ⌊ log( | σ | ) ⌋ , otherwise.It is easy to verify that D is a PSPACE computable martingale and lim sup n →∞ D ( y ↿ n ) =lim n →∞ ⌊ log( n ) ⌋ = ∞ . But this does not imply that y is PSPACE null since D does not win on y fast enough as in condition 5. In fact, we now show that no PSPACE computable martingalesatisfying 5 on y exists since y is PSPACE random. Lemma 9.1. y is PSPACE random and
EXP null.Proof.
We first construct an EXP test h U n i ∞ n =1 such that y ∈ ∩ ∞ n =1 U n . For any n , let U n be thecollection of all 2 n length strings σ such that σ [ i ] = 0 for all i ∈ { , , . . . n } . Clearly, µ ( U n ) ≤ − n and y ∈ ∩ ∞ n =1 U n . It is straightforward to verify that h U n i ∞ n =1 is an EXP test.Now, we show that y is PSPACE random. Assume that there exists D b S n E ∞ n =1 satisfying theproperties in Lemma 6.2 such that y ∈ ∩ ∞ m =1 ∪ ∞ n = m [ b S n ]. Hence, there exists c ∈ N such that forany n , σ ∈ b S n implies | σ | ≤ n c . Let G n be the set of all α ∈ Σ n c such that there exists some σ ∈ b S n with α [ i ] = σ [ i ] for all i
6∈ { j : 1 ≤ j ≤ ⌊ log( n c ) ⌋} . If y ∈ b S n then, by construction itfollows that x ∈ G n . Since µ ( b S n ) ≤ − n , we get µ ( G n ) ≤ ⌊ log( n c ) ⌋ − n . Since for all large enough n , n − ⌊ log( n c ) ⌋ ≥ n/
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