Some Quantitative Aspects of Fractional Computability
aa r X i v : . [ m a t h . G R ] J un SOME QUANTITATIVE ASPECTS OF FRACTIONALCOMPUTABILITY
ILYA KAPOVICH AND PAUL SCHUPP
Abstract.
In this article we apply the ideas of effective Baire categoryand effective measure theory to study complexity classes of functionswhich are “fractionally computable” by a partial algorithm. For thispurpose it is crucial to specify an allowable effective density, δ , of con-vergence for a partial algorithm. The set FC ( δ ) consists of all totalfunctions f : Σ ∗ → { , } where Σ is a finite alphabet with | Σ | ≥ δ ”. The space FC ( δ ) iseffectively of the second category while any fractional complexity class,defined using δ and any computable bound β with respect to an abstractBlum complexity measure, is effectively meager. A remarkable result ofKautz and Miltersen shows that relative to an algorithmically randomoracle A , the relativized class N P A does not have effective polynomialmeasure zero in E A , the relativization of strict exponential time. Wedefine the class UFP A of all languages which are fractionally decidablein polynomial time at “a uniform rate” by algorithms with an oraclefor A . We show that this class does have effective polynomial measurezero in E A for every oracle A . Thus relaxing the requirement of polyno-mial time decidability to hold only for a fraction of possible inputs doesnot compensate for the power of nondeterminism in the case of randomoracles. Introduction
We now know that “worst-case” complexity measures such as polynomialtime do not necessarily give a good overall picture of a particular problemor algorithm since it depends on the difficulty of the hardest instances of theproblem, and these may be very sparse. The famous classic example of thisphenomenon is Dantzig’s Simplex Algorithm for linear programming. Theexamples of V. Klee and G. Minty [11] showing that the simplex algorithmcan be made to take exponential time are very special. A “generic” or “ran-dom” linear programming problem is not “special”, and Dantzig’s algorithmworks quickly. Indeed, later algorithms which are provably polynomial-timehave not replaced the simplex algorithm in practice.Observations of this type led to the development of average-case complex-ity by Gurevich [6] and Levin [12]. There are now different approaches tothe average-case complexity, but they all require computing the expected
Mathematics Subject Classification.
Primary 68Q, Secondary 20P05.Both authors were supported by the NSF grant DMS-0404991. The first author is alsosupported by the NSF grant DMS-0603921. value of the running time of an algorithm with respect to some measure onthe set of inputs. It is often difficult to establish an average-case result sincea basic difficulty of worst-case complexity is still present: one needs a totalalgorithm which solves the problem and some upper bound on its worst-casedifficulty.Kapovich, Myasnikov, Schupp and Shpilrain [7] introduced the notion of generic-case complexity, which deals with the performance of an algorithmon “most” inputs and completely ignores what happens on the “sparse” setof other inputs. They applied the idea to the classic decision problems ofgroup theory - the word and conjugacy problems - and found that the “linearprogramming phenomenon” is extremely widespread there. An importantaspect of generic-case complexity is that it allows us to work with the entireclass of partial computable functions, which is the natural setting of thegeneral theory of computability, and one can often prove generic-case com-plexity results about problems where the worst case complexity is unknown.This paper grew out of our interest in generic-case complexity but herewe are interested in the more general concept of “fractionally computableat an allowable density δ .” The basic idea is essentially the same as forgeneric-case complexity. However, we do not demand that the fraction ofpossible inputs on which a partial algorithm succeeds approaches one, butonly that the algorithm succeeds at the given density δ .Specific questions about fractional complexity are important in cryptog-raphy, where one needs the assumption that problems such as calculatingthe discrete logarithm are generically difficult. Proposition 6 . B p ⊆ Z ∗ p where | B p | ≥ ǫ | Z ∗ p | then there is a probabilistic algorithm that solves the discretelogarithm problem in general with expected running time polynomial in k and 1 /ǫ .In group theory, subgroups of finite index provide natural examples al-gorithms with fractional complexity. Suppose that G is a finitely generatedgroup and N is a normal subgroup of finite index j . Let ψ be the naturalhomomorphism from G onto G/N . Then the algorithm for the word prob-lem of G which simply consists of answering “no” on input w if ψ ( w ) = 1works on the fraction 1 − j of inputs. Note that there is no assumptionabout the complexity of the word problem for G . (The same result holds forsubgroups H which are not normal by using coset diagrams.) There are nowseveral suggestions for using problems about various groups for the purposesof cryptography. Fractional computability issues, such as those coming fromsubgroups of finite index, may pose difficulties for security.The main results of this paper are (see Sections 2 and 3 below for precisedefinitions): UANTITATIVE ASPECTS 3
Theorem 1.1.
For every Blum complexity measure Φ , for every allowabledensity δ and for every effective bound β , the fractional complexity class Φ[ β, δ ] is effectively meager in the space F C ( δ ) . Theorem 1.2.
For every oracle A the set U F P A has effective polynomial-time measure zero with respect to A in E A . In Theorem 1.1, δ ( n ) is an effective density of convergence for a partialalgorithm. Roughly speaking, δ ( n ) specifies the fraction of all inputs oflength n in which a partial function under consideration is required to bedefined. The set F C ( δ ) consists all total functions f : Σ ∗ → { , } where Σ isa finite alphabet with | Σ | ≥ δ ”. The function β ( n ) is an effectively computable resource bound for someabstract Blum complexity measure (e.g. time). Informally, the class Φ[ β, δ ]consists of all partial computable functions that can be computed on a -fraction of the inputs of length n which is at least δ ( n ) with a resourcebound β ( n ). The space F C ( δ ) is effectively of the second category whileany fractional complexity class, defined using δ and resource bound β , iseffectively meager.In Theorem 1.2, the space E A consists of all total functions computablein strict exponential time with an oracle for A . The space U F P A consistsof those functions in E A that are partially calculable by partial computablefunctions that are uniform and are computable in polynomial time. Herea partial function φ from Σ ∗ to { , } is uniform if there exists a positiveinteger k such that for every w ∈ Σ ∗ with | w | ≥ k there is some z with | z | ≤ k log | w | such that φ ( wz ) is defined. Thus being uniform can be viewedas a version of “fractional computability”.A remarkable result of Kautz and Miltersen [10] shows that for an al-gorithmically random set A ⊆ Σ ∗ the class N P A does not have effectivepolynomial-time measure zero in E A . Thus Theorem 1.2 above shows thatfractional polynomial-time computability does compensate for the power ofnondeterminism.The main lines of our considerations are directly taken from known resultsin the theory of effective category and measure. The contribution of thispaper consists in showing that such results apply to the study of fractionalcomplexity. classes. We are particularly indebted to the book “Computa-tional Complexity: A Quantitative View” by Marius Zimand [18] and thearticles by Calude [5] and by Kautz and Miltersen [10].2. Allowable Densities and Fractional Computability
Convention 2.1. We fix a finite alphabet Σ with k ≥ ∗ denotes the set of allwords over Σ. If w ∈ Σ ∗ then the length , | w | , of w is the number of lettersin w . We denote the empty word by λ . The canonical or shortlex orderingof Σ ∗ lists words in order of increasing length and within a given length by I. KAPOVICH AND P. SCHUPP the lexicographical order induced by the given alphabetical ordering of Σ.Thus for Σ = { a, b } the list is λ, a, b, aa, ab, ba, bb, aaa, .... We take the listing w , w , w , ... as defining a bijection between Σ ∗ and the natural numbers N = { , , , ..... } .Using this bijection, we can consider functions from Σ ∗ to { , } as functionsfrom N to { , } . In this article F denotes the set of total functions fromΣ ∗ to { , } .A language L over Σ is a subset of Σ ∗ . We can identify a language L ⊆ Σ ∗ with its characteristic function χ L where χ L ( n ) = ( w n ∈ L, w n L. This identification gives a bijection between the set L of all languagesover Σ and the set F and we take these sets as being essentially the same.A function f from Σ ∗ = { w , w , ..., w n , ... } to { , } is an infinite sequence ( b n ) of 0’s and 1’s. If f ∈ F takes the value1 infinitely often we can regard f as the unique binary expansion of a realnumber in the half-open unit interval (0 ,
1] which is not all 0’s from somepoint onwards.Suppose that we have a partial algorithm
Ω for a set S ⊆ Σ ∗ . In par-ticular, this means that Ω is correct: If Ω converges on an input w then Ωgives the correct answer as to whether or not w ∈ S . We again point outthat we completely ignore the performance of Ω on words not in S and thecomplexity classes we consider will generally contain functions f which arenot computable. Indeed, note that a single partial algorithm Ω genericallycomputes uncountably many different functions if the set D on which thepartial algorithm converges is generic while its complement D is infinite. Let f ′ be the partial function defined by Ω. Then we can choose values on theset D in a totally arbitrary way to complete f ′ to a total function f whichis generically computed by the given algorithm. Convention 2.2.
We want to fix an effective enumeration ( M i ) of all Turingmachines with input alphabet Σ and with a special output tape consistingof a single square in which a machine can print either 0 or 1. Let φ i bethe partial function from Σ ∗ to { , } which is computed by M i . We write φ i ( x ) ↓ if φ i produces a value on input x .A major concern of [7] was the rate of convergence of a given generic-case algorithm. It turns out that this is not an accident and that a generaldiscussion of fractional complexity classes requires providing an effectivedensity function which specifies a lower bound on how many values must UANTITATIVE ASPECTS 5 be defined at a given stage. We now regard the functions in F as functions f : N → { , } . We need “the acceptable density so far” to be defined ateach input n . Definition 2.3. An allowable density function is a computable function δ : N → Q ∩ [0 ,
1] such that lim inf n →∞ δ ( n ) > . Given an allowable δ , if φ i is a p.c. function, we write ∆( φ i ) if thecondition |{ m : m ≤ n, φ i ( w m ) ↓}| /n ≥ δ ( n )holds for all n ≥ F C ( δ ) of functions which are fractionallycomputable at density δ . We assume that an allowable density function δ isnow fixed. Notation 2.4. If φ is a partial function and f ∈ F is a total function, wewrite φ ⊑ f if φ ( x ) = f ( x ) at all arguments for which φ is defined. Definition 2.5.
Let δ be an allowable density function. We define F C ( δ )to be the space of all functions f ∈ F such thatthere exists a partial computable function φ i such that ∆( φ i ) and φ i ⊑ f .In order to define the appropriate topology on F C ( δ ) we consider finitesequences τ = ( s , . . . , s n ) where each s i is from the three-letter alphabet { , , ⊥} . The symbol ⊥ represents an undefined value. If τ has length n asa sequence we write | τ | = n . We also write τ ( j ) for s j .The set of positions for which τ is defined is def ( τ ) = { j : j ≤ | τ | , τ ( j ) = ⊥} . As for partial functions we write ∆( τ ) if |{ j : j ≤ l, j ∈ def ( τ ) | /l ≥ δ ( l )for all l ≤ | τ | . Definition 2.6.
A finite sequence τ is δ - allowable if τ contains at leastone defined entry and ∆( τ ). Let T denote the set of all δ -allowable finitesequences. If τ and τ are allowable sequences we write τ ⊑ τ if τ agreeswith τ at all positions for which τ is defined. If τ ∈ T and f ∈ F we write τ ⊑ f if f agrees with τ at all positions at which τ is defined.If τ ∈ T is an allowable sequence then the basic neighborhood defined by τ is N ( τ ) = { f : f ∈ F C ( δ ) , τ ⊑ f } . Note that if τ ⊑ τ then N ( τ ) ⊆ N ( τ ) since τ specifies more informationthan τ . I. KAPOVICH AND P. SCHUPP
Regarding sequences as words over the three-letter alphabet {⊥ , , } wecan effectively enumerate all δ -allowable sequences as τ , τ , ..., τ n , ... by considering all finite sequences over {⊥ , , } in the canonical order andsuccessively listing only those sequences which are δ -allowable. It is easy toshow that the collection { N ( τ ) : τ ∈ T } is a system of basic neighborhoodsand we use the topology generated by this system. Proposition 2.7.
For every τ i , τ j ∈ T with N ( τ i ) ∩ N ( τ j ) = ∅ there exists τ ∈ T with N ( τ ) ⊆ N ( τ i ) ∩ N ( τ j ) .Proof. Since N ( τ i ) ∩ N ( τ j ) = ∅ , it follows that the sequences τ i and τ j agreeat all positions where both are defined. Thus the following sequence τ oflength r = max {| τ i | , | τ j |} is well-defined. For x ≤ r let τ ( x ) = τ i ( x ) if x ∈ def ( τ i ) ,τ j ( x ) if x ∈ def ( τ j ) , ⊥ if τ i ( x ) = τ j ( x ) = ⊥ . Since τ is defined where either of the δ -allowable sequences τ i or τ j aredefined, the sequence τ is δ -allowable and N ( τ ) ⊆ N ( τ i ) ∩ N ( τ j ) by defini-tion. (cid:3) Blum[3] gave a very general definition of an abstract complexity measureand we work in that context since the specific nature of the complexitymeasure is not important.
Definition 2.8. A Blum Complexity Measure of partially computable func-tions is a partially computable function Φ( i, x ) satisfying the following twoaxioms:(1) Φ( i, x ) ↓ ⇐⇒ φ i ( x ) ↓ . (2) The cost predicate Cost ( i, x, y ) = ( i, x ) ≤ y, . is computable.The standard measures of deterministic time or space are certainly Blumcomplexity measures. For the remainder of this section we assume that someBlum Complexity Measure Φ is fixed.We can now define fractional complexity classes using the complexitymeasure Φ and the density δ . Recall that { φ i } i is an effective enumerationof partial computable functions from Σ ∗ to { , } and that we think ofsuch functions as being given by Turing machines which can print only thesymbols 0 and 1 on their special output tape. We now need to considerfunctions from Σ ∗ to {⊥ , , } and think that such functions are given byTuring machines which can print 0 , ⊥ on their output tape. UANTITATIVE ASPECTS 7
Definition 2.9.
Let β be any total computable function, which we will referto as the effective bound .The function φ i strictly bounded by β , which we denote by φ [ β ] i , is definedas follows. We take the Turing machine M for φ i and obtain the Turingmachine M ′ by adding an initial subroutine which, on input x , calculates Cost ( i, x, β ( x )). If this value is 0, then either Φ( i, x ) is undefined (and hence φ i ( x ) is undefined) or Φ( i, x ) is defined and the complexity Φ( i, x ) on input x exceeds β ( x ). In either case, if Cost ( i, x, β ( x )) = 0, M ′ prints the value ⊥ . If Cost ( i, x, β ( x )) = 1 (so that both φ i ( x ) and Φ( i, x ) are defined and,in addition, Φ( i, x ) is bounded by β ( x )), M ′ prints the value calculated by M on input x .This construction gives us an effective enumeration of all the functions φ [ β ] i . Note that if these are considered as functions from Σ ∗ to { , , ⊥} thenthey are total computable functions. Finally we have Definition 2.10.
The fractional complexity class, Φ[ β, δ ], defined by Φ , β and δ is Φ[ β, δ ] = { f ∈ F C ( δ ) : ∃ i [∆( φ [ β ] i ) and φ [ β ] i ⊑ f ] } We now turn to the notion of effective Baire category. The requirementfor a set S ⊆ F C ( δ ) to be effectively nowhere dense is that there is a uni-form effective method which, when given any basic open neighborhood N ,produces another basic neighborhood N ′ ⊆ N such that S ∩ N ′ = ∅ . Fora meager set, that is, a countable union of nowhere dense sets, we requirethat the method be uniform over all the members of the union. Recall that T denotes the set of all δ -allowable finite sequences. Definition 2.11.
A set X ⊆ F C ( δ ) is effectively nowhere dense in F C ( δ )if there exists a total computable witness function α : T → T such that:(1) τ ⊑ α ( τ ) for all τ ∈ T . (2) X ∩ N ( α ( τ )) = ∅ . A set X ⊆ F C ( δ ) is effectively meager if there exist a sequence of nowheredense sets ( X i ) i and a total computable witness function α : N × T → T oftwo variables such that:(1) X = S ∞ i =1 X i (2) τ ⊑ α ( i, τ ) for all ( i, τ ) ∈ N × T . (3) X i ∩ N ( α ( i, τ )) = ∅ . A set is effectively ample (effectively of the second category) if it is noteffectively meager.It is now easy to prove the desired result that any complexity class Φ[ β, δ ]is effectively meager while the entire space
F C ( δ ) is effectively of the secondcategory. Indeed, we have the following result. (Compare [5].) Lemma 2.12.
For every meager set X and for every τ ∈ T , there is a totalcomputable function f ∈ N ( τ ) − X . I. KAPOVICH AND P. SCHUPP
Proof.
Since X is effectively meager we can write X = S ∞ i =1 X i where X is effectively meager via the witness function α ( i, τ ). We define a totalcomputable function f iteratively by a simple diagonalization argument.For a given τ let σ be the sequence of length | τ | + 1 agreeing with τ at allplaces where τ is defined and having 0 in all places where τ is undefined,and with 0 as the last entry of the sequence σ . Then τ ⊑ σ .Let η = α (1 , σ ). Let σ be the sequence of length | η | + 1 which agreeswith η in all places where η is defined and which has 0 in all places where η is undefined, and with 0 as the last entry of the sequence σ . So | σ | > | η | and all entries in σ are defined. Since η ⊑ σ , N ( σ ) ∩ X = ∅ .We continue in the same fashion. Let η = α (2 , σ ). Let σ be thesequence of length | η | + 1 which agrees with η in all places where η isdefined and which has 0 in all places where η is undefined, and with onemore defined position with entry 0 at the end of η . Thus | σ | = | η | + 1and all entries in σ are defined. Since σ ⊑ η , we have N ( σ ) ∩ X = ∅ .By this process, we iteratively define a sequence ( σ i ) i of δ -allowable in-tervals σ i in which all entries are defined such that N ( σ i ) ∩ X i = ∅ andsuch that σ i ⊑ σ i +1 and | σ i | < | σ i +1 | for every i . Let σ = σ (1) , σ (2) , . . . be the infinite binary sequence such that for every i the initial segment of σ of length | σ i | is σ i . Note that every initial segment of σ is a δ -allowablesequence and that | σ i | ≥ i for every i .Consider the function f defined as f ( j ) = σ ( j ) for every j . Clearly, f is a total computable function, since for every n we have n ≤ | σ n | and f ( n ) = σ n ( n ).Now f ∈ N ( τ ), since it agrees with τ at all places where τ is defined, and f / ∈ X i for all i . (cid:3) The theorem immediately yields the following corollary.
Corollary 2.13.
The set R of total effectively computable functions from Σ ∗ to { , } is not meager in the space F C ( δ ) . Theorem 2.14.
For every Blum complexity measure Φ , for every allowabledensity δ and for every effective bound β , the fractional complexity class Φ[ β, δ ] is effectively meager in the space F C ( δ ) .Proof. We have an effective enumeration ( φ [ β ] i ) i of all strictly β -boundedpartial functions. Let C i = ( { f ∈ F C ( δ ) : φ [ β ] i ⊑ f } if ∆( φ [ β ] i ) ∅ otherwise . It is clear that Φ[ β, δ ] = S i C i so we need only specify an effective witnessfunction α . Given an index i and a δ -allowable sequence τ compute φ [ β ] i onthe first | τ | inputs in the canonical order. If the computed sequence σ oflength | τ | is not δ -allowable then C i = ∅ and we set α ( i, τ ) = τ . Suppose UANTITATIVE ASPECTS 9 now that σ is δ -allowable. If σ has a defined value v on an input w j with v = τ ( j ) again set α ( i, τ ) = τ . Suppose now that σ is allowable and thatfor all j ≤ | τ | with a defined value σ ( j ) we have σ ( j ) = τ ( j ).We claim that there exists r > | τ | such that either φ [ β ] i has a definedvalue φ [ β ] i ( r ) or the sequence σ r of the values of φ [ β ] i on the first r inputsis non-allowable. This follows from the assumption lim inf n →∞ δ ( n ) > δ -allowable sequence. We continue computing values of φ [ β ] i until we findthe smallest r > | τ | with the above property.If the sequence σ r is not allowable then C i = ∅ and we again set α ( i, τ ) = τ . If σ r is allowable and the r -th entry of σ r is a defined value v , we set α ( i, τ ) to be the sequence agreeing with σ r at all the positions j < r andhaving value 1 − v at position r . In either case we have α ( i, τ ) ∩ C i = ∅ . (cid:3) Note that, in general, a fractional complexity class Φ[ δ, β ] contains un-countably many functions while the nonmeager set R is countable.3. Nondeterminism versus fractional polynomial-timecomputability
It should be expected that partial computability at a fixed density cannotmake great inroads into the power of nondeterminism. A nondeterministicmachine can guess on every input, while in considering fractional complexity,we still have a deterministic machine which is required to actually do thedesired calculation on a non-negligible set of inputs.Turing himself [17] introduced the idea of Turing machines with an oracle.We think of an oracle Turing machine as a Turing machine with a specialhardware slot and any set A ⊆ Σ ∗ can be “plugged into” the slot. Themachine has a special query tape and a “branching instruction” in additionto the standard Turing machine instructions. The branching instructionhas the form q i , σ l → q j , q k . It is crucial that all oracle machines are stillspecified by finite programs of instructions of the two types, so we still havean effective enumeration of all oracle Turing machines. In a Turing machine M A with an oracle for A , an instruction q i , σ l → q j , q k works as follows. Ifthe machine M A is in state q i reading the symbol σ l on its work tape thenthe machine goes to state q j if the word written on the query tape belongsto the set A and goes to state q k if the word on the query tape is not in theset A .“Classical” results of computability theory “relativize” in the followingstrong sense. For example, take the proof of the unsolvability of the HaltingProblem. Not only the statement of the theorem but the given proof remaincorrect if one everywhere replaces the words “Turing machine” by the words“Turing machine with an oracle for A”. One could take this relativizationproperty as a definition of “classical”. However, the well-known theorem of Baker, Gill and Solovay, [2] showedthat the question of P versus N P does not relativize. It is easy to con-struct an oracle A such that P A = N P A . Indeed, any set A which iscomplete for P SP ACE will do. But there are many oracles A for which P A = N P A .Indeed, Bennett and Gill [1] showed that P A = N P A withrespect to a “random” oracle. This means that the set of A such that P A = N P A has Lebesgue measure one in the space of all languages over Σ.Later results show that for a random oracle the separation between P A and N P A is indeed very strong. Our approach in this section is inspired by theremarkable result of Kautz and Miltersen [10] which we will explain below.We use this approach to show that requiring polynomial time computationto succeed only on a “reasonable fraction” of the inputs does not signif-icantly improve our computing power when compared to nondeterminismfor “algorithmically random” oracles.First of all, the ideas of generic-case computability, and indeed fractionalcomputability at an allowable density δ certainly relativize without anyproblem. All definitions are exactly the same except that we now considerTuring machines with an oracle for A .For this section we work inside the class of functions E A = [ c DT IM E A (2 cn + c ) , computable in strict exponential time by Turing machines with an oraclefor A . Note that if we are working with respect to an oracle A then theelements of E A are total functions Σ ∗ → { , } .Effective measure theory was formulated by Lutz [14] building on earlierwork of Schnorr [16]. Recall that in discussing P and N P we are consideringsets of languages over an alphabet Σ. As mentioned earlier, we identifya language L with the infinite binary sequence specifying its characteristicfunction. We have the canonical enumeration of all words w , ..., w n , ... of allwords in Σ ∗ . We think of L as the infinite binary sequence L (0) , ..., L ( n ) , ... where L ( n ) = 1 if w n ∈ L and L ( n ) = 0 otherwise. We use the formulationof effective measure theory in terms of computable martingales, which arestrategies for betting on the values of successive bits of an infinite binarysequence. Formally, Definition 3.1. A martingale is a function d : { , } ∗ → R such that forall σ ∈ { , } ∗ ( ∗ ) d ( σ ) = d ( σ
0) + d ( σ d ( λ ) of d on the empty word is greater than 0.The martingale d succeeds on a sequence α ∈ { , } ∞ iflim sup n →∞ d ( α [1 , . . . , n ]) = ∞ . where α [1 , . . . , n ] is the initial segment of α of length n . The martingale d succeeds on a set S ⊆ { , } ∞ if it succeeds on all sequences in S . UANTITATIVE ASPECTS 11
We can think that we start with one dollar and double the bet eachtime, splitting the bet between the two possible next values according tothe strategy d . We succeed on the set S if we win an infinite amount ofmoney on every sequence in S . If we think of { , } ∞ as the unit intervalone can show that a set C ⊆ [0 ,
1] has Lebesgue measure 0 if and only thereexists some martingale which succeeds on C .For effective measure theory one imposes a condition on the difficultyof computing a martingale. We are interested in martingales which arecomputable in polynomial time with respect to a fixed oracle A . Definition 3.2. An A -polynomial-time martingale is a function d : { , } ∗ → Q which satisfies the martingale equation (*) and which is computable inpolynomial time by by some Turing machine with an oracle for the set A .A set S ⊆ { , } ∞ has effective polynomial-time measure zero with respectto A if there exists an A -polynomial-time martingale d which succeeds onall sequences in S . We write “ S has effective P A measure zero”.Recall that we are working inside a space E A of functions computablein strict exponential time by Turing machines with an oracle for A . Theargument given in Zimand [18] relativizes to give: Theorem 3.3. [18]
The set E A does not have effective P A -measure zero.Proof. For every A -polynomial time martingale d we define a language L ∈E A on which d does not succeed. The martingale equation 2 d ( w ) = d ( w
0) + d ( w
1) implies that either d ( w ≤ d ( w ) or d ( w ≤ d ( w ). We put the emptyword λ in L and then iteratively define L . If σ = L [1 , ..n ] has already beendefined, then L [1 , ..., n + 1] = σ d ( σ ≤ d ( σ ) and L [1 , ..., n + 1] = σ d does not succeed on L since d ( L [1 , ..., n ]) ≤ d ( λ )for all n .We need only check that L ∈ E A . Given an arbitrary w ∈ { , } ∗ , with | w | = n , we possibly need to calculate d on all words of length of length( n − c such that on inputs of length r d is calculablein time r c + c by a Turing machine with an oracle for A . Thus the entirecalculation can be done in time 2 n − [( n − c ] so L ∈ E A . (cid:3) (cid:3) In their remarkable article, Kautz and Miltersen [10] use the concept ofsets which are “algorithmically random” in the sense of Martin-Lof [13].The precise details of that definition need not to be given here and theimportant point for us is that it yields a large class of sets for which thefollowing theorem of Kautz and Miltersen holds.
Theorem 3.4 (Kautz, Miltersen [10]) . If A ⊆ Σ ∗ is an algorithmicallyrandom set then the set N P A does not have effective P A -measure zero in E A . In order to discuss fractional polynomial time computability we againneed to impose a suitable effective density condition which now becomes“uniformity”.
Definition 3.5.
A partial function φ from Σ ∗ to { , } is k - uniform if forall w ∈ Σ ∗ with | w | ≥ k , there exists a z with | z | ≤ k log ( | w | ) such that φ ( wz ) ↓ . Thus for every w there is a “reasonably short” z such that φ i converges on wz .A partial function φ is uniform if it is k -uniform for some positive integer k . We write U ( φ ) if φ is uniform.Note that if we have an algorithm Ω which generically solves a decisionproblem, then for every w there is some z such that Ω converges on wz .This is because any cylinder C = { wu } consisting of all words with prefix w is not a negligible set. Convention 3.6.
From now on, we will assume that Σ = { , } althoughall the arguments below work for an arbitrary finite alphabet Σ.In general, a superscript A for a function, such as φ A , indicates that φ A is a partial function computable by a Turing machine with an oracle for A .Similarly, a superscript A for a Turing machine, such as M A , indicates that M A is a Turing machine with an oracle for A . Definition 3.7. If φ Ai is a partial computable function, computed by the i -th Turing machine M Ai with an oracle for A , the function φ Ai [ i ] is thefunction computed as follows. We modify M Ai to a Turing machine Q Ai byadding a subroutine to force the the function obtained to be i -uniform withits computation time bounded by n ci on inputs | w | with | w | ≥ i , where c isa constant independent of i and w .In detail, on an input w , Q Ai prints ⊥ if | w | < i .Suppose now that | w | ≥ i . Then Q Ai carries out the computation of M Ai for n i steps. If M Ai calculates a value from { , } , then Q Ai prints thatvalue. If not, Q Ai considers, in the canonical order, the extensions wz with | z | ≤ i log( | w | ) and carries out the computation of M Ai on wz for | w | i steps.If M Ai calculates a value on such an extension, then the condition that we arecalculating a i -uniform function is verified for the input w and Q Ai outputsthe value ⊥ for input | w | . If M Ai does not calculate a value on any of theseextensions, then Q Ai outputs the value 0 for input w , again ensuring thatthe calculated function is i -uniform.The number of words z with | z | ≤ i log( | w | ) is 2 i log( | w | ) = 2 i | w | . Itfollows that for every w with | w | ≥ i the machine Q Ai prints a value 0 , ⊥ in at most | w | i + | w | i i | w | steps. Recall, that if | w | ≤ i −
1, then Q Ai prints the value ⊥ in the input w . Thus for every w ∈ Σ ∗ the machine Q Ai computes a value from { , , ⊥} on the input w in ≤ | w | ci steps where c > i and w .Since we uniformly effectively obtain Q Ai from M Ai , there is an effectiveenumeration of all the functions φ Ai [ i ]. Note that since any particular partialcomputable function has infinitely many indices, for any partial computablefunction φ A which is k -uniform for some k and whose computation time on UANTITATIVE ASPECTS 13 inputs for which it calculates a value is bounded by a polynomial, there is alarge enough index i such that φ Ai [ i ]( w ) = φ A ( w ) for all inputs with | w | ≥ i .Recall that we are identifying languages with their characteristic func-tions. Definition 3.8.
We consider the set
U F P A of all those languages (func-tions) in E A which are partially calculable by partial computable functionswhich are uniform with computation time strictly bounded by a polynomial n j on some Turing machine with an oracle for A . Formally, U F P A = { f : f ∈ E A , and there is some i such that φ Ai [ i ] ⊑ f } For the next theorem we essentially use the proof in section 3.4 of Zi-mand [18] that polynomial time P has effective polynomial-time measurezero, noting that it applies to U F P A . Theorem 3.9.
For every oracle A , the set U F P A has effective P A - measurezero in E A .Proof. First of all, as noted above, we can give an effective enumeration { Q iA } of all Turing machines with an oracle for A such that Q iA calculates φ Ai [ i ]. This means that we have one Turing machine Q A ( i, w ) such thatfor every w ∈ Σ ∗ Q A ( i, w ) simulates Q iA on input w in time bounded by(log i ) c | w | c i , where c > , c > i and | w | .Let S i be the set of functions S i = { f ∈ E A : φ Ai [ i ] ⊑ f } . Then
U F P A = S i ∈ N S i . We define a martingale which succeeds on U F P A in three stages.First, we need to define a martingale d i which succeeds on on the set S i . We use the variable x to denote arguments to a martingale. Since amartingale is betting on characteristic sequences of languages, the position x ( n ) is supposed to tell us the value f ( w n ) for the functions f in S i . It isimportant to keep in mind that | w n | ≤ log n . (By log n we mean ⌊ log n ⌋ ).) Let x ∈ Σ ∗ and let n = | x | .If | w n | < i −
1, we put d i ( x ) = 1.Suppose that | w n | ≥ i −
1. We set: d i ( x
0) := d i ( x ) if φ Ai [ i ]( w n +1 ) = 0 , φ Ai [ i ]( w n +1 ) = 1 ,d i ( x ) if φ Ai [ i ]( w n +1 ) = ⊥ , and d i ( x
1) := φ Ai [ i ]( w n +1 ) = 0 , d i ( x ) if φ Ai [ i ]( w n +1 ) = 1 ,d i ( x ) if φ Ai [ i ]( w n +1 ) = ⊥ . It is easy to see that d i is a martingale.Suppose that j < · · · < j m are indices such that | w j s | ≥ i and φ Ai [ i ]( w j s )is defined for s = 1 , . . . m . Then for any x ∈ Σ ∗ with | x | = j m such that φ Ai [ i ][1 , . . . , j m ] ⊑ x we have d i ( x ) = d i ( x (1) . . . x ( j ) . . . x ( j ) . . . x ( j m )) ≥ m . Hence d i succeeds on S i . There is a Turing machine D A ( i, x ) with an oraclefor A , which, given i and x ∈ { , } ∗ with | x | = n , computes d i ( x ) in timebounded by n (log i ) c (log n ) c i . If | x | = n ≥ i , this time is at most( † ) n (log n ) c (log n ) c i ≤ n (log n ) c i where c > i, n . Similarly, if n ≥ | x | and n ≥ i then d i ( x ) is computed in time bounded by the estimate ( † ).Second, in order to obtain a global martingale which is calculable in poly-nomial time we need to exponentially inflate indices. Let e S i = S i , and let e S j = ∅ if j does not have the form 2 i .Let e d j be the constant martingale assigning 1 to all inputs if S j = ∅ . Let e d j = d i if j = 2 i . In the this case, e d j ( x ) can be calculated in time ≤ n (log n ) c log(log j ) for | x | = n ≥ j. We now need the inequality( ‡ ) (log n ) log log j ≤ n for n ≥ j ≥ . Since log is an increasing function, if we fix n , it suffices to prove the in-equality for j = n . Taking logs of both sides of the inequality and setting j = n we need (log(log n )) ≤ log n which holds for n ≥ † ) and ( ‡ ) imply that e d j ( x ) can be calculated for | x | = n ≥ j in time n c , where c is independent of j, n . The same is true if n ≥ | x | and n ≥ j .Third, we now need to define another martingale b d j which dampens e d .If x ∈ { , } ∗ is nonempty, let pref ( x ) denote the prefix of x of length | x | −
1. Let δ j ( x ) be defined for nonempty x by e d j ( x ) = δ j ( x ) e d j ( pref ( x )),provided e d j ( pref ( x )) = 0. If e d j ( pref ( x )) = 0 , we put δ j ( x ) = 1.Note that if e d j ( pref ( x )) = 0 then e d j ( x ) = 0 by the martingale equationfor e d j . Thus in this case we also have e d j ( x ) = δ j ( x ) e d j ( pref ( x )).Note also that δ j takes values in { , , } .We set b d j ( x ) = ( the constant value 2 − j if | x | < jδ j ( x ) b d j ( pref ( x )) if | x | ≥ j UANTITATIVE ASPECTS 15
From the martingale equation for e d j we have:2 e d j ( x ) = e d j ( x
0) + e d j ( x
1) = [ δ j ( x
0) + δ j ( x e d j ( x ) , and so δ j ( x
0) + δ j ( x
1) = 2 for all x . Thus for | x | ≥ j − b d j ( x
0) + b d j ( x
1) = δ j ( x ) b d j ( x ) + δ j ( x ) b d j ( x ) = 2 b d j ( x ) . Similarly, it follows from the definition that for | x | < j − b d j ( x
0) + b d j ( x
1) = 2 b d j ( x ). Thus b d j satisfies the martingale equation.It is easy to see that b d j succeeds on e S j . To calculate b d j we need tocompute e d j ( y ) and δ j ( y ) on the prefixes y of x and this can be done in time n c on inputs x with | x | = n ≥ j .We put these martingales together in the “global” martingale b d ( x ) = ∞ X j =1 b d j ( x )(1) = | x | X j =1 b d j ( x ) + ∞ X j = | x | +1 − j (2) = | x | X j =1 b d j ( x ) + 2 −| x | (3)Then b d is a martingale which is calculable in polynomial time by a Turingmachine with an oracle for A . For each j and x with | x | ≥ j we have b d ( x ) > b d j ( x ), so since b d j ( x ) succeeds on e S j then b d succeeds on e S j . Thisimplies that b d succeeds on U F P A = ∪ j e S j = ∪ i S i and hence U F P A haseffective P A -measure zero, as claimed. (cid:3)
Corollary 3.10.
We have:
N P A − U F P A = ∅ . Thus partial complexity cannot compensate for nondeterminism in thepresence of a random oracle and it is reasonable to suppose that some similarseparation remains true without an oracle. For example, let GP be theclass of languages which are generically decidable in polynomial time. Theassumption that N P − GP 6 = ∅ , would say that there are languages in N P which require nondeterminism on a nonnegligible set of inputs and iscertainly a stronger hypothesis than just assuming that
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