On Process Complexity

Transcription

1 On Process Coplexity Ada R. Day School of Matheatics, Statistics and Coputer Science, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand, Eail: Abstract Process coplexity is one of the basic variants of Kologorov coplexity. Unlike plain Kologorov coplexity, process coplexity provides a siple characterization of randoness for real nubers in ters of initial segent coplexity. Process coplexity was first developed in (Schnorr 1973). Schnorr s definition of a process, while siple, can be difficult to work with. In any situations, a preferable definition of a process is that given by Levin in (Levin & Zvonkin 1970). In this paper we define a variant of process coplexity based on Levin s definition of a process. We call this variant strict process coplexity. Strict process coplexity retains the ain desirable properties of process coplexity. Particularly, it provides siple characterizations of Martin-Löf rando real nubers, and of coputable real nubers. However, we will prove that strict process coplexity does not agree within an additive constant with Schnorr s original process coplexity. One of the basic properties of prefix-free coplexity is that it is subadditive. Subadditive eans that there is soe constant d such that for all strings σ, τ the coplexity of στ (σ and τ concatenated) is less than or equal to the su of the coplexities of σ and τ plus d. A fundaental question about any coplexity easure is whether or not it is subadditive. In this paper we resolve this question for process coplexity by proving that neither of these process coplexities is subadditive. 1 Introduction The basic concept behind Algorithic Inforation Theory is that the inherent coplexity of a string can be quantified by the iniu nuber of bits of inforation needed to describe it. This concept was first foralized by defining the coplexity of a string σ, as the length of the shortest description of σ with respect to soe universal interpreter. This is known as the plain Kologorov coplexity of σ. However, plain Kologorov coplexity, while useful for any purposes, does not truly capture this basic concept (Downey & Hirschfeldt to appear, Li & Vitányi 1997). The proble is that the universal interpreter can use not only the bits within the description, but also the length of the description to deterine its output. There are several variants of Kologorov coplexity that attept to overcoe this proble. Each variant provides different insight into the nature of Copyright c 2009, Australian Coputer Society, Inc. This paper appeared at the Fifteenth Coputing: The Australasian Theory Syposiu (CATS2009), Wellington, New Zealand. Conferences in Research and Practice in Inforation Technology (CRPIT), Vol. 94, Rod Downey and Prabhu Manye, Ed. Reproduction for acadeic, not-for profit purposes peritted provided this text is included. coplexity. These variants include the widely used prefix-free coplexity as well as the process coplexity introduced in (Schnorr 1973). One reason prefixfree coplexity has been extensively studied is that the Kraft-Chaitin theore and the coding theore provide an easy eans of constructing prefix-free achines. No siilar theores exist for process coplexity and this coplexity sees genuinely ore difficult to deal with. In order to give a foral definition of the coplexities we will study, first we will define coplexity with respect to a general function. Take any F : 2 <ω 2 <ω (where 2 <ω is the set of all finite binary strings). The coplexity of the string σ with respect to F is: in{ τ : F (τ) = σ} if τ 2 <ω, C F (σ) = F (τ) = σ otherwise In the above definition, τ refers to the length of the string τ. We define the plain Kologorov coplexity of a string σ to be C(σ) = C U (σ) where U is a fixed universal Turing achine. We call a finite string σ rando if C(σ) σ. Note that U is a universal achine for the whole class of Turing achines. One way to produce variants of Kologorov coplexity is to use a universal achine for a subclass of Turing achines. Two iportant subclasses of Turing achines are the class of all prefix-free achines and the class of all process achines. To understand these achines, first note that 2 <ω has the following natural partial ordering. If τ 1, τ 2 2 <ω then we define τ 1 τ 2 to hold if τ 2 is an extension of τ 1. A subset A of 2 <ω is called prefix-free if it is anti-chain with respect to this partial ordering, or alternatively if for all distinct τ 1, τ 2 in A, τ 1 τ 2. A prefix-free achine is a partial coputable function 2 <ω 2 <ω whose doain is a prefix-free set. It can be shown that there exists a universal prefixfree achine and by fixing U this tie as a universal prefix-free achine we can siilarly define the prefixfree coplexity of σ by K(σ) = C U (σ). Another natural subclass is the class of all process achines. The key idea about a process is that it preserves the ordering on 2 <ω aking its action continuous. Definition 1.1. A process achine is a partial coputable function M : 2 <ω 2 <ω such that if τ, τ do(m), and τ τ, then M(τ ) M(τ). A natural exaple of a process is given by the recordings of oves in a gae of chess e.g. 1. e4 e5, 2. Nf3 Nc6 etc. Given the descriptions of the first n oves, it is possible to replay the gae up until that point. If we extend the description, we can extend the replay of the gae by adding new oves, but we cannot change any oves already defined.

2 Again we can take a universal process achine U and define the process coplexity K MD of a string σ by K MD (σ) = C U (σ). This definition of a process achine and related coplexity was given in (Schnorr 1973). We follow the notation of (Downey & Hirschfeldt to appear) by using K MD to denote process coplexity. Schnorr s definition of a process differs slightly fro that given by Levin in (Levin & Zvonkin 1970). We will use the ter strict process achine for Levin s definition. It is defined as follows. Definition 1.2. A strict process achine is a partial coputable function M : 2 <ω 2 <ω such that if τ do(m) and τ τ, then τ do(m) and M(τ ) M(τ). While Levin defined strict process achines, he did not use the to define a easure of coplexity in the sae way as Schnorr. Instead he used strict process achines to construct a universal seieasure fro which he defined another variant of Kologorov coplexity (Levin & Zvonkin 1970, Levin 1973). Both of these definitions of process achines have erit. Schnorr s definition corresponds to a hooorphis of the doain of M. Levin s definition has the following very natural odel. This odel is alost identical to one described in the first paper on algorithic randoness (Soloonoff 1964). Take a three-tape Turing achine M with a read-only oneway input tape, a one-way write-once output tape, and a work tape. The first square of the input table is blank and the input head starts on that square. Let the achine run. If at any stage M wants to ove the input head of the tape, first we define M(τ) = σ, where τ is the input string read so far and σ is the current output on the output tape. Levin s definition of a process can be easier to deal with. This becoes apparent when attepting any gae based proof where the opponent is developing a process achine. If the opponent ust keep the doain of the process achine closed under substrings, then the gae becoes uch sipler. This is our ain otivation for introducing the following definition. Definition 1.3. Given any σ 2 <ω, the strict process coplexity K MS (σ) is defined by C U (σ) where U is a universal strict process achine. The existence of a universal strict process achine is established by (Levin & Zvonkin 1970). Note that a prefix-free achine can be considered as a strict process achine. This is done as follows. Suppose M : 2 <ω 2 <ω is a prefix-free achine. We take λ to be the epty string and we define a strict process achine M as follows: M(σ) M λ (σ) = undefined if σ do(m) if there exists σ > σ and σ do(m) otherwise It can be shown that if M is partial coputable then so is M. Additionally, with the exception of the epty string, the coplexities generated by the two achines agree. As a strict process achine is a type of process achine it follows that for all σ, K MD (σ) K MS (σ) + O(1) K(σ) + O(1). It is not iediately apparent that K MD and K MS differ in any significant way. The first result that we will prove in this paper is that the coplexities K MD and K MS are different, that is they do not agree within any additive constant. In fact we will go further and show that for any a R with 0 < a < 1, there are infinitely any σ such that K MS (σ) K MD (σ) > a log log σ. In the opening paragraph, we stated that plain Kologorov coplexity did not capture the basic concept behind Algorithic Inforation Theory. This becae apparent during attepts to define algorithic randoness for real nubers. 1 The intuition is that a real α should be rando if all of its initial segents are rando strings (though we allow deviation by soe constant aount). Thus we would like to be able to define α as rando if C(α n) n O(1) (where α n is the first n bits of α). The proble with this attept at a definition is that Martin-Löf showed: given any d, then for any sufficiently long string υ there is an initial segent σ of υ such that C(σ) < σ d (Downey & Hirschfeldt to appear). In particular, this proves that there are no reals α with the property that C(α n) n O(1). Martin-Löf s proof ade direct use of the fact that a Turing achine can use the bits of a description τ and additionally another log τ bits fro the length of τ to deterine its output. Prefix-free coplexity or process coplexity can be used to define randoness for a real nuber based on initial segent coplexity (Schnorr 1973, Chaitin 1987). If we say that a real α is rando if for all n, Q(α n) n O(1) where Q is either K, or K MD, then we get a non-epty class of rando reals. Now it does not atter which of the two we choose Q to be because they both give rise to the sae class of rando reals. The class is in fact the class of Martin- Löf rando reals, the ost coonly used notion of randoness in algorithic inforation theory. This shows that prefix-free achines and process achines are unable to use those extra log τ bits. As the coplexity K MS lies between K and K MD it follows that if we replace Q by K MS, we still get the sae class of rando reals. Hence strict process coplexity retains the desirable property of process coplexity by providing a siple characterization of the Martin-Löf rando reals. Additionally, like process coplexity, strict process coplexity provides a siple characterization of coputable reals. A real α is coputable if and only if K MS (α n) K MS (n) + O(1) where K MS (n) = K MS (1 n ) (1 n is the string fored by repeating 1 n ties). The non-trivial direction follows because K MS (n) log(n) + O(1) and C(α n) K MS (α n) + O(1). Hence if for soe α, K MS (α n) K MS (n) + O(1), then C(α n) log(n) + O(1) and so α is coputable by a theore of Chaitin s (Downey & Hirschfeldt to appear). Martin-Löf s proof also showed that plain Kologorov coplexity is not subadditive. That is to say, for any d there exists strings σ, τ such that C(στ) > C(σ) + C(τ) + d where στ is the string fored by appending τ to the end of σ. It does this because we can take a rando finite string υ that has an initial segent σ with C(σ) < σ d. Now if τ is chosen so that στ = υ then C(στ) σ + τ > C(σ) + d + C(τ) i where i is the length of the index of the identity function in U. As i is fixed we can ake d i arbitrarily large. Thus we have that plain Kologorov coplexity is not subadditive. On the other hand, prefix-free coplexity is an exaple of a coplexity that is subadditive. The second result that we will prove is that both process coplexity and strict process coplexity are not subadditive. The proof Martin-Löf used for plain coplexity cannot be adapted to process coplexity. This is because given a rando real α, it is true that 1 A real is identified as an infinite binary string.

3 K MD (α n) n O(1) i.e. there are no arbitrary drops in initial segent coplexity. Thus the question as to whether these coplexities are subadditive requires new techniques. In particular we need to use non-rando strings. The new techniques used for building and analyzing process achines introduced here ay well have wider application. 1.1 Conventions The set of all binary strings of length n, the set of all finite binary strings, and the set of all infinite binary strings will be denoted by {0, 1} n, 2 <ω, and 2 ω respectively. The epty string will be represented by λ. The relation on 2 <ω (2 <ω 2 ω ) is defined by σ τ if σ is an initial segent of τ. We say σ < τ if σ τ and τ σ. If σ τ and τ σ, then τ and σ are said to be incoparable written σ τ. The operation of appending a string τ to the end of a finite string σ, will be represented by στ. If σ 2 <ω let σ be the length of σ and if i N with 1 i σ let σ(i) be the ith bit of σ. The ain proofs in this paper will be cobinatorial in nature. They will ake use of soe basic properties of Cantor space. Cantor space is the topology on 2 ω defined by taking {[σ] : σ 2 <ω }, where [σ] = {σα : α 2 ω } for each σ 2 <ω, as a basis of open sets. If X 2 <ω, then [X] = σ X [σ]. The Lebesgue easure on Cantor space µ is the outer easure obtained by defining µ([σ]) = 2 σ for all open sets [σ] in the basis. The ain properties of Cantor space that we need are as follows. Firstly µ(2 ω ) = µ([λ]) = 1. Secondly if τ 1 and τ 2 are incoparable eleents of 2 <ω then [τ 1 ] [τ 2 ] =. This iplies that if A 2 <ω is a prefix-free set, then µ([a]) = σ A 2 σ. Given a partial coputable function M : X Y and x X, we write M(x) if x is an eleent of the doain of M and M(x) otherwise. Further, if we are regarding M as a Turing achine, we will write M(x)[s] if M halts on input x within s coputational steps, and M(x)[s] otherwise. Logs used are all base 2 and are rounded up to the nearest integer value. By convention log 0 = 0. 2 Strict process coplexity and process coplexity In this section we will show that strict process coplexity and process coplexity are in fact different. As the universal strict process achine is a process achine, there is soe constant d such that for all σ 2 <ω : K MD (σ) K MS (σ) + d We want to show that this inequality cannot be reversed and so strict process coplexity and process coplexity do not agree within an additive constant. To show this, we will ake use of the fact that the universal strict process achine has a coputable approxiation. Let U be a universal strict process achine. For all s N, define: { U(τ) if U(τ)[s] U s = otherwise Because U is a strict process achine, we can take our approxiation to have the property that if U s (τ) = σ and τ < τ then there is soe σ σ such that U s (τ ) = σ. In lea 2.3, we will show that for any i N, we can construct a process achine f i such that there exists a string σ i with C fi (σ i ) + i < K MS (σ i ). Once we have done this, it will not be too difficult to cobine these achines to prove that strict process coplexity and process coplexity do not agree within an additive constant. To understand the ideas behind the proof of lea 2.3, let us take the case i = 1 as an exaple. We will construct a process achine f 1. When we construct this achine, we are able to first define f 1 for all strings of length 3. Then at a later stage, we have the option of defining f 1 for strings of length 2 and even later for strings of length 1. This option is not available to the universal strict process achine U. Once a string τ is added to the doain of U, all initial segents of τ ust be added as well. Our definition of f 1 starts as follows. First let τ = abc be any binary string of length 3, i.e. a, b, and c are the first, second and third bits of τ respectively. We define f 1 by f 1 (abc) = a 8 b 16 c e.g. f 1 (010) = We can consider this as stretching all but the last bit of the the string abc. Now we wait until soe stage s 1, when for all τ {0, 1} 3, C Us 1 (f 1 (τ)) 4. If this never happens then we have finished because for soe τ {0, 1} 3, K MS (f 1 (τ)) = li s CUs (f 1 (τ)) > 4 = τ + 1 = C f1 (f 1 (τ)) + 1 So assue such a stage s 1 occurs. For all τ {0, 1} 3, let ρ τ be soe string in the doain of U s1 such that ρ τ 4 and U s1 (ρ τ ) = f 1 (τ). Let A 1 be the set of all such ρ τ. Note that A 1 ust be a prefixfree set and A 1 = 2 3. If it is not, then there is soe τ, υ {0, 1} 3 with: τ υ; ρ τ, ρ υ A 1 ; and ρ τ ρ υ. But this would ean that f 1 (τ) = U s1 (ρ τ ) U s1 (ρ υ ) = f 1 (υ) which contradicts our definition of f 1. It follows that µ([a 1 ]) A = 1 2. We will now define f 1 for all binary strings of length 2. Given any two bit binary string ab, there ust be soe string a 8 b k with 1 k 16 such that C Us 1 (a 8 b k ) > 3. This is true because there are at ost 15 strings whose coplexity is less than or equal to 3 (as there are only 15 such descriptions). Now we define f 1 (ab) = a 8 b k. Now consider for a oent how the universal strict process achine can respond to this. Because U is a strict process achine, if τ is any initial segent of a string in A 1, then U(τ)[s 1 ]. So if τ 3, then U(τ) f 1 (ab) for any a, b {0, 1}. This eans that in order to reduce the coplexity of f 1 (ab) to 3 or less, U needs to find a short description that is not an initial segent of an eleent of A 1. Again we wait until soe stage s 2, when for all τ {0, 1} 2, C Us 2 (f 1 (τ)) 3. If this never happens then again our objective is achieved. If this stage does occur then for all τ {0, 1} 2, let ρ τ be soe string in the doain of U s2 such that ρ τ 3 and U s2 (ρ τ ) = f 1 (τ). Let A 2 be the set of all such ρ τ. Again µ([a 2 ]) A = 1 2. We want to show that [A 1 ] [A 2 ] =. Take any ρ 1 A 1 and ρ 2 A 2. U(ρ 1 ) > U(ρ 2 ) so we know that ρ 1 ρ 2. Now let us show that U(ρ 2 )[s 1 ]. If U(ρ 2 )[s 1 ], then C Us 1 (U(ρ 2 )) ρ 2 3. So by our construction of f 1, f 1 (ab) U(ρ 2 ) for any a, b {0, 1}. This is a contraction and so U(ρ 2 )[s 1 ]. As U(ρ 1 )[s 1 ] so it ust be that ρ 2 ρ 1 because U is a strict process achine. Hence ρ 1 and ρ 2 are incoparable and so µ([do(u s2 )]) µ(a 1 ) + µ(a 2 ) = 1. Finally we define f 1 (0) = 0 k for soe 1 k 8

4 such that C Us 2 (0 k ) > 2 (there ust be soe k as there are only 7 possible descriptions of length less than or equal to 2). Consider any υ 2 <ω. As µ(a 1 A 2 ) = 1, either υ is an initial segent, or an extension, of soe eleent ρ A 1 A 2. If υ > ρ and U(υ) then U(υ) U(ρ) and so U(υ) f 1 (0). If υ ρ, then U(υ)[s 2 ] so if U(υ) = f 1 (0) then it ust be that υ > 2 (as we chose f 1 (0) so that C Us 2 (f 1 (0)) > 2). Hence: C f1 (f 1 (0)) + 1 = < C U (f 1 (0)) = K MS (f 1 (0)) The ain idea is that if the universal strict process achine attepts to respond each tie strings are added to the doain of f 1, then it will run out of easure. The key point is that during the construction of f 1, we can reuse easure by using initial segents of strings already in the doain of f 1. On the other hand, U needs to add new easure into its doain at each response. To adapt this arguent to hold for any i, let us start with a lea giving a lower bound on the easure of the doain of a strict process achine. Lea 2.1. If {(τ 1, σ 1 ),..., (τ n, σ n )} is a set of ordered pairs such that for all i, j N, 1 i, j n, U(τ i ) = σ i ; and if i j then: 1. σ i σ j ; and 2. σ i > σ j iplies that there exists an s such that U s (τ i ) and U s (τ j ) ; then µ([do(u)]) n i=1 2 τi. Proof. We will show that the τ i for an prefix-free set. Choose any i j. If σ i σ j then as U is a process achine, τ i τ j. If σ i > σ j then there exists an s such that U s (τ i ) and U s (τ j ), so as U is a strict process achine τ j is not an initial segent of τ i. Further τ j cannot extend τ i as this would iply that σ j σ i. Hence τ i τ j. Siilarly if σ j > σ i then τ i τ j as well. We will foralize the notion of stretching a string as follows. Definition 2.2. If g : N N, then ĝ : 2 <ω 2 <ω is defined by: ĝ(τ) = τ(1) g(1) τ(2) g(2)... τ( τ ) g( τ ) For exaple if g is the identity function, then ĝ(0101) = = Lea 2.3. For all i N, there exists a process achine f i and a string σ i such that: C fi (σ i ) + i < K MS (σ i ). Proof. Again we take U to be the universal strict process achine. We looked at the case f 1 as an exaple. In this case we started by defining f 1 for strings of length 3, and then for strings of progressively shorter lengths. For the general case, we will start by defining f i for strings of length 2 i + 1. If necessary we will define f i for strings of length 2 i, 2 i 1, 2 i 2 and so on. Our propt to extend the doain of f i is if at soe stage s, the universal strict process achine has ade C Us (σ) C fi (σ) + i for all eleents σ in the doain of f i at stage s. We define g i : N N by g i (n) = 2 n+i+1. We will use g i in order to stretch strings. g i is defined like this because for any n, there are only 2 n+i+1 1 descriptions of length less than or equal to n + i. By aking g i (n) larger than this, we know that at any stage s, we can always find initial segents of the stretched string without descriptions of length n+i in U s. The construction of f i proceeds as follows. At stage 0, first set l 0 = 2 i + 1. Then for all τ {0, 1} l0 set f i (τ) = ĝ i (τ ( τ 1))τ( τ ) i.e. we use g i to stretch all but the last bit of τ. At stage s + 1, if there exists soe τ {0, 1} ls such that C Us (f i (τ)) > τ + i, then set l s+1 = l s and go to the next stage. Otherwise, we know that for all τ {0, 1} ls, C Us (f i (τ)) τ + i. Now we will extend the doain of f i. We set l s+1 = l s 1. For all τ {0, 1} ls+1 and for all k N, 1 k g i ( τ ). Let σ τ,k = ĝ i (τ (τ 1))τ( τ ) k. As there are 2 τ +i+1 possible values of k, it follows that for any τ, there ust be soe k such that: C Us (σ τ,k ) > τ + i because there are only 2 τ +i+1 1 descriptions of length less than or equal to τ + i. For all τ {0, 1} ls+1, let σ τ = σ τ,k for such a k and set f i (τ) = σ τ. To verify the construction we will first show that for all s, l s > 0. We will prove this by showing that the alternative iplies that U runs out of easure. If l s = 0 for soe s, then for all τ 2 <ω such that 1 τ 2 i + 1, K MS (f i (τ)) τ + i because this is the condition to extend the doain of f i. So for all such τ we can choose a string ρ τ such that (i) ρ τ τ + i, (ii) U(ρ τ ) = f i (τ) and such that no other string with properties (i) and (ii) halts before U(ρ τ ) halts. Now take the set A = {(ρ τ, f i (τ)) : τ 2 <ω, 1 τ 2 i + 1}. Consider any τ 1, τ 2 2 <ω, 1 τ 1, τ 2 2 i +1, and τ 1 τ 2. First f i (τ 1 ) f i (τ 2 ) as f i is injective. If f i (τ 1 ) < f i (τ 2 ) then by construction this iplies that τ 1 < τ 2. Now f i is defined for τ 1 after f i is defined for τ 2. Further if f i (τ 1 ) is defined at stage t + 1, it ust be that: 1. C Ut (f i (τ 2 )) τ 2 + i; and 2. C Ut (f i (τ 1 )) > τ 1 + i. (1) follows as this is required to extend the doain of f i to strings of length < τ 2. (2) follows by our choice of f i (τ 1 ). Now by (i) and (ii), (1) iplies that U t (ρ τ2 ) and (2) iplies that U t (ρ τ1 ). The set A therefore eets the conditions of lea 2.1 and this iplies that: µ([do(u)]) 2 ρτ τ 2 <ω,1 τ 2 i +1 τ 2 <ω,1 τ 2 i +1 = 2 i (2 i + 1) > 1 2 τ i A contradiction and so for all s, l s > 0. Let n = in{l s : s N}. It follows that for soe τ {0, 1} n, for all s, C Us (f i (τ)) > τ + i and hence K MS (f i (τ)) > τ + i. Thus we can take σ i = f i (τ), and we have that C fi (σ i ) + i = τ + i < K MS (σ i ). Finally, f i is a process achine because consider any τ 1, τ 2 do(f i ) and τ 1 < τ 2. We have that: f i (τ 1 ) ĝ i (τ 1 ) ĝ i (τ 2 (τ 2 1)) f i (τ 2 ). Note that lea 2.3 is unifor.

5 Theore 2.4. K MD and K MS do not agree within an additive constant. Proof. Define a process achine f by f(0 i 1τ) = f 2i (τ). Let c be the length of the index of f in the universal process achine. Now given any d, take i = 2(c + d + 1). By lea 2.3, there exists soe σ i such that C fi (σ i ) + i < K MS (σ i ), so: K MD (σ i ) + d C f (σ i ) + d + c C fi (σ i ) + d + c + i = C fi (σ i ) + i < K MS (σ i ) In fact we can go further and prove that this constant can be replaced with a function of order log log(n) where n is string length. First note that we can deterine an upper bound on the length of the σ i fro lea 2.3. Because σ i = f i (τ) for soe τ with τ 2 i + 1, and as we stretch all but the last bit of τ we know that: σ i 1 + g i (n) 2 i n=1 = i n=1 2 n+i+1 2 i 1 = i+2 n=0 2 n = i+2 (2 2i 1) < 2 2i +i+2 It will be easier to express this as log σ i < 2 i + i + 2. Theore 2.5. Given any a R, 0 < a < 1, then there exist infinitely any σ such that: K MS (σ) K MD (σ) > a log log σ. Proof. Given any such a choose k N >0 such that a < 1 1 k. Now define a process f : 2<ω 2 <ω by f(0 i 1τ) = f i (τ). Let c be the length of the index of f in the universal process achine. Choose any d N such that 1 divides c + d + 1. Now let i = (c+d+1) 1. Hence i is a positive integer, and i = c + d i i. This iplies that is a positive integer too. Let σ d = σ i fro lea 2.3. We know that: K MD (σ d ) + d C f (σ d ) + c + d Also we know that: = C fi (σ d ) + c + d i = C fi (σ d ) + i < K MS (σ d ) log σ d < 2 (c+d+1) (c + d + 1) = c d + c d k ( 1)( 2). where c 1 and c 2 are constant. Let j = As there are infinitely any d that we can choose, we can consider those d such that 2 jd ax(c d, c 2, 2). So we have that: log σ d < 2 jd + 2 jd 2 1 d = 2 jd ( d ) 2 jd (2 1 d+jd ) = 2 1 d+2jd = 2 k k 1 d The last step follows because: 1 + 2j = So log log σ d < 1 + 4k 2 = ( 1)( 2) = k k 1 a log log σ d < k 1 k So for these such d: ( 1)( 2) k k 1d and hence we have that: log log σ d < d K MD (σ d ) + d < K MS (σ d ) K MD (σ d ) + a log log σ d < K MS (σ d ) K MS (σ d ) K MD (σ d ) > a log log σ d There are infinitely any d that eet the conditions we require so the result follows. Theore 2.5 gives a lower bound on the difference between K MD and K MS. A basic upper bound on the difference between these coplexities is that for all σ, K MS (σ) K MD (σ) 2 log σ + O(1). This holds because the difference between onotone coplexity K M (a related coplexity) and prefix-free coplexity is bounded above by the sae aount and both K M (σ) K MD (σ) + O(1) and K MS (σ) K(σ)+O(1)(Li & Vitányi 1997). This leaves an open question with respect to the difference between these two coplexities which of these two bounds can be iproved? 3 Process coplexity is not subadditive Our objective for this section is to prove that process coplexity and strict process coplexity are not subadditive. We will present the proof for process coplexity but it holds without odification for strict process coplexity as well. Theore 3.1. Let U be a universal process achine. For any d N, there exists a σ, τ such that: K MD (στ) > K MD (σ) + K MD (τ) + d. We will prove this theore by giving short descriptions to a lot of strings. We will argue cobinatorially that one of these strings σ, ust have an extension στ such that the desired property holds. The following lea expresses a basic cobinatorial fact about process achines.

6 Lea 3.2. If A 2 <ω is a prefix-free set, then: 2 K M (σ) D 1 σ A Proof. Consider B = {τ σ : σ A} where τ σ is a shortest description of σ with respect to the universal process achine U. If τ 1, τ 2 are distinct eleents of B, then U(τ 1 ) and U(τ 2 ) are incoparable (as they are both in A). Therefore because U is a process achine we have that τ 1 τ 2. Hence B is an prefixfree set as well and the result follows because: 2 τ = µ([b]) 1 σ A 2 KMD (σ) = τ B Proof of theore 3.1. The identity function is a process so there is soe constant c i such that for all σ 2 <ω, K MD (σ) σ + c i. Now given any d, choose e = d + c i + c g. Fro the previous lea, there exists soe, i, υ with υ Bi such that K MD (υ) > υ i + e. Now set σ = υ 2i so σ A i and thus K MD (σ) σ i + c g. Choose τ so that στ = υ. Now K MD (τ) τ + c i. Cobining these results gives us that: K MD (σ) + K MD (τ) + d σ i + c g + τ + c i + d = στ i + e < K MD (στ) Let g : N N be the constant function g(x) = 2. Now the function ĝ : 2 <ω 2 <ω is a process (recall the definition of ĝ in section 2). For all i, we can define A i = {ĝ(τ) : τ {0, 1} i }. So A 0 = {λ}, A 1 = {00, 11}, A 2 = {0000, 0011, 1100, 1111} etc. As ĝ is a process, there is soe constant c g such that for all ρ A i, K MD (ρ) i+c g = ρ i+c g. The A i s are our sets of strings with short descriptions. For all, i such that 2i + 2, we define Bi = {σ {0, 1} : ρ A i (ρ01 σ or ρ10 σ)}. The Bi s are the sets of extensions that we will exaine. It is iportant to note that we have constructed the Bi s so that if i j then B i Bj =. We will explain the reason for this using an exaple. Consider σ = Because σ extends an eleent of A 2, 0011, we want to place σ in B2 8 and not in B1 8 even though σ extends 00 as well. This is because if we break σ into the strings 0011 and 0100 then the difference between the length of the first string 0011, and the length of its ĝ description is 2 (ĝ(01) = 0011). This is larger than the difference between the length of 00 and the length of its ĝ description. We need to use the larger difference for the following arguent to hold. Note that Bi = A i 2 2 2i 2 = 2 i 1. Now we will use the following lea to find the extension we want. Lea 3.3. Given any e N,, i, σ with σ B i such that K MD (σ) > σ i + e. Proof. Take = 2 e+3 and assue that no such i, σ exist. If so, then: σ {0,1} 2 KMD (σ) = = σ B i σ B i 2 K M D (σ) 2 σ +i e Bi 2 +i e i 1 2 +i e = ( 2 )2 e 1 > 1 This is a contradiction by lea 3.2 as {0, 1} is a prefix-free set. Acknowledgents The author would like to thank his PhD supervisor, Rod Downey, for helpful discussions in the developent of this paper. The author would also like to thank the anonyous referees for their helpful coents. The author also acknowledges the support of the New Zealand Tertiary Education Coission. References Chaitin, G. (1975), A theory of progra size forally identical to inforation theory, J. ACM 22(3), Chaitin, G. (1987), Incopleteness theores for rando reals, Adv. Appl. Math. 8(2), Downey, R. & Hirschfeldt, D. (to appear), Algorithic Randoness and Coplexity, Springer-Verlag. Gács, P. (1974), On the syetry of algorithic inforation, Soviet Math. Dokl. 15. Kologorov, A. (1965), Three approaches to the quantitiative definition of inforation, Probles of Inforation Transission 1. Levin, L. (1973), On the notion of a rando sequence, Soviet Math. Dokl. 14(5), Levin, L. (1974), Laws of inforation conservation (non-growth) and aspects of the foundation of probability theory, Probles of Inforation Transission 10. Levin, L. & Zvonkin, A. (1970), The coplexity of finite objects and the developent of the concepts of inforation and randoness of eans of the theory of algoriths, Russian Math. Surveys 25(6). Li, M. & Vitányi, P. (1997), An introduction to Kologorov coplexity and its applications (2nd ed.), Springer-Verlag New York, Inc., Secaucus, NJ, USA. Schnorr, C. (1973), Process coplexity adn effective rando test, Journal of Coputer and Syste Sciences 7. Schnorr, C. (1977), A survey of the theory of rando sequences, in: Basic Probles in Methodology and Linguistics, D. Reidel, Dordrecht, Holland, pp Soloonoff, R. (1964), A foral theory of inductive inference, Inforation and Control 7.