Foundations of Online Structure Theory II; The Operator Approach
aa r X i v : . [ m a t h . L O ] J u l FOUNDATIONS OF ONLINE STRUCTURE THEORY II: THEOPERATOR APPROACH
ROD DOWNEY, ALEXANDER MELNIKOV, AND KENG MENG NG
Abstract.
We introduce a framework for online structure theory. Our ap-proach generalises notions arising independently in several areas of computabil-ity theory and complexity theory. We suggest a unifying approach using op-erators where we allow the input to be a countable object of an arbitrarycomplexity. Thus we will begin a view that online algorithms can be viewedas a sub-area of computable analysis. We will give a new framework which (i)ties online algorithms with computable analysis, (ii) shows how to use modi-fications of notions from computable analysis, such as Weihrauch reducibility,to analyse finite but uniform combinatorics, (iii) show how to finitize reversemathematics to suggest a fine structure of finite analogs of infinite combinato-rial problems, and (iv) see how similar ideas can be amalgamated from areassuch as EX-learning, computable analysis, distributed computing and the like.Conversely, we also get an enrichment of computable analysis from ideas fromthe analysis of classical online algorithms.
Contents
1. Introduction 21.1. Our Goal 21.2. The [8] Punctual Model 31.3. Turing computable mathematics 31.4. Online combinatorics 41.5. Online vs. Turing computable 51.6. Our goal, revisited 51.7. The models 52. The Uniform/Operator Model 82.1. Why primitive recursion? 93. The main definition 113.1. Representation spaces 113.2. Representations 113.3. Online problems 123.4. Taking the completion of an online problem 123.5. The definition 133.6. The robustness lemma 133.7. Generalisations and refinements of the main definition. 153.8. General computable and efficient solutions 163.9. Multiple solutions 16
Mathematics Subject Classification.
Primary 03D45, 03C57. Secondary 03D75, 03D80.The first two authors were partially supported by Marsden Fund of New Zealand. The thirdauthor is partially supported by the grant MOE2015-T2-2-055 and RG131/17.
4. Oracle computation and uniformity 174.1. Graph oracles do not help 174.2. Interactions with punctual structure theory 195. Weihrauch reduction and online algorithms 205.1. Weihrauch reduction and incremental computation 215.2. Weihrauch reduction and online graph colouring 226. ∆ processes, finite reverse mathematics, and Weihrauch reduction 287. Real functions. 30References 321. Introduction
Our Goal.
Imagine you are tasked with putting objects of differing sizes intobins of a fixed size. Your goal is to minimize the number of bins you need. This isthe famous
Bin Packing problem which we know is NP complete (see Karp [52]).But imagine that we change the rules and you are only given the objects one at atime and you must choose which bin to put the object into before being given thenext object. You are in an online situation and this is the
Online Bin Packing problem. The “first fit” method is well-known to give a 2-approximation algorithmfor this problem (definitions given in detail below). Alternatively imagine you area scheduler, and your goal is to schedule requests within a computer for memoryallocation amongst users. Again you are in an online situation, but here you mightwant to change the order of allocation depending on priorities of the requests. Orfrom algorithmic randomness, you have a (computable) KC-set of requests of theform (2 − n i , σ i ) with P ∞ i =1 − n i ≤
1, and need to build a prefix-free Turing machine M with strings τ i such that | τ i | = n i and M ( τ i ) = σ i . Then the proof from e.g.Downey and Hirschfeldt [28], Theorem 3.6.1 is online in the sense that for eachrequest at step i we generate the string τ i .Thus an online algorithm is one which acts on a input which is given piece bypiece in a serial fashion. In the case where the input is finite, Karp [53] suggestedthis as a sequence of “requests” r , r , . . . with the algorithm f specifying an ac-tion f ( r ) , f ( r r ) , . . . . The natural model for this would be a database where arequest would be an update. Note that this is quite distinct from the offline versionwhere (in the finite case) the whole input is known in advance. It is important torealize that in practical online algorithms arising in computer science, the actionneeds to be specified before the next request is given. Occasionally this is variedwith a lookahead or delay where typically we might get k further bits of input so r , . . . , r n + k determines the next action .A brief thought on this will reveal that there are potentially hundreds of situa-tions where we are dealing with combinatorial algorithms for tasks where we onlyhave partial evolving information about the the input data, or perhaps the data isso large that we cannot see it in total. This is the reason that there are so manyalgorithms for online tasks. Classical examples include insertion sort, perceptron, As we will see below, it is possible to generalize this further perhaps to have delay or lookahead g ( n ) for action n with g some function of n (or even r n ) but this is not what happens in practice.Most natural examples work from r n to define f ( r n ) and indeed if r n is a structure generated by { , . . . , n } of some kind, then f ( r n ) is some value h ( n ). OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 3 paging, job shop echeduling, ski rental, navigation with only local understanding,etc.; see [4]. On the other hand, there seems no general theory which we can use as aconceptual basis for the theory of (finite and infinite) online algorithms, and onlinestructures. Books in this area, such as Albers [4], all tend to be taxonomies of al-gorithms. Our goal is to give a theoretical basis for the theory of online algorithmsand structures.As a final comment, the reader might wonder why we would need model. On-line algorithms are everywhere without such a model. We see the situation asbeing akin to the development of, for example, asymptotic counting of computa-tion steps as a measure of computational complexity by Hartmanis and Stearns in1965 [45] (amongst others). Before this people ran algorithms and had an intu-itive understanding of complexity, but the development of a framework allows forformal analysis. Similar comments apply to, for example, the development of pa-rameterized complexity by the first author and Fellows [26, 27]. People already hadbeen pre-processing in algorithmics, but the explicit identification of a complexitytheory based on the multivariable contribution of the various aspects of the input(in place of simply the overall size) hastened the introduction of standard tools forsuch algorithmics ([26, 23]). We have similar hopes for online algorithmics.1.2.
The [8]
Punctual Model.
In [8] a related project was started aiming atproviding a model-theoretical foundation to this theory. In that paper we focussedupon the intuition that online decisions in practice have lack of delay.
That is, weneed to pack the object into some bin immediately, before the next one is presentedto us (in the
Bin Packing example). This led to a theory of online structures andalgorithms we generally referred to as punctual structure theory.What is the most general reasonable form of “punctual”? In [8] we gave severalpages of analysis as to why there we chose to interpret punctuality as primitiverecursive . That is, we chose primitive recursive as a unifying abstraction of thenotion of lack of delay.To keep this paper self-contained we will repeat the arguments of [8] here, so thereader familiar with [8] might choose to move on to § Turing computable mathematics.
The general area of computable or effec-tive mathematics is devoted to understanding the algorithmic content of mathemat-ics. The roots of the subject go back to the introduction of non-computable methodsinto mathematics at the beginning of the 20th century as discussed in Metakidesand Nerode [66]. Early work concentrated on developing algorithmic mathematicsin algebra, e.g. Grete Hermann [46], analysis such as Bishop’s constructive analysis,(implicitly) using algorithmic methods to understand randomness (Borel [12], vonMises [77], Church [16]), understanding effective procedures in finitely presentedgroups such as Dehn [24], and most notably Hilbert’s programme seeking to givea decision procedure for first order logic. We know all of these historical rootsled to the development of, for example, computability theory, complexity theory,and algorithmic randomness (see e.g. Downey [25]). The modern version of effec-tive mathematics utilizes the tools developed in these areas, as well as classicaltools in algebra, analysis, etc. to calibrate the algorithmic content of many areas ofmathematics.The standard model for such investigations is a (Turing) computable presentationof a structure. By this we mean a coding of the structure with universe N , and ROD DOWNEY, ALEXANDER MELNIKOV, AND KENG MENG NG the relations and functions coded Turing computably. For example, a computablepresentation of a group would be either a finite group, or one where the universe wasconsidered as N and the group operation was represented as a (Turing) computablefunction. Note that this framework uses the general notion of a Turing computablefunction. In particular, we put no resource bound on our computation.1.4. Online combinatorics.
A hallmark of the majority of algorithms on finitestructures is that the algorithm “knows all about the structure”. In other words, thewhole structure is given to the algorithm at once. For example, when a complexitytheorist talks about the Hamiltonian path problem, they have in mind algorithmsthat given a description of a finite graph (say, a matrix-presentation of it) outputssuch a path. This is sometimes not true for large data sets, and several
Logspace algorithms, but we are using this to refer to those students which would learn in abasic algorithm course. What happens to such algorithms if the graph is not givenat once, but rather is given to us step-by-step and vertex-by-vertex? This situationis an abstraction to an “online” computation in which the input data is too massiveto be given as an input at once. Now, as seen in the introduction, there are many problems in computer science where we can safely assume that universe is infiniteand thus we need an online algorithm. For example, a scheduler which assigns usersto access shared memory is a classic example.In the “online” setup the situation becomes quite a bit harder. Consider thefollowing example. Every tree is 2-colourable, but to achieve this colouring youneed to know the whole of the tree. Suppose we are given a vast tree one vertexat a time, so that G = ∪ s G s , an online presentation of G . When we give youthe vertex v we promise to tell you all of the vertices given so far to which v isjoined; that is the induced subtree of v , . . . , v s . Your goal is to colour the vertex v s , before we give you v s +1 . We are in an online situation. For a tree with n vertices, the sharp lower bound is O (log n ) many colours. It follows that there areonline presentations of infinite (computable) trees which cannot be online colouredwith any finite number of colours. We see that switching to the online case affectsnot only the running time, but also the best solution that we can hope for. Weremark that online algorithms can be quite complex.Beginning in the 1980’s there has been quite a lot of work on online infinitecombinatorics, particularly by Kierstead, Trotter, Remmel and others ([57, 58, 59,64, 71]). Some results were quite surprising. For example, Dilworth’s theorem saysthat a partial ordering of width k can be decomposed into k chains. Szemerediand others showed that there is a computable partial ordering of width k thatcannot be decomposed into k computable chains. But in 1981, Kierstead provedthat there is an online algorithm that will decompose any online presentation ofa computable partial ordering into k − many (computable) chains. Only in 2009was this result improved by Bosek and Krawczyk who demonstrated that it can bedone with k
14 log k many chains. In the case of finite structures most work comesfrom comparing offline vs online performance. In this area, the typical setting is tobuild some kind of function which is measured relative to some size, and the goal ofonline algorithm design is to improve what is called the Competitive PerformanceRatio of online divided by offline. For example, first fit gives a competitive ratio of2 for the classical
Bin Packing problem (see Garey and Johnson [39]).
OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 5
Online vs. Turing computable.
The notion of an “online” algorithm inthe results mentioned above is rather specific. One may complain that, rather thansaying that we must make a decision before the next vertex shows up, it is fine towait for a bit more of a graph to be shown to us. But how much more exactly?Maybe we can wait for 17 more vertices to show up before we make a decision.Perhaps, at stage s we could ask for log( s ) more vertices, etc. It is not hard to seethat various answers to this question will lead to a proper hierarchy – rather, a zoo– of “online” computability notions. It is natural to ask: What is the most general notion of an online algorithm?
Understanding the online content of mathematics so far has no general theory,there are only algorithms or proofs that no algorithm exists. Note that the lackof theory for online mathematics stands in stark contrast with the infinite off-line case described by the computable structure theory [5, 34]. However, as wenoted above, computable structure theory relies on the most general notion of acomputable process that we know today – a Turing computable process. Turingcomputability provides us with many tools, such as the universal Turing machineand the Recursion Theorem, that are useful in proving theorems about algorithms .However, Turing computability in its full generality is not an adequate model inthe online situation, because Turing computable algorithms can use an unboundedsearch. For instance, recall the example in which we had to online colour a tree.A Turing computable algorithm would just wait until a node gets connected tothe root of the tree via a path and then will make a decision. There is no a priori bound on how long it may take for the path to be revealed, but a Turing computablealgorithm does not care. More importantly, Turing computability fails to capturethe “impatient” nature of an online algorithm which has to make a decision “now”.1.6.
Our goal, revisited.
Recall that our goal is to give a general abstract foun-dation for online algorithms. As we will soon see, our approach is based on onenatural interpretation of “online” involving primitive recursive structures.In [8], using some the techniques and intuition coming from the mentioned above(Turing) computable structure theory [5, 34] we developed a theory contrastingand comparing classical computable structure theory with an online “punctual”framework. In [8], we discussed the following models.1.7.
The models.
We will concentrate on infinite structures. Still to do is todevelop an appropriate model theory for online finite structures as asked for byDowney and McCartin [31]. In § A given in stages f (1) , f (2) , . . . , where f is a computable function representingtimestamps. At stage f ( n ) we would enumerate n into the partial structure A f ( n ) and give complete information about how n relates to { , . . . , n − } .Now the question is: What kinds of structures and time functions should beallowed?
Different choices will result in different theories. Our goal is to give ageneral setting that also reflects the common online structures encountered. Weexamine some approaches from the literature:1.7.1.
Automatic structures.
Khoussainov and Nerode [56] initiated a systematicstudy into automatically presentable algebraic structures; but these seem quite
ROD DOWNEY, ALEXANDER MELNIKOV, AND KENG MENG NG rare. For example, the additive group of the rationals is not automatic [Tsa11].The approach via finite automata is highly sensitive to how we define what we meanby automatic. For example treating a function as a relation yields quite a differentkind of automatic presentation. See [33] for an alternate approach to automaticgroups. Although the theory of automatic structures is a beautiful subject, a finiteautomaton is definitely not a general enough model for an online algorithm.1.7.2.
Polynomial time computable structures.
Cenzer and Remmel, Grigorieff, Ala-ev, and others [20, 42, 1, 2] studied polynomial time presentable structures. Weomit the formal definitions, but we note that they are sensitive to how exactly wecode the domain. In many common algebraic classes we can show that all Turingcomputable structures have polynomial-time computable copies. One attractiveresult is that every computably presentable linear ordering has a copy in lineartime and logarithmic space [42]. Similar results hold for broad subclasses of Booleanalgebras [18], some commutative groups [19, 17], and some other structures [18].1.7.3.
Fully primitive recursive structures.
As was noted in [51], many known proofsfrom polynomial time structure theory (e.g., [18, 19, 17, 42]) are focused on makingthe operations and relations on the structure primitive recursive , and then observingthat the presentation that we obtain is in fact polynomial-time.The restricted Church-Turing thesis for primitive recursive functions says thata function is primitive recursive iff it can be described by an algorithm that usesonly bounded loops. For example, we need to eliminate all instances of WHILE . . .
DO, REPEAT . . .
UNTIL, and GOTO in a PASCAL-like language.As we noted above, primitive recursion plays a rather important intermediaterole in transforming (Turing) computable structures into polynomial-time struc-tures. Furthermore, to illustrate that a structure has no polynomial time copy,it is sometimes easiest to argue that it does not even have a copy with primitiverecursive operations, see e.g. [19]. In [8] the intuition above led us to systematicallyinvestigate into those structures that admit a presentation with primitive recursiveoperations, as defined below. Kalimullin, Melnikov, and Ng [51] proposed that an“online” structure must minimally satisfy:
Definition 1.1 ([51]) . A countable structure is fully primitive recursive (fpr) ifits domain is N and the operations and predicates of the structure are (uniformly)primitive recursive.The main intuition is that we need to define more of the structure “without de-lay”. Here “delay” really means an instance of a truly unbounded search. Weinformally call fpr structures punctually computable . We could also agree that allfinite structures are also punctual by allowing initial segments of N to serve as theirdomains Remark 1.2.
The word “fully” in “fully primitive recursive” emphasises that the domain mustbe the whole of N and not merely a primitive recursive subset of N ; these are provably non-equivalent assumptions. If the domain could be merely a primitive recursive subset of N thenwe can delay elements from appearing in the structure; this way one can easily show that eachTuring computable graph has a primitive recursive copy ([8]). We decided that structures in whichelements can be delayed are not really online. Although the definition above is not restricted to finite languages, we will never considerinfinite languages in the paper.
OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 7
Our goal is to give a most general setting that also reflects the common onlinestructures encountered. From a logician’s point of view, where do computablestructures come from? One of the fundamental results of computable structuretheory is that:
A decidable theory has a decidable model .The proof of this elementary fact is to observe that the Henkin construction iseffective, in that if the theory is decidable then the constructed model is decidableas a model. Many standard computable structures come from decidable theories.Most natural decidable theories are elementary decidable in that the decisionprocedures are relatively low level. We have to go out of our way to have naturaldecidable theories whose decision procedures are not primitive recursive. In [8] weobserved that a theory with a primitive recursive decision procedure has a modelwhich is decidable in a primitive recursive sense.1.7.4.
The upshot.
Thus in [8], we chose fully primitive recursive structures as ourcentral model. Primitive recursiveness gives a useful unifying abstraction to com-putational processes for structures with computationally bounded presentations.In such investigations we only care that there is some bound. Furthermore, thesemodels arise quite naturally through standard decision procedures.In [8], we also noted that many results we stated in terms of primitive recur-sion, can likely be pushed to polynomial time structures. Furthermore, some of ourcounterexamples can in fact be stated in terms of any class with sufficiently nice clo-sure properties; e.g., for a class of total computable functions having a uniformlycomputable enumeration and closed under composition and primitive recursion.However, this does not mean that our choice of primitive recursive algorithms asa central model is fairly arbitrary. The mentioned above generalisation to a classof total functions can be viewed as a version of the subrecursive relativisation ofprimitive recursion. The study of relativised versions of our results is interesting onits own right, but it is not really beyond the primitive recursive model. Kalimullin,Melnikov and Montalb´an (in progress) have recently announced a number of unex-pected results connecting relativised primitive recursive presentations with syntaxin the spirit of Ash and Knight [5]. Also, as we see in the present paper, an expertin computable structure theory would know that relativisation is tightly connectedwith uniformity. Generalisations to polynomial time classes seem to require signifi-cant effort in some instances. Alaev [1, 2] has recently initiated a research programfocused on extending these ideas to polynomial time algebra. Dealing with polyno-mial time algorithms requires specific techniques and counting combinatorics; thisis something we do not have to worry in our more “relaxed” model. In contrastwith, e.g., automatic algorithms or polynomial-time algorithms, there is a highlyconvenient and clear version of Turing-Church thesis for primitive recursive func-tions (see above). We will use the thesis throughout the article without explicitreference. It will allow to simplify our proofs and proof sketches. Irrelevant count-ing combinatorics is stripped from such proofs, thus emphasising the effects relatedto the existence of a bound in principle (rather than specifying the bound). Theseeffects are far more significant than it may seem at first glance.Models such as automata-based structure theory such as Khoussainov and Nerode [56],but noted that the approach is highly sensitive to presentations of the structures;for example treating the algorithms as generated by transducers yields a completelydifferent theory to that obtained by treating functions as relations, as can be seen
ROD DOWNEY, ALEXANDER MELNIKOV, AND KENG MENG NG by comparing the approach of Khoussainov and Nerode, with that of Epstein etal. [33]. Also it would seem that although we can incorporate automatic processesin our theories, they are really no general enough for online algorithms in general.Similarly, polynomial time structures such as Cenzer and Remmel, Grigorieff, Ala-ev, and others [1, 2, 3, 20, 42] are rather presentation dependent. Finally, primitiverecursive has a nice Church-Turing thesis, in that it models computable processeswithout unbounded loops.2.
The Uniform/Operator Model
Whilst the [8] model is a natural model, as we observed in § not covered by the model.Imagine we need to build a colouring of a graph G which is given online. Thus,in the very simplest case, we would be given the graph G = lim s G s , where G s has s vertices. When the vertex s is introduced, we are also given at the same timeprecisely which vertices amongst { , . . . , s − } has an edge with s (and this cannotchange later). (This is the “request set” in Karp’s paper.) Our task is to colour s so that no two vertices which are connected have the same colour, before theopponent presents us with G s +1 . Although in practice the task will be finite, since we have no idea how largethe graph is, we can construe this as an infinite process. If we imagine this asan infinite process and we need to colour the whole of an infinite graph G . Wecan think of each possible version of G as being a path through an infinite treeof possibilities. Each node σ of length s of the tree will represent some graph G σ with s vertices, and if σ ≺ σ ′ then G σ is the subgraph of G σ ′ induced by vertices { , . . . , s } . Note that there are only primitively recursively many non-isomorphicgraphs with s vertices.Then this view of an online algorithm differs from that given in [8] for thefollowing core reason:Although G can be viewed a path on an infinite primitive recur-sive tree of possibilities, there is no a priori reason that we shouldonly consider a primitive recursive graph G . There are continuummany such paths and the online graph colouring problem can beconsidered for an infinite countable graph of any complexity.The reader will quickly realise that the key point about online algorithms is oneof continuity or uniformity . If we have a colouring of G σ and we add a new vertex s + 1, the next G will be one of the possible extensions G τ of G σ with the vertex s + 1 added. For each such G τ the colouring χ G τ must be compatible with thecolouring χ G σ on G σ . Computable Analysis.
The conclusion is that whilst online algorithms appearto be combinatorial algorithms on finite objects and possibly infinine ones, in factthey should be formulated as a branch of computable analysis . One of the goals ofthe present paper is to give such a formulation. We need to specify what kinds ofspaces are of relevance and what kinds of operators correspond to online algorithms.We believe that this view will allow a discourse between the discrete and the contin-uous which could prove fruitful. Similar relationships between the continuous andthe discrete have yielded powerful results such as in the Furstenberg view of Sze-meredi’s Theorem. Our unifying abstraction also means that computable analysisis shown to be important in finite combinatorics (see also Avigad [6]). We will also
OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 9 see that our abstraction means that we can relate the proof theory of finite com-binatorics with classical proof theory, obtaining refinements of result for ReverseMathematics, and we can relate the theory of incremental computation with ideasfrom computable analysis such as Weihrauch reducibility. Thus although this isnot the most technically difficult paper, we see it as a conceptual advance showingthat many ideas can be combined into a single unifying abstraction.
Immediate actions.
Since we want the action to be immediate, following theabstraction of [8] and for the reasons above, we will also demand that the actionworks primitive recursively, or perhaps even running in polynomial time. Thus, forthe example above, the most general online colouring algorithm must satisfy thefollowing two features:- χ G τ must agree with χ G σ for σ ≺ τ .- The map τ → χ G τ must be (minimally) primitive recursive .These will all soon be made precise and general using representations (i.e, namingsystems) for online problems. In fact, there are at least three possible interpreta-tions of the second clause above. For example, should we allow lookahead or delay in the computation of f ? In Lemma 3.5 we will prove that all three potential defi-nitions are equivalent up to a primitive recursive change of notation, and thereforeour definition is robust .2.1. Why primitive recursion?
Should we instead require the solution to bepolynomial time? As we have already mentioned above, this notion would betoo notationally dependent to be unambiguous. Also, it is well-known that inpractice some useful algorithms are (provably) not polynomial-time; yet they seemto perform well enough on most inputs. It is therefore not even clear if polynomial-time is the right abstraction for efficiency in the online situation. So we want ourfunction to be in a nice complexity class but we are not yet sure what exactly thisclass should be. It makes sense to develop as much structural “punctual” theoryas possible and then see how much of it is preserved when we restrict ourselves tosome narrow complexity class. And if something fails, we will have a better ideawhat goes wrong in the worst possible scenaria; e.g., we will compare the positiveTheorem 7.3 and the analogous negative result in polynomial-time analysis [61].Perhaps, it will help to define generic-case online algorithms in the future.As with classical complexity theory, there is usually a natural representation fora problem we are interested in. The reader will note that in our definitions below,the actual representation does affect what we will regard as online. Nonetheless, oneof the main advantages of our rather general primitive recursive approach is thatwe still can prove a number of notation-independent results; e.g., the “robustness”Lemma 3.5 and the above-mentioned Theorem 7.3. Such results focus more on theeffects related to online-ness and less on the pathologies related to a specific choiceof representation. Analogous results usually fail if we restrict ourselves to, say,polynomial-time algorithms because passing from one representation to anothercan be computationally too hard.On the other hand, we will also prove several results (e.g., Theorem 4.6) whichshow that sometimes all pathologies come from presentation because the onlynotation-independent online solutions are the trivial ones. Such results of the sec-ond kind will typically hold for polynomial-time or exponential (etc.) algorithms too, and via essentially the same proof. Primitive recursion serves here as a unifyingabstraction rather than an idealisation.We will also see that, modulo subrecursive relativisation, the earlier approachto online algorithms by Kiersetead, Trotter et al. [58, 64, 59] and Borodin andEl-Yaniv [10] can be viewed as a special case of our framework. According to thisearlier approach, the map τ → χ G τ just needs to be total and does not even haveto be computable. So we see that primitive recursion is not that general whencompared to some other definitions in the literature.Should we perhaps use (a use-restricted form of) general Turing computabilityin place of primitive recursion? It is more general, and some analogy of the above-mentioned robustness lemma (Lemma 3.5) will still hold. The key difference hereis that it would hold for a completely different reason. A recursion theorist willbe well-aware of how much unbounded search is abused in many such proofs. Forexample, we can use compactness of the representation space and wait for it to becovered by open sets. There will perhaps be no bound on how long we will haveto wait, but from the point of view of Turing computability it will not make anydifference.However, primitive recursion seems just general enough for many structural re-sults to hold, but often via a different, more subtle argument which takes intoaccount punctuality of our procedure. A fine example of such a theorem will begiven in Section 7 where we prove an online version of a the well-know theorem ofWeierstrass. It is very easy to show using a compactness argument that the the-orem holds (Turing) computably. But it requires some thought and a completelydifferent argument to see why it holds punctually.As mentioned above, we realise that this material also has a connection with computable and feasible analysis , and also with the complexity theory for operatorsin analysis along the lines of Kawamura and Cook [54], Melhorn [65], Ko andFriedman [38], and others. We will also note connections with reverse mathematics,computational learning theory, and even algorithmic randomness. We will also seethat, in this setting, the finiteness of the objects being given is not an essentialrestriction. In the online case, finite objects are only revealed one bit at a time,and for all intents and purposes, we may as well treat all inputs as arbitrarilylarge finite structures. We will prove that under the uniform operator framework,working with arbitrarily large finite structures and infinite structures are indeedthe same for our setting. This allows for example, for a formal approach in whichone can study finite combinatorics in reverse mathematics.As a final remark, we mention that we see this work as an extension of [8] in thefollowing way.[8] considered online computation of primitive recursive structures,with primitive recursive functions. This is akin to The Turing-Markov [76] view of computable analysis as effective processes onthe countable field of computable reals. The Gregorczyk-Kleene [41]views, called type II computability, which views computable analy-sis as effective operators acting on the continuum of all reals. It isalso akin to the bifurcation between computable structure theoryand uniform computable structure theory. OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 11 The main definition
In the following sections, we will work up to the main definition. Rememberthat we want to simultaneously generalize online algorithms on finite and infinitestructures, and in a general setting where the domains might be any kind of struc-ture. Because of this we will need to tour through representations (computationalways of naming infinite objects), and carefully argue why choices, such as that ofprimitive recursion, are made.3.1.
Representation spaces.
It could be argued that for relational structures wecould consider (isomorphism types of) any structure A with universe N , and wecould consider A n to be the induced substructure of A with universe { , . . . , n } .Naturally, we need to assume that this has meaning: and such substructures existin all finite cardinalities. Also, if we choose to add function symbols we would needto only allow a small extension of the structure based on { , . . . , n } . For simplicity,we will stick to relational structures and use the following terminology.A class C of relational structures is called inductive if A ∈ C implies A has a filtration A = ∪ s A s where each A n is finite, has universe { , . . . , n } , and for all n ′ > n the substructure induced by { , . . . , n } in A n ′ is A n . More generally, fora fixed computable function g , we say that C is g - inductive if it has a g -filtrationmeaning that each A n has universe { , . . . , g ( n ) } . Here we will sometimes write O ( h ( n ))-inductive for the case where g is O ( h ). Our language will typically befinite and relational .We refer to the substructure of A based on { , . . . , n } the substructure of height h ( n ) = n . In the example discussed above, the height n structures are the graphswith n vertices. Another example is considered by Khoussainov [55] with a heightfunction in his work on random infinite structures. Natural online structures havenatural height functions.By abusing notation, we will let C <ω denote the class of finite substructures of C . There is also the natural induced topology. For example, in the graph case thiswould be compact and have the totally disconnected topology with basic open setsbeing the extensions of graphs of height n .3.2. Representations. A representation of an inductive class C of structures isa computable surjective function δ : ω <ω → C <ω , which acts faithfully (and con-tinuously) in the sense that δ ( σ ) = C n for | σ | = n and h ( C n ) = n , and if σ (cid:22) τ then δ ( σ ) is an induced substructure of F ( τ ) . We can also extend this in the nat-ural way to g -filtrations. Thus such a δ induces a map δ from ω ω → C , namelylim { δσ | σ ≺ x } . We will call x ∈ ω ω a name for C ∈ C if δ ( x ) = C. Note that it ispossible that each structure C to have a number of different names .For the time being, we will regard δ as being injective. When it is possible,we will replace ω <ω with 2 <ω . We will consider functions f : C → C and these Richard Shore observed that the punctual case focusses attention upon functions and func-tionallanguages, whereas the operator approach seems to tie itself to relational ones. We needsome care if the language has function symbols, see [49]. The point of such naming systems is to allow computational comparison of problems in differ-ing domains. For example, think about the classical situation where names are classical Cauchysequences in computable analysis. One domain might be the totally disconnected Cantor Space,and the other might be R ∩ [0 , are represented by functions F acting on representations Q i of C i , so taking finitestrings to finite strings.We emphasise that the function F is acting on strings which are finite objects .These represent, e.g., graphs. It is the continuity of the action will induce a map F which is the completion of the finite maps. If x is a member of that completion;such as an infinite graph represented by a path through the relevant representingtree, then x = lim n δ − ( G n ). Where the representation is obvious, we will suppressthe explicit mention of this in the below to encourage readability.3.3. Online problems.
Although our objects of study are not strings, we willimplicitly identify them with their representations, in accordance with the previoussubsection. In particular, if the representation space is compact then our objectscan be identified with strings over a finite alphabet.Intuitively, to solve a problem we need to find a function f which, on input i ,chooses an admissible solution from the finite set s ( i ) of “correct” solutions. Definition 3.1.
A online problem is a triple (
I, S, s ), where I is the space ofinputs (i.e. the filtration) viewed as finite strings in a finite or infinite computablealphabet, S is the space of outputs viewed as finite strings in (perhaps, some other)alphabet, and s : I ⇒ S <ω is a (multi-)function which maps each σ ∈ I to the set s ( σ ) of admissible solutions of σ in S .Note that the multi-valued function s does not have to be computable in general.For instance, for a colouring problem I will be codes for finite graphs and S forfinite coloured graphs. Then s ( σ ) will correspond to the collection of all admissiblecolourings; e.g., such that adjacent vertices are distinctly coloured. These colouringswill form the space of admissible solutions.Most natural problems from finite structures will obey the following convention,which we will consider in this section. Only in Section 7 will be consider moregeneral cases. Convention 3.2.
Unless explicitly mentioned, I and S are compact with a prim-itive recursive modulus of compactness; i.e., it is primitively recursively branchingwhen viewed as a tree of strings. Thus, there is a natural primitive recursive wayto transform I into 2 ω (typically not height preserving).3.4. Taking the completion of an online problem.
A solution f to an onlineproblem ( I, S, s ) induces a solution for the (topological) completion of the initialproblem (
I, S, s ), in the sense that f can be uniquely extended to a functional¯ f : [ I ] → [ S ]. Here [ I ] consists of infinite strings ξ such that for every i , ξ ↾ i ∈ I ,and similarly for [ S ].In general, in Definition 3.1 we may also require ¯ f to satisfy some global prop-erty which cannot be always captured by s from Definition 3.1. For example, inSection 4 a solution must be an isomorphism between two presentations of the sameinfinite graph. In general, even if at every stage f ( σ ) may be extendable to someisomorphism, the map associated with ¯ f may fail to be surjective in the limit. Also,in another example in Section 4 we will require our solution to work only if the inputis a presentation of some fixed infinite graph, which is also a property of ¯ f ratherthan of any finite approximation to it. In particular, in this case admissibility of ¯ f cannot be captured by s in Definition 3.1; at least not in general. OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 13
Convention 3.3.
We will refer to such properties of ¯ f as global and will notincorporate them into Definition 3.1.3.5. The definition.
Recall Conventions 3.2 and 3.3.
Definition 3.4.
A punctual solution to (a representation of) an online problem(
I, S, s ) is a computable function f : I → S with the properties:(O1) f ( σ ) ∈ s ( σ ) for every σ ∈ I ;(O2) If σ ≺ τ then f ( σ ) (cid:22) f ( τ ).(O3) f is primitive recursive.Condition (O1) says that the output of f is an admissible solution. In (O2) weask for is that each increment of the input yields an increment in the output, inthe sense that f ( σ ) must be a solution to σ .The reader should note that (O3) is somewhat ambiguous as stated because itmay be interpreted in at least two different ways, namely f could be primitiverecursive either as a function or as a functional:( O ′ the computation of f ( σ ) is bases solely on σ , or( O ′′ the computation of σ may ask for an extension τ of σ before it halts.Indeed, for online computations it would be natural to demand that we have aprimitive recursive timestemp function g and to compute f ( σ ) we would look at σ ′ of length g ( | σ | ) extending σ . In practical computations lookahead will typically be g ( | σ | ) = | σ | + k for some constant k . On the other hand, for a recursion theorist itwould be more natural to consider Turing functionals acting on the representationspaces and demand that the are primitive recursive. By that we mean adding thecharacteristic function for the infinite string in the completion of the problem ( § f ; to be clarified in Subsection 3.6. These twogeneral definitions of lookahead (via timestamp and via oracle) are not equivalentwhen, say, I ∼ = ω <ω . Thus, we have three natural versions of ( O
3) which arefurthermore provably not equivalent in general.Luckily, in the next subsection we will prove that, under Convention 3.2, thesethree versions of the main definition are equivalent up to a primitive recursivechange of notation, and therefore Definition 3.4 is robust.3.6.
The robustness lemma.
As we mentioned above, there are two naturalways of interpreting what it means for f in ( O
3) to be primitive recursive with alookahead. We give more details.In the first definition, we require that it is a Turing functional that possessesa primitive recursive time-function t which, on every input σ outputs the numberof steps which f takes to compute f ( σ ). In particular, t ( σ ) bounds the use of theoperator, that is, the length of τ extending σ which may be used in the computationof f ( σ ). The length of the output f ( σ ) will also be bounded by t ( σ ).The seemingly more general definition of a primitive recursive functional saysthat, for each infinite path x through the space of inputs, f is primitive recursiverelative to x = lim s { σ | σ ≺ x } . The latter can be formally defined by adding thecharacteristic function for x to the primitive recursive schema, and hence wouldpotentially entail that f ( σ ) could be arbitrarily long for various extensions of σ .These two notions are equivalent in our framework. (Recall Convention 3.2.) Lemma 3.5.
For a primitively recursively branching I , a Turing functional f : I → S possesses a primitive recursive time-function iff f is a primitive recursive functional. Moreover, if f possesses a primitive recursive time-function, then thereis an equivalent online problem ( I ′ , S ′ , s ′ ) with a primitive recursive solution withoutlookahead.Proof. In a different terminology the proof will appear in [49]. A similar formalargument can be found in the appendix of [8].Suppose the Turing functional f possesses a primitive recursive time function t .Using t as a universal bound on all the searches which may occur in a computationwith any oracle x extending σ , we can transform the general recursive scheme(augmented with the characteristic function χ x for x ) into a primitive recursivescheme augmented with χ x . This implication holds in general, i.e., without anyextra assumption on I .Now, assuming I is primitively recursively branching, suppose f is a primitiverecursive functional with functional oracle g in the most general relativised sense.For the base of induction consider the following cases: Φ g = o , Φ g = s , Φ g = I nm ,and Φ g = g . The first three cases are evident since they do not refer to g , while inthe case when Φ g = g take t = b , where b is the primitive recursive branching of I .Take t ( x ) = P i ≤ x b ( i ) to make t monotonically increasing in its input.The inductive step splits into two different cases depending on whether the lastiteration is composition or an instance of primitive recursion.Suppose it is composition,Ψ g (¯ x ) = Φ g (Θ g (¯ x ) , . . . , Θ gm (¯ x )) , where Ψ , Θ , . . . , Θ m there are primitive recursive operators with primitive recur-sive time bounds t , t , . . . , t m . As usual, we identify a tuple ¯ x = h x , . . . , x m i with its primitive recursive code. The time functions t , t , . . . , t m can be assumedmonotonically increasing in their inputs.Define a primitive recursive time bound for Ψ by the rule t (¯ x ) = t ( h t (¯ x ) , . . . , t m (¯ x ) i ) + X i t i (¯ x ) , which can be rewritten into a primitive recursive schema using the standard tech-niques. Now suppose that Ψ is defined using an instance of primitive recursion,more specifically Ψ g (¯ x,
0) = Θ g (¯ x );Ψ g (¯ x, y + 1) = Φ g (¯ x, y, Ψ g (¯ x, y )) , where Φ and Θ are primitive recursive operators which have corresponding primitiverecursive time functions t and t . Define t by the rule t (¯ x,
0) = t (¯ x ); t (¯ x, y + 1) = t (¯ x, y, t (¯ x, y )) + t (¯ x, y );assuming that t is monotonically increasing in its input this gives the desired upperbound.The last part of the lemma, recall that the tree I is primitive recursively branch-ing. Thus, we can inductively form a new tree I ′ whose level n nodes are in a(primitive recursive) 1-1 correspondence with the nodes at level t ( n ) = max { t ( σ ) : | σ | = n } in I . Then the algorithm f on I works on I ′ as a strict algorithm. (cid:3) OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 15
Of course, if f is primitive recursive functional without lookahead then f canbe viewed as simply a primitive recursive function mapping finite strings to finitestrings. More formally, we have: Fact 3.6.
Suppose P = ( I, S, s ) is an online problem. Then the following areequivalent: P has a solution witnessed by a primitive recursive function f . The completion of P has a solution witnessed by a primitive recurasiveoperator f without lookahead. Generalisations and refinements of the main definition.
It is importantto understand that primitive recursion smoothens many difficulties related to no-tation. In particular, the robustness lemma from the previous section will typicallyfail for polynomial-time operators. Thus, different interpretations of ( O
3) in Defini-tion 3.4 will potentially lead to different refinements of the main definition to morenarrow complexity classes. On the other hand, different versions of relativisation(such as general Turing and sub-recursive) will lead to potentially non-equivalentgeneralisations of the main definition. We will not develop these topics in toomuch detail, but some notions and notation introduced in this subsection will beimportant in the later sections.3.7.1.
Strict solutions, obT operators, and totality.
We may want to stick with agiven notation (
I, S, s ) because it may be either inconvenient to do so or perhapscomputationally too hard. If the space I is not primitively recursively branchingor not even compact, then Lemma 3.5 no longer holds. Thus, in this case the mostgeneral version of Definition 3.4 becomes ambiguous. In contrast, Fact 3.6 does notrely on compactness of I , let alone its primitive recursive branching, and thereforethe stronger strict version of Definition 3.4 still makes sense even for non-compact I . Thus, the situation described in Fact 3.6 deserves a special attention. Definition 3.7.
In the simpler situation that no lookahead is allowed in Definition3.4 we will call f a strict punctual solution.By Fact 3.6, this situation can be considered an analog of a classical ibT-reduction, but acting on compact spaces with primitive recursive branchings withthe branches of level n being the structures of height n , instead of 2 ω . Classically,ibT refers to an oracle procedure Γ B = A with the use γ ( x ) = x for all x , and herewe are identifying sets with their characteristic functions as usual. ibT functionalsand the induced reduction have been studied quite intensively [22, 7, 28, 30, 75]and even used in (classical) differential geometry [22, 68]. Definition 3.8.
For a fixed filtration representing I we will call such a procedureΓ induced by a strict online solution an obT (online bounded Turing) reduction.Of course, the classical ibT reduction is usually viewed as working on 2 ω . If ourspace does not have primitive recursive branching then we no longer can transformit effectively into a copy of 2 ω . But as mentioned earlier, we see this aspect as afeature of the model, and not a flaw. One should expect online-ness to be generallyrepresentation dependent, at least to some extent.Although there are notions of a polynomial-time functional in the literature [61],Definition 3.7 is much more convenient if we want to define what it means for apunctual solution to be polynomial time. We can also use Definition 3.7 to give an explicit connection of our definitionwith the above-mentioned approach in [58, 10] which relies on total (not necessarilycomputable) functions. As has been observed in [49], a total function can be viewedas a function primitive recursive relative to some oracle. More formally, we have:
Fact 3.9.
For an online problem, the following are equivalent: (1)
The problem has a total strict solution f (in the sense of [58, 10] ); (2) The problem has a strict solution primitive recursive relative to some oracle(in the sense of [49] ). In computability theory many arguments tend to be uniform enough to be rela-tivizable to any oracle. The relativisation phenomenon partially explains why theseemingly crude approach via totality [58, 10] often captures some features of onlinecomputation.3.8.
General computable and efficient solutions.
Many of our results are validfor computable online solutions, and in fact for any total solutions. For example,certainly a result showing that no computable solution is possible is very strong.For example, the proof that online colouring forests from Gasarch [40] requiresΩ(log n ) many colours shows that no computable f is possible. We arrive at thefollowing generalisation of the main definition. Definition 3.10.
A computable solution to (a representation of) an online problem(
I, S, s ) is a
Turing computable function f : I → S with the properties ( O
1) and( O
2) of Definition 3.4.One potential extra feature which is captured by this generalisation is that italso covers partial solutions and, potentially, partial representations. We will notmake it overly formal and leave this to the reader. For example, in the case that therepresentations are partial, f would perhaps only need to work well on a valid input(cf. Convention 3.3). For example, in the case when ω ω is representing Cauchy se-quences, imagine we are seeing an online way to compute some continuous function.Then we would only need to produce a solution for those sequences which actuallycorresponded to convergent sequences. We could then require this solution be insome sense punctual when restricted to valid inputs.We could on the other hand make the definition more efficient, for example: Definition 3.11.
A polynomial-time solution to (a representation of) an onlineproblem (
I, S, s ) is a punctual strict solution which is furthermore polynomial-time.The definition above is of course heavily notation-dependent. For the look-aheadcase the situation becomes even more complex. Although there are definitions ofa polynomial-time functional in the literature [61] they tend to be unconvincinglytechnical. Also, recall that the robustness lemma fails for polynomial-time simplybecause changing notation tends to be exponential time. Therefore the definitionwill also depend on which version of the main punctual definition we choose tomake polynomial-time; recall there were three such versions.3.9.
Multiple solutions.
Notice that in actual practice, we might also need afurther generalisation of the above. Sometimes we might compute a (bounded)collection of solutions at least one of which is correct at any stage and at height
OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 17 n . This occurs in, for example, using automata to compute minimization problemsfor graphs of bounded pathwidth (or k -interval graphs, see section 5.2) given thepath decomposition. We will be computing a table of f ( k ) many solutions at eachlevel n . For example, for finding maximal clique you would have a collection of 2 k many possible solutions. However, it appears that a suitable choice of the space ofoutputs S can cover this seemingly more general case too.4. Oracle computation and uniformity
Graph oracles do not help.
The main goal of this subsection is to showthat a graph-oracle cannot significantly help in computing a function online. Forthat, we consider online functionals and online oracle computations.
Definition 4.1.
We say that f : 2 ω → ω is online computable if f has a repre-sentation F : 2 <ω → <ω , which is online computable in the sense above, so thatfor all α ∈ ω , F ( α ↾ u ( n )) = F ( α ) ↾ n , where F ( α ) = lim { F ( σ ) | σ ≺ α } and u isprimitive recursive.The space 2 ω can be replaced with a primitively recursively branching totallydisconnected space. Identifying f with its representation F , we can unambiguouslywrite this as f ( α ↾ u ( n )) = f ( α ) ↾ n, and (in view of Lemma 3.5) this should causeno problems in the case of primitively recursively branching spaces of strings. Wemay also allow more than one input in f . Notation 4.2.
It is natural to write f α ↾ u ( i ) ( i ) instead of f ( α ↾ u ( i )) and view α as an oracle. The output of f α ↾ u ( i ) ( i ) can also be interpreted as a natural number,when necessary. Remark 4.3.
There are obvious refinements of this. For example, it is natural torestrict ourselves to functionals f whose running time is a polynomial in the lengthof α . Also, having in mind some particularly nice primitive recursive function u , f is u - online computable if f ( α ↾ u ( n )) = f ( α ) ↾ n . An obvious case is when u ( n ) = n + k , which would be online with delay k . An illustration of this notion canbe seen from Section 7, where we look at online real valued functions. We note thataddition of reals is online computable with delay 2, meaning that to compute thesum of x and y to within 2 − n needs x and y with precision 2 − ( n +2) . Similar delayconsiderations come from other procedures in polynomial time analysis such asintegration (see [61]). When u ( n ) = n then the notions can be restated in terms ofstrict (ibT primitive recursive) functionals, while online with delay k corresponds toLipschitz reducibility. Computable Lipschitz reducibility comes from algorithmicrandomness ([28], Chapter 9) where it is shown that if f is online computableLipschitz acting on 2 ω , then it preserves the Kolmogorov complexity of all sequencesin the sense that for all n , K ( α ↾ n ) ≥ + K ( f ( α ) ↾ n ); that is K ( α ↾ n ) ≥ K ( f ( α ) ↾ n ) ± O (1).We will consider online functionals acting on algebraic or combinatorial struc-tures, e.g., α could be viewed as a description of a finite segment of an infinitestructure of some fixed finite relational signature, e.g., a graph. The extensions of α ↾ n are the finitely many possible relational structures on n + 1 elements extend-ing the structure described by α ↾ n . The intuition is that f α ↾ u ( i ) ( i ) is expectedto compute correctly only if α is an initial segment of a graph G . This is a globalproperty; see Convention 3.3. Definition 4.4.
A function h : N → N is online computable from the isomorphismtype of a structure G if there is an online f such that, whenever α is a descriptionof G , h ( i ) = f α ( i ).In other words, h is allowed to use any presentation of some fixed G as its (online)oracle. Example 4.5.
To see how much extra computational power algebraic oracles cangive, consider the following example. Let X be an arbitrary subset of N , and define A ( X ) to be an algebraic structure in the language of one unary function s , oneunary predicate p , and one constant o , and which has the following isomorphismtype. When restricted to s and o , it is just N with s ( x ) = x + 1 and o interpretedas 0. Now define p ( x ) ⇐⇒ x ∈ X . Given any presentation α of A ( X ), we candecide X . So, in particular, computation from an isomorphism type is potentiallyas powerful as just the usual oracle computation.In view of the example above, the reader will likely find the theorem belowunexpected. Its proof is however not difficult; it can be viewed as a variation of anargument in Kalimullin, Melnikov, and Montalb´an [49]. Theorem 4.6.
A function h is online computable from the isomorphism type ofan infinite graph G if, and only if, h is primitive recursive. Remark 4.7.
It will be clear from the proof below that the result has a naturalpolynomial time version. The exact definition of a polynomial time functional is abit lengthy; see [38, 54] . We leave the polynomial time case to the reader. Proof.
By Ramsey’s theorem, G either has an infinite clique or an infinite anti-clique; without loss of generality, suppose it is a clique. Since g ( i ) = f α ↾ u ( i ) ( i ),where α is any representation of G , we can assume that the first u ( i ) bits of α describe a clique. Since the space of all presentations of G is primitively recursivelybranching, the use u is primitive recursive (see Lemma 3.5). Thus, the oracle canbe completely suppressed and the trivial description of an infinite clique can beincorporated into a new procedure f which does not use any oracle. On input i the procedure produces a string of length u ( i ) which describes a finite clique, andthen refers to this finite string (viewed as a partial function) whenever it needs touse the characteristic function of the oracle. This procedure is easily seen to beprimitive recursive (as a function). (cid:3) Informally, the result says that, from the perspective of online computation,graphs cannot code any non-trivial information into their isomorphism type; i.e.,up to a change of their presentation. Both the theorem above and the main resultin [29] imply that graphs are not universal for punctual computability – a notionwhich we will not formally define here (see [8]). See Kalimullin, Melnikov, andMontalb´an [49] for a generalisation of Theorem 4.6 to structures in an arbitraryfinite relational language. The point is that care is needed with which representations are allowed. Polynomial timefunctionals for (0 ,
1) typically use the so-called signed digit representation, but even for R there issome problem with the notion of the size of the input as discussed in, for instance, [54]. However,for any reasonable representation of graphs of size n this becomes relatively straightforward using,e.g, the standard matrix representation as in [39]. OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 19
Interactions with punctual structure theory.
In [8] we described thefoundations of online structure theory. The main objects in this theory are infinitealgebraic structures in which operations and relations are primitive recursive. Aswe argued in [8], there are natural strong connections of this new theory and thetheory of polynomial-time algebraic structures (see also Alaev [1] and Alaev andSelivanov [3]) with applications to automatic structures [9]. Earlier we arguedthat this kind of punctual structure theory is akin to Turing-Markov computableanalysis, in the the objects are given effectively. In this paper structures themselvesdo not have to be primitive recursive. However, the frameworks are closely relatedvia, e.g., Theorem 4.9 below.A presentation of a countably infinite algebraic structure in a finite language isan isomorphic copy of the structure upon the domain N . For simplicity, we mayassume that the structures in this section are all relational. In this case it becomesconsistent with our framework; in particular, the space of all presentations I of afixed structure in a finite relational language is primitively recursively branching.Each such presentation α ∈ [ I ] can be viewed as an isomorphic copy of thestructure upon the domain of N . Some of these presentations will be computable in the sense that the relations on α will be computable predicates over N . Itis well-known that a structure may have non-computably isomorphic computablepresentations. When we restrict ourselves to primitive recursive presentations andprimitive recursive isomorphisms the situation becomes even more complex becausethe inverse of a primitive recursive function does not have to be primitive recursive.See [8] for a detailed exposition of the theory of punctually categorical structures.The following notion is not restricted to primitive recursive presentations. Amore general version of the definition below was first discussed briefly in [51] andthen also mentioned in [50]. An even more general model-theoretic version of thedefinition can be found in [49]. Definition 4.8.
A structure G is strongly online categorical if there is an onlinestrict operator f which, on input α and β arbitrary representations of G outputsan isomorphism from α onto β . In other words, there exists a primitive recursive functional f α ; β with both usesbeing the identity function, such that the associated function h ( i ) = f α ↾ i ; β ↾ i (whoseoutput is interpreted as a natural number) induces an isomorphism from α onto β ;recall the latter two are isomorphic copies of G upon the domain N . Equivalently,we could replace the functional by a primitive recursive function of three inputs σ, τ, i where | σ | = | τ | = i and finite strings are identified with their indices (undersome fixed natural enumeration).The theorem below can be viewed as a variation of another result of Kalimullin,Melnikov, and Montalb´an [49] on punctual categoricity, but in our strongly onlinecase the proof will be significantly simpler. Recall that a structure G is homoge-neous if for any tuple ¯ x in G and any pair of elements y, z ∈ G , we have that y isautomorphic to z over ¯ x . Theorem 4.9.
A structure in a finite relational language is strongly online cate-gorical if, and only if, it is homogeneous.Proof.
Each homogeneous structure is trivially strongly online categorical. Nowsuppose G is strongly online categorical. Suppose the structure is not homogeneous,and let ¯ x be shortest (of length n ) such that for some z, y we have that z is not in the same automorphism orbit as y over ¯ x . Construct α and β as follows. First,copy ¯ x into both and calculate the online isomorphism f from α ↾ n to β ↾ n . Ifwe identify α ↾ n and β ↾ n with ¯ x , then f induces a permutation of β ↾ n ; by thechoice of n any permutation of ¯ x can be extended to an automorphism of the wholestructure. Adjoin z to α and find a y ′ which plays the role of y over β ↾ n underany automorphism extending the permutation β ↾ n ↔ f ( α ↾ n ). Then necessarily f ( z ) = f ( y ′ ), because f has already shown its computation on the first n bits.However, by the choice of z and y ′ , f cannot be extended to an isomorphism nomatter how we extend the presentations further. (cid:3) Note that we used only totality of the strict functional in the proof. In the casewhen the language has functional symbols the theorem no longer holds. Of course,the notion of strongly online and of a presentation will have to be adjusted. Butregardless, strong homogeneity will no longer capture the property (whatever itmay be exactly).
Example 4.10.
Consider the structure in the language of only one unary func-tional symbol s , and which consists entirely of disjoint 2-cycles. Here a 2-cycleis of course a component of the form { x, s ( x ) } where s ( s ( x )) = x and x = s ( x ).According to any reasonable definition of (strong) online categoricity for functionalstructures, this structure has to be (strongly) online categorical. However, it is nothomogeneous.We leave open: Question 1.
Is it possible to find a reasonable algebraic description of (strongly)online categorical algebraic structures in an arbitrary finite language?We suspect that such a description exists, and that the solution will likely boildown to setting the definitions right. If we replace strict with primitive recursiveoperators in Definition 4.8 we will obtain the more general notion of (uniform)online categoricity. With quite a bit of effort Theorem 4.9 can be extended [49] tothis more general notion, and even beyond.5.
Weihrauch reduction and online algorithms
Weihrauch reduction is one of the central notions in computable analysis. Itwas named by Brattka and Gherardi [14], before that it had been called recur-sive reducibility. Weihrauch reducibility can be viewed as a natural generalisationof computable Wadge reducibility [78]. Henceforth will use f ≤ W g to denoteWeihrauch reducibility. f ≤ W g has the following intuition. We have some prob-lem we wish to solve by computing an instance f ( x ) some function f . To do thiswe produce another problem x ′ and solve g ( x ′ ) for g , and then convert g ( x ′ ) backto of f ( x ). In more detail, we for functions f and g on ω ω -represented spaces X and Y , f ≤ W g , is defined to mean that there are computable A and B on ω ω , suchthat for any p x , and any representation G of g , A ( p x , G ( B ( p x )))realizes f (i.e. is a name for f ( x )). (This is defined here for single-valued functions,but does have a multy-valued version we won’t need.) This should be thought ofas follows for the archetypal case of a computable metric space. We computablemetric space, we take a Cauchy sequence converging to x , use B to convert this OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 21 into a one converging to B ( x ), and hence one converging to g ( B ( X )), and finallyusing the one converging to x and this one, to one converging to A ( x, g ( B ( x ))) . The definition has a number of natural variations; some of these will be discussedbelow.5.1.
Weihrauch reduction and incremental computation.
In this subsectionwe establish a formal connection between computable analysis and computer sci-ence. More specifically, we show that a version of Weihrauch reduction borrowedfrom computable analysis [79] is equivalent to incremental reduction between onlineproblems suggested in Miltersen et al. [67].We first state Weihrauch reductions in the online setting. Suppose P , Q areonline problems. Definition 5.1.
We say that P is strongly Weihrauch reducible to Q , written P ≤ sW Q , if there exist Turing functionals Φ and Ψ such that, whenever σ ∈ I P isan instance of P , Φ σ = τ ∈ I Q is an instance of Q , and whenever ρ ∈ s (Φ σ ) is asolution to Φ σ then θ = Φ ρ ∈ s ( σ ) is a solution to σ . Here the reduction is strong in the sense that there is a provably more generaldefinition of (plain) Weihrauch reduction which will be given in due course. Notethat, according to the definition above, all functionals involved are strict, but thiscondition can be relaxed giving a less tight reduction.
Notation 5.2.
We write
P ≤
CsW Q if both strict functionals (in our sense) Φ andΨ in the definitions above belong to a complexity class C having sufficiently strongclosure properties (e.g., polynomial-time, polylogspace, primitive recursive, etc.). Remark 5.3.
The reader might wonder why we will restrict ourselves to strict-typereductions, or slight variations, for the online setting. The reason is the following.Suppose that we have two (represented) online problems I and I . In an onlineway we want to use I to solve I . Now suppose that we have some online algorithmfor I . We could take a σ of length n representing an instance G n of height n of I ,and convert it into an instance σ ′ of I , and use it to produce a solution s ( σ ′ ) of I ,which could be converted back into a solution s ( σ ) = A ( s ( σ ′ )) of I . The key issuewe will investigate is how tight the relationships of sizes of the representations are.Ideally | σ ′ | = | σ | .A problem P = ( I, O, s ) is a decision problem if O = { , } and s is merely apredicate on I . This is the same as to say that any solution simply decides whethera predicate holds on a string or not. We say that σ ∈ I is a positive instance of I if s ( σ ) = 1. Milterson et.al. [67] analysed complexity classes for online algorithms,and in a slightly more general situation than our monotone one where, for example,the objects only get bigger. Miltersen et. al. [67] investigate online algorithms inwhich input data may change with time. For example, in a graph a vertex or anedge can disappear. Their reduction takes into account the potential changes ofthe input. Definition 5.4.
Let C be a complexity class. A decision problem P is C -incrementallyreducible to another decision problem R , denoted P ≤
Cincr R , if the following twoconditions hold:1. There is a transformation T : I P → I R in C which maps instances of P to instances of R such that s P ( σ ) = s R ( T ( σ )) (i.e, σ is a positive instance iff itsimage is a positive instance).
2. There is a transformation Q in C which, given σ ∈ I P and the incremen-tal change δ to σ , where δ changes σ to σ ′ of the same length , constructs theincremental change δ ′ to T ( σ ) (where δ ′ changes T ( σ ) to T ( σ ′ ) ). Remark 5.5.
We will here only consider C to be the class of polynomial timecomputable functions, and hence use ≤ Pincr accordingly. Milterson et. al. [67] alsoconsidered e.g C to be Logspace . In [67] the authors specify the exact time boundsfor all computations involved. This is the reason why they need the seeminglyredundant part 2 of the definition above. Also, they look at auxiliary data structuregenerated for each instance and at the changes induced to the structure. However,from the perspective of general (e.g.) polynomial time computation this extrainformation is not necessary since these auxiliary bounds are evidently polynomialtime.The proposition below shows that P ≤ Pincr Q is a variation of Weihrauch reduc-tion from computable analysis which was independently rediscovered by computerscientists. Recall that strong Weihrauch reduction is witnessed by a pair of func-tionals Φ and Ψ. Fact 5.6.
Suppose P and Q are online decision problems. Then P ≤
Pincr Q iff P ≤
PsW Q with Ψ = Id { , } .Proof. Suppose P ≤ Pincr Q . Then the transformation T from the definition ofincremental reduction can be used as Φ in the definition of ≤ PsW . Since σ is apositive instance iff T ( σ ) is, Ψ = Id { , } .Conversely, suppose P ≤ PsW Q via (Ψ , Id { , } ), where Ψ is a polynomial func-tional from the space of inputs I P of P to the space of inputs I Q of Q . Then thefirst part of the definition of incremental reduction follows from the assumptionthat Ψ is a functional in C . By the continuity of Ψ and the fact that we used Id asthe second functional, it suffices to deduce a polynomial time bound on the changesin the inputs of Ψ( σ ) based on the changes in σ . But this bound is just a big-O ofthe bound given by Ψ. (cid:3) Following [67], we can impose specific bounds on the number of steps requiredfor example, calculating δ ′ based on δ . The expectation is that it should be easierto make the change than to simply recompute T ( σ ′ ) “from scratch”. All thesespecialised bounds can also be expressed in terms of strong Weihrauch reduction;we omit details. As an application of Theorem 5.6 and various results in [67], wecan obtain a number of polynomial time and polylogtime Weihrauch reductions inthe study of online algorithms.5.2. Weihrauch reduction and online graph colouring.
Before we discuss therole of Weihrauch reduction in online colouring problem we give a brief overview ofthe latter.5.2.1.
Online graph colouring.
Many problems can be re-cast as colouring prob-lems, for example
Bin Packing . Indeed, colouring can be thought of as avoidingconfigurations . In basic graph colouring, we are simply avoiding an edge connect-ing vertices of the same colour, but we could instead avoid, for example, trianglesor any finite set of configurations in some kind of constraint satisfaction problem. That is, δ is the difference between σ and σ ′ . OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 23
However, as this is an introductory paper we will stick to basic graph colouring.There is a large literature on this area such as Kierstead [58]. Graph colouring isquite a flexible tool, and many algorithmic meta-theorems such as for monadic sec-ond order logic (like Courcelle’s Theorem (see [26, 44])) can be viewed as colouringwith constraints. We believe that this material has great online potential.We will mention some of this in this subsection. As an illustrative example, wewill online colour finite or infinite trees (or more accurately forests). So the objectsof interest are trees being presented one vertex at a time so at height n we wouldhave a forest of n vertices. Then we can use a compact representation T ⊂ ω ω ofthe collection of forests of of height n . The result below is an easy (restated in ournotation) result from the folklore essentially following from Bean [11]. It works forany (not necessarily primitive recursive) general computable online procedure. Proposition 5.7.
For every online algorithm A there is a σ in T of length t − such that the tree A acting on T F ( σ ) needs at least t colours. We will write χ A ( G σ ) for the number of colours used to colour G when processedby online algorithm A . The above is nearly optimal, in that we have the following: Theorem 5.8 (Lovasz, Saks and Trotter [64]) . There exists an online algorithm A such that for every 2-colourable graph G , if G has n vertices then χ A ( G ) ≤ n . This brings us to the notion of a performance ratio . Most combinatorial algo-rithms we would teach in a standard combinatorics class are offline . This meansthat for a finite structure H , say, the algorithm has H as part of the input andcalculates using the global structure of H .Consider the situation of an inductive problem in a class C , and suppose wehave an optimisation problem. Then associates with a σ ∈ I, τ ∈ S will be a cost function c ( σ, τ ). This is the cost of solution τ for problem instance σ . Now the performance ratio of an algorithm f is the ratio of c ( σ, o ( σ )) with c ( σ, f ( σ )) where o ( σ ) is an optimal solution for σ ; the offline solution.We illustrate with colourings. The offline chromatic number of G σ will be de-noted by χ off( G σ ) and for forests, we would have χ off( T σ ) = 2 . Definition 5.9 (Sleator and Tarjan [74]) . The performance ratio is defined as be r ( σ ) = χ A ( T σ ) χ off( T σ ) . Here we are stating the definition for colouring but the definition applies toany online optimisation problem, as above. In the case of colouring forests, wesee that the approximation ratio is O (log( | σ | )) . In the infinite case, the relevantapproximation ratio is the growth rate of r ( σ ) for all paths in the tree T representingthe problem.For example, a graph is called d - inductive (or d - degenerate ) if the vertices of G can be ordered as { v , . . . , v n } so that for every i ≤ n , |{ j > i | v i v j ∈ E }| ≤ d .For example, by Euler’s formula, all planar graphs are 5-inductive. For those whoknow some graph theory, d -inductive graphs also include all graphs of treewidth d , and extremely important class in algorithmic graph theory (see Downey andFellows [26], for example). Again note that d -inductive graphs have a compactrepresentation (space). Theorem 5.10 (Irani [47, 48]) . Let σ represent a d -inductive graph of height n .Then first fit will use at most O ( d log n ) many colours to colour G σ . Moreover, forany online algorithm A , there is a d -inductive G σ such that χ A ( G σ ) is Ω( d log n ) . Sometimes, this growth rate reaches a limit, as in problems with constant ap-proximation ratios.The classical example is
Bin Packing . We can think of bins as colours, and theobjects having sizes and the constraint being that we cannot have more objects of aspecific colour than the bin constraint. That is,
Bin Packing takes as input sizes a i ∈ N and a parameter V for simplicity, and colours a i with colour c ( a i ) subjectto P c ( a i )= c a i ≤ V , and seeks to minimize the number of colours. Theorem 5.11 (see [39]) . First fit gives a performance ratio of for online BinPacking . Notice that
Bin Packing is another example of colouring with constraints.5.2.2.
Online reduction.
In this subsection we define a new version of Weihrauchreduction, and we also give a non-trivial example of such a reduction between twodistinct online problems.For convenience we use 2 ω as the ambient totally disconnected space, but oth-erwise use appropriate names. Let X and Y be spaces represented by 2 ω . Againwe think of f and g being solutions for minimisation X and Y -problems respec-tively. Thus, for example, we are thinking of X and Y as inductive structures withfiltrations { X n | n ∈ N } and { Y n | n ∈ N } respectively. Then the strings of length n represent the structures of height n , and f ( σ ) will represent a solution to theproblem represented by σ . Thus they will have an associated cost which in the caseof colouring is the number of colours, denoted c ( · ). We will denote f off and g off asoffline solutions. That is f off( σ ) would be the solution to the minimisation problem X n of height n with δ ( σ ) = X n , and similarly g off.We state the below for single valued functions, but again there is an analogousmulti-valued version, where the solution produced for g should be within the correctratio. The idea of the following is that on input α ↾ n , we want to compute(a representation of) f ( α ↾ n ) To to this we will apply (a representation of) B togenerate an input to (a representation of) an input for g , and then use the algorithm A to translate this back to give f ( α ↾ n ). Again we emphasis that this is all workingwith representations, and should be read this way. Definition 5.12.
Let f, g be functions on 2 ω . Then f is called ratio preservingonline reducible to g , f ≤ rO g , if there are (type II) online computable functions A and B with and a constant d , such that for all n , f ( α ↾ n ) = A ( α ↾ n, g ( B ( α ↾ n )) , and the ratio of c ( f ( α ↾ n )) to c ( f off( α ↾ n )) is at most d times the ration of c ( g ( B ( α ↾ n ))) to c ( g off( B ( α ↾ n ))).The fact below isolates the most important feature of the reduction. Fact 5.13. If f ≤ rO g then , for some d > , c ( f ↾ n ) c ( f off ↾ n ) ≤ d c ( g ↾ n ) c ( g off ↾ n ) . We want to avoid explicit representations, but of course we should have F representing f with F acting on 2 ω , and for any α ∈ ω , lim n F ( α ↾ n ) realizes (represents) f ( α ). OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 25
To give a non-trivial example of an online reduction we need several definitions.In classical colouring, Kierstead investigated online colouring of
Interval Graphs .A graph G = ( V, E ) is called a k -interval graph if each vertex v of G can be repre-sented by a closed subinterval of [0 ,
1] such that if I v represents v and I w represents w , then if vw ∈ E , I v ∩ I w = ∅ , such that the largest number of intersecting intervals(the cutwidth ) is ≤ k . These are exactly the graphs which have Pathwidth ≤ k , agraph metric coming from the Robertson-Seymour minors project (see [72, 26]). Definition 5.14.
Let
ColInt k denote the online problem of colouring a k -intervalgraph. (We leave the precise representation of the problem to the reader.)The other online problem is on covering of an interval partial ordering by chains.A partial ordering ( P, ≤ ) is called an interval ordering if P is isomorphic to ( I, ≤ )where I is a set of intervals of the real line and x ≤ y iff the right point of x isleft of the left point of y . Interval orderings can be characterised by the followingtheorem. Theorem 5.15 (Fishburn [36]) . Let P be a poset. Then the following are equiva-lent. (a) P is an interval ordering. (b) P has no subordering isomorphic to + which is the ordering of fourelements with { a, b, c, d } with a < b , c < d and no other relationshipsholding. The width of an interval ordering ( P, ≤ ) is defined naturally to be the minimumover all presentations of the maximum number of intervals covering some point of[0 , P, ≤ ) of width k , our goal is to cover it with asfew chains as possible; the chains do not have to be disjoint. Definition 5.16.
Let
ChInt k denote the online problem of covering an intervalordering ( P, ≤ ) of width k by not necessarily disjoint chains. (We leave the preciserepresentation of the problem to the reader.)The theorem below gives a non-trivial example of an online ratio-preservingreduction between online problems. The proof of the theorem below is essentiallyan analysis of the clever argument given in Kierstead and Trotter [59]. Theorem 5.17.
For any positive k ∈ N there is an online solution g to ChInt k with a constant performance ratio which can be transformed into an online solution f to ColInt k with the property f ≤ rO g via a constant d = 1 . Corollary 5.18 (Kierstead and Trotter [59]) . There is an online algorithm tocolour k interval graphs with a constant competitive ratio.Proof of Corollary. Kierstead and Trotter [59] showed that every ( P, ≤ ) (online)interval ordering of width k can be online covered by 3 k − chain in a partial ordering is a ≤ -linearly ordered subset. A collectionof chains { C , . . . , C q } covers P, ≤ ) if each element of P lies in one of the chains.An antichain is a collection of pairwise ≤ -incomparible elements. We will see that ColInt k ≤ rO ChInt k is witnessed via a reduction with constant d = 1. It remainsto apply Fact 5.13. (cid:3) Proof of Theorem 5.17.
The basic idea is quite simple. Take our online k intervalgraph, turn it into an online interval ordering of width k , and then consider that chain covering as a colouring. However, to see that this idea works, we need toargue that there is an online solution g to the interval chain covering problemwhich uses only the information about comparability of various elements, and nottheir ordering.We first prove the following. Suppose that ( P, ≤ ) is a online interval orderingof width k . Then P can be online covered by 3 k − P , and subsets S, T , we can define S ≤ T iff for each x ∈ S there is some y ∈ T with x ≤ y .(Similarly S | T etc.) Lemma 5.19. If P is an interval order and S, T ⊂ P are maximal antichains theeither S ≤ T or T ≤ S . The algorithm for chain covering uses induction on k . We consider the verticesas 1 , , . . . with p added at step p . If k = 1 then P is a chain, and there is nothingto prove. Suppose the result for k , and consider k = 1. We define B inductively by B = { p ∈ P : width( B p ∪ { p } ) ≤ k } . Here B p denotes the amount of B constructed by step p of the online algorithm.Then B is a maximal subordering of P or width k . By the inductive hypothesisthe algorithm will have covered B by 3 k − A = P − B . Now it willsuffice to show that A can be covered by 3 chains, and then these will be coveredby the greedy algorithm.To see this it is enough to show that every elements of A is incomparable withat most two other elements of A . Then the greedy algorithm will cover A , as wesee elements not in B . Lemma 5.20.
The width of A is at most 2.Proof. To see this, consider 3 elements q, r, s ∈ A . Then there are antichains Q, R, S in P of width k with q | Q , r | R and s | S . Moreover these can be taken as maximalantichains. Applying Lemma 5.19, we might as well suppose Q ≤ R ≤ S . Supposethat r | q and r | s . Then we prove that q < s . Since q | r and width( P ) ≤ k + 1, thereis some r ′ ∈ R with q and r ′ comparable. Since q | Q , r ′ Q . Since the width of B is ≤ k , there is some q ′ ∈ Q q ′ and r ′ comparable. Since Q ≤ R , there is some r ∈ R with q ′ ≤ r . Since e R is an antichain, q ′ ≤ r ′ . Since q | q ′ , q ≤ r ′ . Similarly,there exists r ′′ ∈ R with r ′′ ≤ s . Since P does not have any ordering isomorphic to + , we can choose r ′ = r ′′ , and hence q < s. (cid:3) Now we suppose that r, q, s, t are distinct elements of A with q |{ r, s, t } . Thenwithout loss of generality r < s < t since the width of A is at most 2. Since s ∈ A there is an antichain S ⊂ B of length k with s | S . Since s | q , and width( P ) ≤ k + 1, q is comparable with some element s ′ ∈ S . If s ′ < q , then s ′ | r and hence thesuborder { s ′ , q, r, s } is isomorphic to + . Similarly, q < s ′ implies s ′ | t and thenthe subordering { q, s ′ , s, t } is isomorphic to + . Thus there cannot be 4 elements r, q, s, t of A with q |{ r, s, t } . Hence A can be covered by 3 chains.It is easily see that the procedure above uses only comparability of intervals.Thus, the theorem follows. (cid:3) Problem 5.21.
Investigate online reduction between online algorithms in the lit-erature.
OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 27
We also expect that the online reduction may lead to new online algorithmsbased on the already existing ones.Also graphs with constrained decompositions such as those of bounded treewidth,pathwidth, clique-width, etc have been extensively studied in the literature, andparticularly combine well with algorithmic meta-theorems (see e.g. Downey-Fellows[26], Flum and Grohe [37], Grohe [44] for a sample).One example is given by k -interval graphs met above which are those of path-width ≤ k. . A G of pathwidth k has a path decomposition which is a collection ofsets of vertices V , . . . , V n all of size ≤ k + 1 such that for all vertices v ∈ V ( G ),there is at least one i with v ∈ V i , if xy ∈ E ( G ), then for some i , { x, y } ⊆ V i andfinally if x ∈ V i and x ∈ V j (with i < j ) the for all q ∈ [ i, j ], x ∈ V q . The lastproperty is called the interpolation property , and says that pathwidth is kind of ameasure of how far you are from being either a grid or a clique.Now given such a path decomposition, and some optimisation property we wantto solve (such as for the largest clique), if the property is definable in monadicsecond order logic (even with counting), then we can solve the problem by dynamicprogramming (actually using special automata) beginning at V and finishing at V n by the methods of Courcelle [26, 37, 44]. Problem 5.22.
Investigate the extent to which this dynamic programming is on-line. Presumably, it will be online for properties defined by monadic second ordercounting logic with counting modulo some kind of delay.
Moreover, as we have seen above for the special case of colouring above, we geta constant ratio approximation algorithm, for a graph of pathwidth k , no matterhow we are given the online presentation. The difference is that if we are a given apath decomposition as the presentation, then k + 1 colours will suffice. But perhapsthe methods for colouring are more general. The point is that graphs of boundedpathwidth have very constrained structure. Problem 5.23.
Investigate the approximability of monadic second order definableproperties on graphs of bounded pathwidth, but given as arbitrary online presenta-tions.
The same can be asked for graphs of bounded treewidth which has the samedefinition as pathwidth, but the structure of the decomposition is a tree and nota path. These also have dynamic programming algorithms, but are always leaf toroot , whereas even given a tree decomposition as an online root to leaf structure,presumably some kind of algorithm will work, but it will no longer be automatic.This seems a great area to pursue.Also related seems the idea of online parameterised problems [31, 32], wherewe want an online solution to a problem with a fixed parameter. For example, k - Vertex Cover asks for a collection of vertices where each edge of a graph includesat least one of the vertices, and this is polynomial time for a fixed k . This is alsoonline polynomial time for a fixed k by the following simple algorithm (so long as weare allowed 2 k many possible solutions). We can use the following simple method ofbuilding a tree of height 2 k by taking an edge, and branching on that edge, and thendeleting the covered edges, and repeating. This process is also online. In [31, 32]Downey and McCartin showed that the online view brings to the other parameterssuch as what they call persistence which characterises the extent to which a pathdecomposition does not resemble a fuzzy ball. The point is that online algorithms point at new parameters of a problem which deserve attention, in the same waythat parameterised complexity showed that parameters allow a more fine grainedunderstanding of the computational complexity of a combinatorial problem.6. ∆ processes, finite reverse mathematics, and Weihrauchreduction Imagine we are in a situation where the data we are dealing with is so largethat we cannot see it all. At each stage s our goal is to build a solution f to someproblem. But there might be no hope of giving a fixed solution at each stage n , andlike a Triage Nurse making an ordering for patients to obtain medical attention,we would update our solution as more information becomes available. So for each n ≤ s we would be computing f ( n, s ) from the finite information σ with | σ | = n. For simplicity we state the next definition for combinatorial problems with totallydisconnected representations, and take 2 ω as the representing example. Definition 6.1. A limiting online algorithm on ω is a computable function A suchthat for each s , A ( α ↾ s ) computes a string { f A ( n, s ) | n ≤ s } such that lim s f A ( n, s ) exists for each n .As usual we would have A ( α ↾ g ( s )) for the g -online version. We can then compare combinatorial problems by how fast their limits converge.
Definition 6.2.
We say that algorithm A ≤ O,lim B if there is an online Weihrauchreduction of A to B such that f B ( n, s ) = f B ( n, t ) for all t ≥ s implies f A ( n, s ) = f A ( n, t ) for all t ≥ s . This gives a fine grained measure of the complexity of combinatorial problems.For example, consider the “theorem” that every finite binary tree of height n has apath of length n . Then we can consider the existence of a uniform function A whichtakes a given binary tree of height n to a path. This is an online limit problemwhere the underlying space X is that with nodes generated by the collection ofbinary trees of height n at level n . The completion of this will represent pathsthrough infinite binary trees. Remark 6.3.
We could argue that the Reverse Mathematics principle
W KL which states that every infinite binary tree has a path, is equivalent to the statementthat there is a limiting online algorithm for finding paths which works on X . Wecall this limiting online paths. A binary tree T of height n is called separating if for each j ≤ n −
1, for any node σ on T of height j , and i ∈ { , } , if σ ∗ i does not have an extension in T of height n , then for all τ of length j , neither does τ ∗ i . Let X S be the totally disconnectedspace representing the collection of all separating finite trees. The following is aonline interpretation and refinement of the classical fact that Weak K¨onig’s Lemmais equivalent to Weak K¨onig’s Lemma for separating classes. Proposition 6.4.
There is a (2 n +1 − -limiting online reduction which finds lim-iting online paths in X from those in X S .Proof. We remind the reader of how this proof works. Suppose we have a tree T s of height s . In an online fashion, we will generate a tree H of height 2 s +1 . Thisis done inductively. At step 1, we can think of the nodes labeled 0 and 1 in T asbeing represented by 0 and 1 in H . At step 2, in T it is possible for us to have OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 29 , , ,
11 and these are represented by 4 levels in H , with height 2 representing00 , level 3 01 , level 4 10 , and level 5 11 . Now we continue inductively. This makeslevel n of T correspond to trees of height 2 + 4 + · · · + 2 n = 2 n +1 −
2. As theconstruction proceeds, if some σ fails to have an extension at length s , in T s , therewill be some shortest σ ′ (cid:22) σ which fails to have a length s extension in T s . Thenin H s , we don’t extend to length s (from length s −
1) all paths corresponding to ν ∗ j with j representing σ ′ in H s − . Consider any limiting online algorithm for finding a path for path α correspond-ing to H , in X , This naturally and in a online way allows us from level 2 s +1 togenerate an online path in T s , and is clearly a limiting online reduction. (cid:3) Problem 6.5.
Figure out the smallest g in place of s +1 in the reduction above,which would give a precise measure of how tight the reverse mathematics relation-ship is. There seems a whole research programme available here. For example, we couldbe given an online bipartite graph B σ for σ ≺ α . We either have to build acomplete matching or demonstrate that Hall’s condition fails. One representationof this problem will involve a compact space where then nodes are bipartite graphsof height 2 n , say, and where the paths all represent graphs which obey Hall’scondition. The online operator will act on this compact tree of representations forgraphs B σ . Now as the process goes along, we might have to update the solutionat hand. That is, the online process has B σ M σ ,One intriguing example is that of finding a basis in a vector space. In the casethat the vector space is over the rationals, then presumably this will correlate tosome principle like ACA . But consider a finite field such as GF(2). We knowthat RCA proves that we can find a basis for a vector space for this field. Butit is not hard to construct an online vector space over GF(2) for which there isno online algorithm to do this, unless we have a computable delay. Comparingthe online complexity of such problems with such computable delay would seeto give significant insight into the fine structure of reverse mathematics. In thisparticular case, we also note that a polynomial time algorithm for finding a basis ofa polynomial time vector space was proven to be equivalent to P = N P suggestingintriguing connections with complexity theory.We remark that there are many processes that have been investigated and fallunder the model we have introduced. One such example is algorithmic learningtheory, such as EX -learning (Gold [43]). Here one is presented with a , a , . . . values for a function f (0) , f (1) , . . . , and we need to eventually print out an index for ϕ e = f from some point onwards. This is clearly an example of an online algorithm,and fits into this section as a limiting algorithm. Perhaps there are connectionswith this and reverse mathematics, and this remains to be explored. Another areawhich could be incorporated would be asynchronous computing. Here we havea series of agents A , . . . , A k communicating through asynchronous channels, andattempting to compute a set of functions f , . . . , f k , where there might be e.g.some kind of crash failure meaning that one of the agents dies and stops sendingsignals. For example, the Consensus problem asks for all the f i ’s which have notcrashed to give the same value. A run could be represented in a space of possiblecommunications and failures. There are a number of reductions which have beenproduced in this area, showing that Consensus is a certain kind of minimal failure,and other problems can be solved if Consensus can (Chandra and Toueg [21]). It would be interesting to see if these results can be placed in the hierarchy of onlinelimiting reductions, since they appear to look like online limiting reductions.Finally, one exciting possibility would be to include randomization in this set-ting. Randomized online algorithms are quite common in practice (see e.g. Albers[4]). For us we could use the theory of algorithmic randomness (see [28, 63, 69])easily. For example, an online algorithm with randomized advice (i.e. representinga coin toss at each stage) could be done via (using 2 ω as a representative space)by considering online algorithms from 2 ω × ω → S , with S some solution space,with the first copy of 2 ω representing the problem, the second representing “advice”strings and S the solution space. The online algorithm could take ( σ, τ ) → s n , andwould run on extensions of τ provided that [ τ ] avoids some algorithmic randomnesstest, such as a Martin-L¨of test. Using oracles we could also tie this to the theory ofalgorithmic randomness using the “ fireworks ” method of Shen (see Bienvenu andPatey [13]). These ideas remain to be explored.7. Real functions.
So far all objects of study have been discrete and spaces compact. However,there is a perfectly reasonable extension of these ideas to continuous objects suchas the space of continuous functions on the unit interval. There has been a lotof work on complexity theory of real functions; see, e.g., Ko [61]. In terms ofapplications, a natural object of study would be online analysis; analytic processeswhich run quickly and only use local knowledge of the precision of the inputs. Aswe observe below with natural representations, addition of reals x, y with precision2 − n only needs x and y to within 2 − ( n +2) . Integration and other standard processeshave similar commentry, but we leave this to a later paper. Also there are otheronline processes on non-compact spaces, such as EX-learning, or the KC theoremdiscussed earlier. We also defer discussion of such topics for later papers, and herestick to analysis.The main goal of this section is to demonstrate the role of primitive recursionas a useful abstraction. The content of this section is not technically hard, but onecan easily imagine a much deeper general framework that could emerge from thesebasic ideas.Recall that a Cauchy sequence ( r i ) i ∈ N of rationals is fast if | r i − r i +1 | < − i , forevery i . These are the names which represent the space. A function f : [0 , → R is computable if there is a Turing functional Φ such that, for each x ∈ [0 ,
1] andfor every fast Cauchy sequence χ converging to x , the functional Φ enumeratesa fast Cauchy sequence for f ( x ) using χ as an oracle. In particular, using theterminology, we would be generating a representation of the function via namesof Cauchy sequences in such a way that it is representation independent. Thatis, (Φ χ ( n )) n ∈ N is a fast Cauchy sequence for f ( x ). This in particular means that,on input ( r i ) i ∈ N , the use of Φ ( r i ) i ∈ N ( j ) corresponds to δ when ǫ = 2 − j +1 in thestandard ǫ - δ definition of a continuous function.It is well-known that Weierstrass approximation theorem is effectivisable in thesense of Turing computability [70]. This means that a function f : [0 , → R iscomputable iff there is a computable sequence of polynomials ( p i ) i ∈ N with rationalcoefficients with the propertysup x ∈ [0 , | f ( x ) − p i ( x ) | < − i , OUNDATIONS OF ONLINE STRUCTURE THEORY II: THE OPERATOR APPROACH 31 for every i. We have seen that the most general definition of being online for combinatorialstructures involves being g -online for some primitive recursive function g . That is,there is a translation between using g ( n ) many bits of α to compute n bits of f ( α ).We have also seen that for most natural online situations, we can translate this toa wider tree where α ′ ↾ n represents α ↾ g ( n ), so we can use strict (ibT primitiverecursive) procedures. It is not completely clear if this is natural in the setting ofanalysis, since we might wish to stick to standard representations of the spaces, like2 ω and ω ω , as above.We first consider the most general setting where we allow g -online for a primitiverecursive g , so using g ( n ) bits to decide the output for length n . We will call this punctually computable. In this case, there are two natural definitions of what itwould mean for such an f to be “online” computable in the most general senseof primitive recursion. The first notion is the most straightforward sub-recursiveversion of the standard definition. Definition 7.1.
A function f : [0 , → R is punctually computable if there is aprimitive recursive functional Φ such that, for each x ∈ [0 ,
1] and for every fastCauchy sequence χ converging to x , the functional Φ enumerates a fast Cauchysequence for f ( x ) using χ as an oracle.By restricting ourselves to dyadic rationals, we can assume that fast Cauchysequences come from a compact totally disconnected space of the names of dyadicrationals in [0 , f as aprimitive recursive point in the metric space ( C [0 , , sup) rather than as a func-tional. Definition 7.2.
A function f : [0 , → R is uniformly punctually computable ifthere is a primitive recursive function which on input i outputs (the index of) apolynomial p i with rational coefficients such that sup x ∈ [0 , | f ( x ) − p i ( x ) | < − i .Clearly, there is a natural polynomial-time modification of the definition abovewhich we omit.Every uniformly punctually computable f is punctually computable. Are thesetwo definitions equivalent? It is not completely evident why Weierstrass approxi-mation theorem should hold primitively recursively. Indeed, in the standard Turingcomputable proof we would wait for a cover of [0 ,
1] by δ i -balls B i such that f ( B i )has diameter < ǫ , for every i . It seems that even when f is punctual this searchcould be unbounded.Nonetheless, the theorem below shows that these definitions are equivalent. Thisresult is not really new. With some effort its proof can be extracted from book [61],but the book is mainly focused on polynomial time and exponential versions of thedefinitions above. There is much combinatorics specific to complexity theory whichsignificantly obscures the idea behind the proof. Primitive recursion strips awaycomplex counting combinatorics thus clarifying the idea. Theorem 7.3.
Every punctually computable f : [0 , → R is uniformly punctuallycomputable.Proof sketch. The idea here is similar to that in the proof of Lemma 3.5. Fix n and consider the functional Ψ xn = Φ x ( n ) which uniformly primitively recursivelyoutputs the fest few bits of f ( x ) up to error 2 − n , for any input x . Since Ψ n isgiven a primitive recursive scheme (with parameter n ), we can work by inductionon the complexity of the scheme and emulate all its possible computations at once,as in Lemma 3.5. Since the space of dyadic presentations of rationals is primitivelyrecursively compact, this will lead to a primitively recursively branching tree ofpossible computations whose height is determined by the syntactical complexity ofthe primitive recursive scheme. By the choice of Ψ n , one of these computations mustwork for an arbitrary x ∈ [0 , ,
1] by basic open intervals J , . . . , J k , such that whenever x, y ∈ J i we have | f ( x ) − f ( y ) | < − n +1 . If z i is the center of J i , then define (the graphof a) piecewise linear function h n by connecting points ( z i , Ψ z i n ) and ( z i +1 , Φ z i +1 n ), i = 1 , . . . , n -1. Note that the values of the Φ z i n have already been calculated. Sincethe intervals are overlapping, this piecewise linear function h n approximates f withprecision 2 − n +2 . We can primitively recursively smoothen h n by replacing it witha polynomial p n such that sup x ∈ [0 , | p n ( x ) − f ( x ) | < − n +3 . (cid:3) See Chapter 8 of [61] for a detailed analysis of the polynomial-time versionsof Weierstrass approximation theorem. Recall that in the proof sketch above wegenerated the tree of possible computations. For a polynomial-time operator thistree may be exponentially large at worst. This difficulty cannot be circumventedand the polynomial-time analogy of the theorem above fails as explained in greatdetail in [61].We see that punctual analysis fits somewhere in-between computable analysisand polynomial-time analysis, and there is likely much depth in the subject. Sucha theory could provide us with a stronger technical link between computable andfeasible analysis. Nonetheless, is seems there has been no dedicated study of prim-itive recursive continuous functions and punctual presentations of analytic spaces.
Problem 7.4.
Develop primitive recursive (“punctual”) analysis.
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