Decision times of infinite computations
aa r X i v : . [ m a t h . L O ] N ov DECISION TIMES OF INFINITE COMPUTATIONS
MERLIN CARL, PHILIPP SCHLICHT, AND PHILIP WELCH
Abstract.
The decision time of an infinite time algorithm is the supremum of itshalting times over all real inputs. The decision time of a set of reals is the least decisiontime of an algorithm that decides the set; semidecision times of semidecidable sets aredefined similary. It is not hard to see that ω is the maximal decision time of sets ofreals. Our main results determine the supremum of countable decision times as σ andthat of countable semidecision times as τ , where σ and τ denote the suprema of Σ - and Σ -definable ordinals, respectively, over L ω . We further compute analogous supremafor singletons. Contents
1. Introduction 12. Preliminaries 33. Decision times 33.1. The supremum of countable decision times 43.2. Quick recognising 53.3. Gaps in the decision times 54. Semidecision times 64.1. The supremum of countable semidecision times 64.2. Semirecognisable reals 74.3. Cosemirecognisable reals 85. Sets with countable decision time 96. Open problems 9References 101.
Introduction
Infinite time Turing machines (ittm’s) were invented by Hamkins and Kidder as anatural machine model beyond the Turing barrier and are by now well studied. Themachines and the sets that they define link computability and descriptive set theory. Inhigher recursion theory, Π sets were characterised by transfinite processes bounded by ω ck1 in time. This can be reformulated more concretely by taking infinite time Turingmachines to model transfinite processes.Hamkins and Lewis [HL00] showed that these sets fall strictly between the Π and ∆ sets studied in descriptive set theory. Previously, there had been a lack of natural classesin this region and this class of sets forms a natural test case for properties of Wadge classes.The computational properties of infinite time Turing machines lead to some new phe-nomena that do not occur in Turing machines. For instance, there are two natural notions Date : November 11, 2020.This project has received funding from the European Union’s Horizon 2020 research and innovationprogramme under the Marie Skłodowska-Curie grant agreements No 794020 of the second author (Project
IMIC: Inner models and infinite computations ). The second author was partially supported by FWFgrant number I4039. of halting – besides a program halting in a final state, one can consider stabilisation ofthe output tape. Another new phenomenon is the accidental appearance of new outputsbeyond all stabilisation times. In fact, letting λ , ζ and Σ denote the suprema of ordinalscoded by writable, eventually writable and accidentally writable reals, respectively, wehave λ ă ζ ă Σ (see [Wel00, Section 3]).We give a brief sketch how these machines work (for more details see [HL00].) Anittm-program is a just a regular Turing program. The “hardware” of an ittm consists (forconvenience) of an input, work and output tape, each of length ω , and a single head thatmoves along the tapes. At any stage, it can thus access the n -th cell of each tape, if thehead is situated in position n . Each cell contains or . While the successor stage of anittm works like that of a Turing machine, at any limit stage the head is set to the leftmostcell, the state to a special limit state, and the contents of each cell to the inferior limit ofthe earlier contents of this cell.These machines define the following classes of sets. For a subset A of P p ω q , let χ A denote the characteristic function of A . A is called ittm-decidable if and only if there isan ittm-program p such that, for all subsets x of ω , p p x q halts with output χ A p x q . Wewill often omit the prefix ittm . A is called semidecidable if and only if there is an ittm-program p such that p p x q halts precisely if x P A . We say that A is cosemidecidable ifits complement is semidecidable. For singletons A “ t x u , we call x recognisable if A isdecidable, following [HL00]. Moreover, x is called semirecognisable if A is semidecidableand cosemirecognisable if A is cosemidecidable.By a result of the third author [Wel00, Theorem 1.1], the supremum of ittm-haltingtimes on empty input equals the supremum λ of writable ordinals, i.e. those ordinals forwhich an ittm can compute a code. This solved a well known problem posed by Hamkinsand Lewis. We consider an extension of Hamkins’ and Lewis’ problem by allowing allreal inputs. More precisely, we consider the problem which decision times of sets andsingletons are possible. This is defined as follows. Definition 1.1. (a) The decision time of a program p is the supremum of its halting times for arbitraryinputs.(b) The decision time of a decidable set A is the least decision time of a program thatdecides A .(c) The semidecision time of a semidecidable set A is the least decision time of a programthat semidecides A .(d) The cosemidecision time of a cosemidecidable set A is the semidecision time of itscomplement.Since halting times are always countable, it is clear that these ordinals are always at most ω . It is also not hard to see that the bound ω is attained (see Lemma 3.1 below). Thequestions are then: Which countable ordinals can occur as decision or semidecision timesof sets of real numbers? Which ordinals occur for singletons? As there are only countablemany programs, there are only countably many semidecidable sets of real numbers, so thesuprema of the countable (semi-)decision times must be countable. These suprema willdetermined in the course of this paper (we call a supremum strict when it is not attained): Definition 1.2. (a) σ denotes the first Σ -stable ordinal, i.e the least α with L α ă Σ L ω . Equivalently,this is the supremum of the Σ L ω -definable ordinals (see Lemma 2.2). For ease of notation, we will always write ω for ω V . We mean that for an ordinal α , the set t α u is Σ -definable over L ω , or equivalently in V . The reasonfor writing L ω here is to make the analogy with the definition of τ clear. ECISION TIMES OF INFINITE COMPUTATIONS 3 (b) τ denotes the supremum of the Σ L ω -definable ordinals.While the value of σ is absolute, we would like to remark that τ is sensitive to theunderlying model of set theory. For instance, τ ă ω L holds in L , but τ ą ω Ln if ω Ln iscountable in V . Note that τ equals the ordinal γ studied in [KMS89] by a result of[CSW20].The following are our main results: Theorem. (see Theorem 3.4) The (strict) supremum of countable decision times for setsof reals equals σ . Theorem. (see Theorem 4.3) The (strict) supremum of countable semidecision times forsets of reals equals τ . Theorem. (see Theorems 3.3, 4.4 and 4.7) The (strict) suprema of decision times, semide-cision times and cosemidecision times for singletons equal σ .We also prove the existence of semidecidable and cosemidecidable singletons that arenot recognisable. The latter answers [Car19, Question 4.5.5].Since there are gaps in the clockable ordinals (see [HL00]), it is natural to ask whetherthere are gaps below σ in the countable decision times. We answer this in Theorem 3.6by showing that gaps of arbitrarily large lengths less than σ exists.2. Preliminaries
We fix some notation. We write p p x qÓ if a program p with input x halts as all, and p p x qÓ ď α if it halts at or before time α .When α is an ordinal, let α ‘ denote the least admissible ordinal that is strictly above α .When α is an ordinal which is countable in L , then x α denotes the ă L -least real coding α ;conversely, if a real x codes an ordinal via Gödel’s pairing function on ω , then we denotethis ordinal by α x . Definition 2.1. σ ν denotes the supremum of Σ L ω -definable ordinals with parametersin ν Y t ν u . Lemma 2.2. L σ ν ă L ω for any countable ordinal ν . Proof.
Assume ν “ for ease of notation. Let ˆ σ denote the least α with L α ă L ω . Itsuffices to show that every element of L ˆ σ is Σ L ω -definable. If not, then the set N of Σ L ω -definable elements of L ω is not transitive, so the collapsing map π : N Ñ ¯ N moves someset x . Assume that x has minimal L -rank and x is Σ L ω -definable by ϕ p x q . It is easy tocheck that N ă Σ L ω , so ¯ N |ù ϕ p π p x qq . Since ¯ N is transitive, this implies V |ù ϕ p π p x qq .Since π p x q ‰ x , this contradicts the assumption that ϕ p x q has x as its unique solution. (cid:3) Note that Σ -statements in H ω are equivalent to Σ -statements and conversely [Jec03,Lemma 25.25]. In particular, L σ is Σ -correct in V . This will be used several timeswithout a specific reference. 3. Decision times
In this section, we study the problem what the decision times of ittm-semidecidablesets are.
MERLIN CARL, PHILIPP SCHLICHT, AND PHILIP WELCH
The supremum of countable decision times.
By a standard condensation argu-ment, halting times of ittm’s on arbitrary inputs are always countable, so it is clear thatdecision times are always at most ω . It is also easy to see that their supremum equals ω : Lemma 3.1.
Every set with countable decision time is Borel. Hence any non-Borel Π set has decision time ω . Proof.
Suppose that an ittm-program p semidecides a set A within a countable time α .Note that p p x qÓ ď α can be expressed by Σ and Π formulas in any code for α . By Lusin’sseparation theorem, it is Borel. (cid:3) For instance, the set WO of wellorders on the natural numbers is Π -complete and hencenot Borel. Since all Π sets are ittm-decidable by [HL00, Corollary 2.3], WO is decidablewith decision time ω (cf. [Car20, Prop. 32]).It remains to study sets with countable decision times, and in particular, the followingproblem: Problem.
What is the supremum of countable decision times of sets of reals?We need two auxiliary results to anwer this. The next lemma shows that no x R L α ‘ issemirecognisable with decision time at most α . Lemma 3.2. If p semirecognises x and p p x qÓ ď α , then x P L α ‘ . Proof.
We first claim that for any admissible set M , any forcing P P M and mutually P -generic filters f and g over M , we have M r f s X M r g s “ M . Towards a contradiction,take a set x of least rank in p M r f s X M r g sqz M and let x “ σ f “ τ g . We will identify P with each factor in P and thus interpret σ and τ as P -names. By the forcing theorem for ∆ -formulas, some p p, q q P f ˆ g forces σ “ τ . Since σ f R M , there is some y P M such that p doesn’t decide whether y P σ . Now take generics f , f with p P f , f and σ f ‰ σ f .Moreover, let h be generic over M r f s and M r f s with q P h . Then σ f “ τ h “ σ f ,contradicting the assumption on f and f .Let M “ L α ‘ and take any Col p ω, α q -generic filter g P V over M . Since Col p ω, α q is aset forcing in M , M r g s is admissible if g is taken to be sufficiently generic. Let y P M r g s be a real coding g . Then M r g s “ M r y s “ L ω y r y s . By the above claim, it suffices to showthat x P M r g s . To see this, note that α is countable in M r y s and hence t x u is ∆ p y q bythe assumption of the lemma. Therefore x P L ω y r y s “ L α ‘ r y s “ M r y s by [BGM71] (see[Hjo10, Section 5]). (cid:3) The next result will be used to provide a lower bound for decision times of sets.
Theorem 3.3.
The supremum of decision times of singletons equals σ . Proof.
To see that the supremum is at least σ , take any α ă σ . Pick some β with α ă β ă σ such that some Σ -sentence ϕ holds in L β for the first time. By a standardargument, the ă L -minimal code x for L β ‘ is recognisable. Since β ‘ is admissible, x R L β ‘ .By Lemma 3.2, the decision time of t x u is thus larger than β .It remains to show that any recognisable real x is recognisable with a uniform timebound strictly below σ . To see this, suppose that p recognises x . We shall run p and anew program q synchronously, and halt as soon as one of them does. q will ensure thatthe run time is small.We now describe q . q p y q simulates all ittm-programs with input y synchronously. Foreach halting output on one of these tapes, we check whether it codes a linear order. By [Mat15, Theorem 10.17], it suffices that the generic filter meets every dense class that is a unionof a Σ -definable with a Π -definable class. ECISION TIMES OF INFINITE COMPUTATIONS 5
In this case, run a wellfoundedness test and save the wellfounded part, as far as it isdetected. (These routines are run synchronously for all tapes, one step at a time.) Awellfoundedness test works as follows. We begin by searching for a minimal element; thisis done by a subroutine that searches for a strictly decreasing sequence x , x , . . . . If thesequence cannot be extended after some finite stage, we have found a minimal elementand add it to the wellfounded part. The rest of the algorithm is similar and proceeds bysuccessively adding new elements to the wellfounded part. Each time the wellfounded partincreases to some α ` by adding a new element, we construct a code for L α ` . (Notethat the construction of L α takes approximately ω ¨ α many steps.) We then search for z such that p p z qÓ ď α with output in L α ` . We halt if such a z is found and x ‰ z .By Lemma 3.2, x P L λ x . So for any y with λ y ě λ x , some L α satisfying p p x qÓ ď α withoutput appears in q p y q in ă λ x steps. Otherwise λ y ă λ x , so p p y q will halt in ă λ y andtherefore ă λ x steps. Clearly λ x ă σ . (cid:3) We call an ittm-program total if it halts for every input. We are now ready to provethe main results of this section.
Theorem 3.4.
The suprema of countable decision times of total programs and of decidablesets equal σ . Proof.
Given Theorem 3.3, is remains to show that σ is a strict upper bound for countabledecision times of total programs. Suppose that p is total and has a countable decision time.Since D α ă ω @ x p p x qÓ ď α is a Σ statement, this holds in L by Shoenfield absoluteness.Since L σ ă Σ L , there is some α ă σ such that @ x p p x qÓ ď α holds in V , as required. (cid:3) Quick recognising.
The lost melody theorem, i.e., the existence of recognisable,but not writable reals in [HL00, Theorem 4.9] shows that the recognisability strength ofittm’s goes beyond their writability strength. It thus becomes natural to ask whether thisresult still works with bounds on the time complexity. If a real x can be written in α many steps, then it takes at most α ` ω ` many steps to recognise x by simply writing x and comparing it to the input. Can it happen that a writable real can be semirecognisedmuch quicker than it can be written? The next lemma shows that this is impossible. Lemma 3.5.
Suppose that p recognises x and p p x q halts at time α . Then:(1) x P L β for some β ă α ‘ .(2) If x P L β and β is clockable, then x is writable in time less than β ‘ . Proof.
The first claim holds by Lemma 3.2. For the second claim, note that there is analgorithm that writes a code for β in β steps by the quick writing theorem [Wel09, Lemma48]. One can therefore write codes for L β and any element of L β in less than β ‘ manysteps. (cid:3) Gaps in the decision times.
It is well known that there are gaps in the set ofhalting times of ittm’s (see [HL00, Section 3]). We now show that the same is true forsemidecision times of total programs and thus of sets.A gap in the semidecision times of programs is an interval that itself contains no suchtimes, but is bounded by one.
Theorem 3.6.
For any α ă σ , there is a gap below σ of length at least α in the semide-cision times of programs. Proof.
Consider the Σ -statement there is an interval r β, γ q strictly below ω of length α such that for all programs p , there is (i) a real y such that p p y q halts later than γ , or (ii)for all reals y such that p p y q halts, it does not halt within r β, γ q . This statement holds,since its negation implies that any interval r β, γ q strictly below ω of length α containsthe decision time of a program. Since L σ ă Σ L by Lemma 2.2, such an interval existsbelow σ . (cid:3) MERLIN CARL, PHILIPP SCHLICHT, AND PHILIP WELCH
Note that we similarly obtain gaps below τ of any length α ă τ by replacing σ by σ α .4. Semidecision times
In this section, we shall determine the supremum of the countable semidecision times.We then study semidecision times of singletons and their complements and show thatundecidable singletons of this form exists.4.1.
The supremum of countable semidecision times.
We will need the followingauxiliary result.
Lemma 4.1.
The supremum of Π L ω -definable ordinals equals τ . Proof.
Suppose that α ˚ ă τ is Σ L ω -definable by ψ p u q “ D v @ w ϕ p u, v, w q . Let γ ˚ denotethe unique ordinal γ such that for some α ă β ă γ :(a) @ w ϕ p α, β, w q .(b) p α, β q is the lexically least pair p ¯ α, ¯ β q with ¯ α, ¯ β ă γ and L γ |ù @ w ϕ p ¯ α, ¯ β, w q .(c) γ is the least ¯ γ ą α, β such that (b) holds for ¯ γ in place of γ .Note that (a) determines α uniquely. Given α , (b) and (c) determine β and γ uniquely.Let π : ω ˆ ω Ñ ω denote Gödel’s pairing function and η ˚ “ π p α ˚ , γ ˚ q ě α ˚ . Since π is ∆ L ω -definable, η ˚ is Π L ω -definable as required. (cid:3) We will use the effective boundedness theorem:
Lemma 4.2 (folklore) . The rank of any Σ p x q wellfounded relation is strictly below ω ck ,x .In particular, any Σ p y q subset A of WO is bounded by ω ck ,y .We quickly sketch the proof for the reader. The proof of the Kunen-Martin theoremin [Kec12, Theorem 31.1] shows that the rank of R is bounded by that of a computablewellfounded relation S on ω . Since L ω ck ,x is x -admissible, the calculation of the rank of S takes place in L ω ck ,x and hence the rank is strictly less than ω ck ,x .While the second claim follows immediately from the first one, we give an alternativeproof without use of the Kunen-Martin theorem for the reader. Fix a computable enu-meration ~p “ x p n | n P ω y of all Turing programs. Let N denote the set of n P N suchthat p yn is total and the set decided by p n codes an ordinal. By standard facts in effectivedescriptive set theory (for instance the Spector-Gandy theorem [Spe60, Gan60], see also[Hjo10, Theorem 5.3]), N is Π p y q -complete. In particular, it is not Σ p y q . Towards acontradiction, suppose that A is unbounded below ω ck ,y . Then n P N if and only if thereexist a decided by p n , a linear order b coded by a and some c P A such that b embeds into c . Then N is Σ p y q . Theorem 4.3.
The supremum of countable semidecision times equals τ . Proof.
To see that τ is a strict upper bound, note that the statement there is a countableupper bound for the decision time is Σ L ω . Moreover, it is easy to see that any true Σ L ω -statement has a witness in L τ .It remains to show that the set of semidecision times is unbounded below τ . In thefollowing proof, we call an ordinal β an α -index if β ą α and some Σ L ω fact withparameters in α Y t α u first becomes true in L β . Thus σ α is the supremum of α -indices.Suppose that ν is Π L ω -definable. (There are unboundedly many such ν below τ byLemma 4.1). Fix a Π -formula ϕ p u q defining ν . We will define a Π subset A “ A ν of WO . A will be bounded, since for all x P A , α x will be a ¯ ν -index for some ¯ ν ď ν andhence α x ă σ ν . ECISION TIMES OF INFINITE COMPUTATIONS 7
For each x P WO , let ν x denote the least ordinal ¯ ν ă α x with L α x |ù ϕ p ¯ ν q , if this exists.Let ψ p u q state that ν u exists and α u is a ν u -index . Let A denote the set of x P WO whichsatisfy ψ p x q . Clearly A is Π . Claim.
The decision time of A ν equals σ ν . Furthermore, for any ittm that semidecides A ν , the order type of the set of halting times for real inputs is at least σ ν . Proof.
The definition of A ν yields an algorithm to semidecide A ν in time σ ν . Now supposethat for some γ ă σ ν , there is an ittm-program p that semidecides A with decision time γ . Let g be Col p ω, γ q -generic over L σ ν and x g P L σ ν r g s a real coding g . A is Σ p x g q , since x P A holds if and only if there is a halting computation p p x q of length at most γ . ByLemma 4.2, A is bounded by ω ck ,x g . By the definition of A , it is unbounded in σ ν . We nowshow that ω ck ,y ă σ ν , contradicting the previous facts. We can take g to be sufficientlygeneric so that L σ ν r g s is admissible. Since σ ν is a limit of admissibles and g is set genericover L σ ν , σ ν is a limit of x g -admissibles. Hence ω ck ,y ă σ ν .For the second claim, construct a strictly increasing sequence of halting times of length σ ν by Σ -recursion in L σ ν . It is unbounded in σ ν by the first claim, hence its length is σ ν . (cid:3) This proves Theorem 4.3. (cid:3)
Semirecognisable reals.
We have essentially already done the calculation of thesupremum of semidecision times of reals. The upper bound follows from Lemma 2.2 andthe lower bound from Lemma 3.2.
Theorem 4.4.
The supremum of semidecision times of reals equals σ .To see that this is of interest, note that semirecognisable, but not recognisable singletonsexist by [Car19, Theorem 4.5.4]. In fact, we obtain the following stronger result: Theorem 4.5.
Suppose that α is countable in L and x “ x α .(1) (Cf. [CS17, Lemma 11]) No real in L Σ x z L λ x is recognisable. (2) No real in L ζ x z L λ x is semirecognisable.(3) All reals in L Σ z L ζ are semirecognisable. Proof. (1) Since L λ x “ L λ x r x s , it suffices to show that any recognisable real y P L Σ x iswritable from x . To see this, suppose that p recognises y . We run a universal ittm q withoracle x and run p p z q on each tape contents z produced by q . Once p is successful, wehave found y and shall write it on the output tape.(2) Since L λ x “ L λ x r x s , it suffices to show that every semirecognisable real y P L ζ x is writable from x . Suppose that p eventually writes y from x and q semirecognises y .We run q and in parallel p p z q , where z is the current content of the output tape of q .Whenever this changes, the run of p p z q is restarted. When p p z q halts, output z and halt.To see that this algorithm writes y , note that the output of q eventually stabilises at y , so p p y q is run and y is output when this halts.(3) Let x P L Σ z L ζ . Note that we can write a code for ζ from x . To see this, accidentallywrite codes for L -levels and write a code for α as soon as x P L α . Since α is writablefrom x , ζ is as well. We therefore have λ x ą ζ . Thus λ x ą Σ by the proof of [HL02,Theorem 3.11] or [CS17, Lemma 13]. Take a program p p x q that writes the L -least codefor L Σ and let n denote the place where x appears in this code. The following algorithmsemirecognises x . For an input y , run p p y q . If p p y q halts with output z , we can decide By [Mat15, Theorem 10.17], it suffices that the generic filter meets every dense class that is a unionof a Σ -definable with a Π -definable class. The proofs show that the real is not recognisable from x in (1) and not semirecognisable from x in(2). MERLIN CARL, PHILIPP SCHLICHT, AND PHILIP WELCH whether y equals the L -least code for L Σ . Diverge if they are different. If z equals thiscode, we obtain x in its n th place. If x “ y converge, and diverge otherwise. (cid:3) Note that Proposition 4.5 holds relative to oracles. In particular, the same patternoccurs for λ Σ , ζ Σ and Σ Σ , where λ α is defined as λ x α , with similar definitions of ζ α and Σ α . Moreover, x α is semirecognisable if Σ ď α ă λ Σ by virtually the same argument asfor Proposition 4.5 (3), but we do not know if this is the case for all reals in L λ Σ z L Σ .4.3. Cosemirecognisable reals.
Here we study semidecision times for the complementsof cosemirecognisable reals. We shall call them cosemidecision times . To see that anycountable cosemidecision time of a program p is strictly below σ , note that the Σ -statement there is a real x and a countable ordinal α such that for all y ‰ x , P p y qÓ ď α reflects to L σ .We now determine the supremum of cosemidecision times of reals, i.e. of singletons. Tothis end, we shall need an analogue to Lemma 3.2. Lemma 4.6. If p cosemirecognises x and p p x q has an endless loop ending at or beforetime α , then x P L α ‘ . Proof.
Let M “ L α ‘ and take any Col p ω, α q -generic filter g P V over M . Since Col p ω, α q is a set forcing in M , M r g s is admissible. Let y P M r g s be a real coding g . Then M r g s “ M r y s “ L ω y r y s . As in the proof of Lemma 3.2, it suffices to show that x P M r g s .To see this, note that α is countable in M r y s and hence t x u is ∆ p y q by the assumptionof the lemma. Therefore x P L ω y r y s “ L α ‘ r y s “ M r y s by [BGM71] (see [Hjo10, Section5]). (cid:3) Theorem 4.7.
The supremum of cosemidecision times of reals equals σ . Proof.
We first show that σ is an upper bound. Suppose that p cosemirecognises x . Wedefine a program r by simultaneously running p and the following program q , and halt assoon as p or q halts. The definition of q p y q is based on the machine considered in [FW07],which writes the Σ p y q -theories of J α r y s in its output, successively for α . Note that the Σ p y q -theory of J α r y s appears in step ω ¨ p α ` q . (This is the reason for the choice ofthis specific program.) q p y q searches for two writable reals relative to y , a real z and areal coding an ordinal α such that p p z q has an endless loop that ends at or before α . Notethat such a loop occurs by time Σ z , if it occurs at all (see [Wel00, Main Proposition] or[Wel09, Lemma 2]).Clearly r cosemirecognises x . It suffices to show that its semidecision time is at most p Σ x q ‘ ă σ . To this end, we consider two cases. If λ y ď Σ x , then p p y q halts at time Σ x or before. Now suppose that λ y ą Σ x . By Lemma 4.6, x P L p Σ x q ‘ . By the definition of q , the statement there exists x such that p p x q loops endlessly before α appears in q p y q instrictly less than p Σ x q ‘ steps.It remains to show that σ is a lower bound. Towards a contradiction, suppose that β ă σ is a strict upper bound for the cosemidecision times. We can assume that x β isrecognisable, for instance by taking β to be an index. Suppose that x β is recognised byan algorithm with decision time α . Note that α ă σ by Theorem 3.3. Take γ ě p α ` β q ‘ such that x γ is recognisable. Then x γ ‘ x β is recognisable.We claim that x β ‘ x γ is not cosemirecognisable by an algorithm with decision timestrictly less than β . So suppose that p such an algorithm. We shall describe an algorithm q that semirecognises x γ in at most α ` β ` steps. This contradicts Lemma 3.2. Notethat x β is coded in x γ by a natural number n . The algorithm q extracts the real x codedby n from the input y . It then decides whether x “ x β , taking at most α steps, anddiverges if x ‰ x β . If x “ x β , we run p p x ‘ y q for β steps and let q halt if and only if p p x ‘ y q fails to halt before time β . Then q p y q halts if and only if y “ x γ . (cid:3) ECISION TIMES OF INFINITE COMPUTATIONS 9
Given Theorem 3.3, the previous result is only interesting if cosemirecognisable, butnot recognisable reals exists. We now prove the existence of such reals, answering [Car19,Question 4.5.5].
Lemma 4.8. If x is cosemirecognisable, then x P L σ . Proof.
Suppose that p cosemirecognizes x . Then x is the unique y such that p p y q loops.The statement that p p y q loops for some y is a true Σ -statement and it therefore holds in L σ . By uniqueness, x P L σ . (cid:3) Theorem 4.9.
The cosemirecognisable, but not recognisable, reals are cofinal in L σ . Proof.
Let ξ be an index, x “ x ξ and y “ x λ x . Since y P L ζ x z L λ x , it is not semirecognisableby Lemma 4.5 (2). We claim that y is cosemirecognisable. We shall assume that λ x “ λ ;the general case is similar.For an input z , first test whether it fails to be a code for a wellfounded L α ; if it issuch a code, then check if it has an initial segment which itself has a Σ -substructure(equivalently α ě Σ ); if it fails this test, then check if it has a Σ substructure (if it doesthen α ‰ λ ). At this point, z is a code for an L α with α ď λ . Check if z fails to bethe L -least code for L α . Furthermore, run a universal machine and check whether someprogram halts beyond α . (cid:3) Sets with countable decision time
It is easy to see that any set with countable semidecision time α is Σ in any code for α . Similarly, any set with a countable decision time is Borel. The next result shows thatboth converse implications fail. This also shows that the condition in Lemma 3.1 that A is Σ is not necessary. Theorem 5.1.
There is a cocountable open decidable set A that is not semidecidable incountable time. Proof.
Let ~ϕ “ x ϕ n | n P ω y be a computable enumeration of all Σ -formulas with onefree variable. Let B denote the discrete set of all n a x y a x , where x is the L -least codefor the least L α where ϕ n p x q holds. Let further p denote an algorithm that semidecides B as follows. First test if the input equals ω and halt in this case. Otherwise, test if theinput is of the form n a x y a x , run a wellfoundedness test for x , which takes at least α steps for codes for L α , and then test whether α is least such that ϕ n p x q holds in L α . Thedecision time of p is at least σ . Moreover, it is countable since B is countable.Let A denote the complement of B . Towards a contradiction, suppose that q semide-cides A in countable time. Let r be the decision algorithm for B that runs p and q simultaneously. Then r has a countable decision time α and by Σ -reflection, we have α ă σ . But this is clearly false, since p ’s decision time is at least σ . (cid:3) Open problems
It is natural to ask about the suprema of decision and semidecision times for sets ofnatural numbers instead of reals; it is easy to see that these equal λ , the supremum ofclockable ordinals. We would further like to stress that the results in this paper aboutsets of reals also hold for Turing machines with ordinal time and tape with virtually thesame proofs, while the results about singletons do not.In the main results, we determined the suprema of various decision times, but we havenot characterised the underlying sets. Question 6.1.
How are sets and singletons with countable decision, semidecision andcosemidecision times characterised precisely in the L -hierarchy? I.e. their L -ranks are cofinal in σ . It is natural to ask what are the optimal results regarding the interplay of recognisingspeed, writing speed and L -levels in Lemma 3.2 and Section 3.2. In particular: Question 6.2. If x is semirecognisable in time α , is x P L ω ¨ α ? What is the optimal L -level?Regarding Section 3.3, we know from [HL00, Theorem 8.8] that admissible ordinals arenever clockable and from [Wel09, Theorem 50] that any ordinal that begins a gap in theclockable ordinals is always admissible. We ask if an analogous result holds for decisiontimes. Question 6.3.
Is an ordinal that begins a gap of the (semi-)decision times always admis-sible?We would like to draw a further connection with classical results in descriptive settheory. A set of reals is called thin if it does not have an uncountable closed subset. It isnot hard to show that t x | x P L λ x u is the largest thin semidecidable set. We ask if thesame characterisation holds for eventually semidecidable sets. Question 6.4. Is t x | x P L λ x u the largest thin eventually semidecidable set?In particular, is every eventually semidecidable singleton an element of L λ x (equivalently L Σ x )? An indication that these statements might be true is that one can show an analogousstatement for null sets instead of countable sets: the largest ittm-semidecidable null setequals the largest ittm-eventually semidecidable null set. Related to Theorem 5.1, it isnatural to ask whether a thin semidecidable set can have halting times unbounded in ω . References [BGM71] J. Barwise, R.O. Gandy, and Y.N. Moschovakis. The Next Admissible Set.
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