Some results on $mathbb{R}$-computable structures

SOME RESUL TS ON R -COM PUT ABLE STRUCTU RES W. CAL VER T AND J. E. POR TER Contents 1. Int ro duction 1 1.1. Basic Definitions 2 1.2. Plan o f the Pap er 4 2. Basic Results 4 2.1. R -computable Ordinals 4 2.2. The Complexity of Satisfaction 5 2.3. F orcing as a Constr uction T ec hnique 7 3. Effective Categor ic it y 7 4. Geometry and T op ology 8 4.1. Classifying Compact 2-Manifolds 9 4.2. Computing Homotopy Groups 9 5. Relations with Other Mo dels 10 5.1. Lo c a l Computability 10 5.2. Σ-Definabilit y 13 5.3. F-Parameterizability 14 6. Conclusion 15 References 15 1. Introduction The theor y of effectiveness pr oper ties o n co un table structures who se ato mic di- agrams are T uring computable is well-studied (se e , for insta nce, [1, 1 4]). Typical results describ e which structures in v arious classes are computable (or hav e iso- morphic copies that are ) [18], o r the p otential degre e of unsolv abilit y of v ar ious definable subsets of the struc tur e [1 5]. The goal o f the pres en t pap er is to survey some initial r esults inv estigating similar c o ncerns o n s tr uctures which are effective in a differen t sense. A r ather severe limitation of the T uring model of computability is its traditional restriction to the co untable. Of course , man y succes sful ge neralizations hav e b e e n made (see, for instance, [27, 11, 1 2, 22, 2 3, 25] and the o ther pap ers in the present volume). The genera liz a tion that will b e treated here is bas ed on the observ a tion that while there is obviously no T uring machine for a ddition a nd multiplication of re al num bers, there is s trong intuition tha t these oper ations a re “co mputable.” The BSS mo del of co mputation, fir st in tro duced in [4], approximately takes this to b e the definition o f computatio n on a given r ing (a mor e formal definition is The first author is grateful for the support of Gran t #13397 from the T empleton F oundation. 1 2 W. CAL VER T AND J. E. POR TER forthcoming). This allows several problems of computation in n umerical analysis and co n tin uous geometr y to b e tr eated rig orously . The monogra ph [3] gives the examples of the “decision problem” of the p oin ts for which Newton’s metho d will conv erge to a ro ot, and determining whether a given p o in t is in the Mandelbrot set. 1.1. Basic Definitio ns. The definition of a BSS machine comes from [3]. Suc h a ma c hine should be thought of as the analog ue o f a T uring mach ine (indeed, the t wo notions co incide where R = Z ). Let R be a ring with 1. Let R ∞ be the set o f finite sequences of elements from R , and R ∞ the bi-infinite direct sum M i ∈ Z R. Definition 1.1. A machine M ov er R is a finite co nnected directed gr aph, co n- taining five types o f no des: input, computation, branch, shift, and output, with the following prop erties: (1) The unique input no de has no inco ming edge s and only one outgoing edge. (2) Each co mputation and shift no de has exac tly one output edge and p ossibly several input bra nc hes. (3) Each output no de has no output edges and p ossibly several input edge s. (4) Each branch no de η ha s exactly tw o o utput edg es (lab eled 0 η and 1 η ) and po ssibly several input edges. (5) Asso ciated with the input node is a linear map g I : R ∞ → R ∞ . (6) Asso ciated with each computation no de η is a r ational function g η : R ∞ → R ∞ . (7) Asso ciated with ea ch br a nc h no de η is a p olynomial function h η : R ∞ → R . (8) Asso ciated with each shift no de is a map σ η ∈ { σ l , σ r } , where σ l ( x ) i = x i +1 and σ r ( x ) i = x i − 1 . (9) Asso ciated with ea c h output no de η is a linea r map O η : R ∞ → R ∞ . A machine may b e under stoo d to compute a function in the following wa y: Definition 1.2. Let M b e a ma c hine ov er R . (1) A p ath through M is a s equence of node s ( η i ) n i =0 where η 0 is the input no de, η n is an output no de, and for each i , we hav e an edg e fro m η i to η i +1 . (2) A c omputat ion on M is a se quence o f pairs (( η i , x i )) n i =0 with a n um ber x n +1 , where ( η i ) n i =0 is a path thro ugh M , where x 0 ∈ R ∞ , and where, for each i , the following hold: (a) If η i is an input no de, x i +1 = g I ( x i ). (b) If η i is a computation no de, x i +1 = g η i ( x i ). (c) If η i is a branch no de, x i +1 = x i and η i +1 determined b y h η i so that if h η i ( x i ) ≥ 0, then η i +1 is connected to η i by 1 η i and if h η i ( x i ) < 0, then η i +1 is connected to η i by 0 η i . (Note that in all other c a ses, η i +1 is uniquely determined b y the definition of path.) (d) If η i is a shift no de, x i +1 = σ η i ( x i ) (e) If η i is an output no de, x i +1 = O η i ( x i ). The pro of of the following lemma is an obvious fro m the definitions. Lemma 1 . 3. Given a machine M and an element z ∈ R ∞ , ther e is at most one c omputation on M with x 0 = z . SOME RES UL TS ON R -COMPUT ABLE S TR UCTURES 3 Definition 1.4 . The function ϕ M : R ∞ → R ∞ is defined in the following wa y: F or each z ∈ R ∞ , let ϕ M ( z ) b e x n +1 , wher e ((( η i , x i )) n i =0 , x n +1 ) is the unique computation, if any , where x 0 = z . If there is no such computation, then ϕ M is undefined on z . Since a machine is a finite ob ject, in volving finitely many real num be r s as pa- rameters, it may b e c o ded by a member of R ∞ . Definition 1.5. If σ is a co de for M , we define ϕ σ = ϕ M . W e can now say that a set is computable if and only if its characteris tic function is ϕ M for some M . Example 1 .6. Let R = Z . Now the R -computable functions are exactly the classical T uring - computable functions. Example 1. 7. Let R = R . Then the Mandelbrot s et is not R -c omputable (see Chapter 2 of [3 ]). Definition 1 .8. A machine over R with or acle X is exa ctly like a machine over R , except tha t it has an additional type o f no des, the o r acle no des. Ea c h or acle no de is exa ctly like a computation node, except that g η = χ X . Co mputations in oracle machines are de fined in the o bvious way . W e say that a set S is decidable (resp ectiv ely , X -de c idable) ov er R if and only if S is b oth the halting set of an R -machine (resp ectiv ely , with o r acle X ) and the complement of the halting set o f an R -ma c hine (resp ectively , with oracle X ). W e also say that S is semi-decidable if and only if S is the domain of an R -computable function (if R is a real closed field, it is equiv alent to s a y that S is the ra nge o f an R -computable function [3]). Ziegler [29] gives a sp ecialized but recent sur v ey of results on R -co mputatio n. The following result, fir st presented by Michaux, but prov ed in detail in [9], is useful in c haracter izing the decidable and s emi-decidable sets: Prop osition 1.9. L et S ⊆ R ∞ . Then S is semi-de cidable if and only if S is the u nion of a c ountable family of semialgebr aic sets define d over a single finitely gener ate d ext ension of Q . The “ o nly if ” part of this statement is the upshot o f an earlier theor em describ ed in [3], called the Path Deco mpositio n Theorem. W e can now pro ceed to define computable structures. Definition 1.10 . Let L = ( { P i } i ∈ I P , { f i } i ∈ I f , { c i } i ∈ I C ) be a language with relation symbols { P i } i ∈ I P , function symbols { f i } i ∈ I f , and constant symbols { c i } i ∈ I C . Let A be an L -structur e with universe A ⊆ R ∞ . (1) W e say that L is R -computable if the sets of relations, functions, and con- stants are ea c h decidable ov er R , and if, in addition, there are R -ma c hines which will tell, given P i (resp ectiv ely , f i ), the arity of P i (resp ectiv ely , f i ). (2) W e identify A with its atomic diagr am; in particular , (3) W e s ay that A is co mputable if and only if the atomic diagr am of A is decidable. The obstructions to a dire ct pa rallel be tween the theor y of R -comptuable str uc- tures a nd that of T uring co mputable structures which we have enco un tered so far are tw o in n um ber (one fo r the parsimonious): 4 W. CAL VER T AND J . E. POR TER (1) The r e al num b ers do not admit an ω -like w ell-ordering to fa cilitate sear c hing or priority constructions , and in particular (2) There exist R - computable injective functions who se inv erses ar e not R - computable. 1.2. Plan of the P ap er. In the pr e sen t pap er, we will survey recent work on the theory of R -c omputable s tructures. In Sectio n 2, we give some basic calculatio ns, showing some para llels with the classical theory , including computable ordina ls (Section 2.1), satisfaction of co mputable infinitar y fo rm ulas (Section 2.2 ), and the use o f forcing to ca rry out a simple prio rit y construc tio n (Section 2.3). In Section 3, we explor e effective categoricity , using vector spaces as an example. In Section 4, we describ e some r ecen t results in effective geo metr y and top ology from the p ersp ectiv e of R -computation. In Section 5 we address the r elationship of R -co mputation with other mo dels of effectiv e mathematics for uncountable structur es. In Sectio n 6 we summarize the state o f R -computable mo del theory a nd describ e some directions for future resear c h. 2. Basic Resul ts 2.1. R -computable Ordinals . The T uring co mputable ordina ls constitute a prop er initial segment of the countable ordina ls [28, 19]. This initial s egmen t includes , for instance, the ordinal ω ω ω . . . . In the present sectio n, we will esta blish the following theorem: Theorem 2.1. A wel l-or dering ( L, < ) has an isomorphic c opy which is R -c omputable if and only if L is c ountable. Prop osition 2.2. Every c ountable wel l-or dering ( M , ≺ ) has an isomorphic c opy ( L, < ) which is R -c omputable. Pr o of. Since ( M , ≺ ) is co un table, it has a n isomorphic copy with universe ω . Now D ( M ) = { ( a, b ) ∈ M 2 | a ≺ b } is a subset of ω 2 . Now we define a r eal num ber ℓ in the following wa y : ℓ = X i ∈ ω 10 − i χ D ( M ) ( i ) . There is a R machine which, given a pair ( a, b ) ∈ ω 2 will re tur n the 10 −h a,b i place of ℓ if tha t place is 1 and will diverge if that place is 0. This shows that D ( M ) is the halting set of a R -computable function, a s required.  Prop osition 2.3. Supp ose ( L, < ) is a R -c omput able wel l-or dering. Then | L | ≤ ℵ 0 . Pr o of. Since ( L, < ) is R -computable, the set L < := { ( a, b ) ⊆ L 2 : a < b } is the halting set of a R -machine. By Path Decomp osition, it must b e a disjoint union of s emialgebraic sets, and conse quen tly Bo rel. By the Kunen-Martin Theorem (Theorem 31 .5 of [17]), a nalytic (and hence Bo r el) well-orderings are count able.  A rather differen t pro of o f Pr opo s ition 2.3, using F ubini’s Theor em, is p ossible and enlig h tening. Pr o of. Since L < is uncountable and Borel, | L < | = 2 ℵ 0 . This implies L is Bo rel with | L | = 2 ℵ 0 . Without loss o f generality , we supp ose that L is or der isomorphic to the cardinal 2 ℵ o ; o therwise, an initia l segment of L which was is o morphic to SOME RES UL TS ON R -COMPUT ABLE S TR UCTURES 5 this cardinal would also b e R -co mputable. In par ticular, L contains a Cantor set C . Fix a Bo rel measure µ on C such that µ ( C ) = 1 and extend µ to L by setting µ ( L \ C ) = 0. W e define tw o a uxiliary sets: L x = { b ∈ L : ( x, b ) ∈ L < } L y = { a ∈ L : ( a, y ) ∈ L < } Each of these is a Borel s et. F o r any y , we hav e | L y | < 2 ℵ 0 , since 2 ℵ 0 is a ca r dinal and L y is isomorphic to a n or dina l less than 2 ℵ 0 . Since L y is Borel, w e hav e | L y | = ℵ 0 . This implies the set L y is co -countable for a n y y . Since L < is B o rel, we can apply F ubini’s theorem to calc ula te R L < 1 dλ , wher e λ is the product mea sure µ × µ . On the one hand, Z L < 1 dλ = Z L Z L x 1( dµ )( dµ ) = Z L µ ( L x ) dµ = Z L 1 dµ = µ ( L ) = 1 since L x is co-countable for each x , and th us o f full measur e . On the other hand, since L y is countable for each y , we hav e R L y 1 dy = 0. Hence Z L < 1 dλ = Z L Z L y 1( dµ )( dµ ) = Z L µ ( L y ) dµ = Z L 0 dµ = 0 , which is a contradiction.  2.2. The Com plexit y of Satisfaction. W e define the clas s of R -co mputable in- finitary formulas. The definition is by analogy with the (T uring) computable infini- tary formulas already in broad usag e, describ ed in [1]. The choice of computable infinitary for mulas is nontrivial, s ince there are uncount ably many R -ma chines. One natural approach, not pursued her e, would be to work in the R -computable frag- men t of L (2 ℵ 0 ) + ,ω . T his w ould certainly b e an interesting lo g ic to under stand, but the present author s fo und it more des irable a t first to understand the more familiar R -computable frag men t of L ω 1 ω . A t issue is which conjunctions and disjunctions are allowed in a “co mputable” formula. The lo gic L ω 1 ω allows co un table conjunc- tions and dis junctions , while L (2 ℵ 0 ) + ,ω allows any of size at most 2 ℵ 0 . How ever, the difficult y of desc ribing what is meant by , for insta nce , an interv al of fo r m ulas is a motiv ation (beyond the av o idance of set-theor etic independence) to consider fir st the countably long formulas. Definition 2.4. Let L b e an R -co mputable langua ge. (1) The Σ 0 formulas of L are exactly the finita r y qua n tifier-free for m ulas. The Π 0 formulas are the s a me. (2) F or any or dinal α = β + 1, the Σ 0 α formulas are those of the form _ i ∈ S _ ∃ ¯ y [ ϕ i ( ¯ x ¯ y )] where S is countable and is the halting set of an R -machine, and there is a finitely genera ted field F ⊂ Q such that all para meters in φ i are in F . (3) F or any or dinal α = β + 1, the Π 0 α formulas are those of the form ^ i ∈ S ^ ∀ ¯ y [ ϕ i ( ¯ x ¯ y )] 6 W. CAL VER T AND J . E. POR TER where S is countable and is the halting set of an R -machine, and there is a finitely genera ted field F ⊂ Q such that all para meters in φ i are in F . (4) Suppo s e α = lim n β n where β n is a b ounded R -computable sequence o f ordi- nals, and there is a finitely gener ated field F ⊂ Q such that all par ameters in φ i are in F . (a) The Σ α formulas are those of the form _ n ∈ S _ ϕ n , where for eac h n the fo rm ula ϕ n is a Σ β n formula and S is coun table and is the halting set of a n R -ma c hine. (b) The Π α formulas are those of the form ^ n ∈ S ^ ϕ n , where for each n the form ula ϕ n is a Π β n formula and S is co un table and is the halting set of a n R -ma c hine. The R -computable infinitar y for m ulas will b e ex a ctly the formulas which b elong to either Σ α or Π α for some countable (i.e. R -computable) α . Ash showed that T uring computable Σ α formulas defined sets which were Σ 0 α [1]. W e will say that a set is semantic al ly R - Σ α if a nd only if it is the set o f solutions to an R -computable Σ α formula, a nd similar ly for Π α . W e will say that a set is top olo gic al ly Σ 0 α if it is o f that level in the standar d Borel hierarchy using the order top ology on R . Theorem 2. 5. We char acterize the t op olo gic al structu re of s ets in t he semantic hier ar chy: (1) The semantic al ly R - Σ 0 sets ar e t op olo gic al ly ∆ 0 2 . (2) If 0 < α < ω , then the s emant ic al ly R - Σ α sets ar e include d among t he top olo gic al ly Σ 0 α +1 sets. (3) If α ≥ ω , then the semantic al ly R - Σ α sets ar e include d among the top olo g- ic al ly Σ 0 α sets. Pr o of. Since A is a R - computable structure, the semantically R - Σ 0 sets ar e all countable unions of semialgebraic sets, and the completes of s e ma n tically R -Σ 0 are all countable unions of semia lgebraic s e ts. Since all semialg ebraic sets a r e top ologically ∆ 0 2 (that is, b oth top ologica lly Σ 0 2 and Π 0 2 ), the co un table unions of them are all top ologic ally Σ 0 2 . Now if the statement holds for n ≤ k , it clearly holds for n = k + 1 by the definitions of the v ar ious cla sses inv olved. T ow ard the final statement , notice that the semantically R -Σ ω sets a re countable unions of se ts at low er lev els, and are all top ologically Σ ω . Ab ov e that level, the induction follows exactly as b efore.  A t the finite levels, Cuc ker proved [9] that the union of all the s eman tically R - Σ n for n < ω is the clas s o f Borel sets of finite o r der. Cuck er [9] defined another arithmetical hier arch y: we call a set c omputational ly Σ α +1 if it ca n be enumerated by a real machine with a computationally Σ α oracle. In particular, the semi- decidable s ets a re the computationally Σ 1 sets. Cuc ker pr oved that for all k < ω , the co mputationally Σ k sets are exactly the sema ntically R - Σ k sets. It seems likely SOME RES UL TS ON R -COMPUT ABLE S TR UCTURES 7 that this res ult could b e g eneralized for transfinite α , but w e do not hav e a pro of of this. 2.3. F orcing as a Construction T ec hni que. Aside from the lack of in verse func- tions, the most difficult part of class ical computability theor y to g et by without is the prior it y construction. Unless this niche ca n b e filled, we are not optimistic concerning the pa rallel b etw een T uring-computable structur es and R -computable structures. Co nsequen tly , although there are a d-hoc metho ds to co nstruct R - incomparable sets [21], we give an example in this section that is p otent ially more easily g eneralized. Prop osition 2.6. The r e exist sets A o and A 1 such that n either is c omputable by a R - machine using the other as an or acle. Pr o of. The pro of will clo sely fo llow the sec o nd pro of g iv en for the classical case b y Lerman [20]. Let ( F , ≤ ) b e the set of pairs of pa r tial functions fro m R to 2 whose complement co n tains an interv al, partially ordered by extensio n in the sense that ( p 0 , p 1 ) ≤ ( q 0 , q 1 ) if a nd only if p i extends q i for each i . It suffices to s atisfy the following requirements for every e ∈ R ∞ : P e,i : M A i e 6 = A 1 − i . W e say that ( p 0 , p 1 )  P e,i if there is s o me x such that either M p i e ( x ) ↓6 = p 1 − i ( x ) and the latter is defined, o r for a ny p extending P i , we hav e M p e ↑ . Lemma 2.7 (Density Lemma) . F or any p air ( e, i ) , the set { p ∈ F : p  P e,i } is dense. Pr o of. Let q = ( q 0 , q 1 ) ∈ F . W e will show that there is some p ≤ q such that p  P e,i . Let x be outside the domain of q 1 − i . If there is no pair s, r such that r extends q i and M r e,s ( x ) ↓ , then q  P e,i , so ass ume that such an s exists. No w we extend q 1 − i by setting p 1 − i = q 1 − i ∪ { ( x, 1 − M q i e ( x )) } , and set p i = q i .  The following Lemma is the only part of the constr uction whic h b ecomes gen- uinely mo r e difficult in the uncountable ca se. Lemma 2.8 (Existence of a Generic) . L et C b e the c ol le ct ion of al l sets of the form { p ∈ F : p  P e,i } . Ther e ex ists a C -generic set; that is, a p air of functions G = ( G 0 , G 1 ) wher e G i : R → 2 such that for e ach p air ( e, i ) , the function G extends s ome element of F which for c es P e,i . Pr o of. Let G 0 := ( ∅ , ∅ ), and well-order the requirements. W e define G α +1 to b e the extension of G α which forces the α th requirement. F or limit ordinals γ , we define G γ := S β <γ G β . The union of a ll the G α is a C -generic.  Of co urse, we may take G to b e total, by setting a ll undefined v alues to 0. Now we take A 0 to b e the set whose characteristic function is G 0 and A 1 the set with characteristic function G 1 .  3. Effective Ca tegoricity It is o ften p ossible to pro duce tw o classically co mputable structures which are isomorphic, but for which the is omorphism is not witnessed by a computable func- tion. Any theory of effective mathematics must ta k e ac coun t of this phenomenon. 8 W. CAL VER T AND J . E. POR TER Definition 3. 1. A co mputable str uc tur e M is said to b e c omputably c ate goric al if and only if for any computable s tructure N ≃ M ther e is a computable function f : N ≃ → M . The n um ber of eq uiv alence clas ses under computable isomo r phism contained in a n iso morphism type is called its c omputable dimensio n . In the present section, w e describ e progr ess tow a r d a par allel to the following classical result: Theorem 3 .2 (see [26], altho ugh it was almost cer tainly k no wn ea rlier) . If V is a c ount able ve ctor sp ac e over Q , (1) Ther e is a T uring-c omput able c opy of V , (2) The c ate goricity pr op erties ar e as fol lows: (a) If dim ( V ) is fin ite, then V is c omputably c ate goric al, and (b) If dim ( V ) = ω , then t he c omputable dimension of V is ω . The existence part of the theor em is still true without serious mo dification. Prop osition 3. 3. L et n ∈ ℵ 0 ∪ {ℵ 0 , 2 ℵ 0 } . Then ther e is a R -c omputable ve ctor sp ac e V n of dimension n . F urther, V n has a R -c omputable b asis. Pr o of. Consider the language of real v ector spaces (addition, plus one scaling op- eration for ea c h element o f R ). Let { b i : i ∈ I } b e a R -computable set of constants, where | I | = n . The set of closed terms with constants from { b i : i ∈ I } , modulo prov able equiv alence (in the theo ry o f vector spaces) is a mo del of the theor y of vector spaces, and has dimension n .  Of cours e , the ca tegoricity result highlights an additional co nc e r n with R - comp- utation: It may happ en that ther e is a R - c omputable iso morphism with no R - computable inv erse. Thus, while the following res ult establishes, acco rding to Def- inition 3 .1, something very clo s e to par t 2a of Theor em 3.2, it falls short o f full analogy . Prop osition 3.4. L et n < ℵ 0 . Then for any r e al ve ctor sp ac e W of dimension n , ther e is a c omput able isomorphism f : V n → W . Pr o of. Let { a 1 , . . . , a n } b e a ba sis of W . Each member of V n is a R -linea r co mbi- nation n P i =1 λ i b i . W e map n P i =1 λ i b i to n P i =1 λ i a i .  The classical wa y to prov e part 2b of Theorem 3.2 is to pro duce a computable vector s pa ce with a co mputable basis, and an isomor phic (i.e. same dimension) vector space with no computable basis. Without recours e to priority constructio ns, this str ategy s e e ms, for the prese nt, very difficult in the R -co mputable context. 4. Geometr y and Topolo gy In the talk by the first a uthor at EMU 2008, an early slide asked for a con text in which one could form ulate effectiveness questions for results lik e Tho m’s Theo- rem o n co bordis m or the classificatio n o f co mpact 2-manifolds . So me work in the int ervening mo n ths, which b egan at that mee ting , has yielded interesting results in R -computable top ology . An n -manifold is a to p olog ical space which is lo cally homeomor phic to R n , sat- isfying some fair ly o b vious r egularity co nditions o n the intersections of the neigh- bo rho ods o n w hich homeomo r phism holds . The following definition is given in [8]. SOME RES UL TS ON R -COMPUT ABLE S TR UCTURES 9 Definition 4.1. A r e al-c omputable d -manifold M co nsists of rea l-computable i , j , j ′ , k , the inclus ion functions , satisfying the fo llo wing c o nditions for all m, n ∈ ω . • If i ( m, n ) ↓ = 1, then φ j ( m,n ) is a total r eal-computable homeo morphism from R d int o R d , and φ j ′ ( m,n ) = φ − 1 j ( m,n ) , and k ( m, n ) ↓ = k ( n, m ) ↓ = m . • If i ( m, n ) ↓ = 0, then k ( m, n ) ↓ = k ( n, m ) ↓∈ ω with i ( k ( m, n ) , m ) = i ( k ( m, n ) , n ) = 1 and for all p ∈ ω , if i ( p, m ) = i ( p, n ) = 1, then i ( p, k ( m, n )) = 1, and for all q ∈ ω , if i ( m, q ) = i ( n, q ) = 1, then i ( k ( m, n ) , q ) = 1 with range( φ j ( m,q ) ) ∩ r a nge( φ j ( n,q ) ) = range( φ j ( k ( m,n ) ,q ) ) . • If i ( m, n ) / ∈ { 0 , 1 } , then i ( m, n ) ↓ = i ( n, m ) ↓ = − 1 , and ( ∀ p ∈ ω )[ i ( p, m ) 6 = 1 or i ( p, n ) 6 = 1] , and for all q ∈ ω , if i ( m, q ) and i ( n, q ) b oth lie in { 0 , 1 } , then range( φ j ( k ( m,q ) ,q ) ) ∩ r a nge( φ j ( k ( n,q ) ,q ) ) = ∅ . • F or a ll q ∈ ω , if i ( m, n ) = i ( n, q ) = 1 , then i ( m, q ) = 1 and φ j ( n,q ) ◦ φ j ( m,n ) = φ j ( m,q ) . In essence, each na tural num ber m represents a chart U m . The functions i ( m, n ) tell whether U m is a subset of U n and whether U n is a subs e t o f U m . The function j ( m, n ) is the index for a computable map g iving the inclusion o f U m in U n . 4.1. Classifying Comp act 2-Manifolds . Classifica tion of n -manifolds up to home- omorphism in genera l is quite difficult. Ho wev er , a well-kno wn theory of disputed priority offers the following classifica tion of compact connected 2-manifolds . Theorem 4. 2. L et X b e a c omp act c onne cte d 2-manifold. Then X is home omor- phic to a c onne cte d sum of 2 -spher es, c opies of RP 2 , c opies of S 1 × S 1 , and c opies of the Klein Bottle. Unpublished work by the first autho r and Montalban, inspired in part by dis - cussions with R. Miller, gives an effective version of this result. Theorem 4.3 (Calvert–Montalban) . L et M and N b e R -c omputable c omp act 2 - manifolds. Then ther e is a R -c omputable home omorphism f : M → N . Pr o of outline. W e can triangula te each o f M a nd N to form a finite simplicial complex. The function f co nsists of a mapping on the complexes, with a smo othing effect.  Corollary 4.4. L et X b e a c omp act c onne cte d R -c omputable 2 -manifold. Then X is home omorphic by a R -c omputable function to a c onne cte d su m of 2 -spher es, c opies of RP 2 , c opies of S 1 × S 1 , and c opies of the K lein Bott le. 4.2. Computing Homotopy Groups . One standar d set of top ological inv a riants for a manifold M is the s equence of g roups ( π n ( M )) n ∈ ω , w he r e π n ( M ) is the gr oup of contin uous ma ppings fro m S n to M , up to homotopy equiv alence. Under the classical mo del of co mputation, manifolds ar e often repr e s en ted by simplicial com- plexes in order to discuss the p ossibility o f co mputing v arious top ological inv ar ian ts. Brown show e d [5] that there is a pro cedure whic h will, given a finite simplicial com- plex M , compute a set of gener ators and r elations for each of the g roups π n ( M ). It is na tural to ask, no w that we hav e a notion of co mputation tha t gives us algo- rithmic access to the manifolds themselves, whether this ca n be computed directly 10 W. CAL VE R T AND J. E. POR TER from the manifolds. W e restrict attention her e to the case of π 1 , studied in detail in [8], although it is likely that s imilar r e sults could b e establishe d for π n . Lemma 4.5 (Calvert–Miller [8]) . Every lo op f in a c omputable manifold M is homotopic to a c omputable lo op in M whose only r e al p ar ameters ar e the b ase p oint and t he inclusion functions ne c essary to define M . Nevertheless, the answer to the q uestion of c o mputing a fundamental gr oup from a ma nifold is lar gely negative: Theorem 4.6 (Calvert–Miller [8]) . L et M b e a R -c omputable manifold which is c onne cte d but not simply c onne cte d. Then ther e is no algorithm to de cide whether a given lo op is nul lhomotopic. Theorem 4. 7 (Calvert–Miller [8]) . Ther e is no R -c omputable function which wil l de cide, given a R - c omputable manifo ld, whether that manifold is simply c onne cte d. Nevertheless, there is a canonical family o f lo ops, sufficient to represent the whole (but not re c o verable by a uniform pro cedure) from which we could make the necessary co mputations for a fundamental gro up. Lemma 4.8 (Calvert–Miller [8]) . L et M b e a R -c omputable manifold. Then ther e is a R -c omput able function S M , define d on the n atu r als, su ch t hat t he set S M ( n ) c onsists of a set of indic es for lo ops and c ontains exactly one r epr esentative fr om e ach homotop y e qu ivale nc e typ e. While w e cannot effectiv ely pas s from an index for M to an index for S M , this step includes all of the difficulty in computing π 1 ( M ): Theorem 4.9 (Ca lv ert–Miller [8]) . L et M b e a R -c omput able manifold. Then ther e is a uniform pr o c e dur e to p ass fr om an index for S M to an index for a r e al- c omputable pr esentation of the gr oup π 1 ( M ) . 5. Rela tions with Other Models 5.1. Lo cal Com putabilit y. Let T b e a ∀ -ax iomatizable theory in a languag e w ith n sy m bols. Definition 5. 1. A simple c over of S is a (finite or countable) collection U of finitely generated mo dels A 0 , A 1 , ... of T , such that: - every finitely gener ated substr ucture of S is is omorphic to some A i ∈ U ; and - every A i ∈ U embeds isomorphically into S . A simple cov er U is c omputable if every A i ∈ U is a computable str ucture whose domain is a n initia l segment o f ω . U is uniformly c omputable if the sequence h ( A i , a i ) i i ∈ ω can be given uniformly: there must exist a computable function which, on input i , o utputs a tuple of elements h e 1 , ..., e n , h a 0 , ..., a k ii ∈ ω n × A <ω i such that { a 0 , ..., a k i } generates A i and φ e j computes the j -th function, relation, or c o nstan t in A i . SOME RES UL TS ON R -COMPUT ABLE S TR UCTURES 11 Definition 5.2 . An embedding f : A i ֒ → A j lifts to the inclusion B ⊂ C , via isomorphisms β : A i ։ B and γ : A j ։ C , if the diag ram b elow commut es: B − − − − → ⊆ C β x   ∼ = γ x   ∼ = A i f − − − − → A j with γ ◦ f = β A c over of S consists of a simple cov er U = {A 0 , A 1 , ... } of S , along with sets I U ij (for a ll A i , A j ∈ U ) of injective homo morphisms f : A i ֒ → A j , such that: (1) for all finitely gener a ted substruc tur es B ⊆ C of S , ther e exists i, j ∈ ω a nd an f ∈ I U ij which lifts to B ⊆ C via some iso mo rphisms β : A i ։ B and γ : A j ։ C ; and (2) for every i and j , every f ∈ I U ij lifts to an inclusio n B ⊆ C in S via some isomorphism β a nd γ . This cov er is uniformly c omputable if U is a unifor mly computable simple cov er of S and there exists a c.e. set W such that for all i, j ∈ ω I U ij = { φ e ↾ A i : h i, j, e i ∈ W } . A structure B is lo c al ly c omputable if it has a uniformly co mputable cov er. Prop osition 5. 3 ([22]) . A structur e S is lo cally computable if and only if it has a u niformly c omput able simple c over. Prop osition 5.4 ([22]) . The or der e d field of r e al numb ers is not lo c al ly c omputable. How ever, the order ed field o f rea l num bers is trivia lly R -c omputable. It app ears at fir st that the or dering might b e essential in escaping lo cal computability . Definition 5.5. A R -mac hine is said to b e e quational if and only if each bra nc h no de is decided b y a p olynomial e quation . W e call a structure e quational ly R - c omputable if its diag ram is computable by an e q uational R -machine. Lemma 5 . 6 (Path Decompo sition for Equationa l Machines) . L et M b e an e qua- tional R - m achi ne. Then the halting set of M is a c ountable disjoint union of alge- br aic sets. Pr o of. The pro of is exa ctly the same as for nor mal R -machines.  Corollary 5.7. The or der e d fi eld of r e al nu mb ers is not e quational ly R -c omputable. Theorem 5.8. Ther e is an e qu ationally R -c omputable structure which is not lo c al ly c omputable. Pr o of. Let S b e a noncomputable set of natur al n um ber s, and denote by C n a cyclic graph o n n vertices (i.e. an n -gon). Now let G b e the str ucture given by [ n ∈ S · C 2 n ! [ · [ n / ∈ S · C 2 n +1 ! . T o show tha t G is equationa lly R -computable, we obse rv e that the disjo in t union of t wo R -computable structure s is R -co mputable (since the same is true of the ca r- dinal sum). How ever, ea c h of the graphs C k has a R - computable copy by Lagr ange int erp olation. 12 W. CAL VE R T AND J. E. POR TER Suppo se f is a uniform computable enumeration of the finitely generated sub- structures of G . Then we could compute whether n ∈ S by sea rc hing the structures indexed by f ( t ) for successive t until we see a substructure o f type C 2 n or of t ype C 2 n +1 . Since S is no ncomputable, no such f can exist, so that G is not lo cally computable.  Theorem 5 .9. Ther e is a lo c al ly c omputable struct ur e which is not R -c omputable. Pr o of. Let X b e the set o f all countable g raphs with universe ω , and let E be the isomorphism rela tion o n X . Now E is complete ana ly tic [13], so F = E c is co mplete co-analy tic. Now for any x ∈ X , we hav e ¬ xF x , and for a n y x, y ∈ X we hav e xF y if and only if y F x . Thus, F defines the a djacency relatio n of a gr aph on X . Let X denote the graph ( X , F ). Now X is not real-co mputable, since its diag r am is c omplete co-ana lytic (con- tradicting path decomp osition). W e will show that X is loca lly computable. No w the finitely generated substructures o f X are a ll finite graphs, and it only r emains to deter mine which finite gr aphs a r e included. Let T be the following graph: r r r Let G b e a finite T - free graph. W e will show that G em beds in X . Let G = ( { 0 , . . . , n } , G ). W e will define an eq uiv alence structure R with universe N = { 0 , . . . , n } . F o r x, y ∈ N , w e say that xRy if a nd only if ¬ xGy . This r elation R will be re fle x iv e and symmetric. Since G is T -free, R will also b e trans itiv e. Now s ince the iso mo rphism relation is Bo r el complete [13], there is a function f : N → X such that xRy if and only if f ( x ) E f ( y ). This function can also be required to b e injectiv e [1 6]. Now let Φ b e a computable F riedb erg enumeration of finite gr aphs up to iso - morphism (i.e. a total computable function whose range consists of a n index for exactly one repr e s en tative from each isomor phis m class of finite graphs). Such an enum eration was given in [7]. W e will define a F riedb erg enumeration Ψ of finite T - free gr aphs up to isomorphism as follows: Ψ( x ) will b e Φ( x ′ ) for the lea s t x ′ such that Φ( x ′ ) is T - fr ee and Φ( x ′ ) / ∈ r an (Ψ | x ). Since all of the graphs are finite, we can effectively chec k whether each is T -free, so that Ψ is computable. Now Ψ provides a unifo r m simple computable co ver for X .  Corollary 5. 10. Ther e is another stru ctur e ˜ X with t he same un ifo rm s imple c om- putable c over as X , such that ˜ X is R -c omputable. Pr o of. Let ˜ X be the disjoint union [ x ∈ ω · Ψ( x ) . Now ˜ X is countable, a nd so is trivia lly R -co mputable.  One can s a y more ab out the structure describ ed in Theo rem 5.9. The structure satisfies a stronger condition called p erfe ctly lo c al c omputability . W e recall the definition of pe r fectly lo cally co mputable and leave the deta ils to the rea der. SOME RES UL TS ON R -COMPUT ABLE S TR UCTURES 13 Definition 5.11. Let U b e a uniformly computable cov er for a structure S . A Set M is a c orr esp ondenc e system for U and S if it sa tisfies all of the fo llo wing: (1) Each element of M is a n embedding of some A i ∈ U into S ; and (2) Every A i ∈ U is the domain of some β ∈ M ; a nd (3) Every generated B ⊂ S is the ima g e of some β ∈ M ; and (4) F or every i and j and ev ery β ∈ M with domain A i , ev ery f ∈ I U ij lifts to an inclusion β ( A i ) ⊂ γ ( A i ) via β and s ome γ ∈ M ; and (5) F or every i , every β ∈ M with doma in A i , a nd every finitely genera ted C ⊂ S cont aining β ( A i ), there exist a j and an f ∈ I U ij which lifts to β ( A i ) ⊂ C via β and some γ : A j ։ C ∈ M . The co rresp ondence system is p erfe ct if it also satisfie s 6. F or ev ery finitely g e nerated B ⊂ S , if β : A i ։ B and γ : A j ։ B bo th lie in M and have imag e B , then γ − 1 ◦ β ∈ I U ij . If a p erfect corresp ondence system exis ts , then its elements are called p erfe ct matches betw een their do mains a nd their images. S is then said to b e p erfe ctly lo c al ly c om- putable with p erfe ct c over U . 5.2. Σ -Definability. The fo llowing definition is standard, a nd a ppear s in equiv a- lent for ms in [2] and [10]. Definition 5.12. Given a str ucture M with universe M , we define a new structure H F ( M ) as follows. (1) The universe o f H F ( M ) is the union of the chain H F n ( M ) defined a s follows: (a) H F 0 ( M ) = M (b) H F n +1 ( M ) = P <ω ( M ∪ H F n ( M )), where P <ω ( S ) is the set of all finite subsets of S (2) The language for H F ( M ) c o nsists of a unary predicate U for H F 0 ( M ), as well as a predica te ∈ in terpreted as members hip, plus a symbol σ ∗ for each symbol σ of the language of M , given the interpretation of σ on M = H F 0 ( M ). Ershov gave a definition [10] of a notion genera lizing computability to structur es other than N . W e will first give B a rwise’s definition [2 ] of the cla s s of Σ-formulas. Definition 5.13. The cla ss of Σ-formulas are defined by induction. (1) Each ∆ 0 formula is a Σ-for m ula. (2) If Φ and Ψ are Σ-formulas, then so ar e (Φ ∧ Ψ) and (Φ ∨ Ψ). (3) F or each v a riable x and each term t , if Φ is a Σ-formula, then the following are also Σ-formulas: (a) ∃ ( x ∈ t ) Φ (b) ∀ ( x ∈ t ) Φ, and (c) ∃ x Φ. A predicate S is called a ∆ -pr e dic ate if bo th S a nd its c o mplemen t a re defined by Σ-formulas. Definition 5.14 . Let M a nd N = ( N , P 0 , P 1 , . . . ) be structures. W e say that N is Σ-definable in H F ( M ) if and only if ther e are Σ-formulas Ψ 0 , Ψ 1 , Ψ ∗ 1 , Φ 0 , Φ ∗ 0 , Φ 1 , Φ ∗ 1 , . . . such that 14 W. CAL VE R T AND J. E. POR TER (1) Ψ H F ( M ) 0 ⊆ H F ( M ) is nonempty , (2) Ψ 1 defines a congruence r elation on  Ψ H F ( M ) 0 , Φ H F ( M ) 0 , Φ H F ( M ) 1 , . . .  , (3) (Ψ ∗ 1 ) H F ( M ) is the relative complement in (Ψ H F ( M ) 0 ) 2 of Ψ H F ( M ) 1 , (4) F or each i , the set (Φ ∗ i ) H F ( M ) is the relative complement in Ψ H F ( M ) 0 of Φ H F ( M ) i , and (5) N ≃  Ψ H F ( M ) 0 , Φ H F ( M ) 0 , Φ H F ( M ) 1 , . . .  / Ψ H F ( M ) 1 . Theorem 5.15 (Calvert [6]) . The st ructur es which have isomorp hic c opies Σ - definable over H F ( R ) ar e exactly the ones which have isomorphic c opies which ar e R -c omputable. An interesting consequense of this (an immediate co rollary o f Theore m 5.15 and a r esult of Mor ozov a nd K o rovina [23]) giv es a sense in which so me R - computable structures ca n b e appr oximated by clas sically computable structures . Definition 5.1 6. Let A and B b e str uctures in a common signature. W e wr ite that A ≤ 1 B if A is a substructure of B , and for all existen tial for m ulas ϕ ( ¯ x ) and for all tuples ¯ a ⊆ A , w e hav e B | = ( ϕ (¯ a ) ⇒ A | = ϕ (¯ a ) . Corollary 5.17. F or any R -c omputable structu r e M whose defining machine in- volves only algebr aic r e als as p ar ameters, ther e is a c omputable stru ct ur e M ∗ such that M ∗ ≤ 1 M . 5.3. F-P arameterizabilit y. Morozov int ro duced a co ncept that he called F -p ar- ameterizability in o rder to under stand the elementary substructure relation on b oth automorphism gr o ups and the structure o f hereditarily finite sets ov er a given struc- ture [2 5]. In a talk a t Stanford University , though, he identified this notion as one “which generalize s the notion o f computable” [24]. Definition 5. 1 8 ([2 5]) . Let M be a str ucture in a finite rela tional la ng uage  P k n n ) n ≤ k  . W e say that M is F -parameterizable if and only if there is an in- jection ξ : M → ω ω with the following prop erties: (1) The image of ξ is a nalytic in the Baire spa ce, and (2) F or each n , the set n ( ξ ( a i )) i ≤ k n : M | = P n (¯ a ) o is a nalytic. The function ξ is called a n F -parameteriza tio n of M . Mor ozov a lso introduced the following strong er co ndition, es sen tially r equiring that M b e able to define its own F -par ameterization. Definition 5.19 ([25]) . Let M b e an F -parameteriza ble structure . W e s a y that M is weakly selfpar ameterizable if a nd only if there a r e functions Ξ , p : M × ω → ω , bo th definable without pa rameters in H F ( M ), with the following prop erties: (1) F or all x ∈ M a nd all m ∈ ω , we have Ξ( x, m ) = ξ ( x )[ m ], and (2) F or a ll f ∈ ω ω there is some x ∈ M such that for all n ∈ ω we have p ( x, n ) = f ( n ). In making sense of effectiveness o n uncountable structures, a ma jor motiv ation is to descr ibe a sense in which real num b er ar ithmetic — an op eratio n that, while not T uring computable, do es not seem horr ibly ineffective — can b e considered to be effectiv e. SOME RES UL TS ON R -COMPUT ABLE S TR UCTURES 15 Prop osition 5.20 (Moroz o v [25]) . The r e al field is we akly F -selfp ar ameterizable. Outline of pr o of. Define a function ξ : R → ω ω maps x to its decimal ex pansion. This function is definable without parameters in H F ( R ), in the sense r equired by Definition 5.19.  Theorem 5.21 (Calv ert [6]) . Every R -c omputable structure is F -p ar ameterizable. On the other hand, the stru ctur e ( R , + , · , 0 , 1 , e x ) is we akly F - s elfp ar ameterizable but not R -c omputable. 6. Conclusion W e state here s ome o pen pro blems a rising fro m issues discussed in the present pap er. The first is p erhaps the most vital. Problem 6.1. Develop a subs titute fo r the pr iority method which is capable of handling constructions with injury . Question 6. 2. Is it true that for a n y R -c o mputable finite dimensiona l R -vector spaces M a nd N with the same dimension, there is a R -computable isomo rphism from M to N ? Conjecture 6 .3. A R -c omputable R -ve ct or sp ac e of dimension gr e ater than ℵ 0 is not R - c omputably c ate goric al. W e would also like to know ab out the categ oricity of vector spa ces o f dimension ℵ 0 , but are not re ady to hazar d a c o njecture at this time. Question 6.4. Does there exist a R -co mputable Ba nac h space of infinite dimen- sion in the langua ge of vector spa ces, aug men ted by a sort for R and a function int erpreted as the norm? Question 6. 5. Do es there ex ist a R -computable Hilber t space of infinite dimension in the languag e of vector spa c e s, augmented by a s o rt for R a nd a binary function int erpreted as the inner pro duct? 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Dep ar tment of Mathema tics & St a tistics, F acul ty Hall 6C, M urra y St a te University, Murra y, Ken tucky 420 71 E-mail addr ess : wesley.calvert@ murraystate.edu