Characteristic Properties of Majorant ... - Semantic Scholar
Characteristic Properties of Majorant-Computability over the Reals? M. V. Korovina1 and O. V. Kudinov2 1
Insitute of Informatics Systems, Lavrent'ev pr., 6, Novosibirsk, Russia, email: [email protected] 2 Institute of Mathematics, University pr., 4, Novosibirsk, Russia, email: [email protected]
Abstract. Characteristic properties of majorant-computable real-valu-
ed functions are studied. A formal theory of computability over the reals which satis es the requirements of numerical analysis used in Computer Science is constructed on the base of the de nition of majorant-computability proposed in [13]. A model-theoretical characterization of majorantcomputability real-valued functions and their domains is investigated. A theorem which connects the graph of a majorant-computable function with validity of a nite formula on the set of hereditarily nite sets on IR, HF(IR) (where IR is a proper elementary enlargement of the standard reals) is proven. A comparative analysis of the de nition of majorantcomputability and the notions of computability earlier proposed by Blum et al., Edalat, Sunderhauf, Pour-El and Richards, Stoltenberg-Hansen and Tucker is given. Examples of majorant-computable real-valued functions are presented.
1 Introduction In the recent time, attention to the problems of computability over uncountable structures, particularly over the reals, is constantly raised. The theories proposed by Barwise [1], Scott [18], Ershov [7], Grzegorczyk [9], Moschovakis [15], Freedman [8] got further development in the works of Blum, Shub, Smail [4], Poul-El, Richards [16], Edalat, Sunderhauf [6], Stoltenberg-Hansen, Tucker [19] Korovina, Kudinov [13] and others. This work continues the investigation of the approach to computability proposed in [13]. Developing our approach we took into consideration the following requirements: 1. Our notion of computability should involve only e ective processes. 2. Its de nition should contain minimal number of limitations. 3. The class of computable real-valued functions should have clear and exact classi cations in logical and topological terms. ?
This reseach was supported in part by the RFBR (grant N 96-15-96878) and by the Siberian Division of RAS (a grant for young reseachers, 1997)
4. This class should also contain the classes of computable real-valued functions proposed in earlier works by Blum Shub, Smail; Poul-El, Richards; Stoltenberg-Hansen, Tucker; Edalat, Sunderhauf as subclasses. According to these requirements we construct a notion of computability over the reals with the following properties: 1. In our approach, the computation of a real-valued function is an in nite process that produces approximations closer and closer to the result. 2. This approach does not depend on the way of representing the reals. The de nition of computability does not limit the class of considered functions by the property of continuity and the property of being total. Also uniform convergence of processes that produce approximations to a computed function is not required. The use of nonstandard models of the rst-order theory of the reals enables us to investigate properties of computability of partial real-valued functions. 3. Special attention is paid to de nability of majorant-computable real-valued functions and their domains. A theorem which connects the graph of a majorantcomputable function with validity of a nite formula in the set of hereditarily (where IR is a proper elementary enlargement of the standard nite sets HF(IR) real numbers) is proven. The property of de nability can be considered as a denotational semantics of computational processes. Also we give a characterization of the domains of majorant-computable functions via formulas. 4. A comparative analysis of the de nition of majorant-computability and the notions of computability earlier proposed by Blum et. al., Poer-El and Richards, Edalat and Sunderhauf is given. For continuous total real-valued functions, the class of majorant-computable functions coincides with the class of computable functions introduced by Pour-El, Richards, and with the class introduced by Edalat and Sunderhauf. For partial real-valued functions, the class of computable functions introduced by Edalat and Sunderhauf is contained in the class of majorant-computable functions. In our approach, the majorant-computable functions include an interesting class of real-valued total functions that admit meromorphic continuation onto C. This class, in particular, contains functions that are solutions of well-known differential equations.
2 Basic Notions Throughout the article, < IR; 0; 1; +; ; > is the standard model of the reals, denoted also by IR, where + and are regarded as predicate symbols. Let Th(IR) be the rst order theory of IR. Let IN denote the set of natural numbers and Q the set of rational numbers. We use de nability as one of the basic conceptions. Montague [14] proposed to consider computability from the point of view of de nability. Later, many authors among them Ershov [7], Moschovakis [15] paid attention to properties
of this approach applied to various basic models. We base on the following notion of de nability in a structure M = hM; 0 i, where 0 is a nite rst-order language. Bold face indicates sequences, in particular, x = x1 ; : : : ; xn . A formula (x) is said to de ne the set fr 2 M n jM j= (r)g, and a set of this form is called a de nable subset of M n . A function F : M m ! M is said to be de nable if its graph is de nable (in M). Recall some properties of de nable sets and functions in IR. The following statements are proven in [5, 12]. Denote by IR0 the real closure of the rational numbers. We use standard notations for real intervals (a; b), [a; b], (a; b], [a; b). In addition, ha; bi denotes any of these intervals.
Proposition 1 (0-minimality property). 1. A set B IR is de nable if and only ifS there exist n 2 ! and a ; b ; : : : ;an ; bn 2 IR [ f+1; ?1g such that B = in hai ; bi i : 1
1
0
2. Each de nable subset of IRn is the disjoint union of nitely many cells, each of which is also de nable. (A cell is a space that homeomor c to some IRk , where k n.)
Proof. See [2, 5].
Since the theory Th(IR) admits elimination of quanti ers, it follows that 0-minimality property holds on every model of Th(IR). For a function f let us denote dom(f ) = fx j 9y (f (x) = y)g; im(f ) = fy j 9x (f (x) = y)g; ?f = f(x; y) j [(x; y) 2 dom(f ) im(f )] ^ [f (x) = y]g : A function f : M ! L is called total if dom(f ) = M .
De nition 2. A partial real-valued function f : IR ! IR is said to be algebraic if for some a; b; c; d 2 IR [ f?1; +1g and a polynomial p 2 Q[x; y], 0
the following conditions hold: 1. 2. 3.
dom(f ) = ha; bi; im(f ) = hc; di; ?f = f(x; y) j [(x; y) 2 ha; bi hc; di] ^ [p(x; y) = 0]g.
Proposition 3. A real-valued function f : IR ! IR is de nable if and only if the following conditions hold. 1. There existSn 2 ! and a1 ; b1 ; : : : ; anT ; bn 2 IR0 [ f+1; ?1g such that dom(f ) = in hai ; bi i and haj ; bj i hai ; bi i = ; for i 6= j . 2. f is an algebraic function on the interval hai ; bi i for all i n. Proof. See [2, 12].
The previous proposition implies that a real-valued function f is de nable if and only if f is the nite union of algebraic functions de ned on disjoint intervals with algebraic endpoints. In order to study richer classes of real-valued functions then the above one we need an enlargement of our basic model. Let us construct the set of hereditarily nite sets HF(M ) over a model M. This structure is rather well studied in the theory of admissible sets [1, 7] and permits us to de ne the natural numbers, to code, and to store information via formulas. Let M be a model whose language 0 contains no function symbols and whose carrier set is M . We construct the set of hereditarily nite sets, HF(M ), as follows: 1. S0 (M ) * ) P! (Sn (M )) [ Sn (M ); where n 2 ! and for every ) M; Sn+1 (M ) * set B , P! (S B ) is the set of all nite subsets of B . 2. HF(M ) = n2! Sn (M ).
We de ne HF(M) * ) HF(M ); M; 0 ; ;HF(M) ; 2HF(M) ; where ;HF(M) and the binary predicate symbol 2HF(M) have the set-theoretic interpretation. Denote = 0 [ f;HF(M); 2HF(M)g. The notions of a term and an atomic formula are given in a standard manner. The set of 0 -formulas is the closure of the set of atomic formulas in the language under ^; _; :; 9x 2 t and 8x 2 t, where 9x 2 t ' denotes 9x(x 2 t ^ ') and 8x 2 t ' denotes 8x(x 2 t ! '). The set of -formulas is the closure of the set of 0 formulas under ^; _; 9x 2 t; 8x 2 t; and 9. We de ne -formulas as negations of -formulas.
De nition 4. 1. A set B HF(M ) is -de nable if there exists a -formula (x) such that x 2 B $ HF(M) j= (x): 2. A function f : HF(M ) ! HF(M ) is -de nable if there exists a -formula (x; y) such that f (x) = y $ HF(M) j= (x; y): In a similar way, we de ne the notions of -de nable functions and sets. The class of -de nable functions (sets) is the intersection of the class -de nable functions (sets) and the class of -de nable functions (sets). Note that the sets M and M n are 0 {de nable. This fact makes HF(M) a suitable domain for studying functions from M k to M . To introduce the de nition of majorant-computability we use a class of -de nable real-valued functions as a basic class. So, we recall properties of -de nable real-valued functions and subsets of IRn . The following propositions give important properties of -, -, - de nable sets and functions. Proposition 5. Let IR~ be a model of Th(IR) and 0 be the language of Th(IR). ~ n is -de nable if and only if there exists an e ective se1. A set B IR quence of quanti er-free formulas in the language 0 , fs (x)gs2! , such that W x 2 B $ IR~ j= s2! s (x):
~ n is -de nable if and only if there exists an e ective se2. A set B IR quence of quanti er-free formulas in the language 0 , fs (x)gs2! , such that V x 2 B $ IR~ j= s2! s (x): Proof. The claim is immediate from the properties of the set of hereditarily nite sets and the fact that Th(IR) admits elimination of quanti ers. ut
Proposition 6. Let IR be the standard reals. The following statements hold in HF(IR). 1. The set IN is -de nable. 2. The class of -de nable real-valued total functions is closed under composition. 3. There is an universal real-valued -de nable function in the class of -de nable real-valued functions. Proof. See [7, 12, 11] and Proposition 5.
Proposition 7. A real-valued function f : IR ! IR is -de nable if and only if the following conditions hold. 1. There exist e ective sequences fai gi2! , fbi gi2! , where ai ; bi 2 IR0 [ f+1; ?1g for i 2 !, such that
dom(f ) =
[
i2!
\
hai ; bi i and haj ; bj i hai ; bi i = ; for i 6= j :
2. There exists e ective sequences of algebraic functions fgigi2! such that f coincides with gi on the interval hai ; bi i for all i 2 !. Proof. See [12] and Proposition 5.
~ be a model of Th(IR). Theorem 8 (Uniformization). Let IR n
~ which is {de nable by a {formula there exists For any subset of IR ~ n?1 ! IR ~ such that an {de nable function f : IR ~ j= 9y(x; y)g: 1. dom(f ) = fxjHF(IR) ~ j= (x; f (x)) : 2. for x 2 dom(f ) we have HF(IR) Proof. See [12] and Proposition 7.
To introduce the notion of majorant-computability for partial real-valued functions we need a notion of prime enlargement of IR. De nition 9. A model IR of Th(IR) is called a prime enlargement of IR if there such that t > n for every natural n (we write t > IN) and IR is exists t 2 IR the real closure of the ordered eld IR(t). In addition, we write t > IR if t > r for all r 2 IR.
De nition 10. Let IR be a proper elementary enlargement of IR and let t > IN. ! IR [ f?1; +1g We de ne the following function sp : IR 8 < x 2 IR if for j x ? x j= " the condition 0 < " < IR holds ; sp(x) = : +1 if x > IR; ?1 if x < IR : is nite if and only if ?1 < sp(x) < +1. The set of all nite An element x 2 IR +
is denoted by Fin(IR). elements of IR j= Th(IR), IR is a proper elementary enlargment of IR with the same Since IR true rst-order formulas. Recall properties of prime enlargements and proper elementary enlargements of IR.
Proposition 11. 1. Every two prime enlargements of IR are isomorphic. 2. There exists a nonstandard element t > IN in a proper elementary enlargement of IR. Proof. See [10, 17].
Lemma12. Let p(x; y) be a countable set of rst-order formulas with one variable x and real parameters y. Given proper elementary enlargements IR , IR of IR, p(x; y) holds in IR if and only if p(x; y) holds in IR . Proof. The claim is immediate from 0-minimality of IR and IR . ut Lemma13. Let IR be a prime enlargement of IR. For every -formula '(z; x) there exists a -formula ' (x) such that, j= '(t ; x) $ HF(IR) j= ' (x): for all x 2 IRn and t > IN, HF(IR) Proof. The claim is immediate from 0-minimality of IR. ut Lemma14. Let IR be a model of Th(IR) and let B be a countable subset of IR . then there exists a -de nable function If B is -de nable in HF(IR) h : IN ! IR numbering B . Proof. The claim is immediate from Proposition 1 and Proposition 5. ut 1
1
2
2
1
2
3 Majorant-Computable Functions Let us recall the notion of majorant-computability for real-valued functions presented in [13]. We would use as a basic class the class of -de nable total func n ! IR, where IR is a proper elementary enlargement of IR. tions of type f : IR A real-valued function is said to be majorant-computable if we can construct a special kind of nonterminating process computing approximations closer and closer to the result.
De nition 15. A function f : IRn ! IR is called majorant-computable if there exist e ective sequences of -formulas fs (a; x; y)gs2! and fGs (a; x; y)gs2! ,
of IR such that with a parameter a, a proper elementary enlargement IR the following conditions hold. such that t > IN; 1. There exists t 2 IR 2. For all s 2 !, the formulas s (a; x; y) and Gs (a; x; y) de ne total functions fs and gs as follows: n n ! IR and gs : IR ! IR, a. fs : IR j= s (t; x; y), b. fs (x) = y $ HF(IR) j= Gs (t; x; y); gs (x) = y n$ HF(IR) , the sequence ffs (x)gs2! increases monotonically; the sequ3. For all x 2 IR ence fgs (x)gs2! decreases monotonically ; n, 4. For all s 2 !, x 2 dom(f ), fs (x) f (x) gs (x) and, for all x 2 IR fs (x) gs (x); 5. f (x) = y $ lims!1 sp(fs (x)) = y and lims!1 sp(gs (x)) = y. The sequence ffs gs2! in De nition 15 is called a sequence of lower -approximations for f . The sequence fgs gs2! is called a sequence of upper -approximations for f . As we can see the process which carries out the computation is represented by two e ective procedures. These procedures produce -approximations closer and closer to the result. If the computational process converges to in nity the procedures produce nonstandard elements like ?t or t, where t > IR. So, using a proper elementary enlargement of IR in the De nition 9 we admit only precise computations at nite steps. Since a prime enlargement is e ectively constructed over IR, this approach admit us to consider computability of partial functions in a natural way. The following theorem connects the graph of a majorant-computable func tion with validity of a nite formula in the set of hereditarily nite sets, HF(IR) (where IR is a proper elementary enlargement of the standard real numbers). De nition 16. Let IR be a proper nelementary enlargement of IR. A formula ! IR in the model HF(IR) if the following is said to determine a function f : IR statement holds f (x) = y $ ? j= (x; y) for x 2 IR and fsp(z ) j (x; z )g = fyg for y 2 IR: HF(IR)
Theorem 17. For all functions f : IRn ! IR, the following assertions are equiv-
alent: 1. The function f is majorant-computable. IR and a -formula that determines 2. There exist a prime enlargement IR a function F in HF(IR) with the property F jIR = f . ^ IR 3. There exists a -formula that in any proper elementary enlargement IR determines a function F with the property F jIR = f . Proof. 1 ! 2) Let f be majorant-computable. By Proposition 11, without loss of IR, of generality, we may assume that there exist a prime enlargement IR
a sequence ffsgs2! of lower -approximations for f , and a sequence fgsgs2! of upper -approximations for f . Let t > IN. Denote B = Q [ ftn j n 2 Qg: we construct a sequenUsing ffs gs2! , fgsgs2! , and co nality of B in IR, ce ffs gs2! of new lower -approximations for f and a sequence fgsgs2! of new upper -approximations for f such that, for all s 2 !, the ranges of fs and gs are subsets of B . Denote: D1 (x) = fz 2 B j 9s (fs (x) z )g; D2(x) = fz 2 B j 9s (gs (x) z )g : The sets D1 (x) and D2 (x) are -de nable and countable; so, by Lemma 14, there exist a function h1 (n; x) numbering D1 (x) and a function h2 (n; x) numbering D2 (x). Next, put fs (x) = maxns h1 (n; x); gs (x) = minnsh2 (n; x) : By construction, the sequences ffsgs2! and fgsgs2! are what we seek. Let j= Gs (t; x; y) ; j= s (t; x; y); gs (x) = y * fs(x) = y * ) HF(IR) ) HF(IR) where s , Gs are -formulas. From Lemma 12 it follows that, for all s 2 !, n there exist -formulas s and Gs such that, for every x 2 IR and y 2 IR : j= Gs (t; x; y) $ G j= s (t; x; y) $ HF(IR) s (x; y ); HF(IR) s (x; y ) : Let ; fs (x); gs (x) 2 Fin(IR) g for s 2 !: ds = fx j x 2 Finn (IR) the set ds is -de nable. It is easy to see that By -de nability of Fin(IR), ds !s!! dom(f ). Put (x; y) * ) 8s8y18y2 [(x 2 ds ) ! (( s (x; y1 ) ^ Gs (x; y2 )) ! (y1 y y2 ))] : ! IR. The -formula determines a function F : IR Let us prove that F j IR = f . Let f (x) = y. Suppose the contrary: F (x) 6= y. By De nition 16, there exist s 2 !; y1 ; y2 2 IR such that j= (x 2 ds ) ^ HF(IR) s (x; y1 ) ^ Gs (x; y2 ) ^ (y 62[y1 ; y2 ]) : By the constructions of fs , gs and the de nition of ds , there exist z1 ; z2 2 IR
such that j= s (t; x; z1 ); gs (x) = z2 $ HF(IR) j= Gs (t; x; z2 ) : fs (x) = z1 $ HF(IR) Note that x 2 IRn and z1 ; z2 2 IR. So, we have j= j= G fs(x) = z1 $ HF(IR) s (x; z1 ); gs (x) = z2 $ HF(IR) s (x; z2 ) and j= (x 2 ds ) ^ HF(IR) s (x; z1 ) ^ Gs (x; z2 ) ^ (y 2 [y1 ; y2 ]) :
Note that if a formula G(x; y) de nes the function f in HF(IR) then 6j=G(x; y) for x 2 dom(f ), y62IR. It is easy to see that the formulas HF(IR) s , Gs de ne functions in HF(IR), so y1 = z1 and y2 = z2; a contradiction. If F (x) = y for x 2 IRn and y 2 IR,
fsp(z ) j
^
i2!
(fi(x) z gi (x))g = fyg :
It follows that lims!1 sp(fs (x)) = y and lims!1 sp(gs (x)) = y. So, f (x) = y and the formula is what we seek. 2 ! 3) This is obvious. be a prime enlargement of IR and let a -formula (x; y) deter3 ! 1) Let IR mine the function F such that F jIR = f . From Proposition 5 it follows that there exists an e ective sequence of quanti er-free formulas f'i (x; y)gi2! such that ^ j= (x; y) $ HF(IR) j= 'i (x; y) : HF(IR) i2!
Denote Put
s (x; y) * )
^
is
'i (x; y) :
(
fy j s (x; y)g if inf fy j s (x; y)g exists; fs (x) = inf ?t otherwise, ( fy j s (x; y)g if supfy j s (x; y)g exists; gs (x) = sup t otherwise, where t > IN : Then ffs gs2! and fgsgs2! are the sought sequences. ut We remark that if f is a total function then the sequences ffs gs2! and fgs gs2! can be constructed to converge uniformly to f on IR.
As a corollary from the previous theorem we note that whenever we want to prove some statements to de ne a majorant-computable function, without loss of generality, we can assume that the proper elementary enlargement of IR in De nition 15 is a prime enlargement of IR. Proposition 18. Let f : IRn ! IR be majorant-computable. ? Then dom(f ) is 2 -de nable by a formula of type (8z 2 IR) (z; x) , where is a -formula. Proof. Let ffs gs2! be a sequence of lower -approximations for f , and let fgsgs2! be a sequence of upper -approximations for f . We obtain: x 2 dom(f ) $ HF(IR) j= (8" 2 IR)(9s 2 IN) fs (x) ? gs (x) " : By -de nability of the natural numbers, the set dom(f ) is 2 -de nable. ut
Corollary 19. The domain of a majorant-computable function is the e ective intersection of intervals with algebraic endpoints.
Proof. The claim is immediate from Proposition 5 and Proposition 18.
ut
De nition 20. Let a function f : IRn ! IR be total. The epigraph of f is de ned to be the set U = f(x; y) j f (x) < yg. The ordinate set of f is de ned to be the set D = f(x; y) j f (x) > yg. Corollary 21. Let f : IRn ! IR be a total function. The function f is majorant-computable if and only if the epigraph of f and the ordinate set of f are -de nable sets. Proof. It is clear that (x; z ) 2 U $ HF(IR) j= 9s9y [ (s 2 IN) ^ G (x; y) ^ (z > y)] ; (x; z ) 2 D $ HF(IR) j= 9s9y [ (s 2 IN) ^ F (x; y) ^ (z < y)] ; where the formulas G and F are those in the proof of Theorem 17.
ut
Corollary 22. Let f : IR ! IR.
1. If f is a -de nable real-valued function, then f is majorant-computable. 2. If domf IN and imf IN, then f is majorant-computable if f is a partial recursive function. 3. If f is a majorant-computable total function, then f is a piecewise continuous function.
Proof. The claims follow from the de nition of majorant-computability.
ut
4 Majorant-Computability and PR-, ES-Computabilities over the Reals Let us denote the computability proposed in [16] as PR-computability. Recall the de nition of PR-computability for real-valued functions.
De nition 23. A total function f : IR ! IR is PR-computable if and only if 1. it maps computable sequences to computable sequences, 2. it is e ectively uniformly continuous on intervals [?n; n]. Let us denote the computability proposed in [6] as ES-computability. Recall the de nition of ES-computability for real-valued functions. We recall the de nition of the interval domain I : I = f[a; b] IR j a; b 2 IR; a bg [ f?g : The relation is de ned as follows: ? J for all J 2 I and [a; b] [c; d] if and only if a < c and b > d.
De nition 24. Let I0 = fb1; : : : ; bn; : : :g [ f?g be the e ective enumerated set of all intervals with rationals endpoints. A continuous function f : I ! I is computable, if the relation bm f (bn ) is r.e. in n; m, where bm ; bn 2 I0 . De nition 25. A function f : IR ! IR is ES-computable if and only if there is an enlargement g : I ! I (i.e., g(fxg) = ff (x)g for all x 2 dom(f ) ) which is computable in the sense of the De nition 24. Let us de ne gx;y(= [x;y]; g^x;y = [x;y); if x 2 M; where M (x) = unde ned1 otherwise . De nition 26. A continuous function h : [a; b] ! IR is called a piecewise linear function if there exist x ; : : : ; xn 2 IR such that a = x < x < : : : xn = b, and h j x ;x +1 is a linear function for all 0 i m. We de ne the code [h] of h as follows: [h] = f< xi ; h(xi ) >j 0 i mg: 0
[
i
i
0
1
]
Let us consider functions from < a; b > to IR. Proposition 27. Let f :< a; b >! IR be a continuous majorant-computable function. The function f is PR-computable if and only if the relation Rf = f< x; y; z >2 [a; b]2 IR j x < y ^ f j[x;y] > zgx;y g is -de nable in HF(IR). Proof. !) By the de nition of PR-computability, f is an e ectively uniformly continuous function. It follows that the set Qf = < x; y; z; >2 [a; b] IR2 j jjf ? zgx;y jj[x;y] < is -de nable. For x; y 2 IR such that x < y, the following equivalence holds. ? 3 9 < b ; : : : ; b >2 Q
?1 ?n^ i=0
1
n
jbi ? bi j < = wf ( 3 ) ^ b = x ^ bn = y ^ +1
0
n^ ?1 i=0
jfs (bi ) ? z j < 3 ;
where w is an e ective modulus of continuity for f . It follows that < x; y; z >2 Rf if and only if there exist > 0 and a step-function h : [x; y] ! Q such that h > zga;b + and jjf ? hjj[x;y] < . S ?1 ? For arbitrary x0 < : : : < xi < : : : < xm = y and h = m i=0 zi g^x ;x +1 , jjf ? hjj[x;y] = max0im?1 jjf ? zi g^x ;x +1 jj. So the set Rf is -de nable in HF(IR). i
i
i
i
) Let Rf be -de nable in HF(IR). For every = 1s > 0, where s 2 !, we will e ectively construct piecewise linear functions fs : [a; b] ! IR, gs : [a; b] ! IR
with the following conditions: 1. fs (x) f (x) gs (x) for all x 2 [a; b]; 2. jjfs ? gs jj . In fact, by -de nability of Rf , the set
Tf = < x; y; z; t >2 [a; b] IR j x < y ^ 8u 2 [x; y] f (u) > z + uy(t??xz ) is -de nable. It follows that the set Lf = f< s; [f ]; [g] >j s 2 !; f; g are piecewise linear with the condition 1-2g is -de nable. Using uniformization (see Theorem 8) we can construct the required functions fs ; gs . Obviously, by the code [h] Q2 of a piecewise linear function, we can e ectively construct a function wh : [0; 1] ! IR with the following properties: 1. wh is a linear function; 2. (8 2 (0; 1]) (8x; y 2 [0; 1]) (jx ? yj < wh () ! jh(x) ? h(y)j < ) As we can see, wh is a modulus of continuity for h. Let us de ne the function wf as follows: wf = wf (), where s() = [ 3 ] + 1. It is easy to see that wf is -de nable, and it is a modulus of continuity for f . By the de nition of PR-computability, f is PR-computable. ut It is easy to see that the continuous majorant-computable total function f is PR-computable if and only if the intersection Rf \ Q is -de nable in HF(IR). Proposition 28. If B is an open -de nable set then B is the e ective union 2
2
s
of open intervals. Proof. Let B be an open -de nable set. By Proposition 3, [ B = < i ; i > :
We represent B as follows: [ B = [ i ; i ) [
i2!
[
i2I1
i2I2
i2I1
i2I2
( i ; i ] [
[
i2I3
( i ; i ) [
[
i2I4
[ i ; i ]:
Put I40 = fi 2 I4 j i 6= i ; i 6= j ; j 2 I4 g. Because B is open, we can e ectively choose i ; i ; i , and i as follows
i < i and i 2 ( j ; j ); where i 2 I1 ; j 2 I2 [ I3 [ I40 ; i > i and i 2 ( j ; j ); where i 2 I2 ; j 2 I1 [ I3 [ I40 ; i < i and i 2 ( j ; j ); where i 2 I4 ; j 2 I1 [ I2 [ I3 ; i > i and i 2 ( j ; j ); where i 2 I4 ; j 2 I1 [ I3 : So, [ [ [ [ B = (i ; i ) [ ( i ; i ) [ ( i ; i ) [ (i ; i ): i2I3
i2I4
This means that B is the e ective union of open intervals.
ut
Proposition 29. Let f : [0; 1] ! IR be a majorant-computable continuous function. The set Rf = < a; b; c >j 0 a < b 1 ^ f j[a;b] > c is -de nable in HF(IR). Proof. Without loss of generality, let us consider the case f : [0; 1] ! [0; 1]. Let us de ne x 2 Asi $ jf (x) ? si j < 1s ; for i = 0; : : : ; s. WeShave the following properties: 1. Asi = [0; 1], 2. Asi is an open -de nable set. S j ) for some rational numbers j j . By Proposition 28, Asi = j2! ( ji;s ; i;s i;s i;s Let [a; b] [0; 1]. Let us consider the case cS=s 0. By continuity of f , f j[a;b] > 0 if and only if there exists s such that [a; b] S i=1 Asi . Really, if there exists s such that [a; b] si=1 Asi and x 2 [a; b] then x 2 Asi for some i. By the de nition of Asi , jf (x) ? si j < 1s , so f (x) > 0. s1 . If f j[a;b] > 0, then there exists s1 such that f j[a;b] > s11 . Put s = S i 1 s For some i > 0, we have jf (x) ? s j < s , i.e., x 2 Ai . So, [a; b] si=1 Asi . By compactness of [a; b], S f[a;b] > 0 if and only if there exists s such that S j ), and J is some nite subset of [a; b] si=1 A^si , where A^si = j2J ( ji;s ; i;s i;s !, 0 < i < s, i.e., f[a;b] > 0 if and only if the -condition holds. In the general case, for [a; b] and c, using similar considerations we check the following condition: [ [a; b] Asi : i;s
i:j ?cj 1 i s
s
ut
Theorem 30. Let f : [a; b] ! IR be a continuous total function.
The function f is majorant-computable if and only if f is PR-computable. Proof. !) The claim is immediate from Proposition 27 and Proposition 29. ) The claim is immediate from Corollary 21. ut
Corollary 31. Let f : [a; b] ! IR be a total continuous function.
The function f is majorant-computable if and only if f is ES-computable.
Proof. The claims are immediate from Corollary 30 in [6].
ut
Theorem 32. The class of ES-computable functions is contained in the class of majorant-computable functions.
Proof. Let f : IR ! IR be ES-computable. For n 2 !, we de ne An = fx 2 IR j (f ([x])) < n1 g, where is the natural measure de ned on I . It is easy to see that An is -de nable open set, and T dom(f ) = n2! An . S By Proposition 28, A1 = i2! ( i ; i ), where i ; i 2 Q and i i . By Lemma 14, there exist a -de nable function h : IR IN ! IR numbering the -de nable set fz 2 Q j z < f ([x])g, and a -de nable function H : IR IN ! IR numbering the -de nable set fz 2 Q j z > f ([x])g.
Put
(
S
(
S
maxks h(x; k) if x 2 ns ( n ; n ); fs (x) = ? t otherwise, ks H (x; k ) if x 2 ns ( n ; n ); gs (x) = tmin otherwise, where t > IN : Then ffsgs2! and fgs gs2! are the sought sequences. So, f is majorant-compu-
table. There exist majorant-computable functions that are not ES-computable. For example, a total step-function with a computable set of discontinuities is the sought one. ut
Corollary 33. If f is ES-computable, then the domain of f is the e ective intersection of open intervals.
Proof. The claim is immediate from the proof of Theorem 32.
ut
Corollary 34. For real-valued functions, we have the following inclusions: 1. The class of -de nable functions is contained in the class of majorant-computable functions. 2. The class of computable functions introduced by Moschovakis is contained in the class of majorant-computable functions. 3. The class of functions de nable by nite dimensional machines of Blum et al. without real constants is contained in the class of majorant-computable functions. 4. The class of PR-computable functions is contained in the class of majorantcomputable functions. 5. The class of ES-computable functions is contained in the class of majorantcomputable functions. 6. The class of computable functions introduced by Stoltenberg-Hansen, Tucker is contained in the class of majorant-computable functions.
5 Examples Consider an interesting class of real-valued total functions possessing meromorphic continuation onto C. This subclass contains, for example, solutions of known di erential equations. Let C be the set of complex numbers.
De nition 35. A function f : IR ! IR is said to admit meromorphic continuation onto C if there exists a meromorphic function f : C ! C such that f jIR = f .
We recall a fact concerning meromorphic functions in complex numbers theory [3]. Proposition 36. Let f : C ! C be a total meromorphic function. There exist entire functions a(z ) and b(z ) such that 1. f (z ) = ab((zz)) , 2. a(z ) and b(z ) have Taylor expansions with coecients in IR. Proof. The claim follows from the Schwartz principle [3]. ut Theorem 37. Let f : IR ! IR be a total function and let f be its meromorphic continuation onto C. Suppose that f can be written as f (z ) = ab((zz)) , where functions a(z ) and b(z ) are as in Proposition 36. The function f is majorantcomputable if the following conditions on f hold: 1. There exist -de nable real-valued functions A and B such that maxjzjw j a(z ) j A(w); maxjzjw j b(z ) j B (w); where w 2 IR ; 2. The coecients of the Taylor expansions for a(x); b(x) are majorant-computable as constant functions. Proof. Let f satisfy the conditions of the theorem. We shall prove that the epigraph of f and the ordinate set of f are -de nable. Denote the Taylor expansions for a(z ) and b(z ) at the point 0 as follows: X X a(z ) = ak z k ; b(z ) = bk z k : k2!
k2!
Show that a jIR and b jIR are majorant-computable. Let us consider an arbitrary disk j z j R. From the Cauchy inequality [3], for all k 2 !, we have
ja(z )j : jak j RMk ; where M = jmax zjR
This remark admits us to construct two sequences of required approximations for a jIR . It is easy to see that a jIR is majorant-computable. Similarly, b jIR is majorant-computable. It is easy to proof that if a jIR and b jIR are majorant computable, then jakIR and jbkIR are majorant-computable. From now it is evident that the epigraph and the ordinate set of the function f are -de nable sets. Corollary 2 implies that f is a majorant{computable function. ut Corollary 38. If an analytical function f has Taylor expansion with majorantcomputable coecients and the function j f j is bounded by an -de nable total real function, then f is a majorant{computable function. Since the considered language contains equality, majorant{computable functions can be discontinuous. The natural question is raised describe the situation without equality. Our forthcoming results would be devoted to this problem.
References 1. J. Barwise, Admissible sets and structures, Berlin, Springer{Verlag, 1975. 2. J. Bochnak, M. Coster and M.-F. Roy, Geometrie algebrique reelle Springer{ Verlag, 1987. 3. A. V. Bitsadze, The bases of analytic function theory of complex variables, Nauka, Moskow, 1972. 4. L. Blum and M. Shub and S. Smale, On a theory of computation and complexity over the reals:NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc., (N.S.) , v. 21, no. 1, 1989, pages 1{46. 5. L. van den Dries, Remarks on Tarski's problem concerning (IR + exp), Proc. Logic Colloquium'82, 1984, pages 240{266. 6. A. Edalat, P. Sunderhauf, A domain-theoretic approach to computability on the real line, Theoretical Computer Science. To appear. 7. Yu. L. Ershov, De nability and computability, Plenum, New York, 1996. 8. H. Freedman and K. Ko, Computational complexity of real functions, Theoret. Comput. Sci. , v. 20, 1982, pages 323{352. 9. A. Grzegorczyk, On the de nitions of computable real continuous functions, Fund. Math., N 44, 1957, pages 61{71. 10. N. Jacobson, Basic Algebra II, W.E. Freedman and Company, New York, 1995. 11. M. Korovina, Generalized computability of real functions, Siberian Advance of Mathematics, v. 2, N 4, 1992, pages 1{18. 12. M. Korovina, About an universal recursive function and abstract machines on the list superstructure over the reals, Vychislitel'nye Sistemy, Novosibirsk, v. 156, 1996, pages 3{20. 13. M. Korovina, O. Kudinov, A New Approach to Computability over the Reals, SibAM, v. 8, N 3, 1998, pages 59{73. 14. R. Montague, Recursion theory as a branch of model theory, Proc. of the third international congr. on Logic, Methodology and the Philos. of Sc., 1967, Amsterdam, 1968, pages 63{86. 15. Y. N. Moschovakis, Abstract rst order computability, Trans. Amer. Math. Soc., v. 138, 1969, pages 427{464. 16. M. B. Pour-El, J. I. Richards, Computability in Analysis and Physics, SpringerVerlag, 1988. 17. A. Robinson, Nonstandard Analysis, North-Holland, Studies in Logic and foundation of Mathematics, 1966. 18. D. Scott, Outline of a mathematical theory of computation, In 4th Annual Princeton Conference on Information Sciences and Systems, 1970, pages 169{176. 19. V. Stoltenberg-Hansen and J. V. Tucker, E ective algebras, Handbook of Logic in computer Science, v. 4, Clarendon Press, 1995, pages 375{526. ;
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This article was processed using the LATEX macro package with LLNCS style
Insitute of Informatics Systems, Lavrent'ev pr., 6, Novosibirsk, Russia, email: [email protected] 2 Institute of Mathematics, University pr., 4, Novosibirsk, Russia, email: [email protected]
Abstract. Characteristic properties of majorant-computable real-valu-
ed functions are studied. A formal theory of computability over the reals which satis es the requirements of numerical analysis used in Computer Science is constructed on the base of the de nition of majorant-computability proposed in [13]. A model-theoretical characterization of majorantcomputability real-valued functions and their domains is investigated. A theorem which connects the graph of a majorant-computable function with validity of a nite formula on the set of hereditarily nite sets on IR, HF(IR) (where IR is a proper elementary enlargement of the standard reals) is proven. A comparative analysis of the de nition of majorantcomputability and the notions of computability earlier proposed by Blum et al., Edalat, Sunderhauf, Pour-El and Richards, Stoltenberg-Hansen and Tucker is given. Examples of majorant-computable real-valued functions are presented.
1 Introduction In the recent time, attention to the problems of computability over uncountable structures, particularly over the reals, is constantly raised. The theories proposed by Barwise [1], Scott [18], Ershov [7], Grzegorczyk [9], Moschovakis [15], Freedman [8] got further development in the works of Blum, Shub, Smail [4], Poul-El, Richards [16], Edalat, Sunderhauf [6], Stoltenberg-Hansen, Tucker [19] Korovina, Kudinov [13] and others. This work continues the investigation of the approach to computability proposed in [13]. Developing our approach we took into consideration the following requirements: 1. Our notion of computability should involve only e ective processes. 2. Its de nition should contain minimal number of limitations. 3. The class of computable real-valued functions should have clear and exact classi cations in logical and topological terms. ?
This reseach was supported in part by the RFBR (grant N 96-15-96878) and by the Siberian Division of RAS (a grant for young reseachers, 1997)
4. This class should also contain the classes of computable real-valued functions proposed in earlier works by Blum Shub, Smail; Poul-El, Richards; Stoltenberg-Hansen, Tucker; Edalat, Sunderhauf as subclasses. According to these requirements we construct a notion of computability over the reals with the following properties: 1. In our approach, the computation of a real-valued function is an in nite process that produces approximations closer and closer to the result. 2. This approach does not depend on the way of representing the reals. The de nition of computability does not limit the class of considered functions by the property of continuity and the property of being total. Also uniform convergence of processes that produce approximations to a computed function is not required. The use of nonstandard models of the rst-order theory of the reals enables us to investigate properties of computability of partial real-valued functions. 3. Special attention is paid to de nability of majorant-computable real-valued functions and their domains. A theorem which connects the graph of a majorantcomputable function with validity of a nite formula in the set of hereditarily (where IR is a proper elementary enlargement of the standard nite sets HF(IR) real numbers) is proven. The property of de nability can be considered as a denotational semantics of computational processes. Also we give a characterization of the domains of majorant-computable functions via formulas. 4. A comparative analysis of the de nition of majorant-computability and the notions of computability earlier proposed by Blum et. al., Poer-El and Richards, Edalat and Sunderhauf is given. For continuous total real-valued functions, the class of majorant-computable functions coincides with the class of computable functions introduced by Pour-El, Richards, and with the class introduced by Edalat and Sunderhauf. For partial real-valued functions, the class of computable functions introduced by Edalat and Sunderhauf is contained in the class of majorant-computable functions. In our approach, the majorant-computable functions include an interesting class of real-valued total functions that admit meromorphic continuation onto C. This class, in particular, contains functions that are solutions of well-known differential equations.
2 Basic Notions Throughout the article, < IR; 0; 1; +; ; > is the standard model of the reals, denoted also by IR, where + and are regarded as predicate symbols. Let Th(IR) be the rst order theory of IR. Let IN denote the set of natural numbers and Q the set of rational numbers. We use de nability as one of the basic conceptions. Montague [14] proposed to consider computability from the point of view of de nability. Later, many authors among them Ershov [7], Moschovakis [15] paid attention to properties
of this approach applied to various basic models. We base on the following notion of de nability in a structure M = hM; 0 i, where 0 is a nite rst-order language. Bold face indicates sequences, in particular, x = x1 ; : : : ; xn . A formula (x) is said to de ne the set fr 2 M n jM j= (r)g, and a set of this form is called a de nable subset of M n . A function F : M m ! M is said to be de nable if its graph is de nable (in M). Recall some properties of de nable sets and functions in IR. The following statements are proven in [5, 12]. Denote by IR0 the real closure of the rational numbers. We use standard notations for real intervals (a; b), [a; b], (a; b], [a; b). In addition, ha; bi denotes any of these intervals.
Proposition 1 (0-minimality property). 1. A set B IR is de nable if and only ifS there exist n 2 ! and a ; b ; : : : ;an ; bn 2 IR [ f+1; ?1g such that B = in hai ; bi i : 1
1
0
2. Each de nable subset of IRn is the disjoint union of nitely many cells, each of which is also de nable. (A cell is a space that homeomor c to some IRk , where k n.)
Proof. See [2, 5].
Since the theory Th(IR) admits elimination of quanti ers, it follows that 0-minimality property holds on every model of Th(IR). For a function f let us denote dom(f ) = fx j 9y (f (x) = y)g; im(f ) = fy j 9x (f (x) = y)g; ?f = f(x; y) j [(x; y) 2 dom(f ) im(f )] ^ [f (x) = y]g : A function f : M ! L is called total if dom(f ) = M .
De nition 2. A partial real-valued function f : IR ! IR is said to be algebraic if for some a; b; c; d 2 IR [ f?1; +1g and a polynomial p 2 Q[x; y], 0
the following conditions hold: 1. 2. 3.
dom(f ) = ha; bi; im(f ) = hc; di; ?f = f(x; y) j [(x; y) 2 ha; bi hc; di] ^ [p(x; y) = 0]g.
Proposition 3. A real-valued function f : IR ! IR is de nable if and only if the following conditions hold. 1. There existSn 2 ! and a1 ; b1 ; : : : ; anT ; bn 2 IR0 [ f+1; ?1g such that dom(f ) = in hai ; bi i and haj ; bj i hai ; bi i = ; for i 6= j . 2. f is an algebraic function on the interval hai ; bi i for all i n. Proof. See [2, 12].
The previous proposition implies that a real-valued function f is de nable if and only if f is the nite union of algebraic functions de ned on disjoint intervals with algebraic endpoints. In order to study richer classes of real-valued functions then the above one we need an enlargement of our basic model. Let us construct the set of hereditarily nite sets HF(M ) over a model M. This structure is rather well studied in the theory of admissible sets [1, 7] and permits us to de ne the natural numbers, to code, and to store information via formulas. Let M be a model whose language 0 contains no function symbols and whose carrier set is M . We construct the set of hereditarily nite sets, HF(M ), as follows: 1. S0 (M ) * ) P! (Sn (M )) [ Sn (M ); where n 2 ! and for every ) M; Sn+1 (M ) * set B , P! (S B ) is the set of all nite subsets of B . 2. HF(M ) = n2! Sn (M ).
We de ne HF(M) * ) HF(M ); M; 0 ; ;HF(M) ; 2HF(M) ; where ;HF(M) and the binary predicate symbol 2HF(M) have the set-theoretic interpretation. Denote = 0 [ f;HF(M); 2HF(M)g. The notions of a term and an atomic formula are given in a standard manner. The set of 0 -formulas is the closure of the set of atomic formulas in the language under ^; _; :; 9x 2 t and 8x 2 t, where 9x 2 t ' denotes 9x(x 2 t ^ ') and 8x 2 t ' denotes 8x(x 2 t ! '). The set of -formulas is the closure of the set of 0 formulas under ^; _; 9x 2 t; 8x 2 t; and 9. We de ne -formulas as negations of -formulas.
De nition 4. 1. A set B HF(M ) is -de nable if there exists a -formula (x) such that x 2 B $ HF(M) j= (x): 2. A function f : HF(M ) ! HF(M ) is -de nable if there exists a -formula (x; y) such that f (x) = y $ HF(M) j= (x; y): In a similar way, we de ne the notions of -de nable functions and sets. The class of -de nable functions (sets) is the intersection of the class -de nable functions (sets) and the class of -de nable functions (sets). Note that the sets M and M n are 0 {de nable. This fact makes HF(M) a suitable domain for studying functions from M k to M . To introduce the de nition of majorant-computability we use a class of -de nable real-valued functions as a basic class. So, we recall properties of -de nable real-valued functions and subsets of IRn . The following propositions give important properties of -, -, - de nable sets and functions. Proposition 5. Let IR~ be a model of Th(IR) and 0 be the language of Th(IR). ~ n is -de nable if and only if there exists an e ective se1. A set B IR quence of quanti er-free formulas in the language 0 , fs (x)gs2! , such that W x 2 B $ IR~ j= s2! s (x):
~ n is -de nable if and only if there exists an e ective se2. A set B IR quence of quanti er-free formulas in the language 0 , fs (x)gs2! , such that V x 2 B $ IR~ j= s2! s (x): Proof. The claim is immediate from the properties of the set of hereditarily nite sets and the fact that Th(IR) admits elimination of quanti ers. ut
Proposition 6. Let IR be the standard reals. The following statements hold in HF(IR). 1. The set IN is -de nable. 2. The class of -de nable real-valued total functions is closed under composition. 3. There is an universal real-valued -de nable function in the class of -de nable real-valued functions. Proof. See [7, 12, 11] and Proposition 5.
Proposition 7. A real-valued function f : IR ! IR is -de nable if and only if the following conditions hold. 1. There exist e ective sequences fai gi2! , fbi gi2! , where ai ; bi 2 IR0 [ f+1; ?1g for i 2 !, such that
dom(f ) =
[
i2!
\
hai ; bi i and haj ; bj i hai ; bi i = ; for i 6= j :
2. There exists e ective sequences of algebraic functions fgigi2! such that f coincides with gi on the interval hai ; bi i for all i 2 !. Proof. See [12] and Proposition 5.
~ be a model of Th(IR). Theorem 8 (Uniformization). Let IR n
~ which is {de nable by a {formula there exists For any subset of IR ~ n?1 ! IR ~ such that an {de nable function f : IR ~ j= 9y(x; y)g: 1. dom(f ) = fxjHF(IR) ~ j= (x; f (x)) : 2. for x 2 dom(f ) we have HF(IR) Proof. See [12] and Proposition 7.
To introduce the notion of majorant-computability for partial real-valued functions we need a notion of prime enlargement of IR. De nition 9. A model IR of Th(IR) is called a prime enlargement of IR if there such that t > n for every natural n (we write t > IN) and IR is exists t 2 IR the real closure of the ordered eld IR(t). In addition, we write t > IR if t > r for all r 2 IR.
De nition 10. Let IR be a proper elementary enlargement of IR and let t > IN. ! IR [ f?1; +1g We de ne the following function sp : IR 8 < x 2 IR if for j x ? x j= " the condition 0 < " < IR holds ; sp(x) = : +1 if x > IR; ?1 if x < IR : is nite if and only if ?1 < sp(x) < +1. The set of all nite An element x 2 IR +
is denoted by Fin(IR). elements of IR j= Th(IR), IR is a proper elementary enlargment of IR with the same Since IR true rst-order formulas. Recall properties of prime enlargements and proper elementary enlargements of IR.
Proposition 11. 1. Every two prime enlargements of IR are isomorphic. 2. There exists a nonstandard element t > IN in a proper elementary enlargement of IR. Proof. See [10, 17].
Lemma12. Let p(x; y) be a countable set of rst-order formulas with one variable x and real parameters y. Given proper elementary enlargements IR , IR of IR, p(x; y) holds in IR if and only if p(x; y) holds in IR . Proof. The claim is immediate from 0-minimality of IR and IR . ut Lemma13. Let IR be a prime enlargement of IR. For every -formula '(z; x) there exists a -formula ' (x) such that, j= '(t ; x) $ HF(IR) j= ' (x): for all x 2 IRn and t > IN, HF(IR) Proof. The claim is immediate from 0-minimality of IR. ut Lemma14. Let IR be a model of Th(IR) and let B be a countable subset of IR . then there exists a -de nable function If B is -de nable in HF(IR) h : IN ! IR numbering B . Proof. The claim is immediate from Proposition 1 and Proposition 5. ut 1
1
2
2
1
2
3 Majorant-Computable Functions Let us recall the notion of majorant-computability for real-valued functions presented in [13]. We would use as a basic class the class of -de nable total func n ! IR, where IR is a proper elementary enlargement of IR. tions of type f : IR A real-valued function is said to be majorant-computable if we can construct a special kind of nonterminating process computing approximations closer and closer to the result.
De nition 15. A function f : IRn ! IR is called majorant-computable if there exist e ective sequences of -formulas fs (a; x; y)gs2! and fGs (a; x; y)gs2! ,
of IR such that with a parameter a, a proper elementary enlargement IR the following conditions hold. such that t > IN; 1. There exists t 2 IR 2. For all s 2 !, the formulas s (a; x; y) and Gs (a; x; y) de ne total functions fs and gs as follows: n n ! IR and gs : IR ! IR, a. fs : IR j= s (t; x; y), b. fs (x) = y $ HF(IR) j= Gs (t; x; y); gs (x) = y n$ HF(IR) , the sequence ffs (x)gs2! increases monotonically; the sequ3. For all x 2 IR ence fgs (x)gs2! decreases monotonically ; n, 4. For all s 2 !, x 2 dom(f ), fs (x) f (x) gs (x) and, for all x 2 IR fs (x) gs (x); 5. f (x) = y $ lims!1 sp(fs (x)) = y and lims!1 sp(gs (x)) = y. The sequence ffs gs2! in De nition 15 is called a sequence of lower -approximations for f . The sequence fgs gs2! is called a sequence of upper -approximations for f . As we can see the process which carries out the computation is represented by two e ective procedures. These procedures produce -approximations closer and closer to the result. If the computational process converges to in nity the procedures produce nonstandard elements like ?t or t, where t > IR. So, using a proper elementary enlargement of IR in the De nition 9 we admit only precise computations at nite steps. Since a prime enlargement is e ectively constructed over IR, this approach admit us to consider computability of partial functions in a natural way. The following theorem connects the graph of a majorant-computable func tion with validity of a nite formula in the set of hereditarily nite sets, HF(IR) (where IR is a proper elementary enlargement of the standard real numbers). De nition 16. Let IR be a proper nelementary enlargement of IR. A formula ! IR in the model HF(IR) if the following is said to determine a function f : IR statement holds f (x) = y $ ? j= (x; y) for x 2 IR and fsp(z ) j (x; z )g = fyg for y 2 IR: HF(IR)
Theorem 17. For all functions f : IRn ! IR, the following assertions are equiv-
alent: 1. The function f is majorant-computable. IR and a -formula that determines 2. There exist a prime enlargement IR a function F in HF(IR) with the property F jIR = f . ^ IR 3. There exists a -formula that in any proper elementary enlargement IR determines a function F with the property F jIR = f . Proof. 1 ! 2) Let f be majorant-computable. By Proposition 11, without loss of IR, of generality, we may assume that there exist a prime enlargement IR
a sequence ffsgs2! of lower -approximations for f , and a sequence fgsgs2! of upper -approximations for f . Let t > IN. Denote B = Q [ ftn j n 2 Qg: we construct a sequenUsing ffs gs2! , fgsgs2! , and co nality of B in IR, ce ffs gs2! of new lower -approximations for f and a sequence fgsgs2! of new upper -approximations for f such that, for all s 2 !, the ranges of fs and gs are subsets of B . Denote: D1 (x) = fz 2 B j 9s (fs (x) z )g; D2(x) = fz 2 B j 9s (gs (x) z )g : The sets D1 (x) and D2 (x) are -de nable and countable; so, by Lemma 14, there exist a function h1 (n; x) numbering D1 (x) and a function h2 (n; x) numbering D2 (x). Next, put fs (x) = maxns h1 (n; x); gs (x) = minnsh2 (n; x) : By construction, the sequences ffsgs2! and fgsgs2! are what we seek. Let j= Gs (t; x; y) ; j= s (t; x; y); gs (x) = y * fs(x) = y * ) HF(IR) ) HF(IR) where s , Gs are -formulas. From Lemma 12 it follows that, for all s 2 !, n there exist -formulas s and Gs such that, for every x 2 IR and y 2 IR : j= Gs (t; x; y) $ G j= s (t; x; y) $ HF(IR) s (x; y ); HF(IR) s (x; y ) : Let ; fs (x); gs (x) 2 Fin(IR) g for s 2 !: ds = fx j x 2 Finn (IR) the set ds is -de nable. It is easy to see that By -de nability of Fin(IR), ds !s!! dom(f ). Put (x; y) * ) 8s8y18y2 [(x 2 ds ) ! (( s (x; y1 ) ^ Gs (x; y2 )) ! (y1 y y2 ))] : ! IR. The -formula determines a function F : IR Let us prove that F j IR = f . Let f (x) = y. Suppose the contrary: F (x) 6= y. By De nition 16, there exist s 2 !; y1 ; y2 2 IR such that j= (x 2 ds ) ^ HF(IR) s (x; y1 ) ^ Gs (x; y2 ) ^ (y 62[y1 ; y2 ]) : By the constructions of fs , gs and the de nition of ds , there exist z1 ; z2 2 IR
such that j= s (t; x; z1 ); gs (x) = z2 $ HF(IR) j= Gs (t; x; z2 ) : fs (x) = z1 $ HF(IR) Note that x 2 IRn and z1 ; z2 2 IR. So, we have j= j= G fs(x) = z1 $ HF(IR) s (x; z1 ); gs (x) = z2 $ HF(IR) s (x; z2 ) and j= (x 2 ds ) ^ HF(IR) s (x; z1 ) ^ Gs (x; z2 ) ^ (y 2 [y1 ; y2 ]) :
Note that if a formula G(x; y) de nes the function f in HF(IR) then 6j=G(x; y) for x 2 dom(f ), y62IR. It is easy to see that the formulas HF(IR) s , Gs de ne functions in HF(IR), so y1 = z1 and y2 = z2; a contradiction. If F (x) = y for x 2 IRn and y 2 IR,
fsp(z ) j
^
i2!
(fi(x) z gi (x))g = fyg :
It follows that lims!1 sp(fs (x)) = y and lims!1 sp(gs (x)) = y. So, f (x) = y and the formula is what we seek. 2 ! 3) This is obvious. be a prime enlargement of IR and let a -formula (x; y) deter3 ! 1) Let IR mine the function F such that F jIR = f . From Proposition 5 it follows that there exists an e ective sequence of quanti er-free formulas f'i (x; y)gi2! such that ^ j= (x; y) $ HF(IR) j= 'i (x; y) : HF(IR) i2!
Denote Put
s (x; y) * )
^
is
'i (x; y) :
(
fy j s (x; y)g if inf fy j s (x; y)g exists; fs (x) = inf ?t otherwise, ( fy j s (x; y)g if supfy j s (x; y)g exists; gs (x) = sup t otherwise, where t > IN : Then ffs gs2! and fgsgs2! are the sought sequences. ut We remark that if f is a total function then the sequences ffs gs2! and fgs gs2! can be constructed to converge uniformly to f on IR.
As a corollary from the previous theorem we note that whenever we want to prove some statements to de ne a majorant-computable function, without loss of generality, we can assume that the proper elementary enlargement of IR in De nition 15 is a prime enlargement of IR. Proposition 18. Let f : IRn ! IR be majorant-computable. ? Then dom(f ) is 2 -de nable by a formula of type (8z 2 IR) (z; x) , where is a -formula. Proof. Let ffs gs2! be a sequence of lower -approximations for f , and let fgsgs2! be a sequence of upper -approximations for f . We obtain: x 2 dom(f ) $ HF(IR) j= (8" 2 IR)(9s 2 IN) fs (x) ? gs (x) " : By -de nability of the natural numbers, the set dom(f ) is 2 -de nable. ut
Corollary 19. The domain of a majorant-computable function is the e ective intersection of intervals with algebraic endpoints.
Proof. The claim is immediate from Proposition 5 and Proposition 18.
ut
De nition 20. Let a function f : IRn ! IR be total. The epigraph of f is de ned to be the set U = f(x; y) j f (x) < yg. The ordinate set of f is de ned to be the set D = f(x; y) j f (x) > yg. Corollary 21. Let f : IRn ! IR be a total function. The function f is majorant-computable if and only if the epigraph of f and the ordinate set of f are -de nable sets. Proof. It is clear that (x; z ) 2 U $ HF(IR) j= 9s9y [ (s 2 IN) ^ G (x; y) ^ (z > y)] ; (x; z ) 2 D $ HF(IR) j= 9s9y [ (s 2 IN) ^ F (x; y) ^ (z < y)] ; where the formulas G and F are those in the proof of Theorem 17.
ut
Corollary 22. Let f : IR ! IR.
1. If f is a -de nable real-valued function, then f is majorant-computable. 2. If domf IN and imf IN, then f is majorant-computable if f is a partial recursive function. 3. If f is a majorant-computable total function, then f is a piecewise continuous function.
Proof. The claims follow from the de nition of majorant-computability.
ut
4 Majorant-Computability and PR-, ES-Computabilities over the Reals Let us denote the computability proposed in [16] as PR-computability. Recall the de nition of PR-computability for real-valued functions.
De nition 23. A total function f : IR ! IR is PR-computable if and only if 1. it maps computable sequences to computable sequences, 2. it is e ectively uniformly continuous on intervals [?n; n]. Let us denote the computability proposed in [6] as ES-computability. Recall the de nition of ES-computability for real-valued functions. We recall the de nition of the interval domain I : I = f[a; b] IR j a; b 2 IR; a bg [ f?g : The relation is de ned as follows: ? J for all J 2 I and [a; b] [c; d] if and only if a < c and b > d.
De nition 24. Let I0 = fb1; : : : ; bn; : : :g [ f?g be the e ective enumerated set of all intervals with rationals endpoints. A continuous function f : I ! I is computable, if the relation bm f (bn ) is r.e. in n; m, where bm ; bn 2 I0 . De nition 25. A function f : IR ! IR is ES-computable if and only if there is an enlargement g : I ! I (i.e., g(fxg) = ff (x)g for all x 2 dom(f ) ) which is computable in the sense of the De nition 24. Let us de ne gx;y(= [x;y]; g^x;y = [x;y); if x 2 M; where M (x) = unde ned1 otherwise . De nition 26. A continuous function h : [a; b] ! IR is called a piecewise linear function if there exist x ; : : : ; xn 2 IR such that a = x < x < : : : xn = b, and h j x ;x +1 is a linear function for all 0 i m. We de ne the code [h] of h as follows: [h] = f< xi ; h(xi ) >j 0 i mg: 0
[
i
i
0
1
]
Let us consider functions from < a; b > to IR. Proposition 27. Let f :< a; b >! IR be a continuous majorant-computable function. The function f is PR-computable if and only if the relation Rf = f< x; y; z >2 [a; b]2 IR j x < y ^ f j[x;y] > zgx;y g is -de nable in HF(IR). Proof. !) By the de nition of PR-computability, f is an e ectively uniformly continuous function. It follows that the set Qf = < x; y; z; >2 [a; b] IR2 j jjf ? zgx;y jj[x;y] < is -de nable. For x; y 2 IR such that x < y, the following equivalence holds. ? 3 9 < b ; : : : ; b >2 Q
?1 ?n^ i=0
1
n
jbi ? bi j < = wf ( 3 ) ^ b = x ^ bn = y ^ +1
0
n^ ?1 i=0
jfs (bi ) ? z j < 3 ;
where w is an e ective modulus of continuity for f . It follows that < x; y; z >2 Rf if and only if there exist > 0 and a step-function h : [x; y] ! Q such that h > zga;b + and jjf ? hjj[x;y] < . S ?1 ? For arbitrary x0 < : : : < xi < : : : < xm = y and h = m i=0 zi g^x ;x +1 , jjf ? hjj[x;y] = max0im?1 jjf ? zi g^x ;x +1 jj. So the set Rf is -de nable in HF(IR). i
i
i
i
) Let Rf be -de nable in HF(IR). For every = 1s > 0, where s 2 !, we will e ectively construct piecewise linear functions fs : [a; b] ! IR, gs : [a; b] ! IR
with the following conditions: 1. fs (x) f (x) gs (x) for all x 2 [a; b]; 2. jjfs ? gs jj . In fact, by -de nability of Rf , the set
Tf = < x; y; z; t >2 [a; b] IR j x < y ^ 8u 2 [x; y] f (u) > z + uy(t??xz ) is -de nable. It follows that the set Lf = f< s; [f ]; [g] >j s 2 !; f; g are piecewise linear with the condition 1-2g is -de nable. Using uniformization (see Theorem 8) we can construct the required functions fs ; gs . Obviously, by the code [h] Q2 of a piecewise linear function, we can e ectively construct a function wh : [0; 1] ! IR with the following properties: 1. wh is a linear function; 2. (8 2 (0; 1]) (8x; y 2 [0; 1]) (jx ? yj < wh () ! jh(x) ? h(y)j < ) As we can see, wh is a modulus of continuity for h. Let us de ne the function wf as follows: wf = wf (), where s() = [ 3 ] + 1. It is easy to see that wf is -de nable, and it is a modulus of continuity for f . By the de nition of PR-computability, f is PR-computable. ut It is easy to see that the continuous majorant-computable total function f is PR-computable if and only if the intersection Rf \ Q is -de nable in HF(IR). Proposition 28. If B is an open -de nable set then B is the e ective union 2
2
s
of open intervals. Proof. Let B be an open -de nable set. By Proposition 3, [ B = < i ; i > :
We represent B as follows: [ B = [ i ; i ) [
i2!
[
i2I1
i2I2
i2I1
i2I2
( i ; i ] [
[
i2I3
( i ; i ) [
[
i2I4
[ i ; i ]:
Put I40 = fi 2 I4 j i 6= i ; i 6= j ; j 2 I4 g. Because B is open, we can e ectively choose i ; i ; i , and i as follows
i < i and i 2 ( j ; j ); where i 2 I1 ; j 2 I2 [ I3 [ I40 ; i > i and i 2 ( j ; j ); where i 2 I2 ; j 2 I1 [ I3 [ I40 ; i < i and i 2 ( j ; j ); where i 2 I4 ; j 2 I1 [ I2 [ I3 ; i > i and i 2 ( j ; j ); where i 2 I4 ; j 2 I1 [ I3 : So, [ [ [ [ B = (i ; i ) [ ( i ; i ) [ ( i ; i ) [ (i ; i ): i2I3
i2I4
This means that B is the e ective union of open intervals.
ut
Proposition 29. Let f : [0; 1] ! IR be a majorant-computable continuous function. The set Rf = < a; b; c >j 0 a < b 1 ^ f j[a;b] > c is -de nable in HF(IR). Proof. Without loss of generality, let us consider the case f : [0; 1] ! [0; 1]. Let us de ne x 2 Asi $ jf (x) ? si j < 1s ; for i = 0; : : : ; s. WeShave the following properties: 1. Asi = [0; 1], 2. Asi is an open -de nable set. S j ) for some rational numbers j j . By Proposition 28, Asi = j2! ( ji;s ; i;s i;s i;s Let [a; b] [0; 1]. Let us consider the case cS=s 0. By continuity of f , f j[a;b] > 0 if and only if there exists s such that [a; b] S i=1 Asi . Really, if there exists s such that [a; b] si=1 Asi and x 2 [a; b] then x 2 Asi for some i. By the de nition of Asi , jf (x) ? si j < 1s , so f (x) > 0. s1 . If f j[a;b] > 0, then there exists s1 such that f j[a;b] > s11 . Put s = S i 1 s For some i > 0, we have jf (x) ? s j < s , i.e., x 2 Ai . So, [a; b] si=1 Asi . By compactness of [a; b], S f[a;b] > 0 if and only if there exists s such that S j ), and J is some nite subset of [a; b] si=1 A^si , where A^si = j2J ( ji;s ; i;s i;s !, 0 < i < s, i.e., f[a;b] > 0 if and only if the -condition holds. In the general case, for [a; b] and c, using similar considerations we check the following condition: [ [a; b] Asi : i;s
i:j ?cj 1 i s
s
ut
Theorem 30. Let f : [a; b] ! IR be a continuous total function.
The function f is majorant-computable if and only if f is PR-computable. Proof. !) The claim is immediate from Proposition 27 and Proposition 29. ) The claim is immediate from Corollary 21. ut
Corollary 31. Let f : [a; b] ! IR be a total continuous function.
The function f is majorant-computable if and only if f is ES-computable.
Proof. The claims are immediate from Corollary 30 in [6].
ut
Theorem 32. The class of ES-computable functions is contained in the class of majorant-computable functions.
Proof. Let f : IR ! IR be ES-computable. For n 2 !, we de ne An = fx 2 IR j (f ([x])) < n1 g, where is the natural measure de ned on I . It is easy to see that An is -de nable open set, and T dom(f ) = n2! An . S By Proposition 28, A1 = i2! ( i ; i ), where i ; i 2 Q and i i . By Lemma 14, there exist a -de nable function h : IR IN ! IR numbering the -de nable set fz 2 Q j z < f ([x])g, and a -de nable function H : IR IN ! IR numbering the -de nable set fz 2 Q j z > f ([x])g.
Put
(
S
(
S
maxks h(x; k) if x 2 ns ( n ; n ); fs (x) = ? t otherwise, ks H (x; k ) if x 2 ns ( n ; n ); gs (x) = tmin otherwise, where t > IN : Then ffsgs2! and fgs gs2! are the sought sequences. So, f is majorant-compu-
table. There exist majorant-computable functions that are not ES-computable. For example, a total step-function with a computable set of discontinuities is the sought one. ut
Corollary 33. If f is ES-computable, then the domain of f is the e ective intersection of open intervals.
Proof. The claim is immediate from the proof of Theorem 32.
ut
Corollary 34. For real-valued functions, we have the following inclusions: 1. The class of -de nable functions is contained in the class of majorant-computable functions. 2. The class of computable functions introduced by Moschovakis is contained in the class of majorant-computable functions. 3. The class of functions de nable by nite dimensional machines of Blum et al. without real constants is contained in the class of majorant-computable functions. 4. The class of PR-computable functions is contained in the class of majorantcomputable functions. 5. The class of ES-computable functions is contained in the class of majorantcomputable functions. 6. The class of computable functions introduced by Stoltenberg-Hansen, Tucker is contained in the class of majorant-computable functions.
5 Examples Consider an interesting class of real-valued total functions possessing meromorphic continuation onto C. This subclass contains, for example, solutions of known di erential equations. Let C be the set of complex numbers.
De nition 35. A function f : IR ! IR is said to admit meromorphic continuation onto C if there exists a meromorphic function f : C ! C such that f jIR = f .
We recall a fact concerning meromorphic functions in complex numbers theory [3]. Proposition 36. Let f : C ! C be a total meromorphic function. There exist entire functions a(z ) and b(z ) such that 1. f (z ) = ab((zz)) , 2. a(z ) and b(z ) have Taylor expansions with coecients in IR. Proof. The claim follows from the Schwartz principle [3]. ut Theorem 37. Let f : IR ! IR be a total function and let f be its meromorphic continuation onto C. Suppose that f can be written as f (z ) = ab((zz)) , where functions a(z ) and b(z ) are as in Proposition 36. The function f is majorantcomputable if the following conditions on f hold: 1. There exist -de nable real-valued functions A and B such that maxjzjw j a(z ) j A(w); maxjzjw j b(z ) j B (w); where w 2 IR ; 2. The coecients of the Taylor expansions for a(x); b(x) are majorant-computable as constant functions. Proof. Let f satisfy the conditions of the theorem. We shall prove that the epigraph of f and the ordinate set of f are -de nable. Denote the Taylor expansions for a(z ) and b(z ) at the point 0 as follows: X X a(z ) = ak z k ; b(z ) = bk z k : k2!
k2!
Show that a jIR and b jIR are majorant-computable. Let us consider an arbitrary disk j z j R. From the Cauchy inequality [3], for all k 2 !, we have
ja(z )j : jak j RMk ; where M = jmax zjR
This remark admits us to construct two sequences of required approximations for a jIR . It is easy to see that a jIR is majorant-computable. Similarly, b jIR is majorant-computable. It is easy to proof that if a jIR and b jIR are majorant computable, then jakIR and jbkIR are majorant-computable. From now it is evident that the epigraph and the ordinate set of the function f are -de nable sets. Corollary 2 implies that f is a majorant{computable function. ut Corollary 38. If an analytical function f has Taylor expansion with majorantcomputable coecients and the function j f j is bounded by an -de nable total real function, then f is a majorant{computable function. Since the considered language contains equality, majorant{computable functions can be discontinuous. The natural question is raised describe the situation without equality. Our forthcoming results would be devoted to this problem.
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