Program Schemes over Continuous Algebras ... - Semantic Scholar

Program Schemes over Continuous Algebras (Extended Abstract) Wilson R. de Oliveira, Jr.  Departamento de Informatica UFPE - CCEN Caixa Postal 7851 - CEP 50732-970 Recife - PE, Brazil wrdodi.ufpe.br http://www.di.ufpe.br/~wrdo

Abstract

Motivated by the real numbers representation as lazy lists [4] a notion of continuous algebras was introduced in [9, 7] which generalizes both the ordered [15] and metric approaches [25]. Here we investigate the notion of program scheme acting on this algebras with a view to study notions of computability and complexity of real functions. A byproduct of the present and future work is to provide the scienti c computing programmer with tools from the formal speci cation theory in order to achieve a rigorous development and analysis of numerical programs.

Classi cation of topics covered:

program schemes, abstract data types, program speci cation, exact real arithmetic, lazy functional languages, continuity spaces, computability and complexity of real functions.

1 Introduction In order to amend the relative neglect that the formal methods researchers treat numerical computation in contrast to the seminumerical one, despite the fact that numerical programs are being used in vital activities on our day-life, we have on previous occasions put forward a programme [9, 7, 8] aiming at the speci cation of the reals as an abstract data type. For that we use one of the most widely used and successful approach to the speci cation of data types: the algebraic speci cation method. Here we investigate the semantics of nondeterministic recursive program schemes acting on the continuous algebras as in [9, 7] with a view to study computability and complexity of real functions. Program schemes is a formalization of the control structure of programs [6]. In this work program schemes is the language to express the computable functions over the reals seen as abstract data type. Recent work on the study of restrict notions of computability characterizing complexity classes (see [5] for an excellent survey) can be seen as the study of restricted subclasses of recursive program schemes. The connections with real numbers computations remains to be investigated. Algebraic speci cation methods provide techniques for data abstraction and the structured speci cation, validation and analysis of data structures. The data type being speci ed is then the term algebra quotient out by the equations describing its properties. The initial algebra approach to speci cation of data types has proposed solutions to deal with data of in nity elements ([15, 25]) via completion of the nite ones but their solutions does not cover the lazy reals. The mainstream approaches come then in two avors: ordered and metric. The ordered approach uses continuos cpos and ideal completion [15]; and the metric approach uses ultrametric spaces and Cauchy sequence completion [25]. 

Supported by the Brazilian Research Council CNPq-RHAE, grant no. 610293192.3.

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One drawback of the ordered approach to continuous data types is that domains (Scott domains, algebraic cpos, continuous cpos) are not generally closed under quotients [19]. In its turn the metric approach is not able to deal with partial information. Besides, to date, the metric approach is restricted to ultrametric spaces and then are unable to capture the real line. The author has suggested a common generalization of the above two approaches by using quasimetric spaces [9] and continuity space [7], spaces on which a non symmetric distance function is de ned. Most of the drawbacks of the classical approaches (see below) to continuous data types disappear. As we shall see in Section 2 the space of the lazy reals is not a domain (algebraic nor continuous cpos) of any kind and can not be seen as an ultrametric space. The reals as data type is an important issue in the analysis and semantics of numeric programs [4]. In what follows a variant of the redundant balanced radix notation [17] is employed. In this representation we allow digits to be negative as well as positive. For a more complete study of this representation in contrast to others see [4]. What it is important for the work here is its simplicity. None of the results depend on the chosen representation. The redundant balanced radix notation is also known as signed digits representation. In its simplest form, which we use below, it is also known as tritstreams representation or modi ed binary representation, since the only digits allowed are in the set T = f?1; 0; 1g. We shall, w.l.o.g, limit ourselves to the discussion of reals in [?1; 1]. It is assumed the reader is familiar with the basic notions of functional analysis, domain theory, algebraic and continuous data types as e.g. in (respectively) [22], [23] and [15]. We present the basic ingredients of the approach to semantics via continuity spaces but leaving the interested reader to ll in the gaps in the literature (q.v. [12, 11, 13])

2 The Lazy Reals One of the most widely used representation of the reals in both practical and theoretical computer science is the signed digit representation (q.v. [4, 14, 18, 26, 16]). The reason comes from its simplicity, computability of the arithmetic operations, and easy implementation in lazy functional languages. Here a real number is represented by an in nite list or string. The reals represented in this way are usually called Lazy Reals since they can be implemented in a lazy functional language as a lazy list. Numbers in the interval [?1; 1] can be represented as signed binary digits i.e. string composed of digits in T = f?1; 0; 1g or T = f?; 0; +g. Let T ! be the set of streams (in nite strings) on T = f?; 0; +g. The real number represented by the P stream  = a1 a2 : : : is de ned as [ ] R = i>0 ai  2?i . A string  = a1 : : : an 2 V  represents the set of possible continuations of ; hence, an interval. This de nes a map [ ] I from T  to the closed intervals

P

P

[ a1 : : : an ] I = [( ni=1 ai  2?i ) ? 2?n ; ( ni=1 ai  2?i ) + 2?n ] = [[[a1 : : : an ?! ] R ; [ a1 : : : an +! ] R ]

Notation 2.1 We use [ ] to denote the function which behaves as [ ] I on the strings and as [ ] R on streams.

We have two ways of looking at the lazy reals. As the set of nite and in nite strings on T ; and as the set of closed intervals plus the real line. Actually, the set T 1 (= T  [ T ! ) needs to be quotiented out by the equivalence relation   i [ ] = [ ] . An information order v can then be de ned on the equivalence classes. Indicating by [] the equivalence class containing , [] v [ ] i [ ]  [ ] . The equivalence on strings can be described more operationally as: 2

0+x=+?x 0?x=?+x

(1) (2)

where x 2 T 1, which generalizes the obvious arithmetic fact that 0+ 14 = 12 ? 41 and that 0 ? 41 = ? 12 + 41 : The reader familiar with data type speci cation would suggest the following approach to the speci cation of the \real domain":

 start with nite lists of f?; 0; +g,  subject them to the equations (0) and (1). Quotienting out the (pre x) order on the nite lists by these equations, in a standard way, gives us

in eect the appropriate closed intervals with the intended order. The approach mostly used in the literature coming from the continuous data type theory ([15]), would take the ideal completion of the poset of closed dyadic intervals. This yields a Scott domain [23] but gives us three versions of each dyadic rational: one approximating from the \left", other from the \right" and the third one from \both" directions at the same time. Another approach is to take a continuous quotient and use a notion of normalizer [19]. But, for a start our equivalence relation is not continuous. There obviously exists (q.v. [15]) a least continuous extension of our equivalence which would identify the three copies of each dyadic rational. The trouble is that the resulting space is a partial order which is far from being a domain of any kind. It is not even a continuous cpo. In particular no nite string is compact nor is well below any in nite strings [7]. When dealing with in nite objects one usually works on the nite portions of them and get the results on the in nite via some sort of completion. That is, in a nutshell, the methodology of Computability Theory, Domain Theory, Denotational Semantics, Constructive Mathematics and some approaches to continuous data types. We have seen above that we can not follow this methodology when dealing with the real domain in the cpo approach. It is worth mentioning that the use of metric space is also not adequate, rstly by the fact that we can not represent partial or nite information (metric spaces are necessarily Hausdor) and secondly that the approach to metric data types in the literature (q.v. [25]) uses ultrametrics (those space are necessarily zero dimensional). Our proposal is then to use continuity spaces [9, 7]. Our approach the speci cation of reals as an algebraic data type is new. Despite the use of similar real number representation found in the literature one should stress the dierences. In [4] is shown that no set of sequential primitives are sucient to de ne the reals as an abstract data type but is left open the use of non-sequential ones. [14] restricts to algebraic cpo case (and as a result the reals are obtained indirectly via retracts). [21] uses quasiuniformities and do obtain the real line as the subspace of total elements but no notion of continuous algebras is introduced. The same can be said to [10] which uses continuous cpos and uses the cpos of intervals which, dierently from ours, has an in nite number of generators; it is similar in spirit to [14] (in the sense that extends a ?calculus based language with a type of real numbers). While all the approaches cited above, including mine, obtain the reals as completions those of [3, 26, 16] start with the reals already given in way or another. [3] assumes the all reals are available for manipulations and de nes computations and its complexity over the ordered eld of real numbers by Random Access Machine model; this is a departure from the standard eectivity theory of computations in the sense that assumes non (Turing) computable primitives such as the equality relation over the reals and does no take as computable very well known computable functions such as the exponential and square root functions. [26] by using representations applies his well 3

developed Type 2 Theory of Eectivity to the real numbers represented as in nity strings (exactly the representation used in this work) where computations are performed by oracle Turing machines. [16] proposes a modi ed real RAM model which ends up being polynomially equivalent to the Turing model of [26].

3 Continuity Spaces We quickly survey the main results and de nitions of continuity spaces needed in Section 4. The reader interested in more details may consult [12, 11, 13] The bottom element of a complete lattice V is denoted by 0, the top element by 1 and for A a subset of V , the least upper bound of A is denoted by sup A and the greatest lower bound of A by inf A.

De nition 3.1 A value distributive lattice is a completely distributive lattice V satisfying the following two conditions: 1. 1  0. 2. if p  0 and q  0, then p ^ q  0.

De nition 3.2 A quantale V = hV; ; +i consists of a complete lattice hV; i and an associative and

commutative binary operation + on V satisfying: 1. for all p 2 V , p + 0 = p;

2. for all p 2 V and all families fqi gi2I of elements of V , p + inf i2I qi = inf i2I (p + qi). De nition 3.3 A value quantale is a quantale V = hV; ; +i such that hV; i is a value distributive lattice.

De nition 3.4 A V -continuity space is a pair X = (X; d) consisting of a set X and a function d : X  X ! V satisfying the following conditions: 1. for all x 2 X , d(x; x) = 0; and 2. for all x, y, z 2 X , d(x; z ) + d(z; y)  d(x; y).

If X = (X; d) is a V -continuity space, then the dual of X is the pair X  = (X; d ), where for all x, y 2 X , d (x; y) = d(y; x) and the symmetrization of X is the pair X s = (X; ds ), where for all x, y 2 X , ds(x; y) = d(x; y) _ d(y; x). It easily follows that X  and X s are V -continuity spaces. There is a natural topology on a V -continuity space X = (X; d), which is de ned in a way completely analogous to the de nition of the metric topology on a metric space. And so are all notions such open balls, closed ball, etc. and their decorated version *-open, *-closed, s-open, s-closed, etc. Recall that the specialization order on a topological space (X;  ), denoted by  , is de ned by: x  y i x 2 clfyg. Then  is always re exive and transitive, and it is a partial order on X i X is T0 . A V -continuity space X = (X; d) is said to be T0 if d(x; y) = 0 and d(y; x) = 0 imply x = y, for all x, y 2 X. A number of basic operations on continuity spaces, which are needed to build up complex data types from primitive ones, can be de ned: Cartesian and tensor products, coproduct and function space construction [12]. 4

A map f : X ! Y between V -continuity spaces X and Y is nonexpansive if for all x1 , x2 2 X , dY (f (x1 ); f (x2 )  dX (x1 ; x2 )). De nition 3.5 X is called a V -domain if it is T0 and compact in its symmetric topology. Let V -Dom denote the category with objects the V -domains and morphisms the functions which are continuous relative to the induced topologies. A proof that V -Dom is closed under elementary type forming operations is found in [12]. A net (x )2 in a V -continuity space X is Cauchy if for every "  0 there is a 0 such that for all ,   0 , d(x ; x )  ". X is complete if every Cauchy net in X has a limit in the symmetric topology on X . Cauchy completeness in continuity spaces and its relationship to natural order theoretic notions of completeness are studied in [11]. X is totally bounded if for all "  0 there is a nite F  X such that X = [y2F N"s (y). The proof of the next result is just a translation of the usual argument for the corresponding result on uniform spaces. Theorem 3.6 Assume X = (X; d) is a T0 V -continuity space. Then X is a V -domain i X is complete and totally bounded. Powerdomains provide domain-theoretic analogs of the power set. Their consideration is motivated by the need to model nondeterministic constructs. The standard results of the theory of the upper powerdomain can be adapted to the setting of V -domains. The lower and convex powerdomains can also be adapted to this setting but we do not use them here. Assume X is a V -continuity space. De ne the upper Hausdor distance, dU , on the power set of X as dU (A; B ) = sup ainf d(a; b); A; B  X: 2A b2B

The upper powerdomain of X , U (X ) is de ned to be the collection of nonempty ?closed subsets of X with the upper Hausdor distance. The justi cation for our use of the upper powerdomain lies in the fact that this is the one which matches with the intuition that non-determinism is lack of de nition. The lesser deterministic the more de ned. For us nondeterminism is just the impossibility of eectively specify some relations on the reals such as the equality relation. Notation 3.7 Denote by 2 the value quantale f0; 1g with 0  1 and 1 + 1 = 0 + 0 = 0 and 1 + 0 = 0 + 1 = 1; by 2 the value quantale f2?n j n 2 N g [ f0g with the usual order and max; by I the value quantale of real numbers in [0; 1] with the usal order and the truncated addition x + y = minfx + y; 1g. Example 3.8 On the algebraic cpo T 1, de ne a 2 -valued distance by: d(x; y) = 2? maxfnjx[n]y[n]g where x[n] denotes the n-truncation of x, that is, the result of deleting all terms of x after the rst n. Notice that d(x; y) = 0 i x  y. We see that d induces the Scott topology on T 1 , and is a 2 -domain since T is nite. Example 3.9 Let D be any Scott domain, and r : BD ! N a map (a `rank function') such that r?1 (n) is a nite set for each n 2 N . De ne a 2 -valued distance by: d(x; y) = 2? maxfnjevx)evy for every e of rank  ng Then d is a 2 -domain and induces the Scott topology of D. The two examples above could as well be de ned as a I-valued distance given rise to dierent kinds of spaces, as we shall see in a moment. 5

As we have hinted at above, an important part in the speci cation of an abstract data type is to obtain a quotient of a space. As usual the quotient space X=R of a space X by a given equivalence relation R is de ned in such a way that the structure on X=R is nal with respect to the canonical map x 7! [x] of X onto X=R (in a suitable category). For (quasi)metric spaces the suitable category for constructions has as morphisms the non-expansive maps (q.v. [1]). The situation is surprisingly similar here. De nition 3.10 Given a continuity space (X; d) and an equivalence relation R we de ne the quotient continuity space X=R = (Y; dY ) as the greatest distance on Y for which [:] is non-increasing, i.e. dY ([x]; [y])  d(x; y): Concretely, let  = hx0 ; x1 ;


; x2n+1 i be a path, i.e. x2i?1 Rx2i (i = 1;


; n), whose length is l() = supni=0 d(x2i ; x2i+1 ). The triangle inequality requires that dY ([x0 ]; [x2n+1 ])  l(): Thus we de ne dY by: dY (u; v) = inf fl()j is a path from x to y where [x] = u; [y] = vg:

Observation 3.11 It can easily be shown that the de nition of quotient space above is coherent with

the categorical notion of quotient structure of [1] which, by applying their general de nitions to our case, says that X=R is the quotient continuity space of (X; d) if the canonical map R : (X; d) ! X=R has a co-optimal lift d0 . A function f : (X; d) ! Y has co-optimal lift if there exists a distance d0 such that f : (X; d) ! (Y; d0 ) is non-expansive and whenever (Z; d00 ) and g : Y ! Z are such that g:f : (X; d) ! (Z; d00 ) is non-expansive,

f

- (Y; d0 ) @@ ? ? g:f @@ R 00 ?? g

(X; d)

(Z; d )

then g : (Y; d0 ) ! (Z; d00 ) is non-expansive.

4 Continuity ?algebras The trouble with the discussion in Section 2 is that we have forgotten the quantitative structure. A continuity can be de ned on T 1 as in the Example 3.8 and 0! , +?! are to be identi ed because their distance is (to be made) 0: n n d(0! ; +?! ) = d(nlim !1 0 ; nlim !1 +? )  0

We assume familiarity with the basics notions of [15] and in particular with [20], where the notion of non-deterministic data type is studied. For the sake of simplicity, let us restrict ourselves w.l.o.g. to the one S sorted case. So, in the terminology of [15], hiii2N is a one sorted operator domain and  = i2N i . A nondeterministic operation  : A ! B is equally interpreted as a multivalued function f : A ! 2B or a function f : A  B ! B where B is the at domain of truth values with distance d(x; y) = 0 () x =?.

De nition 4.1 A -tree is a partial function t : N  ?!  such that, for all u 2 N  and i 2 N 1. ui 2 dom(t) ) u 2 dom(t); 2. ui 2 dom(t) ) 9n > 0 : t(u) 2 n and i < n

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Notation 4.2  CT denotes the set of all ?trees.  FT the nite ?trees (i.e. all t such that dom(t) is nite).  N (n) = fu 2 N  jlength(u)  ng = fu[k] j u 2 N  ; k  ng  t(n) = t j N (n) i.e. t(n) : N (n) !  and t(n) (u) = t(u), for all u 2 N (n) De nition 4.3 d : CT  CT ! 2 is de ned as d(t1 ; t2 ) = 2maxfnjt t g : (n) 1

(n) 2

S 1. hCT ; di is a completeS continuity space which is 2 -domain if i2N i is nite (as a set). From now on we assume i2N i is nite. 2. hCT ; di is the completion of FT (with the 2 -valued distance induced from that of CT ).

Observations 4.4

We make CT into a ?algebra in the following way: 1. For  2 0 , let CT = fh; ig;

S

2. For  2 n, n > 0, and t1 ; : : : ; tn 2 CT let CT = fh; ig [ i 3.

FT  f?; 0; +g , with the distance as in Example 3.8. E = f(0 + x; + ? x); (0 ? x; ? + x)g over FT(fxg). CT=E is isomorphic to the continuity space T  of Section 2. 8 > < y [ x]  0  cond(x; y; z) = > z 0  [ x] : w w v y&w v z; otherwise

 The completion of the totally bounded space T=E has as maximal elements a set which is homeomorphic to real line when D = I and the real line with the discrete topology when D = 2 .  The function cond is a parallel test function and cannot be implemented sequentially. In [4] it is proved that no set of sequential primitives are sucient to de ne the reals as an abstract data type. It is given in the speci cation in order to make the representation hidden from the programmer.

Another speci cation of the lazy reals which, by importing results from [10], could be useful is the following:

 0 = f?; 0; +g ; 1 = ftail; headg; 2 = fconsg; 3 = fcondg; i = ;; i > 3.  0 with the distance as in Example 3.8.  E0 = f(cons(0; cons(+; x)); cons(+; cons(?; x))); (cons(0; cons(?; x)); cons(?; cons(+; x)))g over T (fxg).  E1 = f(tail(cons(c; x)); x); (head(cons(c; x)); c)g  cond(x; y; z) de ned as above  FT=E , where E = E0 [ E1 , is isomorphic to the continuity space T  of Section 2 with the operation on continuous word of [10].

5 Program Schemes In order to de ne program scheme we need to slight extend the previous notion of -algebras to term algebras with variables. The notions and de nitions in this section are at most adaptations from [2] for which the reader is referred to for further details. The results concerning the real numbers are new.

De nition 5.1 Let U be a set, whose elements are called variables or indeterminates, disjoint from . The set FT (; U ) of all -terms in the variables of U is de ned inductively as: 1. U [ 0  FT (; U ); and 2. if  2 k and ti 2 FT (; U ), then (t1 ; : : : ; tk ) 2 FT (; U ).

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We de ne the term algebra FT (; U ) of all -terms in the variables of U and the set of -equations as usual [24]. One can readly see the possibility of extending the de nition of the distance function from FT to FT (; U ) and so how to obtain the totally bounded space CT (; U ). We shall denote by }(CT (; U ))) the upper powerdomain of CT (; U ). De nition 5.2 Given a set  = f'1 ; : : : ; 'k g of unknown funtions symbols, where 'i has arity ni , and let or be a special binary symbol. A (nondeterministic) program scheme is a set P = f'i = i j i = 1; : : : ; kg, where i 2 FT ( [  [ forg; Xn ): In the sequel we assume w.l.o.g. that the rst unknown function symbol, '1 above, has arity k and then is called the main program. Recalling that (t) is the sort of the term t, we denote by (P ) the sequence (x1 )


(xk ) and (P ) the sort of '1 . We then say that (P ) is the arity of P , that (P ) is its sort and that (P ) ! (P ) is its pro le. An ?interpretation D is a continuous ?algebra. A semantical mapping associates with a program scheme P of pro le s1


sn ! s and with and ?interpretation D the function computed by P in D, denoted by PD : Ds 


 Ds ! Ds . Out of several distinct semantical mappings which can be de ned (call-by-name, call-by-value, call-by-need, etc, see [6]) we shall only need for our purposes the call-by-need one. De nition 5.3 Given a program scheme P , we associate with it a rewriting system, also denoted P : 'i ! i ; i = 1; : : : ; k or(x1 ; x2 ) ! x1 or(x1 ; x2 ) ! x2 De nition 5.4 A rewrite pattern  for a nite tree t 2 FT ( [  [ forg; Xn ) is a set of pairs ho; ri where o is an occurrence of a symbol of  [ forg and r is a rule for rewriting this symbol. If the symbol is or, the corresponding rule is one of the ori : or(x1 ; x2 ) = xi , i 2 f1; 2g. Let us denote by R(t) the set of rewriting patterns for t. Formally  2 R(T ) i  is a nite set fhoi ; ri ig such that i

1

k

1. oi 2 dom(t) and t(oi ) 2  [ forg; ( tj if t(oi ) = 'j ; 2. ri = 'orj ! 1 or or2 if t(oi ) = or Applying the rules of P to t according to a rewriting pattern  2 R(t), we get a tree (t) which is obtained by simultaneously applying each rule ri to the occurrence oi in t.

De nition 5.5

A nite computation from t is a nite sequence  = t0 ; 1 ; t1 ; : : : ; n ; tn such that 1. t0 = t; 2. i 2 R(ti?1 ) and ti = i (ti?1 ) for i = 1; : : : ; n. A in nite computation from t is a in nite sequence  = t0 ; 1 ; t1 ; : : : ; n ; tn ; : : : such that 1. t0 = t; 2. i 2 R(ti?1 and ti = i (ti?1 ) for all i  1. We denote by
 (t)(resp.
1) the set of nite (resp. nite and in nite) computations from t.

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With the above de nitions in the full version of the present paper we prove consistency, soundness and completeness results of program schemes acting on the data types speci ed in Section4.

6 Conclusions and further directions A natural follow up to this work is to investigate the notion of eective continuous algebras and to arrive at results of adequacy and universality. Examing the recent work on the study of restrict notions of computability characterizing complexity classes (see [5] for an excellent survey) we see the prominent role played by restricted subclasses of recursive program schemes. The connections with real numbers computations should be investigated.

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