Lambda Calculus and Products e : t e1 : s! t e2 : s e : s t e1 : s e2 : t x s : s x s :e : s! t e1 e 2 : t () : unit 1 e : s 2 e : t (e1 ; e 2) : s t

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1 Conservativity of Nested Relational Calculi with Internal Generic Functions Leonid Libkin and Limsoon Wong Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA , USA. flibkin, Abstract It is known that queries in nested relational calculus are independent of the depth of set nesting in the intermediate data and this remains true in the presence of aggregate functions. We prove that this continues to be true if the calculus is augmented with any internal generic family of functions. 1 Introduction Paredaens and Van Gucht [7] proved that the nested relational calculus is no more expressive than the traditional relational calculus when input and output are at relations. That is, every query on input and output whose depth of set nesting is at most 1 (or at relations) can be expressed without using any intermediate data whose depth of set nesting is greater than 1. This result was generalized by Wong [10] who showed that every query on input and output whose depth of set nesting is at most k can be expressed without using any intermediate data whose depth of set nesting is greater than k. Hence the expressive power of the nested relational calculus is independent of the depth of set nesting allowed in the intermediate data. This property is called the conservative extension property. Libkin and Wong [5] showed that when the nested relational calculus is endowed with aggregate functions, it retains the conservative extension property. The aim of this paper is to demonstrate that the nested relational calculus continues to retain the conservative extension property even in the presence of any family of functions that are generic and do not invent new values. The strategy is to combine the rewriting technique of Wong [10] and the eective lifting of linear orders of Libkin and Wong [5] to encode any such function into one whose height is minimum. To appear in Information Processing Letters. 1

2 Lambda Calculus and Products e : t e1 : s! t e2 : s e : s t e1 : s e2 : t x s : s x s :e : s! t e1 e 2 : t () : unit 1 e : s 2 e : t (e1 ; e 2) : s t Set Monad fg s : fsg e : s feg : fsg e1 : fsg e2 : fsg e1 [ e2 : fsg e1 : ftg e2 S : fsg fe1 j x s 2 e2g : ftg Booleans true : B false : B if e1 : B 2 : t 3 : t e1 then e 2 else e 3 : t 2 Nested relational calculus We use the nested relational calculus N RC as presented in Breazu-Tannen, Buneman, and Wong [3]. In this section, it is extended with natural numbers, simple arithmetics, and a summation operator. Types. A type in N RC is either a complex object type or is a function type s! t where s and t are complex object types. The complex object types are given by the grammar: s; t ::= b j B j unit j s t j fsg Objects of type B are the two boolean values true and false. The unique object of type unit is denoted by (). Objects of type s t are pairs whose rst (or left) components are objects of type s and second (or right) components are objects of type t. Objects of type fsg are nite sets of objects of type s. We also included some uninterpreted base types b. Expressions. Expressions of N RC are constructed using the rules in the gure. The language also contains some uninterpreted constants c of base type T ype(c) and uninterpreted functions p of function type T ype(p). The type superscripts are omitted in the rest of the paper because they can be inferred [6]. The semantics of N RC was described in [3]. We repeat the meaning of the set monad constructs here. fg is the empty set. feg is the singleton set containing e. e 1 [ e 2 is the union of sets e 1 and e 2. The construct S fe 1 j x 2 e 2 g denotes the set obtained by rst applying the function x:e 1 to elements of the set e 2 and then taking their union. The shorthand fo 1 ; : : : :o n g is used to denote fo 1 g [ : : : [ fo n g, provided o 1,..., o n are distinct objects. As it stands, N RC can merely express queries that are purely structural. It was shown in [3] that endowing N RC with equality test = s : s s! B at all types s elevates N RC to a fully edged nested relational language (in fact, equivalent to classical nested relational algebra of [8]). That is, operations such as nest, membership test 2 s : s fsg! B, subset test s : fsg fsg! B, set intersection, set 2

3 dierence, etc. are expressible in N RC(=) (we write the additional primitive in brackets to distinguish various extensions of the language). Practical database query languages frequently have to deal with queries such as \select maximum of column," \select count from column," etc. To handle this kind of queries, we add natural numbers (whose type is denoted by N) and the following constructs: e 1 : N 2 : N e 1 + e 2 : N e 1 : N 2 : N e 1 e 2 : N e 1 : N 2 : N e 1 : e 2 : N e 1 : N 2 : fsg P fe1 j x s 2 e 2 g : N where +,, and : are respectively addition, multiplication, and modied subtraction on natural numbers. (That is, if n m, then n : m = 0f n > m, then n : m = n m.) The summation construct P fe 1 j x s 2 e 2 g denotes the number obtained by rst applying the function x:e 1 to every item in the set e 2 and then adding the results up. That is, P fe 1 j x 2 Xg is f(o 1 ) + : : : + f(o n ) if f is the function denoted by x:e 1 and fo 1 ; : : : o n g is the set denoted by X. It should be stressed that in the construct P fe 1 j x 2 e 2 g, the fe 1 j x 2 e 2 g part is not an expression; hence this construct is stronger than just adding P : fng! N. The extended language N RC(N; +; ; : ; P ; =), where the additional base types and primitives are explicitly listed between brackets, is capable of expressing many aggregate functions found in commercial databases. Here are two examples: \count the number of records in R" is count(r), P f1j x 2 Rg and \total up the rst column of R" is total(r), P f 1 x j x 2 Rg. Now we formally dene the conservative extension property. The set height ht(s) of a type s is dened by induction on the structure of type: ht(unit) = ht(b) = 0, ht(s t) = ht(s! t) = max(ht(s); ht(t)), and ht(fsg) = 1 + ht(s). Every expression of our language has a unique typing derivation. The set height of expression e is dened as ht(e) = maxfht(s) j s occurs in the type derivation of eg. Let L i;o;h denote the class of functions whose input has set height at most i, whose output has set height at most o, and which are denable in the language L using an expression whose set height is at most h max(i; o). L is said to have the conservative extension property with xed constant k if L i;o;h = L i;o;h+1 for all i, o, and h max(i; o; k). It is known from Wong [10] and Libkin and Wong [5] that Theorem 2.1 N RC(N; +; ; : ; P ; =) has the conservative extension property with xed constant 0. Moreover, N RC(N; +; ; : ; P ; =) endowed with any additional primitive p : s! t also has the conservative extension property with xed constant ht(s! t). 2 3 Lifting of linear orders In this section we present a technique to lift linear order from base types to all types. This technique plays the key role in the encoding used in the next section. The following lemma is a folklore (see Wechler [9]): 3

4 Lemma 3.1 Given a partially ordered set ha, dene an ordering - on its nite powerset P n (A) as follows: X - Y i max((x Y )[(Y X)) Y, or, equivalently, if 8x 2 X Y 9y 2 Y X : x y. Then - is a partial order. Moreover, if is linear, then so is -. 2 This way of lifting linear order can be easily expressed in N RC(N; +; ; : ; P ; =). Therefore, we have Theorem 3.2 Suppose a linear order b is given for each base type b. Then a linear order t is computable by N RC(N; +; ; : ; P ; =) for each type t. Proof. Dene t by induction on types. For a base type b is given. For pairs it is dened lexicographically: x ts y, if 1 x s 1 y then (if 1 x = s 1 y then 2 x t 2 y else true) else false. For set types we use lemma 3.1: X fsg Y, if X v [ s Y then (if Y v[ s X then X [ s Y else true) else false. Here X v [ s Y i 8x 2 X9y 2 Y : x s y and X [ s Y i X Y v[ s Y X. These can be implemented as follows. X v [ s Y, 0 =B fif (fif x s y then 1 else 0 j y 2 Y g) = N 0 then 1 else 0 j x 2 Xg and X [ s Y, (fif x 2s Y then 0 else (if (fif y 2 s X then 0 else (if x s y then 1 else 0) j y Y g) = N 0 then 1 else 0) j x 2 Xg) = N 0. 2 We denote N RC(N; +; ; : P ; ; =) endowed with linear orders at base types by N RC(N; +; ; : P ; ; =; ). Notice that if B and N are the only base types, then this does not add expressive power since the orderings on booleans and naturals are denable: false B true and n N m, (n : m = N 0). Corollary 3.3 A rank assignment is a function sort : fsg! fs Ng such that sortfa 1 ; : : : ; a n g = f(a 1 ; 1); : : : ; (a n ; n)g where a 1 s : : : s a n. Rank assignment is expressible in N RC(N; +; ; : ; P ; =; ). Proof. Dene sort(x), S f(r; fif c s r then 1 else 0 j c 2 xg) j r 2 xg. 2 Finally, we have the following Corollary 3.4 If all orderings on the base types are well-founded, then so are the lifted orderings. Furthermore, if constant min b : b and function succ b : b! b (meaning minimal element and successor) are given for each base type b, they can be dened for any type. Proof. The rst statement follows from the fact that X - Y implies X v [ Y where X v [ Y i 8x 2 X9y 2 Y : x y, and v [ is known to be well-founded if is [2]. Minimal elements are denable as follows: min ts = (min t ; min s ) and min ftg = fg. It is clear how to dene succ for pairs. To dene succ for set types, consider a set X : ftg. Let a be the minimal element of type t which is not in X, and let X 0 = fx 2 X j x t ag and X 1 = fx 2 X j a t xg. We claim X 1 [ a = succ ftg (X). Clearly, X ftg X 1 [ a. Let X ftg Y. Let a 2 Y. Assume x 2 X 1 [ a Y = X 1 Y. Then x 2 X Y and there exists y 2 Y X such that x t y. Since y 6= a, we obtain y 2 Y (X 1 [ a) and X 1 [ a ftg Y. If a 62 Y and x 2 X 1 [ a Y, then Y (X 1 [ a) = Y X 1. Therefore, if x 2 X 1 Y, there exists y 2 Y X 1 such that x t y. Let y m = max(y X 1 ). If y m t a, then Y X 1 X 0 and Y X which contradicts our assumption. Then, since a 62 Y, a t y m and therefore X 1 [ a ftg Y. This proves our claim. Therefore, succ ftg (X) is fmin t g [ X if min t 62 X and X 1 [ a if min t 2 X where a = succ t (x 0 ) and x 0 = minfx 2 X j succ t (x) 62 Xg and X 1 is dened as above. The expressibility of succ ftg follows immediately. 2 4

5 4 The main result We rst dene the notion of internal and generic family of functions. Then we show that the conservative extension property of N RC(N; +; ; : ; P ; =) endowed with well-founded linear orders can be preserved in the presence of any such family of functions. Introduce type variables i and consider nonground complex object types ; ::= j b j B j N j unit j j fg If 1,..., n occur in, then [s 1 = 1 ; : : : ; s n = n ] stands for the type obtained by replacing every occurrence of i in by s i. A complex object type s is an instance of a nonground complex object type if there are complex object types s 1,..., s n such that s = [s 1 = 1 ; : : : ; s n = n ] where 1,..., n are all the type variables in. The minimal height mht() of type is dened as the depth of nesting of set brackets in. That is, mht() is equivalent to ht(s) where s is obtained from by replacing all occurrences of type variables in by unit. Let p 1;:::; n :! stand for a family of functions p s 1;:::;s n : s! t where s = [s 1 = 1 ; : : : ; s n = n ] and t = [s 1 = 1 ; : : : ; s n = n ]. (Note that for each s 1,..., s n, there is exactly one p s 1;:::;s n in the family p 1;:::; n.) The minimal height mht(p) of p 1;:::; n :! is dened as max(mht(); mht()). Let s = [s 1 = 1 ; : : : ; s n = n ; t=]. Let dom s;t ;(o) be the set of subobjects of type t in the object o : s occurring at positions corresponding the the type variable. Formally, dene dom s;t ; : s! ftg as follows: dom s;t b; (x) = fg; doms;t ;(x) = fxg; dom s;t 0 ;(x) = fg, where and 0 are distinct type variables; dom uv;t ;(x; y) = dom u;t ;(x) [ dom v;t ;(y); and dom fsg;t (X) = S fdom s;t fg; ;(x) j x 2 Xg. Denition 4.1 The family of functions p 1;:::; n :! is internal (see [4]) in i if for all complex object types s = [s 1 = 1 ; : : : ; s n = n ], t = [s 1 = 1 ; : : : ; s n = n ], and complex object o : s, it is the case that dom t;s i (p s 1;:::;s n (o)) dom s;s i (o). 2 In other words, p 1;:::; n :! is internal in i if it does not invent new values in positions corresponding to the type variable i. Let s = [s 1 = 1 ; : : : ; s n = n ; t=], r = [s 1 = 1 ; : : : ; s n = n ; t 0 =], and : t! t 0. Let modulate s;t;t0 be the object O 0 : r obtained by replacing every subobject o : t in O : s occurring in positions corresponding to type variable by (o) : t 0. Formally, dene modulate s;t;t0 : s! r as follows: modulate s;t;t0 b (x) = x; modulates;t;t0 (x) = (x); modulates;t;t0 0 (x) = x, where and 0 are distinct type variables; modulate uv;t;t0 (x) j x 2 Xg. fmodulate s;t;t0 (x; y) = (modulateu;t;t0 (x); (modulatev;t;t0 (O) (y)); modulatefsg;t;t0 fg (X) = Denition 4.2 The family of functions p 1;:::; n :! is generic in i if for all complex object types s = [s 1 = 1 ; : : : ; s n = n ], t = [s 1 = 1 ; : : : ; s n = n ], complex object o : s, set R : frg, and : s i! r such that is a bijection from dom s;s i (o) to R and 1 : r! s i is its inverse when restricted to 5

6 dom s;s i (o), it is the case that s modulate s;s i;r ; p s 1;:::;s n - t 6 modulate t0 ;r;s i ; 1 s 0 p s0 1 ;:::;s0 n the above diagram, where s 0 j = s j for j 6= i and s 0 i = r, commutes. 2 - t 0 A family p 1;:::; n :! is called internal generic if it is internal and generic in all type variables. We remark here that our notion of genericity is slightly dierent from that in [4]. In particular, generic does not imply internal, in contrast to [4]. Now we demonstrate that adding a internal and generic family p 1;:::; n to N RC(N; +; ; : P ; ; =; ) does not destroy its conservative extension property. We assume that b : b b! B is a well-founded linear order for every base type b. Consider N RC(N; +; ; : P F ; ; =; ; ) obtained by adding the following construct to N RC(N; +; ; : P ; ; =; ): e 1 : s e 2 : ftg F fe1 j x t 2 e 2 g : s where F fe 1 j x t 2 e 2 g is the greatest element in the set fe 1 j x t 2 e 2 g (it is min s when the set is empty). Note that F fe 1 j x 2 e 2 g is already denable in N RC(N; +; ; : P ; ; =; ) if e 1 : fsg, and can be treated as a syntactic sugar. It is clear that both dom s;t ; and modulate s;t;t0 are denable in N RC(N; +; ; : P F ; ; =; ; ) whenever is. Proposition 4.3 Let p 1;:::; n :! be a family of functions that is internal generic. Then N RC(N; +; ; : P F ; ; =; ; ) endowed with the family of primitives p 1;:::; n has precisely the expressive power of N RC(N; +; ; : P F ; ; =; ; ) endowed with just the primitive p N;:::;N. Proof. For each s = [s 1 = 1 ; : : : ; s n = n ] and o : fs i g, dene (o), x: F fif x = 1 y then 2 y else 0 j y 2 sort(o)g and 1 (o), x: F fif x = 2 y then 1 y else min s j y 2 sort(o)g, where sort : fs i g! fs i Ng is as dened in corollary 3.3. (o) and 1 (o) are functions of type s i! N and N! s i respectively. Clearly, (o) when restricted to o is a bijection whose inverse is 1 (o). Let u i = [N= 1 ; : : : ; N= i1 ; s i = i ; : : : s n = n ] and v i = [N= 1 ; : : : ; N= i1 ; s i = i ; : : : s n = n ]. Note that s = u 1 and t = v 1. Dene i (o), modulate u i;s i ;N ; (dom s;s i (o)) and 6

7 1 i (o), modulate v i+1;n;s i. ; 1 (dom s;s i (o)) Then the following diagram commutes by induction on n and by the assumption that the family p 1;:::; n is internal and generic. o : u 1 1(o) - : u : u n n(o) - : un+1 p s 1;:::;s n : v 1 p N;s 2;:::;s n 1 (o) : v : vn 1 p N;:::N;s n p N;:::;N 1 n (o) : v n+1 Hence p s 1;:::;s n = x: (x) n (x) p N;:::;N n (x) 1 (x). The right hand side is clearly expressible in N P F RC(N; +; ; ; ; ; =; ; ; p N;:::;N ). 2 Next we sketch the proof of the conservativity of N RC(N; +; ; : ; P ; =; ; F ). Proposition 4.4 N RC(N; +; ; : ; P ; =; ; F ) has the conservative extension property with xed constant 0. Moreover, when endowed with any additional primitive p, it retains the conservative extension property with xed constant ht(p). Proof sketch. The proof of theorem 2.1 is based on rewriting expressions of the language to normal forms in which the e 2 in subexpressions of the form S fe 1 j x 2 e 2 g are variables. Such rewrite system was exhibited in Wong [10]. In Libkin and Wong [5] additional rules for P were given which guarantee that e 2 in any subexpression P fe 1 j x 2 e 2 g is a variable. The rewrite system was shown to be strongly normalizing and conservativity was derived by analyzing the normal forms. Now we add the following rewrite rules for F, assuming that the use of the construct F fe 1 j x 2 e 2 g is restricted to the situation when the type of e 1 is not a set type (when e 1 : fsg, it is treated as a shorthand.) F fe j x 2 fgg ; min F fe 1 j x 2 fe 2 gg ; e 1 [e 2 =x] F fe j x 2 e 1 [ e 2 g f F fe j x 2 e1 g F fe j x 2 e 2 g then F fe j x 2 e 2 g else F fe j x 2 e 1 g F fe 1 j x 2 S fe 2 j y 2 e 3 gg ; F f F fe 1 j x 2 e 2 g j y 2 e 3 g F fe j x 2 if e 1 then e 2 else e 3 g f e 1 then F fe j x 2 e 2 g else F fe j x 2 e 3 g F i fe1 j x 2 e 2 g ; S fif e 2 g, when e 1 : fsg. P fif e1 e 1 [y=x] then 1 else 0 j y 2 e 2 g = 1 then i e 1 else fg j x 2 F i fe1 j x 2 e 2 g ; F P fif fif e1 e 1 [y=x] then 1 else 0 j y 2 e 2 g = 1 then i e 1 else fg j x 2 e 2 g, when e 1 is not of set type. 7

8 The extended collection of rewrite rules forms a weakly normalizing rewrite system and conservativity can be derived by induction on the induced normal forms along the lines of Wong [10]. 2 Putting together the two previous propositions, the desired theorem follows straightforwardly. Theorem 4.5 N RC(N; +; ; : P F ; ; =; ; ) endowed with a internal generic family p 1;:::; n :! has the conservative extension property with xed constant mht(p). 2 5 Some corollaries As remarked earlier, F fe 1 j x 2 e 2 g is already denable in N RC(N; +; ; : ; P ; =; ) if e 1 : fsg. Therefore, if every type variable occurs in the scope of some set brackets in and, then the assumption of well-foundedness on b used in proposition 4.3 is not required and the proposition holds for N RC(N; +; ; : ; P ; =; ). Thus, we have Corollary 5.1 N RC(N; +; ; : ; P ; =; ) endowed with a internal generic family p 1;:::; n :!, where each type variable is within the scope of some set brackets, has the conservative extension property at all input and output heights with xed constant mht(p). 2 In particular, any polymorphic function denable in the algebra of Abiteboul and Beeri [1], which is equivalent to N RC(=; powerset), gives rise to a internal generic family of functions for all possible instantiations of type variables. (Observe that this is not true for N RC(N; +; ; : ; P ; =; ) because succ N is denable and therefore can be lifted to any type by corollary 3.4. This also indicates that the notion of internal and generic family is more general than polymorphic functions.) Now we conclude immediately that the following languages possess the conservative extension property (see also [5]): N RC(N; +; ; : ; P ; =; ; t) with xed constant 1, and N RC(N; +; ; : ; P ; =; ; powrst) with xed constant 2. where tc s : fs sg! fs sg is the transitive closure of binary relations. Since the Abiteboul and Beeri algebra has the power of a xpoint logic, a great deal of polymorphic functions can be added to N RC(N; +; ; : ; P ; =; ) without destroying its conservative extension property (but may be increasing the xed constant). Acknowledgements. We thank Peter Buneman, Rick Hull and Dan Suciu for helpful discussions. We are also grateful to the anonymous referees for the useful comments. Leonid Libkin was supported in part by NSF Grant IRI and AT&T Doctoral Fellowship and Limsoon Wong was supported in part by ARO Grant DAAL03-89-C-0031-PRIME. References [1] S. Abiteboul and C. Beeri. On the power of languages for the manipulation of complex objects. In Proc. International Workshop on Theory and Applications of Nested Relations and Complex Objects, Darmstadt,

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