V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with Sets/Bags/Lists. In LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....computations over them. For example, record formation is the introduction operation for records, and projections are the elimination operations. How does one find those introductions and elimination operations Databases work with various kinds of collections. One approach (due to Tannen [BBW92, BTS91] to finding the introduction and elimination operations for those collections is to look for operations naturally associated with them. To do so, one often characterizes the semantic domains of collection types via universality properties, which tell us what the introduction and the elimination ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with Sets/Bags/Lists. In LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991.

....then must be too. For example, the bag cardinality function can be expressed as hom[ x: 1) A, which is well formed, while the similar function for sets hom[ x: 1) A is not (since is commutative but not idempotent) Without this restriction we would have (see also V. Tannen et al. BTS91] 1 = hom[ x: 1) fag) hom[ x: 1) fag [ fag) hom[ x: 1) fag) hom[ x: 1) fag) 1 1 = 2 This restriction also prohibits the conversion of sets into lists (since [ 6 ) In addition, De nition 2 justi es the restriction that non commutative monoids ....

....since the more bulk operations an algebra supports, the more transformation rules it needs and, therefore, the harder the optimization task becomes. Our framework is based on monoid homomorphisms, which were rst introduced as an e ective way to capture database queries by Tannen, et al. BTBN91, BTS91, BTBW92] Their form of monoid homomorphism (also called structural recursion over the union presentation SRU) is more expressive than ours. Operations of the SRU form, though, require the validation of the associativity, commutativity, and idempotence properties of the monoid associated with ....

[Article contains additional citation context not shown here]

V. Breazu-Tannen and R. Subrahmanyam. Logical and Computational Aspects of Programming with Sets/Bags/Lists. In 18th International Colloquium on Automata, Languages and Programming, Madrid, Spain, pages 60-75. Springer-Verlag, July 1991. LNCS 510. 43

....theories db creates a complex object that is the set of all the entires in all the theories available in the running HOL90 system. The reason for this being a function is that this is an extensible collection. An example of the creation of some complex objects is as follows: val a = mksetint [1,3,5]; val a = 1, 3, 5 : int : co val b = mkbaseco (StoredThm thyname = primrec , theorem = LESS0 , theorem primrec LESS0 ) val b = StoredThm theorem = LESS0 , n. 0 SUC n) thyname = primrec ) holtheorydata : co val c = itlist (fn x = fn y = ....

....cartesian product and their counterparts for or sets, see [2, 8] These functions are included in OR SML in the form of a structure called Set. val x1 = mksetint [1,2] val x1 = 1, 2 : int : co smap (pair(id,id) x1; val it = 1, 1) 2, 2) int int) co val x2 = mksetint [3,4]; val x2 = 3, 4 : int : co union(x1,x2) val it = 1, 2, 3, 4 : int : co Set.cartprod(x1,x2) val it = 1, 3) 1, 4) 2, 3) 2, 4) int int) co 3 Normalization As we discussed before, while an object h1; 2; 3i is structurally just a set, conceptually it is a ....

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V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with sets/bags/lists. In LNCS 510: Proc. of ICALP-1991, Springer Verlag, 1991, pages 60--75.

....objects of a given type whereas the elimination operations are used for doing computations over them. For example, record formation is the introduction operation for records, and projections are the elimination operations. Databases work with various kinds of collections. One approach (cf. [8, 4]) to find the introduction and elimination operations for those collections is to look for operations naturally associated with them. To do so, one often characterizes the semantic domains of collection types via universality properties, which suggest what the introduction and the elimination ....

....as ; and [ But if e and u do not supply the range of f with the structure of a semilattice, then f may not be well defined. For example, if e is 0, f is x:1, and u is , then f [e; u] f1g) f [e; u] f1g [ f1g) thus implying 1 = 2. To overcome this problem, originally noticed in [4], one can require that e be interpreted as ; and u as [ Generally, the simplest way to ensure well definedness of f is to require that hX; Omega i be h[ C(s) Omega i for some type s. Thus, we obtain the second diagram in figure 2. The unique completing homomorphism is called ext(f ) the ....

[Article contains additional citation context not shown here]

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with sets/bags/lists. In LNCS 510: Proc. of 18th ICALP, Madrid, Spain, pages 60--75. Springer, 1991.

.... z [u u ORG] 2 The inspiration for CPL came primarily from [6] that presented structural recursion as a query language. However, structural recursion has two difficulties. The first is that not every syntactically acceptable structural recursion program is logically well defined [7]. The second is that structural recursion has too much expressive power because it can express queries that require exponential time and space. While programming languages always take Turing completeness for granted, the attitude in database programming is radically different. In the context of ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with Sets/Bags/Lists. ICALP, 18:60--75, 1991.

....of the universe. There is a large body of research dealing with languages capable of expressing polynomial time generic database queries or transformations. Languages in which such transformations can be expressed include datalog with negation [1] languages based on structural recursion [10], loops [24] nondeterministic reduce operators [21] while queries [1] etc. Hence, our first goal is to produce a generic polynomial time computable transaction that separates WPC(FO) from PR(FO) The transaction exhibited below can be expressed in all languages mentioned in the previous ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with sets/bags/lists. In LNCS 510: Proc. of 18th ICALP, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991. 10

.... ; xn Gamma1 def = f(x 0 ) f(xn Gamma1 ) scan (e,f,g) x 0 ; xn Gamma1 def = e; f(x 0 ) f(x 0 ) f(x 1 ) f(x 0 ) f(xn Gamma2 ) The commutativity and associativity conditions imposed on g are undecidable and hence cannot be checked by a compiler [BTS91] The commutativity condition on g is motivated by our goal of an efficient implementation rather than by the semantics of reduce and scan. 4 We experimented with sequential restructuring functions and observed only negligible changes in execution times. 7 The sequence module is itself an ML ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with Sets/Bags/Lists. In LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991. 21

....to putting a condition on the third line of the definition of hom that the expressions e 1 and e 2 denote disjoint sets. Unfortunately this considerably complicates the operational semantics of the language, and it precludes the possibility of lazy evaluation. For a resolution of this issue, see [BTS91, BTBN91] which disuss the semantic properties of programs with sets and other collection types. In this paper we shall occasionally make use of incorrect applications of hom; however we are confident that the adoption of an alternative semantics will not affect typing issues, which are the ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and Computational Aspects of Programming with Sets/Bags/Lists, Proceedings of the 18th International Colloquium on Automata, Languages, and Programming, Madrid (Spain), July 1991, Springer LNCS 510, pp. 60--75.

.... of queries and related to it indexing, a variety of query transformation and optimization algorithms, and in particular those designed for expensive predicates [CS93, HS93, H94, KMPS94] 4 We adopt a variant of monoid comprehension calculus [FM95] which is based on monoid homomorphisms [BTBN91, BTS91, BTBW92] and monad comprehensions [TW89, W90, CT94] as a query evaluation platform. This decision is motivated by the fact that the monoid calculus is formally, and accurately defined, and readily captures such features as extensible multiple collection types, aggregations, arbitrary compositions of type ....

V. Breazu-Tannen and R. Subrahmanian, Logical and Computational Aspects of Programming with Sets/Bags/Lists, Proc. 18-th International Colloquium on Automata, Languages and Programming, Madrid, pp. 60-75, 1991, LNCS 510.

.... formation, and [ are the introduction operations, we define the elimination operation by prescribing its action in each of the three cases: fun s sru(e; f; u) e j s sru(e; f; u) fxg) f(x) j s sru(e; f; u) X [ Y ) u(s sru(e; f; u) X) s sru(e; f; u) Y ) Following Tannen et al. [6, 7], s sru stands for structural recursion on the union presentation of sets. It has three parameters, e, f , and u. Setting these parameters arbitrarily leads to ill defined programs. It is well known that checking whether a program using s sru is well defined is undecidable [7] Hence, it was ....

....Tannen et al. 6, 7] s sru stands for structural recursion on the union presentation of sets. It has three parameters, e, f , and u. Setting these parameters arbitrarily leads to ill defined programs. It is well known that checking whether a program using s sru is well defined is undecidable [7]. Hence, it was proposed [8] that some syntactic restrictions be imposed on s sru to ensure well definedness. In particular, this is achieved by taking e to be ; and u to be union. Then, the resulting function s sru( f; which is called s ext(f ) has type fsg ftg if f has type s ftg. ....

[Article contains additional citation context not shown here]

V. Breazu-Tannen, R. Subrahmanyam, Logical and computational aspects of programming with sets/bags/lists, in "LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming," Madrid, Spain, July 1991.

....#organism: a.#descr) x DB, f x.#feature, a f.#anno, a.#annoname = organism in let ORG = y.#organism y DB in [ #organism: z, #entries: v.#entry v DB , v. #organism = z ) z [ u u ORG ] 2 The inspiration for CPL came primarily from (Breazu Tannen et al. 1991) that presented structural recursion as a query language. However, structural recursion has two difficulties. The first is that not every syntactically acceptable structural recursion program is logically well defined (Breazu Tannen Subrahmanyam, 1991) The second is that structural recursion ....

....] 2 The inspiration for CPL came primarily from (Breazu Tannen et al. 1991) that presented structural recursion as a query language. However, structural recursion has two difficulties. The first is that not every syntactically acceptable structural recursion program is logically well defined (Breazu Tannen Subrahmanyam, 1991). The second is that structural recursion has too much expressive power because it can express queries that require exponential time and space. While programming languages always take Turing completeness for granted, the attitude in database programming is radically different. In the context of ....

[Article contains additional citation context not shown here]

Breazu-Tannen, V., & Subrahmanyam, R. (1991). Logical and computational aspects of programming with Sets/Bags/Lists. Pages 60--75 of: LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming. Springer Verlag.

....10.4: The mappings tree and inst provide an isomorphism between the set of bisimulation classes of model 1 instances and the equivalence classes of model 2 instances. 11 Query language based on structural recursion In this section we will present an adaption of the query language SRI ([10, 11]) to the model of definition 4.3. The language is based on the mechanism of structural recursion over sets which was described in [10] as a basis for a query language on the nested relational data model. Our choice of this mechanism is because it is semantically well understood and because it is ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with Sets/Bags/Lists. In LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991.

....model [8] Each of these data types are encapsulated within the application programming interface by a collection of ML modules. The core of the collection type modules (that is, those for sets, lists, and bags) are inspired principally by the work of Tannen, Buneman, Naqvi, Subrahmanyam, and Wong [4, 5, 6]. An ML programmer can directly manipulate Kleisli objects via function call to these modules. Each module consists of: a collection of cannonical operators for the particular data type encapsulated by that module, additional operators designed for efficiency, additional operators that are ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with Sets/Bags/Lists. In LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991.

....and Stemple ( IPS91] Theorem 7.8) They also note that dcr is in NC . As part of a larger group of researchers, we became interested in dcr because it fits into a natural hierarchy of query languages that share a common semantic basis, built around forms of structural recursion on collection types [BTS91, BTBN91, BBW92, BNTW95] (see Section 3) Theoretical studies of expressiveness, such as [Won93, LW94a, Suc94, LW94b, SW95] and the present paper help us with the choice and mix of primitives, as well as implementation strategies. In particular, dcr is at the core of a sub language for which we are currently seeking ....

....we are proposing: the semantics of dcr puts it naturally in NC ; there is no need to impose logarithmic bounds on the number of iterations or recursion depth. Moreover, it can be shown that a different kind of recursion on sets, namely structural recursion on the insert presentation of sets ([BTS91]; notation sri ; definitions reviewed in Section 3) together with the relational algebra expresses exactly the PTIME computable queries on ordered databases 1 . This follows from results in [IPS91] we state the corresponding result for complex objects in Proposition 6.7. Hence, at least ....

[Article contains additional citation context not shown here]

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with Sets/Bags/Lists. In LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991.

....data items, the tuples, are unordered, and the relations contain no duplicates, and cannot be nested. Relaxing these assumptions leads to numerous distinct data types, such as the complex objects (nested sets) Jac82, AB87, KV84, KRS85, AG91] the bags (sets with duplicates) BK90, Mum90, Alb91, BS91, GM93, LW94] the lists (internal order) the ordered sets, and the pomsets (partially ordered multisets) Pra84] 2.1 Partially Ordered Multisets Pomsets The pomset type generalizes sets, bags, lists, trees, and other ordered types, and therefore provides a uniform representation for all ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with sets/bags/lists. In Proc. 18th Int. Col. on Automata, Languages and Programming, 1991.

.... ff set ; merge : ff set ff set ff set) The closely related work of [15] for example, adopted the union representation while [4] exploits both viewpoints. Issues of expressiveness are examined by [46] and an in depth comparison of insertion and union representation has been undertaken by [3]. We prefer the insertion presentation because it gets by with two constructors instead of three. Aggregation and quanti cation may be uniformly understood in this model as well. To de ne the summation aggregate sum we employ the algebra sum = num ; 0; which uses the domain of the basic ....

....fold fusion rule. Such rule sets imply the need for a more or less sophisticated rule application strategy. Additionally, 39] accompanies speci c rules with rather complex provisos (e.g. strictness or distributivity of functions) that cannot easily be asserted, let al..one be checked syntactically [3]. Query deforestation is able to deforest the unsafe programs (queries in which a nested subquery traverses partial results computed by an outer query) of [14] Instances of this class of programs, elements of which are aggregate queries that compute running sums or list reversal, are not ....

Val Breazu-Tannen and Ramesh Subrahmanyam. Logical and Computational Aspects of Programming with Sets/Bags/Lists. In Proc. of th 18th Int'l Colloquium on Automata, Languages and Programming, pages 6075, Madrid, Spain, July 1991.

....types, such as integers and booleans. Table 1 presents some examples of collection and primitive monoids. The C I column indicates whether the monoid is a commutative or idempotent monoid. The monoids list, bag, and set capture the well known collection types for linear lists, multisets, and sets [7] (where is list append and ] is the additive union for bags) The monoid sorted[f ] is parameterized by the function f whose range is associated with a partial order . The merge function of this monoid merges two sorted lists into a sorted list. If x appears before y in a sorted[f ] list, then ....

....Updates of the form path : e, where path is a collection, can be translated into: somef true x path, path = x, y e, path = y g 7 Related Work Ourframework is based onmonoidhomomorphisms, whichwere first introduced as an effective way to capture database queries by V. Tannen and P. Buneman [5, 7, 6]. Their form of monoid homomorphism (also called structural recursion over the union presentation SRU) is more expressive than our calculus. Operations of the SRU form, though, require the validation of the associativity, commutativity, and idempotence properties of the monoid associated with ....

[Article contains additional citation context not shown here]

V. Breazu-Tannen and R. Subrahmanyam. Logical and Computational Aspects of Programming with Sets/Bags/Lists. In 18th International Colloquium on Automata, Languages and Programming, Madrid, Spain, pp 60--75. Springer-Verlag, July 1991. LNCS 510.

....(nor is it bounded depth domain independent [5] However it is easy to prove the following: Proposition 4.2 All queries in NRA( Sigma) fix are ef domain independent and continuous. Also, queries expressed with other forms of iterations, like loop of [18] the structural recursions sru; sri of [7, 8], and the divide and conquer recursion dcr of [24] are also ef domain independent and continuous. We take the above proposition as evidence that the notion of ef domain independence is more appropriate for queries with external functions than the notions of em domain independence or boundeddepth ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with Sets/Bags/Lists. In LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991.

....within a database. The importance of algebraic specification for specifying schemas and for supporting a functional style of database computations has been emphasized by previous research, such as that reported in [16, 9, 10] it is also closely related to the topic of collection or bulk types [7, 12]. Object oriented modules support the declarative definition of class hierarchies of objects that can be updated and queried in a distributed fashion by means of messages. 2 System modules, of which object oriented modules are a special case, are not discussed in this paper; see [29] ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with sets/bags/lists. In Proc. ICALP'91, pages 60--75. Springer LNCS 510, 1991.

....iterated powerdomains [18] it becomes sufficiently rich to express a large number of queries on sets and or sets. Since monads arise from adjunctions, above them we have yet another powerful programming tool which is the structural recursion. We do not discuss it here but refer the reader to [2, 4, 3, 21] for discussion on advantage and problems of using the structural recursion. Thus, it is the freeness property of a construction that admits an easy way of being incorporated into the syntax of a programming language. Therefore, if we want to program with approximations, we should look for their ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with sets/bags/lists. In LNCS 510: Proc. of 18th ICALP, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991.

.... in, large part, this lack of support of complex data structures is one of the main reasons explaining the poor impact of commercial database management systems in scientific applications [16] There have been attempts to more formally express the use of collections in programming languages (e.g. [3]) and at least one attempt to interpret collections in the context of relational query languages [5] In the object oriented database context, there exist an algebra calculus (generator free) approach [21] a mixed comprehension algebra approach [13] and a comprehension approach [9, 12] for ....

.... [square x j x [1; 2; 3] odd x] h list;list (x: square x j odd x] 1; 2; 3] making x: square x j odd x] j f , and applying rule (iii) h l;l f merge( 1] 2; 3] merge list (h l;l f [1] h l;l f[2; 3] apply rule (ii) and rule (iii) f [1] merge list (h l;l f [2] h l;l f[3]) apply rule (ii) two times: f [1] f [2] f [3] Developing f [1] f [1] x: square x j odd x] 1] substituting x: square 1 j odd 1] applying rule (vi) if odd 1 then [square 1 j ] else zero list applying rule (iv) unit list [square 1] 1] Similarly f [2] and f ....

[Article contains additional citation context not shown here]

V. Breazu-Tannen and R. Subrahmanyan. Logical and computational aspects of programming with sets/bags/lists. In Proceedings of the 18th Intern. Colloquium on Automata, Languages and Programming, pages 60--75, Madrid, Spain, July 1991.

....it otherwise. Inference steps may not mention the select operator, or use lemmas whose top bit is false, if the bit at the node being refined is true. Details on the actual HOL embedding and some of its practical applications will be given in a forthcoming paper. 5 Related Work, Discussion In [4], Breazu Tannen and Subrahmanyam give a logic for reasoning about programs using structural recursion over data types formed from constructors subject to some equations. Their idea, to make the meaning of a definition by structural recursion if it does not respect the equations, is somewhat ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with sets/bags/lists. In Automata, Languages and Programming: 18 th International Colloquium, Lecture Notes in Computer Science, pages 60--75. Springer-Verlag, 1991.

....a set into a list, can be detected at type checking time. This can be done effectively by extending the parametricity theorem to capture the new datatypes of bags and sets. 3 F db : The Polymorphic Database Calculus Bags and sets can be considered as lists with some additional properties [2, 9]. For example, the list append function is associative but the bag union is both associative and commutative, and the set union is associative, commutative, and idempotent (i.e. the union of a set with itself is the same set) These additional properties make the list representation inappropriate ....

V. Breazu-Tannen and R. Subrahmanyam. Logical and Computational Aspects of Programming with Sets/Bags/Lists. In 18th International Colloquium on Automata, Languages and Programming, Madrid, Spain, pp 60--75. Springer-Verlag, July 1991. LNCS 510.

....must be too. For example, the bag cardinality function can be expressed as hom[ x: 1) A, which is well formed, while the similar function for sets hom[ x: 1) A is not (since is commutative but not idempotent) Without this restriction we would have (see also V. Tannen et al. BTS91] 1 = hom[ x: 1) fag) hom[ x: 1) fag [ fag) hom[ x: 1) fag) hom[ x: 1) fag) 1 1 = 2 This restriction also prohibits the conversion of sets into lists (since [ 6 ) In addition, Definition 2 justifies the restriction that non commutative monoids should be ....

....the more bulk operations an algebra supports, the more transformation rules it needs and, therefore, the harder the optimization task becomes. Our framework is based on monoid homomorphisms, which were first introduced as an effective way to capture database queries by Tannen, et al. BTBN91, BTS91, BTBW92] Their form of monoid homomorphism (also called structural recursion over the union presentation SRU) is more expressive than ours. Operations of the SRU form, though, require the validation of the associativity, commutativity, and idempotence properties of the monoid associated with ....

[Article contains additional citation context not shown here]

V. Breazu-Tannen and R. Subrahmanyam. Logical and Computational Aspects of Programming with Sets/Bags/Lists. In 18th International Colloquium on Automata, Languages and Programming, Madrid, Spain, pages 60--75. Springer-Verlag, July 1991. LNCS 510.

....In this report, the same syntax is given a semantics based on bags in section 2. We use this language as our ambient bag language. This highlights the uniform manipulation of sets and bags using monad as noted by Wadler [36] and structural recursion as noted by Breazu Tannen and Subrahmanyam [4]. Incidentally, the equivalence between nested relational algebra and nested relational calculus in [5] carries over here effortlessly as an equivalence between nested bag algebra and nested bag calculus. The ambient bag language is inadequate in expressive power as it stands. In section 3, ....

....mentioned earlier, although a powerbag primitive increases expressive power considerably, it is difficult to express algorithms that are efficient. While structural recursion does not have this deficiency, it requires the satisfaction of certain preconditions that cannot be automatically verified [4]. In section 8, a bounded loop construct which does not require the verification of any precondition is introduced. It is shown to be equivalent in expressive power to structural recursion over sets, bags, as well as lists. This confirms the intuition that structural recursion is just a special ....

[Article contains additional citation context not shown here]

V. Breazu-Tannen and R. Subrahmanyam. Logical and computational aspects of programming with sets/bags/lists. In LNCS 510: Proceedings of 18th International Colloquium on Automata, Languages, and Programming, Madrid, Spain, July 1991, pages 60--75. Springer Verlag, 1991.

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