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12
Database Query Languages Embedded in the Typed Lambda Calculus
, 1993
"... We investigate the expressive power of the typed calculus when expressing computations over finite structures, i.e., databases. We show that the simply typed calculus can express various database query languages such as the relational algebra, fixpoint logic, and the complex object algebra. In ..."
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Cited by 26 (6 self)
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We investigate the expressive power of the typed calculus when expressing computations over finite structures, i.e., databases. We show that the simply typed calculus can express various database query languages such as the relational algebra, fixpoint logic, and the complex object algebra. In our embeddings, inputs and outputs are terms encoding databases, and a program expressing a query is a term which types when applied to an input and reduces to an output.
Space Usage in Functional Query Languages
 in &quot;LNCS 893: Proceedings of 5th International Conference on Database Theory,&quot; 439454
, 1995
"... We consider evaluation strategies for database queries expressed in three functional query languages: the complex value algebra, the simply typed lambda calculus, and method schemas. Each of these query languages derives its expressive power from a different primitive: the complex value algebra fr ..."
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Cited by 14 (2 self)
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We consider evaluation strategies for database queries expressed in three functional query languages: the complex value algebra, the simply typed lambda calculus, and method schemas. Each of these query languages derives its expressive power from a different primitive: the complex value algebra from the powerset operator, the simply typed lambda calculus from list iteration, and method schemas from recursion. We show that "natural" evaluation strategies for these primitives may lead to very inefficient space usage, but that with some simple optimizations many queries can be evaluated with little or no space overhead. In particular, we show: (1) In the complex value algebra, all expressions with set nesting depth at most 2 can be evaluated in pspace, and this set of expressions is sufficient to express all queries in the polynomial hierarchy; (2) In the simply typed lambda calculus with equality and constants, all query terms of order at most 5 (where "query term" is a syntactic condition on types) can be evaluated in pspace, and this set of terms expresses exactly the pspace queries; (3) There exists a set of secondorder method schemas (with no simple syntactic characterization) that can be evaluated in pspace, and this set of schemas is sufficient to express all pspace queries.
Finite Model Theory In The Simply Typed Lambda Calculus
, 1994
"... Church's simply typed calculus is a very basic framework for functional programming language research. However, it is common to augment this framework with additional programming constructs, because its expressive power for functions over the domain of Church numerals is very limited. In this ..."
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Cited by 8 (5 self)
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Church's simply typed calculus is a very basic framework for functional programming language research. However, it is common to augment this framework with additional programming constructs, because its expressive power for functions over the domain of Church numerals is very limited. In this thesis: (1) We reexamine the expressive power of the "pure" simply typed calculus, but over encodings of finite relational structures, i. e., finite models or databases . In this novel framework the simply typed calculus expresses all elementary functions from finite models to finite models. In addition, many common database query languages, e. g., relational algebra, Datalog : , and the Abiteboul/Beeri complex object algebra, can be embedded into it. The embeddings are feasible in the sense that the terms corresponding to PTIME queries can be evaluated in polynomial time. (2) We examine fixedorder fragments of the simply typed calculus to determine machine independent characterizations of complexity classes. For this we augment the calculus with atomic constants and equality among atomic constants. We show that over ordered structures, the order 3, 4, 5, and 6 fragments express exactly the firstorder, PTIME, PSPACE, and EXPTIME queries, respectively, and we conjecture that for general k 1, order 2 k + 4 expresses exactly the kEXPTIME queries and order 2 k + 5 expresses exactly the kEXPSPACE queries. (3) We also reexamine other functional characterizations of PTIME and we show that method schemas with ordered objects express exactly PTIME. This is a firstorder framework proposed for objectoriented databasesas opposed to the above higherorder frameworks. In summary, this research provides a link between finite model theory (and thus computational complexity), dat...
On the expressive power of simply typed and letpolymorphic lambda calculi
 11th Annual IEEE Symp. on Logic in Computer Science (LICS'96)
, 1996
"... We present a functional framework for descriptive computational complexity, in which the Regular, Firstorder, Ptime, Pspace, kExptime, kExpspace (k 1), and Elementary sets have syntactic characterizations. In this framework, typed lambda terms represent inputs and outputs as well as programs. The ..."
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Cited by 7 (0 self)
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We present a functional framework for descriptive computational complexity, in which the Regular, Firstorder, Ptime, Pspace, kExptime, kExpspace (k 1), and Elementary sets have syntactic characterizations. In this framework, typed lambda terms represent inputs and outputs as well as programs. The lambda calculi describing the above computational complexity classes are simply or letpolymorphically typed with functionalities of fixed order. They consist of: order 0 atomic constants, order 1 equality among these constants, variables, application, and abstraction. Increasing functionality order by one for these languages corresponds to increasing the computational complexity by one alternation. This exact correspondence is established using a semantic evaluation of languages for each fixed order, which is the primary technical contribution of this paper.
An Analysis of the CoreML Language: Expressive Power and Type Reconstruction
 In Proc. 21st Int'l Coll. Automata, Languages, and Programming
, 1994
"... CoreML is a basic subset of most functional programming languages. It consists of the simply typed (or monomorphic) calculus, simply typed equality over atomic constants, and let as the only polymorphic construct. We present a synthesis of recent results which characterize this "toy" ..."
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Cited by 5 (3 self)
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CoreML is a basic subset of most functional programming languages. It consists of the simply typed (or monomorphic) calculus, simply typed equality over atomic constants, and let as the only polymorphic construct. We present a synthesis of recent results which characterize this "toy" language's expressive power as well as its type reconstruction (or type inference) problem. More specifically: (1) CoreML can express exactly the ELEMENTARY queries, where a program input is a database encoded as a term and a query program is a term whose application to the input normalizes to the output database. In addition, it is possible to express all the PTIME queries so that this normalization process is polynomial in the input size. (2) The polymorphism of let can be explained using a simple algorithmic reduction to monomorphism, and provides flexibility, without affecting expressibility. Algorithms for type reconstruction offer the additional convenience of static typing without type declarations. Given polymorphism, the price of this convenience is an increase in complexity from lineartime in the size of the program typed (without let) to completeness in exponentialtime (with let).
Positive HigherOrder Queries
"... We investigate a higherorder query language that embeds operators of the positive relational algebra within the simplytyped λcalculus. Our language allows one to succinctly define ordinary positive relational algebra queries (conjunctive queries and unions of conjunctive queries) and, in addition ..."
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Cited by 4 (2 self)
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We investigate a higherorder query language that embeds operators of the positive relational algebra within the simplytyped λcalculus. Our language allows one to succinctly define ordinary positive relational algebra queries (conjunctive queries and unions of conjunctive queries) and, in addition, secondorder query functionals, which allow the transformation of CQs and UCQs in a generic (i.e., syntaxindependent) way. We investigate the equivalence and containment problems for this calculus, which subsumes traditional CQ/UCQ containment. Query functionals are said to be equivalent if the output queries are equivalent, for each possible input query, and similarly for containment. These notions of containment and equivalence depend on the class of (ordinary relational algebra) queries considered. We show that containment and equivalence are decidable when query variables are restricted to positive relational algebra and we identify the precise complexity of the problem. We also identify classes of functionals where containment is tractable. Finally, we provide upper bounds to the complexity of the containment problem when functionals act over other classes.
A Data Model for Effectively Computable Functions
 PhD Workshop in 7 th International Conference on extending Database Technology
"... Searching for a better data model would continue unless a satisfaction was reached. Improving the productivity and quality of developing traditional and emerging database applications is the motivation driving the research activities. This abstract is to introduce the approach of a data model EP da ..."
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Searching for a better data model would continue unless a satisfaction was reached. Improving the productivity and quality of developing traditional and emerging database applications is the motivation driving the research activities. This abstract is to introduce the approach of a data model EP data model. Its data structure is able to store as a finite set of nodes arbitrary effectively computable functions. Its queries are arbitrary computable functions. Both the data structure and the queries of the EP data model are unified under an extended lambda caculus, which provides a uniform manner to access data (either finite or infinite). Expressiveness and Descriptiveness The less implementation details developers have to deal with,
Reflections on complexity of ML type reconstruction
, 1997
"... This is a collection of some more or less chaotic remarks on the ML type system, definitely not sufficient to fill a research paper of reasonable quality, but perhaps interesting enough to be written down as a note. At the beginning the idea was to investigate the complexity of type reconstruction a ..."
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This is a collection of some more or less chaotic remarks on the ML type system, definitely not sufficient to fill a research paper of reasonable quality, but perhaps interesting enough to be written down as a note. At the beginning the idea was to investigate the complexity of type reconstruction and typability in bounded order fragments of ML. Unexpectedly the problem turned out to be hard, and finally I obtained only partial results. I do not feel like spending more time on this topic, so the text is not polished, the proofs  if included at all  are only sketched and of rather poor mathematical quality. I believe however, that some remarks, especially those of "philosophical" nature, shed some light on the ML type system and may be of some value to the reader interested especially in the interaction between theory and practice of ML type reconstruction. 1 Introduction The ML type system was developed by Robin Milner in the late seventies [26, 3], but was influenced by much ol...
General Terms
"... While relational algebra and calculus are a wellestablished foundation for classical database query languages, it is less clear what the analog is for higherorder functions, such as query transformations. Here we study a natural way to add higherorder functionality to query languages, by adding d ..."
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While relational algebra and calculus are a wellestablished foundation for classical database query languages, it is less clear what the analog is for higherorder functions, such as query transformations. Here we study a natural way to add higherorder functionality to query languages, by adding database query operators to the λcalculus as constants. This framework, which we refer to as λembedded query languages, was introduced in [BPV10]. That work had a restricted focus: the containment and equivalence problems for querytoquery functions, in the case where only positive relational operators are allowed as constants. In this work we take an indepth look at the most basic issue for such languages: the evaluation problem. We give a full picture of the complexity of evaluation for λembedded query languages, looking at a number of variations: with negation and without; with only relational algebra operators, and also with a recursion mechanism in the form of a query iteration operator; in a stronglytyped calculus as well as a weaklytyped one. We give tight bounds on both the combined complexity and the query complexity of evaluation in all these settings, in the process explaining connections with Datalog and prior work on λcalculus evaluation.
A Polymorphic Calculus for Database Languages
, 1997
"... We present a new database calculus, F db , which is based on Reynold's polymorphic lambda calculus. It supports new type constructors that capture precisely the commutativity and idempotence properties of bags and sets. The type checking rules for these extensions accept most valid programs and ..."
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We present a new database calculus, F db , which is based on Reynold's polymorphic lambda calculus. It supports new type constructors that capture precisely the commutativity and idempotence properties of bags and sets. The type checking rules for these extensions accept most valid programs and filter out all inconsistent ones, including for example the programs that convert sets into lists. We give the meaning of these type extensions by using a parametricity theorem and we use this theorem to prove the soundness of the type checking rules. 1 Introduction The simply typed calculus (also called F 1 ) was invented by Church [6]. It differs from the untyped calculus in that it requires type annotations for lambda abstractions, x : t: e, where x is a parameter of type t and e is the function body. A recent work by Hillebrand et al [12, 13] uses the simply typed calculus as a basis for a database language. In that language, collection types, such as lists, are represented by their Chu...