Regular and First-Order List Functions

Bojańczyk, Mikołaj; Daviaud, Laure; Krishna, Shankara Narayanan

doi:10.1145/3209108.3209163

Cited by 11 publications

(24 citation statements)

References 21 publications

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“…In addition to the application areas exhibited in Examples 4.4 and 4.5, we think that procedures for generating implementations in functional languages from implicit definitions should have other applications in programming languages and verification. For example they could be relevant for generating programs transforming structured data in the context of more specialized input structures, such as strings and trees [Bojanczyk et al 2018].…”

Section: Resultsmentioning

confidence: 99%

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Generating collection transformations from proofs

Benedikt

Pradic

2021

Proc. ACM Program. Lang.

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Nested relations, built up from atomic types via product and set types, form a rich data model. Over the last decades the nested relational calculus, NRC, has emerged as a standard language for defining transformations on nested collections. NRC is a strongly-typed functional language which allows building up transformations using tupling and projections, a singleton-former, and a map operation that lifts transformations on tuples to transformations on sets. In this work we describe an alternative declarative method of describing transformations in logic. A formula with distinguished inputs and outputs gives an implicit definition if one can prove that for each input there is only one output that satisfies it. Our main result shows that one can synthesize transformations from proofs that a formula provides an implicit definition, where the proof is in an intuitionistic calculus that captures a natural style of reasoning about nested collections. Our polynomial time synthesis procedure is based on an analog of Craig's interpolation lemma, starting with a provable containment between terms representing nested collections and generating an NRC expression that interpolates between them. We further show that NRC expressions that implement an implicit definition can be found when there is a classical proof of functionality, not just when there is an intuitionistic one. That is, whenever a formula implicitly defines a transformation, there is an NRC expression that implements it.

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Section: Resultsmentioning

confidence: 99%

“…We first review the notion of an interpretation, which has become a common way of defining transformations using logical expressions [Bojanczyk et al 2018;Colcombet and Löding 2007]. Let S and S be multi-sorted vocabularies.…”

Section: Interpretations and Nested Relationsmentioning

confidence: 99%

Generating collection transformations from proofs

Benedikt

Pradic

2021

Proc. ACM Program. Lang.

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“…A closely related paper to our work is [13], where first-order and regular list functions were introduced. Using the basic functions reverse, append, co-append, map, block on lists, and combining them with the function combinators of disjoint union, map, pairing and composition, these were shown to be equivalent (after a suitable encoding) to FO transductions a la Courcelle (extendible to MSO transductions by adding to the basic functions, the prefix multiplication operation on groups).…”

Section: As An Example Consider the Sdrtesmentioning

confidence: 99%

SD-Regular Transducer Expressions for Aperiodic Transformations

Dartois

Gastin

Krishna

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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FO transductions, aperiodic deterministic two-way transducers, as well as aperiodic streaming string transducers are all equivalent models for first order definable functions. In this paper, we solve the problem of expressions capturing first order definable functions, thereby generalizing the seminal SF=AP (star-free expressions = aperiodic languages) result of Schützenberger. Our result also generalizes a lesser known characterization by Schützenberger of aperiodic languages by SD-regular expressions (SD=AP). We show that every first order definable function over finite words captured by an aperiodic deterministic two-way transducer can be described with an SDregular transducer expression (SDRTE). An SDRTE is a regular expression where Kleene stars are used in a restricted way: they can appear only on aperiodic languages which are prefix codes of bounded synchronization delay. SDRTEs are constructed from simple functions using the combinators unambiguous sum (deterministic choice), Hadamard product, and unambiguous versions of the Cauchy product and the k-chained Kleene-star, where the star is restricted as mentioned. In order to construct an SDRTE associated with an aperiodic deterministic two-way transducer, (i) we concretize Schützenberger's SD=AP result, by proving that aperiodic languages are captured by SD-regular expressions which are unambiguous and stabilising; (ii) by structural induction on the unambiguous, stabilising SD-regular expressions describing the domain of the transducer, we construct SDRTEs. Finally, we also look at various formalisms equivalent to SDRTEs which use the function composition, allowing to trade the k-chained star for a 1-star.

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“…The recent work [Bojańczyk et al 2018] gives a characterization of the first-order definable and MSO-definable string-to-string transformations using algebras of functions that operate on objects such as lists, lists of lists, pairs of lists, lists of pairs of lists, and so on. Monitors with finite-state control and unbounded integer registers are studied in [Ferrère et al 2018] and a hierarchy of expressiveness is established on the basis of the number of available registers.…”

Section: Quantitative Automatamentioning

confidence: 99%

Modular quantitative monitoring

Alur

Mamouras

Stanford

2019

Proc. ACM Program. Lang.

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In real-time decision making and runtime monitoring applications, declarative languages are commonly used as they facilitate modular high-level specifications with the compiler guaranteeing evaluation over data streams in an efficient and incremental manner. We introduce the model of Data Transducers to allow modular compilation of queries over streaming data. A data transducer maintains a finite set of data variables and processes a sequence of tagged data values by updating its variables using an allowed set of operations. The model allows unambiguous nondeterminism, exponentially succinct control, and combining values from parallel threads of computation. The semantics of the model immediately suggests an efficient streaming algorithm for evaluation. The expressiveness of data transducers coincides with streamable regular transductions, a robust and streamable class of functions characterized by MSO-definable string-to-DAG transformations with no backward edges. We show that the novel features of data transducers, unlike previously studied transducers, make them as succinct as traditional imperative code for processing data streams, but the structuring of the transition function permits modular compilation. In particular, we show that operations such as parallel composition, union, prefix-sum, and quantitative analogs of combinators for unambiguous parsing, can be implemented by natural and succinct constructions on data transducers. To illustrate the benefits of such modularity in compilation, we define a new language for quantitative monitoring, QRE-Past, that integrates features of past-time temporal logic and quantitative regular expressions. While this combination allows a natural specification of a cardiac arrhythmia detection algorithm in QRE-Past, compilation of QRE-Past specifications into efficient monitors comes for free thanks to succinct constructions on data transducers.

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Regular and First-Order List Functions

Cited by 11 publications

References 21 publications

Generating collection transformations from proofs

Generating collection transformations from proofs

SD-Regular Transducer Expressions for Aperiodic Transformations

Modular quantitative monitoring

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