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Register a new userStrongly-Normalizing Higher-Order Relational Queries
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Added by
Wilmer Ricciotti
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Language-integrated query is a popular and powerful programming construct allowing database queries and ordinary program code to interoperate seamlessly and safely. Language-integrated query techniques rely on classical results about the nested relational calculus stating that its queries can be algorithmically translated to SQL, as long as their result type is a flat relation. Cooper and others advocated higher-order nested relational calculi as a basis for language-integrated queries in functional languages such as Links and F#. However, the translation of higher-order relational queries to SQL relies on a rewrite system for which no strong normalization proof has been published: a previous proof attempt does not deal correctly with rewrite rules that duplicate subterms. This paper fills the gap in the literature, explaining the difficulty with the previous attempt, and showing how to extend the $toptop$-lifting approach of Lindley and Stark to accommodate duplicating rewrites. We also show how to extend the proof to a recently-introduced calculus for heterogeneous queries mixing set and multiset semantics.
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We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kocks synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.
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