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Wand, Mitchell

doi:10.1023/a:1007720632734

Cited by 25 publications

(5 citation statements)

References 13 publications

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“…Nested queries bear a resemblance to evaluation within evaluation, which is one form of computational reflection [Smith 1982]. In a historically damning paper on fexprs (a mechanism for total reification), Wand [1998] proves that too much reification can hinder all optimizations, admitting only łtrivialž equalities. By contrast, nested queries have a richer equational theory.…”

Section: Related Work and Discussionmentioning

confidence: 99%

Reasoning about “reasoning about reasoning”: semantics and contextual equivalence for probabilistic programs with nested queries and recursion

Zhang

Amin

2022

Proc. ACM Program. Lang.

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Metareasoning can be achieved in probabilistic programming languages (PPLs) using agent models that recursively nest inference queries inside inference queries. However, the semantics of this powerful, reflection-like language feature has defied an operational treatment, much less reasoning principles for contextual equivalence. We give formal semantics to a core PPL with continuous distributions, scoring, general recursion, and nested queries. Unlike prior work, the presence of nested queries and general recursion makes it impossible to stratify the definition of a sampling-based operational semantics and that of a measure-theoretic semantics—the two semantics must be defined mutually recursively. A key yet challenging property we establish is that probabilistic programs have well-defined meanings: limits exist for the step-indexed measures they induce. Beyond a semantics, we offer relational reasoning principles for probabilistic programs making nested queries. We construct a step-indexed, biorthogonal logical-relations model. A soundness theorem establishes that logical relatedness implies contextual equivalence. We demonstrate the usefulness of the reasoning principles by proving novel equivalences of practical relevance—in particular, game-playing and decisionmaking agents. We mechanize our technical developments leading to the soundness proof using the Coq proof assistant. Nested queries are an important yet theoretically underdeveloped linguistic feature in PPLs; we are first to give them semantics in the presence of general recursion and to provide them with sound reasoning principles for contextual equivalence.

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Section: Related Work and Discussionmentioning

confidence: 99%

Reasoning about “reasoning about reasoning”: semantics and contextual equivalence for probabilistic programs with nested queries and recursion

Zhang

Amin

2022

Proc. ACM Program. Lang.

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“…This is a rather mysterious construction with unclear semantics that have been the subject of investigation by Friedman and Wand [1984], Wand and Friedman [1988], and Danvy and Malmkjaer [1988]. This line of work eventually concluded with a theorem of Wand [1998], which shows that there are no useful semantic descriptions of the reflective tower. We will discuss Wand's result in §1.3.2.…”

Section: Reflective Programmingmentioning

confidence: 99%

“…This strand of research quickly ran into impossibility results that demonstrate that reflective features are logically ill-behaved: see e.g. [2] for reflection in untyped λ-calculus, or [25] for a more involved example involving the LISP fexpr construct. Our viewpoint allows us to talk about the notion of intensional recursion, which is more general than ordinary extensional recursion, and seems to correspond to a well-behaved form of reflection.…”

Section: Introduction: Intensionality and Intensional Recursionmentioning

confidence: 99%

On the Semantics of Intensionality

Kavvos¹

2017

Lecture Notes in Computer Science

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In this paper we propose a categorical theory of intensionality. We first revisit the notion of intensionality, and discuss its relevance to logic and computer science. It turns out that 1-category theory is not the most appropriate vehicle for studying the interplay of extension and intension. We are thus led to consider the P-categories of Čubrić, Dybjer and Scott, which are categories only up to a partial equivalence relation (PER). In this setting, we introduce a new P-categorical construct, that of exposures. Exposures are very nearly functors, except that they do not preserve the PERs of the Pcategory. Inspired by the categorical semantics of modal logic, we begin to develop their theory. Our leading examples demonstrate that an exposure is an abstraction of well-behaved intensional devices, such as Gödel numberings. The outcome is a unifying framework in which classic results of Kleene, Gödel, Tarski and Rice find concise, clear formulations, and where each logical device or assumption involved in their proofs can be expressed in the same algebraic manner.

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“…In fact, it allows for a certain kind of computational reflection, or reflective programming, of the same kind envisaged by Brian Cantwell Smith [22]. But the programme of Smith's reflective tower involved a rather mysterious construction with unclear semantics [8,29,6], eventually leading to a theorem that-even in the presence of a mild reflective construct, the so-called fexprobservational equivalence of programs collapses to α-conversion [28].…”

Section: Intensional Recursionmentioning

confidence: 99%

Intensionality, Intensional Recursion, and the Gödel-Löb axiom

Kavvos

2017

Preprint

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The use of a necessity-like modality in a typed λ -calculus can be used as a device for separating the calculus in two separate regions. These can be thought of as intensional vs. extensional data: data in the first region, the modal one, are available as code, and their description can be examined, whereas data in the second region are only available as values up to ordinary equality. This allows us to add seemingly non-functional operations at modal types, whilst maintaining consistency. In this setting the Gödel-Löb axiom acquires a novel constructive reading: it affords the programmer the possibility of a very strong kind of recursion, by enabling him to write programs that have access to their own code. This is a type of computational reflection that is strongly reminiscent of Kleene's Second Recursion Theorem. We prove that it is consistent with the rest of the system.

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Cited by 25 publications

References 13 publications

Reasoning about “reasoning about reasoning”: semantics and contextual equivalence for probabilistic programs with nested queries and recursion

Reasoning about “reasoning about reasoning”: semantics and contextual equivalence for probabilistic programs with nested queries and recursion

On the Semantics of Intensionality

Intensionality, Intensional Recursion, and the Gödel-Löb axiom

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