39 research outputs found

    On the mathematical synthesis of equational logics

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    We provide a mathematical theory and methodology for synthesising equational logics from algebraic metatheories. We illustrate our methodology by means of two applications: a rational reconstruction of Birkhoff's Equational Logic and a new equational logic for reasoning about algebraic structure with name-binding operators.Comment: Final version for publication in Logical Methods in Computer Scienc

    Step-Indexing: The Good, the Bad and the Ugly

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    Over the last decade, step-indices have been widely used for the construction of operationally-based logical relations in the presence of various kinds of recursion. We first give an argument that step-indices, or something like them, seem to be required for defining realizability relations between high-level source languages and low-level targets, in the case that the low-level allows egregiously intensional operations such as reflection or comparison of code pointers. We then show how, much to our annoyance, step-indices also seem to prevent us from exploiting such operations as aggressively as we would like in proving program transformations

    A Provably Correct Sampler for Probabilistic Programs

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    Formally Verified Simulations of State-Rich Processes using Interaction Trees in Isabelle/HOL

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    Simulation and formal verification are important complementary techniques necessary in high assurance model-based systems development. In order to support coherent results, it is necessary to provide unifying semantics and automation for both activities. In this paper we apply Interaction Trees in Isabelle/HOL to produce a verification and simulation framework for state-rich process languages. We develop the core theory and verification techniques for Interaction Trees, use them to give a semantics to the CSP and Circus languages, and formally link our new semantics with the failures-divergences semantic model. We also show how the Isabelle code generator can be used to generate verified executable simulations for reactive and concurrent programs

    Interaction Trees: Representing Recursive and Impure Programs in Coq

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    "Interaction trees" (ITrees) are a general-purpose data structure for representing the behaviors of recursive programs that interact with their environments. A coinductive variant of "free monads," ITrees are built out of uninterpreted events and their continuations. They support compositional construction of interpreters from "event handlers", which give meaning to events by defining their semantics as monadic actions. ITrees are expressive enough to represent impure and potentially nonterminating, mutually recursive computations, while admitting a rich equational theory of equivalence up to weak bisimulation. In contrast to other approaches such as relationally specified operational semantics, ITrees are executable via code extraction, making them suitable for debugging, testing, and implementing software artifacts that are amenable to formal verification. We have implemented ITrees and their associated theory as a Coq library, mechanizing classic domain- and category-theoretic results about program semantics, iteration, monadic structures, and equational reasoning. Although the internals of the library rely heavily on coinductive proofs, the interface hides these details so that clients can use and reason about ITrees without explicit use of Coq's coinduction tactics. To showcase the utility of our theory, we prove the termination-sensitive correctness of a compiler from a simple imperative source language to an assembly-like target whose meanings are given in an ITree-based denotational semantics. Unlike previous results using operational techniques, our bisimulation proof follows straightforwardly by structural induction and elementary rewriting via an equational theory of combinators for control-flow graphs.Comment: 28 pages, 4 pages references, published at POPL 202

    Probabilistic Programming with Densities in SlicStan: Efficient, Flexible, and Deterministic

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    Stan is a probabilistic programming language that has been increasingly used for real-world scalable projects. However, to make practical inference possible, the language sacrifices some of its usability by adopting a block syntax, which lacks compositionality and flexible user-defined functions. Moreover, the semantics of the language has been mainly given in terms of intuition about implementation, and has not been formalised. This paper provides a formal treatment of the Stan language, and introduces the probabilistic programming language SlicStan --- a compositional, self-optimising version of Stan. Our main contributions are: (1) the formalisation of a core subset of Stan through an operational density-based semantics; (2) the design and semantics of the Stan-like language SlicStan, which facilities better code reuse and abstraction through its compositional syntax, more flexible functions, and information-flow type system; and (3) a formal, semantic-preserving procedure for translating SlicStan to Stan

    A Kripke Logical Relation Between ML and Assembly

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    There has recently been great progress in proving the correctness of compilers for increasingly realistic languages with increasingly realistic runtime systems. Most work on this problem has focused on proving the correctness of a particular compiler, leaving open the question of how to verify the correctness of assembly code that is hand-optimized or linked together from the output of multiple compilers. This has led Benton and other researchers to propose more abstract, compositional notions of when a low-level program correctly realizes a high-level one. However, the state of the art in so-called “compositional compiler correctness ” has only considered relatively simple high-level and low-level languages. In this paper, we propose a novel, extensional, compilerindependent notion of equivalence between high-level programs in an expressive, impure ML-like λ-calculus and low-level programs in an (only slightly) idealized assembly language. We define this equivalence by means of a biorthogonal, step-indexed, Kripke logical relation, which enables us to reason quite flexibly about assembly code that uses local state in a different manner than the high-level code it implements (e.g., self-modifying code). In contrast to prior work, we factor our relation in a symmetric, languagegeneric fashion, which helps to simplify and clarify the formal presentation, and we also show how to account for the presence of a garbage collector. Our approach relies on recent developments in Kripke logical relations for ML-like languages, in particular the idea of possible worlds as state transition systems. 1