605 research outputs found
Theory Morphisms in Church's Type Theory with Quotation and Evaluation
is a version of Church's type theory with global
quotation and evaluation operators that is engineered to reason about the
interplay of syntax and semantics and to formalize syntax-based mathematical
algorithms. is a variant of that
admits undefined expressions, partial functions, and multiple base types of
individuals. It is better suited than as a logic for
building networks of theories connected by theory morphisms. This paper
presents the syntax and semantics of , defines a notion of
a theory morphism from one theory to another, and gives
two simple examples that illustrate the use of theory morphisms in .Comment: 17 page
Formal verification of higher-order probabilistic programs
Probabilistic programming provides a convenient lingua franca for writing
succinct and rigorous descriptions of probabilistic models and inference tasks.
Several probabilistic programming languages, including Anglican, Church or
Hakaru, derive their expressiveness from a powerful combination of continuous
distributions, conditioning, and higher-order functions. Although very
important for practical applications, these combined features raise fundamental
challenges for program semantics and verification. Several recent works offer
promising answers to these challenges, but their primary focus is on semantical
issues.
In this paper, we take a step further and we develop a set of program logics,
named PPV, for proving properties of programs written in an expressive
probabilistic higher-order language with continuous distributions and operators
for conditioning distributions by real-valued functions. Pleasingly, our
program logics retain the comfortable reasoning style of informal proofs thanks
to carefully selected axiomatizations of key results from probability theory.
The versatility of our logics is illustrated through the formal verification of
several intricate examples from statistics, probabilistic inference, and
machine learning. We further show the expressiveness of our logics by giving
sound embeddings of existing logics. In particular, we do this in a parametric
way by showing how the semantics idea of (unary and relational) TT-lifting can
be internalized in our logics. The soundness of PPV follows by interpreting
programs and assertions in quasi-Borel spaces (QBS), a recently proposed
variant of Borel spaces with a good structure for interpreting higher order
probabilistic programs
Relational Parametricity for Computational Effects
According to Strachey, a polymorphic program is parametric if it applies a
uniform algorithm independently of the type instantiations at which it is
applied. The notion of relational parametricity, introduced by Reynolds, is one
possible mathematical formulation of this idea. Relational parametricity
provides a powerful tool for establishing data abstraction properties, proving
equivalences of datatypes, and establishing equalities of programs. Such
properties have been well studied in a pure functional setting. Many programs,
however, exhibit computational effects, and are not accounted for by the
standard theory of relational parametricity. In this paper, we develop a
foundational framework for extending the notion of relational parametricity to
programming languages with effects.Comment: 31 pages, appears in Logical Methods in Computer Scienc
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