A general theorem on the total correctness of programs in a category

AbstractIn [7], we presented a completeness theorem for proving partial correctness of programs in a large class of categories. This theorem generalized a classical result of S.Cook [5] for the language of while-programs.Here we address the total correctness of programs. Again, we use the semantics based on partially additive categories, which was introduced by M.A.Arbib and E.G.Manes [3,4,6]. Our theorems generalize the non categorical results of K.R.Apt [1,2]. They are valid for a large class of partially additive categories, including the category of sets and partial functions and the category of sets and relations, i.e. for deterministic and nondeterministic programs

A General Framework for Sound and Complete Floyd-Hoare Logics

This paper presents an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. Our abstraction is based on a traced monoidal functor from an arbitrary traced monoidal category into the category of pre-orders and monotone relations. We give several examples of how our theory generalises usual Hoare logics (partial correctness of while programs, partial correctness of pointer programs), and provide some case studies on how it can be used to develop new Hoare logics (run-time analysis of while programs and stream circuits).Comment: 27 page

Data optimizations for constraint automata

Constraint automata (CA) constitute a coordination model based on finite automata on infinite words. Originally introduced for modeling of coordinators, an interesting new application of CAs is implementing coordinators (i.e., compiling CAs into executable code). Such an approach guarantees correctness-by-construction and can even yield code that outperforms hand-crafted code. The extent to which these two potential advantages materialize depends on the smartness of CA-compilers and the existence of proofs of their correctness. Every transition in a CA is labeled by a "data constraint" that specifies an atomic data-flow between coordinated processes as a first-order formula. At run-time, compiler-generated code must handle data constraints as efficiently as possible. In this paper, we present, and prove the correctness of two optimization techniques for CA-compilers related to handling of data constraints: a reduction to eliminate redundant variables and a translation from (declarative) data constraints to (imperative) data commands expressed in a small sequential language. Through experiments, we show that these optimization techniques can have a positive impact on performance of generated executable code

Coinductive Formal Reasoning in Exact Real Arithmetic

In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by coinductive streams. The algorithms work on coinductive stream of M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic. We use the machinery of the Coq proof assistant for the coinductive types to present the formalisation. The formalised algorithms are only partially productive, i.e., they do not output provably infinite streams for all possible inputs. We show how to deal with this partiality in the presence of syntactic restrictions posed by the constructive type theory of Coq. Furthermore we show that the type theoretic techniques that we develop are compatible with the semantics of the algorithms as continuous maps on real numbers. The resulting Coq formalisation is available for public download.Comment: 40 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

Fifty years of Hoare's Logic

We present a history of Hoare's logic.Comment: 79 pages. To appear in Formal Aspects of Computin

Glueability of Resource Proof-Structures: Inverting the Taylor Expansion

A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures