Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finite-dimensional, they cannot accomodate (co)limit-based constructions. For example, they cannot capture protocols such as quantum key distr
ibution, that rely on the law of large numbers. To overcome this limitation, we introduce the notion of a compactly accessible category, relying on the extra structure of a factorisation system. This notion allows for infinite dimension while retaining key properties of compact categories: the main technical result is that the choice-of-duals functor on the compact part extends canonically to the whole compactly accessible category. As an example, we model a quantum key distribution protocol and prove its correctness categorically.
The use of annotations, referred to as assertions or contracts, to describe program properties for which run-time tests are to be generated, has become frequent in dynamic programing languages. However, the frameworks proposed to support such run-tim
e testing generally incur high time and/or space overheads over standard program execution. We present an approach for reducing this overhead that is based on the use of memoization to cache intermediate results of check evaluation, avoiding repeated checking of previously verified properties. Compared to approaches that reduce checking frequency, our proposal has the advantage of being exhaustive (i.e., all tests are checked at all points) while still being much more efficient than standard run-time checking. Compared to the limited previous work on memoization, it performs the task without requiring modifications to data structure representation or checking code. While the approach is general and system-independent, we present it for concreteness in the context of the Ciao run-time checking framework, which allows us to provide an operational semantics with checks and caching. We also report on a prototype implementation and provide some experimental results that support that using a relatively small cache leads to significant decreases in run-time checking overhead.
We present a method for verifying the correctness of imperative programs which is based on the automated transformation of their specifications. Given a program prog, we consider a partial correctness specification of the form ${varphi}$ prog ${psi}$
, where the assertions $varphi$ and $psi$ are predicates defined by a set Spec of possibly recursive Horn clauses with linear arithmetic (LA) constraints in their premise (also called constrained Horn clauses). The verification method consists in constructing a set PC of constrained Horn clauses whose satisfiability implies that ${varphi}$ prog ${psi}$ is valid. We highlight some limitations of state-of-the-art constrained Horn clause solving methods, here called LA-solving methods, which prove the satisfiability of the clauses by looking for linear arithmetic interpretations of the predicates. In particular, we prove that there exist some specifications that cannot be proved valid by any of those LA-solving methods. These specifications require the proof of satisfiability of a set PC of constrained Horn clauses that contain nonlinear clauses (that is, clauses with more than one atom in their premise). Then, we present a transformation, called linearization, that converts PC into a set of linear clauses (that is, clauses with at most one atom in their premise). We show that several specifications that could not be proved valid by LA-solving methods, can be proved valid after linearization. We also present a strategy for performing linearization in an automatic way and we report on some experimental results obtained by using a preliminary implementation of our method.
Software testing is one of the most popular validation techniques in the software industry. Surprisingly, we can only find a few approaches to testing in the context of logic programming. In this paper, we introduce a systematic approach for dynamic
testing that combines both concrete and symbolic execution. Our approach is fully automatic and guarantees full path coverage when it terminates. We prove some basic properties of our technique and illustrate its practical usefulness through a prototype implementation.
We study idempotents in intensional Martin-Lof type theory, and in particular the question of when and whether they split. We show that in the presence of propositional truncation and Voevodskys univalence axiom, there exist idempotents that do not s
plit; thus in plain MLTT not all idempotents can be proven to split. On the other hand, assuming only function extensionality, an idempotent can be split if and only if its witness of idempotency satisfies one extra coherence condition. Both proofs are inspired by parallel results of Lurie in higher category theory, showing that ideas from higher category theory and homotopy theory can have applications even in ordinary MLTT. Finally, we show that although the witness of idempotency can be recovered from a splitting, the one extra coherence condition cannot in general; and we construct the type of fully coherent idempotents, by splitting an idempotent on the type of partially coherent ones. Our results have been formally verified in the proof assistant Coq.
Nondeterminism in scheduling is the cardinal reason for difficulty in proving correctness of concurrent programs. A powerful proof strategy was recently proposed [6] to show the correctness of such programs. The approach captured data-flow dependenci
es among the instructions of an interleaved and error-free execution of threads. These data-flow dependencies were represented by an inductive data-flow graph (iDFG), which, in a nutshell, denotes a set of executions of the concurrent program that gave rise to the discovered data-flow dependencies. The iDFGs were further transformed in to alternative finite automatons (AFAs) in order to utilize efficient automata-theoretic tools to solve the problem. In this paper, we give a novel and efficient algorithm to directly construct AFAs that capture the data-flow dependencies in a concurrent program execution. We implemented the algorithm in a tool called ProofTraPar to prove the correctness of finite state cyclic programs under the sequentially consistent memory model. Our results are encouranging and compare favorably to existing state-of-the-art tools.
We study the sequences of numbers corresponding to lambda terms of given sizes, where the size is this of lambda terms with de Bruijn indices in a very natural model where all the operators have size 1. For plain lambda terms, the sequence correspond
s to two families of binary trees for which we exhibit bijections. We study also the distribution of normal forms, head normal forms and strongly normalizing terms. In particular we show that strongly normalizing terms are of density 0 among plain terms.
In this paper we present several examples of solving algorithmic problems from the Google Code Jam programming contest with Picat programming language using declarative techniques: constraint logic programming and tabled logic programming. In some ca
ses the use of Picat simplifies the implementation compared to conventional imperative programming languages, while in others it allows to directly convert the problem statement into an efficiently solvable declarative problem specification without inventing an imperative algorithm.
We demonstrate that general-purpose memory allocation involving many threads on many cores can be done with high performance, multicore scalability, and low memory consumption. For this purpose, we have designed and implemented scalloc, a concurrent
allocator that generally performs and scales in our experiments better than other allocators while using less memory, and is still competitive otherwise. The main ideas behind the design of scalloc are: uniform treatment of small and big objects through so-called virtual spans, efficiently and effectively reclaiming free memory through fast and scalable global data structures, and constant-time (modulo synchronization) allocation and deallocation operations that trade off memory reuse and spatial locality without being subject to false sharing.
Scripting code may present maintenance problems in the long run. There is, then, the call for methodologies that make it possible to control the properties of programs written in dynamic languages in an automatic fashion. We introduce Lucretia, a cor
e language with an introspection primitive. Lucretia is equipped with a (retrofitted) static type system based on local updates of types that describe the structure of objects being used. In this way, we deal with one of the most dynamic features of scripting languages, that is, the runtime modification of object interfaces. Judgements in our systems have a Hoare-like shape, as they have a precondition and a postcondition part. Preconditions describe static approximations of the interfaces of visible objects before a certain expression has been executed and postconditions describe them after its execution. The field update operation complicates the issue of aliasing in the system. We cope with it by introducing intersection types in method signatures.
