Logic Programming Based on Higher-Order Hereditary Harrop Formulas

Hereditary Harrop formulas are an extension to Horn clauses in which the body of clauses can contain implications and universal quantifiers. These formulas can further be extended by embedding them in a higher-order logic; that is, by permitting quantification over function symbol occurrences and some predicate symbol occurrences, and by replacing first-order terms with simply typed λ-terms. Our justification for considering this rich extension of Horn clause theory as a satisfactory logic programming language is provided by a proof-theoretic notion we call uniform proofs . This notion will be defined and motivated. This extended language can provide very natural and direct implementations of various kinds of abstraction mechanisms. For example, higher-order hereditary Harrop formulas (hohh) can be used to support aspects of modular programming, abstract data types, and higher-order programming. We have designed and built a logic programming system which implements hohh in much the same way Prolog implements first-order Horn clauses. This language and its interpreter, collectively called λProlog, will be described. We will present several example programs where λProlog provides a much more immediate and satisfactory implementation language than first-order Prologs. These examples are taken from theorem proving and program transformation. Finally, we will describe some aspects of our implementation of λProlog

Interactive Programming by Example

As of today, programming has never been so accessible. Yet, it remains a challenge for end-users: students, non-technical employees, experts in their domains outside of computer science, and so on. With its forecast potential for solving problems by only observing inputs and outputs, programming-by-example was supposed to alleviate complex tasks requiring programming for end-users. The initial ideas of macro-based editors paved the way to subsequent practical solutions, such as spreadsheet transformations from examples. Finding the right program is the core of the programming-by-example systems. However, users find it difficult to trust such generated programs. In this thesis, we contribute to proving that some forms of interaction alleviate, by having users provide examples, the problem of finding correct and reliable programs. We first report on two experiments that enable us to conjecture what kind of interaction brings benefits to programming-by-example. First, we present a new kind of game engine, Pong Designer. In this game engine, by using their finger, users program rules on the fly, by modifying the game state. We analyze its potential, and its eventual downsides that have probably prevented its wide adoption. Second, we present StriSynth, an interactive command-line tool that uses programming-by-example to transform string and collections. The resulting programs can also rename or otherwise manage files. We obtained the result that confirms that many users preferred StriSynth over usual programming languages, but would appreciate to have both. We then report on two new exciting experiments with verified results, using two forms of interaction truly benefiting programming-by-example. Third, on top of a programmingby- example-based engine for extracting structured data out of text files, in this thesis we study two interaction models implemented in a tool named FlashProg: a view of the program with notification about ambiguities, and the asking of clarification questions. In this thesis, we prove that these two interaction models enable users to perform tasks with less errors and to be more confident with the results. Last, for learning recursive tree-to-string functions (e.g., pretty-printers), in this thesis we prove that questioning breaks down the learning complexity from a cubic to a linear number of questions, in practice making programming-by-example even more accessible than regular programming. The implementation, named Prosy, could be easily added to integrated development environments

TriCheck: Memory Model Verification at the Trisection of Software, Hardware, and ISA

Memory consistency models (MCMs) which govern inter-module interactions in a shared memory system, are a significant, yet often under-appreciated, aspect of system design. MCMs are defined at the various layers of the hardware-software stack, requiring thoroughly verified specifications, compilers, and implementations at the interfaces between layers. Current verification techniques evaluate segments of the system stack in isolation, such as proving compiler mappings from a high-level language (HLL) to an ISA or proving validity of a microarchitectural implementation of an ISA. This paper makes a case for full-stack MCM verification and provides a toolflow, TriCheck, capable of verifying that the HLL, compiler, ISA, and implementation collectively uphold MCM requirements. The work showcases TriCheck's ability to evaluate a proposed ISA MCM in order to ensure that each layer and each mapping is correct and complete. Specifically, we apply TriCheck to the open source RISC-V ISA, seeking to verify accurate, efficient, and legal compilations from C11. We uncover under-specifications and potential inefficiencies in the current RISC-V ISA documentation and identify possible solutions for each. As an example, we find that a RISC-V-compliant microarchitecture allows 144 outcomes forbidden by C11 to be observed out of 1,701 litmus tests examined. Overall, this paper demonstrates the necessity of full-stack verification for detecting MCM-related bugs in the hardware-software stack.Comment: Proceedings of the Twenty-Second International Conference on Architectural Support for Programming Languages and Operating System

Deterministic stream-sampling for probabilistic programming: semantics and verification

Probabilistic programming languages rely fundamentally on some notion of sampling, and this is doubly true for probabilistic programming languages which perform Bayesian inference using Monte Carlo techniques. Verifying samplers - proving that they generate samples from the correct distribution - is crucial to the use of probabilistic programming languages for statistical modelling and inference. However, the typical denotational semantics of probabilistic programs is incompatible with deterministic notions of sampling. This is problematic, considering that most statistical inference is performed using pseudorandom number generators.We present a higher-order probabilistic programming language centred on the notion of samplers and sampler operations. We give this language an operational and denotational semantics in terms of continuous maps between topological spaces. Our language also supports discontinuous operations, such as comparisons between reals, by using the type system to track discontinuities. This feature might be of independent interest, for example in the context of differentiable programming.Using this language, we develop tools for the formal verification of sampler correctness. We present an equational calculus to reason about equivalence of samplers, and a sound calculus to prove semantic correctness of samplers, i.e. that a sampler correctly targets a given measure by construction

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

Deterministic stream-sampling for probabilistic programming: semantics and verification

Probabilistic programming languages rely fundamentally on some notion of sampling, and this is doubly true for probabilistic programming languages which perform Bayesian inference using Monte Carlo techniques. Verifying samplers - proving that they generate samples from the correct distribution - is crucial to the use of probabilistic programming languages for statistical modelling and inference. However, the typical denotational semantics of probabilistic programs is incompatible with deterministic notions of sampling. This is problematic, considering that most statistical inference is performed using pseudorandom number generators. We present a higher-order probabilistic programming language centred on the notion of samplers and sampler operations. We give this language an operational and denotational semantics in terms of continuous maps between topological spaces. Our language also supports discontinuous operations, such as comparisons between reals, by using the type system to track discontinuities. This feature might be of independent interest, for example in the context of differentiable programming. Using this language, we develop tools for the formal verification of sampler correctness. We present an equational calculus to reason about equivalence of samplers, and a sound calculus to prove semantic correctness of samplers, i.e. that a sampler correctly targets a given measure by construction.Comment: Extended version of LiCS 2023 pape

Verification of systolic arrays: a stream functional approach

Journal ArticleWe illustrate that the verification of systolic architectures can be carried out using techniques developed in the context of verification of programs. This is achieved by a decomposition of the original problem into separately proving the correctness of the data representation and of the individual processing elements in the systolic architecture. By expressing a processing element as a function on a stream of data we are able to utilize standard proof techniques from programming language theory. This decomposition leads to relatively straightforward proofs of the properties of the systolic architecture. We illustrate the techniques via a substantial example, the proof of the correctness of a linear-time systolic architecture for computing the gcd of polynomials. Although this architecture has been designed a few years ago, a formal proof of correctness has not hitherto appeared in the literature