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  • Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width
    arXiv.cs.LO Pub Date : 2020-01-18
    Michał Wrona

    Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP for first-order expansions S of countably infinite homogeneous graphs that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of the CSP not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking. Our main result is that the structures S under consideration have relational width exactly (2, L) where L is the maximal size of a forbidden subgraph of a homogeneous graph under consideration, but not smaller than 3. It beats the upper bound (2m, 3m) where m = max(arity(S)+1, L, 3) and arity(S) is the largest arity of a relation in S, which follows from a sufficient condition implying bounded relational width from the literature. Since L may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures , whose relational width, if finite, is always at most (2,3).

    更新日期:2020-01-22
  • FASiM: A Framework for Automatic Formal Analysis of Simulink Models of Linear Analog Circuits
    arXiv.cs.LO Pub Date : 2020-01-18
    Adnan Rashid; Ayesha Gauhar; Osman Hasan

    Simulink is a graphical environment that is widely adapted for the modeling and the Laplace transform based analysis of linear analog circuits used in signal processing architectures. However, due to the involvement of the numerical algorithms of MATLAB in the analysis process, the analysis results cannot be termed as complete and accurate. Higher-order-logic theorem proving is a formal verification method that has been recently proposed to overcome these limitations for the modeling and the Laplace transform based analysis of linear analog circuits. However, the formal modeling of a system is not a straightforward task due to the lack of formal methods background for engineers working in the industry. Moreover, due to the undecidable nature of higher-order logic, the analysis generally requires a significant amount of user guidance in the manual proof process. In order to facilitate industrial engineers to formally analyze the linear analog circuits based on the Laplace transform, we propose a framework, FASiM, which allows automatically conducting the formal analysis of the Simulink models of linear analog circuits using the HOL Light theorem prover. For illustration, we use FASiM to formally analyze Simulink models of some commonly used linear analog filters, such as Sallen-key filters.

    更新日期:2020-01-22
  • Infinitary Action Logic with Exponentiation
    arXiv.cs.LO Pub Date : 2020-01-19
    Stepan L. Kuznetsov; Stanislav O. Speranski

    We introduce infinitary action logic with exponentiation---that is, the multiplicative-additive Lambek calculus extended with Kleene star and with a family of subexponential modalities, which allows some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut elimination and, in the case where at least one subexponential allows non-local contraction, establish exact complexity boundaries in two senses. First, we show that the derivability problem for this logic is $\Pi_1^1$-complete. Second, we show that the closure ordinal of its derivability operator is $\omega_1^{\mathrm{CK}}$.

    更新日期:2020-01-22
  • Semantics for first-order affine inductive data types via slice categories
    arXiv.cs.LO Pub Date : 2020-01-19
    Vladimir Zamdzhiev

    Affine type systems are substructural type systems where copying of information is restricted, but discarding of information is permissible at all types. Such type systems are well-suited for describing quantum programming languages, because copying of quantum information violates the laws of quantum mechanics. In this paper, we consider a first-order affine type system with inductive data types and present a novel categorical semantics for it. The most challenging aspect of this interpretation comes from the requirement to construct appropriate discarding maps for our data types which might be defined by mutual/nested recursion. We show how to achieve this for all types by taking models of a first-order linear type system whose atomic types are discardable and then presenting an additional affine interpretation of types within the slice category of the model with the tensor unit. We present some concrete categorical models for the language ranging from classical to quantum. Finally, we discuss potential ways of dualising and extending our methods and using them for interpreting coalgebraic and lazy data types.

    更新日期:2020-01-22
  • Quantitative Aspects of Programming Languages and Systems over the past $2^4$ years and beyond
    arXiv.cs.LO Pub Date : 2020-01-20
    Alessandro AldiniUniversity of Urbino

    Quantitative aspects of computation are related to the use of both physical and mathematical quantities, including time, performance metrics, probability, and measures for reliability and security. They are essential in characterizing the behaviour of many critical systems and in estimating their properties. Hence, they need to be integrated both at the level of system modeling and within the verification methodologies and tools. Along the last two decades a variety of theoretical achievements and automated techniques have contributed to make quantitative modeling and verification mainstream in the research community. In the same period, they represented the central theme of the series of workshops entitled Quantitative Aspects of Programming Languages and Systems (QAPL) and born in 2001. The aim of this survey is to revisit such achievements and results from the standpoint of QAPL and its community.

    更新日期:2020-01-22
  • Streaming Transformations of Infinite Ordered-Data Words
    arXiv.cs.LO Pub Date : 2020-01-20
    Xiaokang Qiu

    In this paper, we define streaming register transducer (SRT), a one-way, letter-to-letter, transductional machine model for transformations of infinite data words whose data domain forms a linear group. Comparing with existing data word transducers, SRT are able to perform two extra operations on the registers: a linear-order-based comparison and an additive update. We consider the transformations that can be defined by SRT and several subclasses of SRT. We investigate the expressiveness of these languages and several decision problems. Our main results include: 1) SRT are closed under union and intersection, and add-free SRT are also closed under composition; 2) SRT-definable transformations can be defined in monadic second-order (MSO) logic, but are not comparable with first-order (FO) definable transformations; 3) the functionality problem is decidable for add-free SRT, the reactivity problem and inclusion problem are decidable for deterministic add-free SRT, but none of these problems is decidable in general for SRT.

    更新日期:2020-01-22
  • Modular coinduction up-to for higher-order languages via first-order transition systems
    arXiv.cs.LO Pub Date : 2020-01-20
    Jean-Marie Madiot; Damien Pous; Davide Sangiorgi

    The bisimulation proof method can be enhanced by employing `bisimulations up-to' techniques. A comprehensive theory of such enhancements has been developed for first-order (i.e., CCS-like) labelled transition systems (LTSs) and bisimilarity, based on abstract fixed-point theory and compatible functions. We transport this theory onto languages whose bisimilarity and LTS go beyond those of first-order models. The approach consists in exhibiting fully abstract translations of the more sophisticated LTSs and bisimilarities onto the first-order ones. This allows us to reuse directly the large corpus of up-to techniques that are available on first-order LTSs. The only ingredient that has to be manually supplied is the compatibility of basic up-to techniques that are specific to the new languages. We investigate the method on the pi-calculus, the lambda-calculus, and a (call-by-value) lambda-calculus with references.

    更新日期:2020-01-22
  • Dynamic Epistemic Logic Games with Epistemic Temporal Goals
    arXiv.cs.LO Pub Date : 2020-01-20
    Bastien Maubert; Aniello Murano; Sophie Pinchinat; François Schwarzentruber; Silvia Stranieri

    Dynamic Epistemic Logic (DEL) is a logical framework in which one can describe in great detail how actions are perceived by the agents, and how they affect the world. DEL games were recently introduced as a way to define classes of games with imperfect information where the actions available to the players are described very precisely. This framework makes it possible to define easily, for instance, classes of games where players can only use public actions or public announcements. These games have been studied for reachability objectives, where the aim is to reach a situation satisfying some epistemic property expressed in epistemic logic; several (un)decidability results have been established. In this work we show that the decidability results obtained for reachability objectives extend to a much more general class of winning conditions, namely those expressible in the epistemic temporal logic LTLK. To do so we establish that the infinite game structures generated by DEL public actions are regular, and we describe how to obtain finite representations on which we rely to solve them.

    更新日期:2020-01-22
  • A graph-based spatial temporal logic for knowledge representation and automated reasoning in cognitive robots
    arXiv.cs.LO Pub Date : 2020-01-20
    Zhiyu Liu; Meng Jiang; Hai Lin

    A new graph-based spatial temporal logic is proposed for knowledge representation and automated reasoning in this paper. The proposed logic achieves a balance between expressiveness and tractability in applications such as cognitive robots. The satisfiability of the proposed logic is decidable. A Hilbert style axiomatization for the proposed graph-based spatial temporal logic is given where Modus ponens and IRR are the inference rules. It has been shown that the corresponding deduction system is sound and complete and can be implemented through constraint programming.

    更新日期:2020-01-22
  • Sampling and Learning for Boolean Function
    arXiv.cs.LO Pub Date : 2020-01-21
    Chuyu Xiong

    In this article, we continue our study on universal learning machine by introducing new tools. We first discuss boolean function and boolean circuit, and we establish one set of tools, namely, fitting extremum and proper sampling set. We proved the fundamental relationship between proper sampling set and complexity of boolean circuit. Armed with this set of tools, we then introduce much more effective learning strategies. We show that with such learning strategies and learning dynamics, universal learning can be achieved, and requires much less data.

    更新日期:2020-01-22
  • Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory
    arXiv.cs.LO Pub Date : 2017-11-14
    Mathys Rennela; Sam Staton

    We describe categorical models of a circuit-based (quantum) functional programming language. We show that enriched categories play a crucial role. Following earlier work on QWire by Paykin et al., we consider both a simple first-order linear language for circuits, and a more powerful host language, such that the circuit language is embedded inside the host language. Our categorical semantics for the host language is standard, and involves cartesian closed categories and monads. We interpret the circuit language not in an ordinary category, but in a category that is enriched in the host category. We show that this structure is also related to linear/non-linear models. As an extended example, we recall an earlier result that the category of W*-algebras is dcpo-enriched, and we use this model to extend the circuit language with some recursive types.

    更新日期:2020-01-22
  • Language Preservation Problems in Parametric Timed Automata
    arXiv.cs.LO Pub Date : 2018-07-18
    Étienne André; Didier Lime; Nicolas Markey

    Parametric timed automata (PTA) are a powerful formalism to model and reason about concurrent systems with some unknown timing delays. In this paper, we address the (untimed) language- and trace-preservation problems: given a reference parameter valuation, does there exist another parameter valuation with the same untimed language, or with the same set of traces? We show that these problems are undecidable both for general PTA and for the restricted class of L/U-PTA, even for integer-valued parameters, or over bounded time. On the other hand, we exhibit decidable subclasses: 1-clock PTA, and 1-parameter deterministic L-PTA and U-PTA. We also consider robust versions of these problems, where we additionally require that the language be preserved for all valuations between the reference valuation and the new valuation.

    更新日期:2020-01-22
  • Continuous Ordinary Differential Equations and Transfinite Computations
    arXiv.cs.LO Pub Date : 2019-02-19
    Olivier Bournez; Sabrina Ouazzani

    We consider Continuous Ordinary Differential Equations (CODE) y'=f(y), where f is a continuous function. They are known to always have solutions for a given initial condition y(0)=y0, these solutions being possibly non unique. We restrict to our attention to a class of continuous functions, that we call greedy: they always admit unique greedy solutions, i.e. going in greedy way in some fixed direction. We prove that they can be seen as models of computation over the ordinals and conversely in a very strong sense. In particular, for such ODEs, to a greedy trajectory can be associated some ordinal corresponding to some time of computation, and conversely models of computation over the ordinals can be associated to some CODE. In particular, analyzing reachability for one or the other concept with respect to greedy trajectories has the same hardness. This also brings new perspectives on analysis in Mathematics, by providing ways to translate results for ITTMs to CODEs. This also extends some recent results about the relations between ordinary differential equations and Turing machines, and more widely with (generalized) computability theory.

    更新日期:2020-01-22
  • Axiomatizing first-order consequences in inclusion logic
    arXiv.cs.LO Pub Date : 2019-04-12
    Fan Yang

    Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed-point logic. Inclusion logic is not axiomatizable in full, but its first-order consequences can be axiomatized. In this paper, we provide such an explicit partial axiomatization by introducing a system of natural deduction for inclusion logic that is sound and complete for first-order consequences in inclusion logic.

    更新日期:2020-01-22
  • A Formal Axiomatization of Computation
    arXiv.cs.LO Pub Date : 2019-07-04
    Rasoul Ramezanian

    We introduce an axiomatization for the notion of computation. Based on the idea of Brouwer choice sequences, we construct a model, denoted by $E$, which satisfies our axioms and $E \models \mathrm{ P \neq NP}$. In other words, regarding "effective computability" in Brouwer intuitionism viewpoint, we show $\mathrm{ P \neq NP}$.

    更新日期:2020-01-22
  • Toward a Dempster-Shafer theory of concepts
    arXiv.cs.LO Pub Date : 2019-08-14
    Sabine Frittella; Krishna Manoorkar; Alessandra Palmigiano; Apostolos Tzimoulis; Nachoem M. Wijnberg

    In this paper, we generalize the basic notions and results of Dempster-Shafer theory from predicates to formal concepts. Results include the representation of conceptual belief functions as inner measures of suitable probability functions, and a Dempster-Shafer rule of combination on belief functions on formal concepts.

    更新日期:2020-01-22
  • On Expert Behaviors and Question Types for Efficient Query-Based Ontology Fault Localization
    arXiv.cs.LO Pub Date : 2020-01-16
    Patrick Rodler

    We challenge existing query-based ontology fault localization methods wrt. assumptions they make, criteria they optimize, and interaction means they use. We find that their efficiency depends largely on the behavior of the interacting expert, that performed calculations can be inefficient or imprecise, and that used optimization criteria are often not fully realistic. As a remedy, we suggest a novel (and simpler) interaction approach which overcomes all identified problems and, in comprehensive experiments on faulty real-world ontologies, enables a successful fault localization while requiring fewer expert interactions in 66 % of the cases, and always at least 80 % less expert waiting time, compared to existing methods.

    更新日期:2020-01-17
  • Reward Shaping for Reinforcement Learning with Omega-Regular Objectives
    arXiv.cs.LO Pub Date : 2020-01-16
    E. M. Hahn; M. Perez; S. Schewe; F. Somenzi; A. Trivedi; D. Wojtczak

    Recently, successful approaches have been made to exploit good-for-MDPs automata (B\"uchi automata with a restricted form of nondeterminism) for model free reinforcement learning, a class of automata that subsumes good for games automata and the most widespread class of limit deterministic automata. The foundation of using these B\"uchi automata is that the B\"uchi condition can, for good-for-MDP automata, be translated to reachability. The drawback of this translation is that the rewards are, on average, reaped very late, which requires long episodes during the learning process. We devise a new reward shaping approach that overcomes this issue. We show that the resulting model is equivalent to a discounted payoff objective with a biased discount that simplifies and improves on prior work in this direction.

    更新日期:2020-01-17
  • A Categorical Reconstruction of Quantum Theory
    arXiv.cs.LO Pub Date : 2018-04-06
    Sean Tull

    We reconstruct finite-dimensional quantum theory from categorical principles. That is, we provide properties ensuring that a given physical theory described by a dagger compact category in which one may `discard' objects is equivalent to a generalised finite-dimensional quantum theory over a suitable ring $S$. The principles used resemble those due to Chiribella, D'Ariano and Perinotti. Unlike previous reconstructions, our axioms and proof are fully categorical in nature, in particular not requiring tomography assumptions. Specialising the result to probabilistic theories we obtain either traditional quantum theory with $S$ being the complex numbers, or that over real Hilbert spaces with $S$ being the reals.

    更新日期:2020-01-17
  • Advances in Symmetry Breaking for SAT Modulo Theories
    arXiv.cs.LO Pub Date : 2019-08-02
    Saket Dingliwal; Ronak Agarwal; Happy Mittal; Parag Singla

    Symmetry breaking is a popular technique to reduce the search space for SAT solving by exploiting the underlying symmetry over variables and clauses in a formula. The key idea is to first identify sets of assignments which fall in the same symmetry class, and then impose ordering constraints, called Symmetry Breaking Predicates (SBPs), such that only one (or a small subset) of these assignments is allowed to be a solution of the original SAT formula. While this technique has been exploited extensively in the SAT literature, there is little work on using symmetry breaking for SAT Modulo Theories (SMT). In SMT, logical constraints in SAT theories are combined with another set of theory operations defined over non-Boolean variables such as integers, reals, etc. SMT solvers typically use a combination of SAT solving techniques augmented with calls to the theory solver. In this work, we take up the advances in SAT symmetry breaking and apply them to the domain of SMT. Our key technical contribution is the formulation of symmetry breaking over the Boolean skeleton variables, which are placeholders for actual theory operations in SMT solving. These SBPs are then applied over the SAT solving part of the SMT solver. We implement our SBP ideas on top of CVC4, which is a state-of-the-art SMT solver. Our approach can result in significantly faster solutions on several benchmark problems compared to the state-of-the-art. Our final solver is a hybrid of the original CVC4 solver, and an SBP based solver, and can solve up to 3.8% and 3.1% more problems in the QF_NIA category of 2018 and 2019 SMT benchmarks, respectively, compared to CVC4, the top performer in this category.

    更新日期:2020-01-17
  • Gillian: Compositional Symbolic Execution for All
    arXiv.cs.LO Pub Date : 2020-01-14
    José Fragoso Santos; Petar Maksimović; Sacha-Élie Ayoun; Philippa Gardner

    We present Gillian, a language-independent framework for the development of compositional symbolic analysis tools. Gillian supports three flavours of analysis: whole-program symbolic testing, full verification, and bi-abduction. It comes with fully parametric meta-theoretical results and a modular implementation, designed to minimise the instantiation effort required of the user. We evaluate Gillian by instantiating it to JavaScript and C, and perform its analyses on a set of data-structure libraries, obtaining results that indicate that Gillian is robust enough to reason about real-world programming languages.

    更新日期:2020-01-16
  • Circular Proofs in First-Order Linear Logic with Least and Greatest Fixed Points
    arXiv.cs.LO Pub Date : 2020-01-15
    Farzaneh Derakhshan; Frank Pfenning

    Inductive and coinductive structures are everywhere in mathematics and computer science. The induction principle is well known and fully exploited to reason about inductive structures like natural numbers and finite lists. To prove theorems about coinductive structures such as infinite streams and infinite trees we can appeal to bisimulation or the coinduction principle. Pure inductive and coinductive types however are not the only data structures we are interested to reason about. In this paper we present a calculus to prove theorems about mutually defined inductive and coinductive data types. Derivations are carried out in an infinitary sequent calculus for first order intuitionistic multiplicative additive linear logic with fixed points. We enforce a condition on these derivations to ensure their cut elimination property and thus validity. Our calculus is designed to reason about linear properties but we also allow appealing to first order theories such as arithmetic, by adding an adjoint downgrade modality. We show the strength of our calculus by proving several theorems on (mutual) inductive and coinductive data types.

    更新日期:2020-01-16
  • Approximate Weighted First-Order Model Counting: Exploiting Fast Approximate Model Counters and Symmetry
    arXiv.cs.LO Pub Date : 2020-01-15
    Timothy van Bremen; Ondrej Kuzelka

    We study the symmetric weighted first-order model counting task and present ApproxWFOMC, a novel anytime method for efficiently bounding the weighted first-order model count in the presence of an unweighted first-order model counting oracle. The algorithm has applications to inference in a variety of first-order probabilistic representations, such as Markov logic networks and probabilistic logic programs. Crucially for many applications, we make no assumptions on the form of the input sentence. Instead, our algorithm makes use of the symmetry inherent in the problem by imposing cardinality constraints on the number of possible true groundings of a sentence's literals. Realising the first-order model counting oracle in practice using the approximate hashing-based model counter ApproxMC3, we show how our algorithm outperforms existing approximate and exact techniques for inference in first-order probabilistic models. We additionally provide PAC guarantees on the generated bounds.

    更新日期:2020-01-16
  • De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic
    arXiv.cs.LO Pub Date : 2016-02-19
    Ross Horne; Alwen Tiu; Bogdan Aman; Gabriel Ciobanu

    This paper explores the proof theory necessary for recommending an expressive but decidable first-order system, named MAV1, featuring a de Morgan dual pair of nominal quantifiers. These nominal quantifiers called `new' and `wen' are distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers. The novelty of these nominal quantifiers is they are polarised in the sense that `new' distributes over positive operators while `wen' distributes over negative operators. This greater control of bookkeeping enables private names to be modelled in processes embedded as formulae in MAV1. The technical challenge is to establish a cut elimination result, from which essential properties including the transitivity of implication follow. Since the system is defined using the calculus of structures, a generalisation of the sequent calculus, novel techniques are employed. The proof relies on an intricately designed multiset-based measure of the size of a proof, which is used to guide a normalisation technique called splitting. The presence of equivariance, which swaps successive quantifiers, induces complex inter-dependencies between nominal quantifiers, additive conjunction and multiplicative operators in the proof of splitting. Every rule is justified by an example demonstrating why the rule is necessary for soundly embedding processes and ensuring that cut elimination holds.

    更新日期:2020-01-16
  • Reinforcement Learning of Control Policy for Linear Temporal Logic Specifications Using Limit-Deterministic Büchi Automata
    arXiv.cs.LO Pub Date : 2020-01-14
    Ryohei Oura; Ami Sakakibara; Toshimitsu Ushio

    This letter proposes a novel reinforcement learning method for the synthesis of a control policy satisfying a control specification described by a linear temporal logic formula. We assume that the controlled system is modeled by a Markov decision process (MDP). We transform the specification to a limit-deterministic B\"uchi automaton (LDBA) with several accepting sets that accepts all infinite sequences satisfying the formula. The LDBA is augmented so that it explicitly records the previous visits to accepting sets. We take a product of the augmented LDBA and the MDP, based on which we define a reward function. The agent gets rewards whenever state transitions are in an accepting set that has not been visited for a certain number of steps. Consequently, sparsity of rewards is relaxed and optimal circulations among the accepting sets are learned. We show that the proposed method can learn an optimal policy when the discount factor is sufficiently close to one.

    更新日期:2020-01-15
  • A (Simplified) Supreme Being Necessarily Exists -- Says the Computer!
    arXiv.cs.LO Pub Date : 2020-01-14
    Christoph Benzmüller

    A simplified variant of Kurt G\"odel's modal ontological argument is presented. Some of G\"odel's, resp. Scott's, premises are modified, others are dropped, and modal collapse is avoided. The emended argument is shown valid already in quantified modal logic K. The presented simplifications have been computationally explored utilising latest knowledge representation and reasoning technology based on higher-order logic. The paper thus illustrates how modern symbolic AI technology can contribute new knowledge to formal philosophy and theology.

    更新日期:2020-01-15
  • A circular proof system for the hybrid mu-calculus
    arXiv.cs.LO Pub Date : 2020-01-14
    Sebastian Enqvist

    We present a circular and cut-free proof system for the hybrid mu-calculus and prove its soundness and completeness. The system uses names for fixpoint unfoldings, like the circular proof system for the mu-calculus previously developed by Stirling.

    更新日期:2020-01-15
  • Characterizing Polynomial Ramsey Quantifiers
    arXiv.cs.LO Pub Date : 2016-01-10
    Ronald de Haan; Jakub Szymanik

    Ramsey quantifiers are a natural object of study not only for logic and computer science, but also for the formal semantics of natural language. Restricting attention to finite models leads to the natural question whether all Ramsey quantifiers are either polynomial-time computable or NP-hard, and whether we can give a natural characterization of the polynomial-time computable quantifiers. In this paper, we first show that there exist intermediate Ramsey quantifiers and then we prove a dichotomy result for a large and natural class of Ramsey quantifiers, based on a reasonable and widely-believed complexity assumption. We show that the polynomial-time computable quantifiers in this class are exactly the constant-log-bounded Ramsey quantifiers.

    更新日期:2020-01-15
  • Combinatorial Conversion and Moment Bisimulation for Stochastic Rewriting Systems
    arXiv.cs.LO Pub Date : 2019-04-15
    Nicolas Behr; Vincent Danos; Ilias Garnier

    We develop a novel method to analyze the dynamics of stochastic rewriting systems evolving over finitary adhesive, extensive categories. Our formalism is based on the so-called rule algebra framework and exhibits an intimate relationship between the combinatorics of the rewriting rules (as encoded in the rule algebra) and the dynamics which these rules generate on observables (as encoded in the stochastic mechanics formalism). We introduce the concept of combinatorial conversion, whereby under certain technical conditions the evolution equation for (the exponential generating function of) the statistical moments of observables can be expressed as the action of certain differential operators on formal power series. This permits us to formulate the novel concept of moment-bisimulation, whereby two dynamical systems are compared in terms of their evolution of sets of observables that are in bijection. In particular, we exhibit non-trivial examples of graphical rewriting systems that are moment-bisimilar to certain discrete rewriting systems (such as branching processes or the larger class of stochastic chemical reaction systems). Our results point towards applications of a vast number of existing well-established exact and approximate analysis techniques developed for chemical reaction systems to the far richer class of general stochastic rewriting systems.

    更新日期:2020-01-15
  • Complexity of controlled bad sequences over finite sets of $\mathbb{N}^d$
    arXiv.cs.LO Pub Date : 2019-09-04
    A. R. Balasubramanian

    We provide upper and lower bounds for the length of controlled bad sequences over the majoring and the minoring orderings of finite sets of $\mathbb{N}^d$. The results are obtained by bounding the length of such sequences by functions from the Cichon hierarchy. This allows us to translate these results to bounds over the fast-growing complexity classes. The obtained bounds are proven to be tight for the majoring ordering, which solves a problem left open by Abriola, Figueira and Senno (Theor. Comp. Sci, Vol. 603). Finally, we use the results on controlled bad sequences to prove upper bounds for the emptiness problem of some classes of automata.

    更新日期:2020-01-15
  • Predicate Transformer Semantics for Hybrid Systems: Verification Components for Isabelle/HOL
    arXiv.cs.LO Pub Date : 2019-09-12
    Jonathan Julián Huerta y Munive; Georg Struth

    We present a semantic framework for the deductive verification of hybrid systems with Isabelle/HOL. It supports reasoning about the temporal evolutions of hybrid programs in the style of differential dynamic logic modelled by flows or invariant sets for vector fields. We introduce the semantic foundations of our approach and summarise their Isabelle formalisation as well as the resulting verification components. A series of examples shows our approach at work.

    更新日期:2020-01-15
  • Smarter Features, Simpler Learning?
    arXiv.cs.LO Pub Date : 2019-11-15
    Sarah WinklerUniversity of Verona; Georg MoserUniversity of Innsbruck

    Earlier work on machine learning for automated reasoning mostly relied on simple, syntactic features combined with sophisticated learning techniques. Using ideas adopted in the software verification community, we propose the investigation of more complex, structural features to learn from. These may be exploited to either learn beneficial strategies for tools, or build a portfolio solver that chooses the most suitable tool for a given problem. We present some ideas for features of term rewrite systems and theorem proving problems.

    更新日期:2020-01-15
  • ReluDiff: Differential Verification of Deep Neural Networks
    arXiv.cs.LO Pub Date : 2020-01-10
    Brandon Paulsen; Jingbo Wang; Chao Wang

    As deep neural networks are increasingly being deployed in practice, their efficiency has become an important issue. While there are compression techniques for reducing the network's size, energy consumption and computational requirement, they only demonstrate empirically that there is no loss of accuracy, but lack formal guarantees of the compressed network, e.g., in the presence of adversarial examples. Existing verification techniques such as Reluplex, ReluVal, and DeepPoly provide formal guarantees, but they are designed for analyzing a single network instead of the relationship between two networks. To fill the gap, we develop a new method for differential verification of two closely related networks. Our method consists of a fast but approximate forward interval analysis pass followed by a backward pass that iteratively refines the approximation until the desired property is verified. We have two main innovations. During the forward pass, we exploit structural and behavioral similarities of the two networks to more accurately bound the difference between the output neurons of the two networks. Then in the backward pass, we leverage the gradient differences to more accurately compute the most beneficial refinement. Our experiments show that, compared to state-of-the-art verification tools, our method can achieve orders-of-magnitude speedup and prove many more properties than existing tools.

    更新日期:2020-01-14
  • Deciding the Loosely Guarded Fragment and Querying Its Horn Fragment Using Resolution
    arXiv.cs.LO Pub Date : 2020-01-12
    Sen Zheng; Renate A. Schmidt

    We consider the following query answering problem: Given a Boolean conjunctive query and a theory in the Horn loosely guarded fragment, the aim is to determine whether the query is entailed by the theory. In this paper, we present a resolution decision procedure for the loosely guarded fragment, and use such a procedure to answer Boolean conjunctive queries against the Horn loosely guarded fragment. The Horn loosely guarded fragment subsumes classes of rules that are prevalent in ontology-based query answering, such as Horn ALCHOI and guarded existential rules. Additionally, we identify star queries and cloud queries, which using our procedure, can be answered against the loosely guarded fragment.

    更新日期:2020-01-14
  • Games Where You Can Play Optimally with Finite Memory
    arXiv.cs.LO Pub Date : 2020-01-12
    Patricia Bouyer; Stéphane Le Roux; Youssouf Oualhadj; Mickael Randour; Pierre Vandenhove

    For decades, two-player (antagonistic) games on graphs have been a framework of choice for many important problems in theoretical computer science. A notorious one is controller synthesis, which can be rephrased through the game-theoretic metaphor as the quest for a winning strategy of the system in a game against its antagonistic environment. Depending on the specification, optimal strategies might be simple or quite complex, for example having to use (possibly infinite) memory. Hence, research strives to understand which settings allow for simple strategies. In 2005, Gimbert and Zielonka provided a complete characterization of preference relations (a formal framework to model specifications and game objectives) that admit memoryless optimal strategies for both players. In the last fifteen years however, practical applications have driven the community toward games with complex or multiple objectives, where memory --- finite or infinite --- is almost always required. Despite much effort, the exact frontiers of the class of preference relations that admit finite-memory optimal strategies still elude us. In this work, we establish a complete characterization of preference relations that admit optimal strategies using arena-independent finite memory, generalizing the work of Gimbert and Zielonka to the finite-memory case. We also prove an equivalent to their celebrated corollary of utmost practical interest: if both players have optimal (arena-independent-)finite-memory strategies in all one-player games, then it is also the case in all two-player games. Finally, we pinpoint the boundaries of our results with regard to the literature: our work completely covers the case of arena-independent memory (e.g., multiple parity objectives, lower- and upper-bounded energy objectives), and paves the way to the arena-dependent case (e.g., multiple lower-bounded energy objectives).

    更新日期:2020-01-14
  • Commonly Knowingly Whether
    arXiv.cs.LO Pub Date : 2020-01-12
    Jie Fan; Davide Grossi; Barteld Kooi; Xingchi Su; Rineke Verbrugge

    This paper introduces `commonly knowing whether', a non-standard version of classical common knowledge which is defined on the basis of `knowing whether', instead of classical `knowing that'. After giving five possible definitions of this concept, we explore the logical relations among them both in the multi-agent case and in the single-agent case. We focus on one definition and treat it as a modal operator. It is found that the expressivity of this operator is incomparable with the classical common knowledge operator. Moreover, some special properties of it over binary-tree models and KD45-models are investigated.

    更新日期:2020-01-14
  • Interactive Visualization of Saturation Attempts in Vampire
    arXiv.cs.LO Pub Date : 2020-01-13
    Bernhard Gleiss; Laura Kovacs; Lena Schnedlitz

    Many applications of formal methods require automated reasoning about system properties, such as system safety and security. To improve the performance of automated reasoning engines, such as SAT/SMT solvers and first-order theorem prover, it is necessary to understand both the successful and failing attempts of these engines towards producing formal certificates, such as logical proofs and/or models. Such an analysis is challenging due to the large number of logical formulas generated during proof/model search. In this paper we focus on saturation-based first-order theorem proving and introduce the SATVIS tool for interactively visualizing saturation-based proof attempts in first-order theorem proving. We build SATVIS on top of the world-leading theorem prover VAMPIRE, by interactively visualizing the saturation attempts of VAMPIRE in SATVIS. Our work combines the automatic layout and visualization of the derivation graph induced by the saturation attempt with interactive transformations and search functionality. As a result, we are able to analyze and debug (failed) proof attempts of VAMPIRE. Thanks to its interactive visualisation, we believe SATVIS helps both experts and non-experts in theorem proving to understand first-order proofs and analyze/refine failing proof attempts of first-order provers.

    更新日期:2020-01-14
  • Possibility and prevention of inappropriate data manipulation in Polar Data Journal
    arXiv.cs.LO Pub Date : 2020-01-13
    Takeshi Terui; Yasuyuki Minamiyama; Kazutsuna Yamaji

    Stakeholders in the scientific field must always maintain transparency in the process of publishing research results in journals. Unfortunately, although research misconduct has stopped, certain forms of manipulation continue to appear in other forms. As new techniques of scientific publishing develop, science stakeholders need to examine the possibility of inappropriate activity in these new platforms. The National Institute of Polar Research in Japan launched a new data journal Polar Data Journal (PDJ) in 2017 to review the quality of data obtained in the polar region. To maintain transparency in this new data journal, we investigated the possibility of inappropriate data manipulation in peer reviews before the inception of this journal. We clarified inappropriate activity for the data in the peer review and considered preventive measures. We designed a specific workflow for PDJ. This included two measures: (i) the comparison of hash values in the review process and (ii) open peer review report publishing. Using the hash value comparison, we detected two instances of inappropriate data manipulation after the start of the journal. This research will help improve workflow in data journals and data repositories.

    更新日期:2020-01-14
  • (Newtonian) Space-Time Algebra
    arXiv.cs.LO Pub Date : 2019-12-20
    James E. Smith

    The space-time (s-t) algebra provides a mathematical model for communication and computation using values encoded as events in discretized linear (Newtonian) time. Consequently, the input-output behavior of s-t algebra and implemented functions are consistent with the flow of time. The s-t algebra and functions are formally defined. A network design framework for s-t functions is describe, and the design of temporal neural networks, a form of spiking neural networks, is discussed as an extended case study. Finally, the relationship with Allen's interval algebra is briefly discussed.

    更新日期:2020-01-14
  • An extended quantum process algebra (eQPAlg) approach for distributed quantum systems
    arXiv.cs.LO Pub Date : 2020-01-06
    Salman Haider; Dr. Syed Asad Raza Kazmi

    In this work, we have expounded the communication procedure of quantum systems by means of process algebra. The main objective of our research effort is to formally represent the communication between distributed quantum systems. In this new proposed communication model we have ameliorated the existing rules of Lalire's quantum process algebra QPAlg. We have brought some important modification in QPAlg by introducing the concept of formally specifying the Quantum teleportation protocol. We have further introduced the formal description of protocol by using programs that best explains its working and satisfies the specification. Examples have been provided to describe the working of the improved algebra that formally explain the sending and receiving of both classical as well as quantum data, keeping in mind the principal features of quantum mechanics.

    更新日期:2020-01-14
  • On Concept of Petri Nets Receptors and Effectors
    arXiv.cs.LO Pub Date : 2019-12-18
    Alexander Yu. Chunikhin; Marina D. Sviatnenko

    New subclasses of Petri nets - Petri nets receptors and Petri nets effectors are introduced. The introduction/exclusion of such substructures in the main Petri net may be fulfilled in accordance with the Fusion/Defusion principles. We propose two pairs of entities: position marking receptor (effector) and transition marking receptor (effector), which allow to observe parameters of the main Petri net and, if necessary, to carry out their regulation.

    更新日期:2020-01-14
  • On the linear structure of cones
    arXiv.cs.LO Pub Date : 2020-01-13
    Thomas EhrhardIRIF

    For encompassing the limitations of probabilistic coherence spaces which do not seem to provide natural interpretations of continuous data types such as the real line, Ehrhard and al. introduced a model of probabilistic higher order computation based on (positive) cones, and a class of totally monotone functions that they called "stable". Then Crubill{\'e} proved that this model is a conservative extension of the earlier probabilistic coherence space model. We continue these investigations by showing that the category of cones and linear and Scott-continuous functions is a model of intuitionistic linear logic. To define the tensor product, we use the special adjoint functor theorem, and we prove that this operation is and extension of the standard tensor product of probabilistic coherence spaces. We also show that these latter are dense in cones, thus allowing to lift the main properties of the tensor product of probabilistic coherence spaces to general cones. Last we define in the same way an exponential of cones and extend measurability to these new operations.

    更新日期:2020-01-14
  • Symblicit Exploration and Elimination for Probabilistic Model Checking
    arXiv.cs.LO Pub Date : 2020-01-08
    Ernst Moritz Hahn; Arnd Hartmanns

    Binary decision diagrams can compactly represent vast sets of states, mitigating the state space explosion problem in model checking. Probabilistic systems, however, require multi-terminal diagrams storing rational numbers. They are inefficient for models with many distinct probabilities and for iterative numeric algorithms like value iteration. In this paper, we present a new "symblicit" approach to checking Markov chains and related probabilistic models: We first generate a decision diagram that symbolically collects all reachable states and their predecessors. We then concretise states one-by-one into an explicit partial state space representation. Whenever all predecessors of a state have been concretised, we eliminate it from the explicit state space in a way that preserves all relevant probabilities and rewards. We thus keep few explicit states in memory at any time. Experiments show that very large models can be model-checked in this way with very low memory consumption.

    更新日期:2020-01-14
  • Tabled Typeclass Resolution
    arXiv.cs.LO Pub Date : 2020-01-13
    Daniel Selsam; Sebastian Ullrich; Leonardo de Moura

    Typeclasses provide an elegant and effective way of managing ad-hoc polymorphism in both programming languages and interactive proof assistants. However, the increasingly sophisticated uses of typeclasses within proof assistants has exposed two critical problems with existing typeclass resolution procedures: the diamond problem, which causes exponential running times in both theory and practice, and the cycle problem, which causes loops in the presence of cycles and so thwarts many desired uses of typeclasses. We present a new typeclass resolution procedure, called tabled typeclass resolution, that solves these problems. We have implemented our procedure for the upcoming version (v4) of the Lean Theorem Prover, and we confirm empirically that our implementation is exponentially faster than existing systems in the presence of diamonds. Our procedure is sufficiently lightweight that it could easily be implemented in other systems. We hope our new procedure facilitates even more sophisticated uses of typeclasses in both software development and interactive theorem proving.

    更新日期:2020-01-14
  • A Universal Attractor Decomposition Algorithm for Parity Games
    arXiv.cs.LO Pub Date : 2020-01-13
    Marcin Jurdziński; Rémi Morvan

    An attractor decomposition meta-algorithm for solving parity games is given that generalizes the classic McNaughton-Zielonka algorithm and its recent quasi-polynomial variants due to Parys (2019), and to Lehtinen, Schewe, and Wojtczak (2019). The central concepts studied and exploited are attractor decompositions of dominia in parity games and the ordered trees that describe the inductive structure of attractor decompositions. The main technical results include the embeddable decomposition theorem and the dominion separation theorem that together help establish a precise structural condition for the correctness of the universal algorithm: it suffices that the two ordered trees given to the algorithm as inputs embed the trees of some attractor decompositions of the largest dominia for each of the two players, respectively. The universal algorithm yields McNaughton-Zielonka, Parys's, and Lehtinen-Schewe-Wojtczak algorithms as special cases when suitable universal trees are given to it as inputs. The main technical results provide a unified proof of correctness and deep structural insights into those algorithms. A symbolic implementation of the universal algorithm is also given that improves the symbolic space complexity of solving parity games in quasi-polynomial time from $O(d \lg n)$---achieved by Chatterjee, Dvo\v{r}\'{a}k, Henzinger, and Svozil (2018)---down to $O(\lg d)$, where $n$ is the number of vertices and $d$ is the number of distinct priorities in a parity game. This not only exponentially improves the dependence on $d$, but it also entirely removes the dependence on $n$.

    更新日期:2020-01-14
  • Decisiveness of Stochastic Systems and its Application to Hybrid Models
    arXiv.cs.LO Pub Date : 2020-01-13
    Patricia Bouyer; Thomas Brihaye; Mickael Randour; Cédric Rivière; Pierre Vandenhove

    In [ABM07], Abdulla et al. introduced the concept of decisiveness, an interesting tool for lifting good properties of finite Markov chains to denumerable ones. Later, this concept was extended to more general stochastic transition systems (STSs), allowing the design of various verification algorithms for large classes of (infinite) STSs. We further improve the understanding and utility of decisiveness in two ways. First, we provide a general criterion for proving decisiveness of general STSs. This criterion, which is very natural but whose proof is rather technical, (strictly) generalizes all known criteria from the literature. Second, we focus on stochastic hybrid systems (SHSs), a stochastic extension of hybrid systems. We establish the decisiveness of a large class of SHSs and, under a few classical hypotheses from mathematical logic, we show how to decide reachability problems in this class, even though they are undecidable for general SHSs. This provides a decidable stochastic extension of o-minimal hybrid systems. [ABM07] Parosh A. Abdulla, Noomene Ben Henda, and Richard Mayr. 2007. Decisive Markov Chains. Log. Methods Comput. Sci. 3, 4 (2007).

    更新日期:2020-01-14
  • State diagrams of functional programs
    arXiv.cs.LO Pub Date : 2020-01-10
    Andrew M. Mironov

    In the paper we introduce graphical objects (called state diagrams) related to functional programs. It is shown that state diagrams of functional programs can be used to solve problems of verification of functional programs. The proposed approach is illustrated by an example of verification of a sorting program.

    更新日期:2020-01-14
  • Session Types with Arithmetic Refinements and Their Application to Work Analysis
    arXiv.cs.LO Pub Date : 2020-01-13
    Ankush Das; Frank Pfenning

    Session types statically prescribe bidirectional communication protocols for message-passing processes and are in a Curry-Howard correspondence with linear logic propositions. However, simple session types cannot specify properties beyond the type of exchanged messages. In this paper we extend the type system by using index refinements from linear arithmetic capturing intrinsic attributes of data structures and algorithms so that we can express and verify amortized cost of programs using ergometric types. We show that, despite the decidability of Presburger arithmetic, type equality and therefore also type checking are now undecidable, which stands in contrast to analogous dependent refinement type systems from functional languages. We also present a practical incomplete algorithm for type equality and an algorithm for type checking which is complete relative to an oracle for type equality. Process expressions in this explicit language are rather verbose, so we also introduce an implicit form and a sound and complete algorithm for reconstructing explicit programs, borrowing ideas from the proof-theoretic technique of focusing. We conclude by illustrating our systems and algorithms with a variety of examples that have been verified in our implementation.

    更新日期:2020-01-14
  • A Verified Packrat Parser Interpreter for Parsing Expression Grammars
    arXiv.cs.LO Pub Date : 2020-01-13
    Clement Blaudeau; Natarajan Shankar

    Parsing expression grammars (PEGs) offer a natural opportunity for building verified parser interpreters based on higher-order parsing combinators. PEGs are expressive, unambiguous, and efficient to parse in a top-down recursive descent style. We use the rich type system of the PVS specification language and verification system to formalize the metatheory of PEGs and define a reference implementation of a recursive parser interpreter for PEGs. In order to ensure termination of parsing, we define a notion of a well-formed grammar. Rather than relying on an inductive definition of parsing, we use abstract syntax trees that represent the computational trace of the parser to provide an effective proof certificate for correct parsing and ensure that parsing properties including soundness and completeness are maintained. The correctness properties are embedded in the types of the operations so that the proofs can be easily constructed from local proof obligations. Building on the reference parser interpreter, we define a packrat parser interpreter as well as an extension that is capable of semantic interpretation. Both these parser interpreters are proved equivalent to the reference one. All of the parsers are executable. The proofs are formalized in mathematical terms so that similar parser interpreters can be defined in any specification language with a type system similar to PVS.

    更新日期:2020-01-14
  • One-Clock Priced Timed Games are PSPACE-hard
    arXiv.cs.LO Pub Date : 2020-01-13
    John Fearnley; Rasmus Ibsen-Jensen; Rahul Savani

    The main result of this paper is that computing the value of a one-clock priced timed game (OCPTG) is PSPACE-hard. Along the way, we provide a family of OCPTGs that have an exponential number of event points. Both results hold even in very restricted classes of games such as DAGs with treewidth three. Finally, we provide a number of positive results, including polynomial-time algorithms for even more restricted classes of OCPTGs such as trees.

    更新日期:2020-01-14
  • Datatype defining rewrite systems for naturals and integers
    arXiv.cs.LO Pub Date : 2016-08-22
    Jan A. Bergstra; Alban Ponse

    A datatype defining rewrite system (DDRS) is a ground-complete term rewriting system, intended to be used for the specification of datatypes. First we define two concise DDRSes for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying) or on the successor function. Then we specify arithmetic based on two other views: binary and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a DDRS, and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yield comparable normal forms, which for that view admits expressions that are algorithmically more involved. These DDRSes are incorporated because they are closer to existing literature. For all DDRSes considered, ground-completeness is either proved, or references to a proof are provided.

    更新日期:2020-01-14
  • Semiring Programming: A Declarative Framework for Generalized Sum Product Problems
    arXiv.cs.LO Pub Date : 2016-09-21
    Vaishak Belle; Luc De Raedt

    To solve hard problems, AI relies on a variety of disciplines such as logic, probabilistic reasoning, machine learning and mathematical programming. Although it is widely accepted that solving real-world problems requires an integration amongst these, contemporary representation methodologies offer little support for this. In an attempt to alleviate this situation, we introduce a new declarative programming framework that provides abstractions of well-known problems such as SAT, Bayesian inference, generative models, and convex optimization. The semantics of programs is defined in terms of first-order structures with semiring labels, which allows us to freely combine and integrate problems from different AI disciplines.

    更新日期:2020-01-14
  • Non-idempotent types for classical calculi in natural deduction style
    arXiv.cs.LO Pub Date : 2018-02-15
    Delia Kesner; Pierre Vial

    In the first part of this paper, we define two resource aware typing systems for the {\lambda}{\mu}-calculus based on non-idempotent intersection and union types. The non-idempotent approach provides very simple combinatorial arguments-based on decreasing measures of type derivations-to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the lengths of the head reduction and the maximal reduction sequences to normal-form. In the second part of this paper, the {\lambda}{\mu}-calculus is refined to a small-step calculus called {\lambda}{\mu}s, which is inspired by the substitution at a distance paradigm. The {\lambda}{\mu}s-calculus turns out to be compatible with a natural extensionof the non-idempotent interpretations of {\lambda}{\mu}, i.e., {\lambda}{\mu}s-reduction preserves and decreases typing derivations in an extended appropriate typing system. We thus derive a simple arithmetical characterization of strongly {\lambda}{\mu}s-normalizing terms by means of typing.

    更新日期:2020-01-14
  • An extended type system with lambda-typed lambda-expressions
    arXiv.cs.LO Pub Date : 2018-03-21
    Matthias Weber

    We present the system $\mathtt{d}$, an extended type system with lambda-typed lambda-expressions. It is related to type systems originating from the Automath project. $\mathtt{d}$ extends existing lambda-typed systems by an existential abstraction operator as well as propositional operators. $\beta$-reduction is extended to also normalize negated expressions using a subset of the laws of classical negation, hence $\mathtt{d}$ is normalizing both proofs and formulas which are handled uniformly as functional expressions. $\mathtt{d}$ is using a reflexive typing axiom for a constant $\tau$ to which no function can be typed. Some properties are shown including confluence, subject reduction, uniqueness of types, strong normalization, and consistency. We illustrate how, when using $\mathtt{d}$, due to its limited logical strength, additional axioms must be added both for negation and for the mathematical structures whose deductions are to be formalized.

    更新日期:2020-01-14
  • Comparing Type Systems for Deadlock-Freedom
    arXiv.cs.LO Pub Date : 2018-10-01
    Ornela Dardha; Jorge A. Pérez

    Message-passing software systems exhibit non-trivial forms of concurrency and distribution; they are expected to respect intended protocols among interacting services, but also to never "get stuck". This intuitive requirement has been expressed by liveness properties such as progress or (dead)lock freedom; various type systems ensure these properties for concurrent processes. Unfortunately, very little is known about the precise relationship between these type systems and the classes of typed processes they induce. This paper puts forward the first comparative study of different type systems for message-passing concurrent processes that enforce deadlock-freedom. We compare two classes of deadlock-free typed processes, here denoted L and K. The class L stands out for its canonicity: it results naturally from Curry-Howard interpretations of linear logic propositions as session types. The class K, obtained by encoding session types into Kobayashi's linear types with usages, includes processes not typable in other type systems. We show that L is strictly included in K. We also identify the precise condition under which L and K coincide. One key observation is that the degree of sharing between parallel processes determines a new expressiveness hierarchy for typed concurrent processes. Furthermore, we provide two type-preserving procedures for rewriting processes in K into processes in L. Our two procedures suggest that, while effective, the degree of sharing is a rather subtle criterion for distinguishing typed concurrent processes.

    更新日期:2020-01-14
  • Abstracting Probabilistic Models: A Logical Perspective
    arXiv.cs.LO Pub Date : 2018-10-04
    Vaishak Belle

    Abstraction is a powerful idea widely used in science, to model, reason and explain the behavior of systems in a more tractable search space, by omitting irrelevant details. While notions of abstraction have matured for deterministic systems, the case for abstracting probabilistic models is not yet fully understood. In this paper, we provide a semantical framework for analyzing such abstractions from first principles. We develop the framework in a general way, allowing for expressive languages, including logic-based ones that admit relational and hierarchical constructs with stochastic primitives. We motivate a definition of consistency between a high-level model and its low-level counterpart, but also treat the case when the high-level model is missing critical information present in the low-level model. We prove properties of abstractions, both at the level of the parameter as well as the structure of the models. We conclude with some observations about how abstractions can be derived automatically.

    更新日期:2020-01-14
  • Scaling up Probabilistic Inference in Linear and Non-Linear Hybrid Domains by Leveraging Knowledge Compilation
    arXiv.cs.LO Pub Date : 2018-11-29
    Anton Fuxjaeger; Vaishak Belle

    Weighted model integration (WMI) extends weighted model counting (WMC) in providing a computational abstraction for probabilistic inference in mixed discrete-continuous domains. WMC has emerged as an assembly language for state-of-the-art reasoning in Bayesian networks, factor graphs, probabilistic programs and probabilistic databases. In this regard, WMI shows immense promise to be much more widely applicable, especially as many real-world applications involve attribute and feature spaces that are continuous and mixed. Nonetheless, state-of-the-art tools for WMI are limited and less mature than their propositional counterparts. In this work, we propose a new implementation regime that leverages propositional knowledge compilation for scaling up inference. In particular, we use sentential decision diagrams, a tractable representation of Boolean functions, as the underlying model counting and model enumeration scheme. Our regime performs competitively to state-of-the-art WMI systems but is also shown to handle a specific class of non-linear constraints over non-linear potentials.

    更新日期:2020-01-14
  • Searching for an analogue of ATR in the Weihrauch lattice
    arXiv.cs.LO Pub Date : 2018-12-04
    Takayuki Kihara; Alberto Marcone; Arno Pauly

    There are close similarities between the Weihrauch lattice and the zoo of axiom systems in reverse mathematics. Following these similarities has often allowed researchers to translate results from one setting to the other. However, amongst the big five axiom systems from reverse mathematics, so far ATR_0 has no identified counterpart in the Weihrauch degrees. We explore and evaluate several candidates, and conclude that the situation is complicated.

    更新日期:2020-01-14
  • Typal Heterogeneous Equality Types
    arXiv.cs.LO Pub Date : 2019-07-17
    Andrew M. Pitts

    The usual homogeneous form of equality type in Martin-L\"of Type Theory contains identifications between elements of the same type. By contrast, the heterogeneous form of equality contains identifications between elements of possibly different types. This paper introduces a simple set of axioms for such types. The axioms are equivalent to the combination of systematic elimination rules for both forms of equality, albeit with typal (also known as "propositional") computation properties, together with Streicher's Axiom K, or equivalently, the principle of uniqueness of identity proofs.

    更新日期:2020-01-14
  • Revisiting the duality of computation: an algebraic analysis of classical realizability models
    arXiv.cs.LO Pub Date : 2019-10-07
    Étienne MiqueyGALLINETTE, LS2N

    In an impressive series of papers, Krivine showed at the edge of the last decade how classical realizability provides a surprising technique to build models for classical theories. In particular, he proved that classical realizability subsumes Cohen's forcing, and even more, gives rise to unexpected models of set theories. Pursuing the algebraic analysis of these models that was first undertaken by Streicher, Miquel recently proposed to lay the algebraic foundation of classical realizability and forcing within new structures which he called implicative algebras. These structures are a generalization of Boolean algebras based on an internal law representing the implication. Notably, implicative algebras allow for the adequate interpretation of both programs (i.e. proofs) and their types (i.e. formulas) in the same structure. The very definition of implicative algebras takes position on a presentation of logic through universal quantification and the implication and, computationally, relies on the call-by-name $\lambda$-calculus. In this paper, we investigate the relevance of this choice, by introducing two similar structures. On the one hand, we define disjunctive algebras, which rely on internal laws for the negation and the disjunction and which we show to be particular cases of implicative algebras. On the other hand, we introduce conjunctive algebras, which rather put the focus on conjunctions and on the call-by-value evaluation strategy. We finally show how disjunctive and conjunctive algebras algebraically reflect the well-known duality of computation between call-by-name and call-by-value.

    更新日期:2020-01-14
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