E-Cyclist: Implementation of an Efficient Validation of FOLID Cyclic Induction Reasoning

Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

In call-by-value languages, some mutually-recursive value definitions can be safely evaluated to build recursive functions or cyclic data structures, but some definitions (let rec x = x + 1) contain vicious circles and their evaluation fails at runtime. We propose a new static analysis to check the absence of such runtime failures. We present a set of declarative inference rules, prove its soundness with respect to the reference source-level semantics of Nordlander, Carlsson, and Gill (2008), and show that it can be (right-to-left) directed into an algorithmic check in a surprisingly simple way. Our implementation of this new check replaced the existing check used by the OCaml programming language, a fragile syntactic/grammatical criterion which let several subtle bugs slip through as the language kept evolving. We document some issues that arise when advanced features of a real-world functional language (exceptions in first-class modules, GADTs, etc.) interact with safety checking for recursive definitions.

First-order logic (FOL) is evidently insufficient for the many applications of logic in computer science, mainly due to its inability to provide inductive definitions. Therefore, only an extension of FOL which allows finitary inductive definitions can be used as a framework for automated reasoning. The minimal logic that is suitable for this goal is Ancestral Logic (AL), which is an extension of FOL by a transitive closure operator. In order for AL to be able to serve as a reasonable (and better) substitute to the use of FOL in computer science, it is crucial to develop adequate, user-friendly proof systems for it. While the expressiveness of AL renders any effective proof system for it incomplete with respect to the standard semantics, there are useful approximations. In this paper we show that such a Gentzen-style approximation is both sound and complete with respect to a natural, computationally-meaningful Henkin-style semantics for AL.

CLKID\(^{\omega }\) is a sequent-based cyclic inference system able to reason on first-order logic with inductive definitions. The current approach for verifying the soundness of CLKID\(^{\omega }\) proofs is based on expensive model-checking techniques leading to an explosion in the number of states.

Separation Logic (SL) with inductive definitions is a natural formalism for specifying complex recursive data structures, used in compositional verification of programs manipulating such structures. The key ingredient of any automated verification procedure based on SL is the decidability of the entailment problem. In this work, we reduce the entailment problem for a non-trivial subset of SL describing trees (and beyond) to the language inclusion of tree automata (TA). Our reduction provides tight complexity bounds for the problem and shows that entailment in our fragment is EXPTIME-complete. For practical purposes, we leverage from recent advances in automata theory, such as inclusion checking for non-deterministic TA avoiding explicit determinization. We implemented our method and present promising preliminary experimental results.

The calculation of thermal endurance and the estimation of lifetime at certain temperatures based on thermogravimetric data for the tungstophosphoric acid, H3[PW12O40]·H2O (H3PW) and its Cs acidic salts, pure or doped with Pd, were done. The calculation of catalyst’s lifetime with high degree of confidence needs reliable values for activation energy. For this reason, the activation energy values were calculated with ASTM E 1641-04 “Standard Test Method for Decomposition Kinetics by Thermogravimetry” based on Flynn–Wall isoconversional method and with Friedman isoconversional method (differential model-free) for checking, because the last provides accurate values of activation energies compared with the integral model-free methods. The thermal decomposition of the H3PW and some of its acidic Cs salts, pure or doped with Pd, occurs through the constitutional water release [the water formed of the protons and the oxygen from the [PW12O40]3−] having as result the loss of acid sites (protons) and its catalytic activity diminution at the same time. The activation energies are reliable values for the conversion range of 5 ≤ α ≤ 20, but the value for α = 10 % is recommended for the calculation of the lifetime. The estimation of lifetimes was done with ASTM E 1877-00 “Standard Practice for Calculating Thermal Endurance of Materials from Thermogravimetric Data” method. The estimated thermal life of the H3PW and some of its acidic Cs salts pointed out a very good thermal endurance at temperatures below 573 K. The Pd doping decreases drastically the thermal endurance in all cases.

By Gentzen's famous Hauptsatz the cut-elimination theorem every proof in sequent calculus for first-order logic with cuts can be transformed into a cut-free proof; cut-free proofs are analytic and consist entirely of syntactic material of the end-sequent the proven theorem. But in systems with induction rules, cut-elimination is either impossible or does not produce proofs with the subformula property. One way to overcome this problem is to formulate induction proofs as infinite sequences of proofs in a uniform way and to develop a method, which yields a uniform description of the corresponding cut-free proofs. We present such a formalism, as an alternative to systems with induction rules, and define a corresponding cut-elimination method based on the CERES-method for first-order logic. The basic tools of proof theory, such as sequent- and resolution calculi are enriched with inductive definitions and schemata of terms, formulas, proofs, etc. We define a class of inductive proofs which can be transformed into this formalism and subjected to schematic cut-elimination.

We present a novel tool for parameter synthesis of piecewise multi-affine dynamical systems from specifications expressed in a hybrid branching-time temporal logic. The tool is based on the algorithm of parallel semi-symbolic coloured model checking that extends standard model checking methods to cope with parametrised Kripke structures. The tool implements state-of-the-art techniques developed in our previous research and is primarily intended to be used for the analysis of dynamical systems with uncertain parameters that frequently arise in computational systems biology. However, it can be employed for any dynamical system where the non-linear equations can be sufficiently well approximated by piecewise multi-affine equations.

State-based formal methods [e.g. Event-B/RODIN (Abrial in Modeling in Event-B—system and software engineering. Cambridge University Press, Cambridge, 2010; Abrial et al. in Int J Softw Tools Technol Transf (STTT) 12(6):447–466, 2010)] for critical system development and verification are now well established, with track records including tool support and industrial applications. The focus of proof-based verification, in particular, is on safety properties. Liveness properties, which guarantee eventual, or converging computations of some requirements, are less well dealt with. Inductive reasoning about liveness is not explicitly supported. Liveness proofs are often complex and expensive, requiring high-skill levels on the part of the verification engineer. Fairness-based temporal logic approaches have been proposed to address this, e.g. TLA Lamport (ACM Trans Program Lang Syst 16(3):872–923, 1994) and that of Manna and Pnueli (Temporal verification of reactive systems—safety. Springer, New York, 1995). We contribute to this technology need by proposing a fairness-based method integrating temporal and first-order logic, proof and tools for modelling and verification of safety and liveness properties. The method is based on an integration of Event-B and TLA. Building on our previous work (Mery and Poppleton in Integrated formal methods, 10th international conference, IFM 2013, Turku, Finland, pp 208–222, 2013. doi: 10.1007/978-3-642-38613-8_15 ), we present the method via three example population protocols Angluin et al. (Distrib Comput 18(4):235–253, 2006). These were proposed as a theoretical framework for computability reasoning about Wireless Sensor Network and Mobile Ad-Hoc Network algorithms. Our examples present typical liveness and convergence requirements. We prove convergence results for the examples by integrated modelling and proof with Event-B/RODIN and TLA. We exploit existing proof rules, define and apply three new proof rules; soundness proofs are also provided. During the process we observe certain repeating patterns in the proofs. These are easily identified and reused because of the explicit nature of the reasoning.

We report the results of a simulation and reconstruction of observations of a young stellar object (YSO) jet with the LINC-NIRVANA (LN) interferometric instrument, which will be mounted on the Large Binocular Telescope (LBT). This simulation has been performed in order to investigate the ability of observing the weak diffuse jet line emission against the strong IR stellar continuum through narrow band images in the H and K atmospheric windows. In general, this simulation provides clues on the image quality that could be achieved in observations with a high dynamic range. In these cases, standard deconvolution methods, such as Richardson-Lucy, do not provide satisfactory results: we therefore propose here a new method of image reconstruction. It consists in considering the image to be reconstructed as the sum of two terms: one corresponding to the star (whose position is assumed to be known) and the other to the jet. A regularization term is introduced for this second component and the reconstruction is obtained with an iterative method alternating between the two components. An analysis of the results shows that the image quality obtainable with this method is significantly improved with respect to standard deconvolution methods, reducing the number of artifacts and allowing us to reconstruct the original jet intensity distribution with an error smaller than 10%.

On the expressive power of counting

A common problem associated with large vacuum vessels used in magnetic confinement fusion experiments is that leak checking is hampered by the inaccessibility to most of the vacuum vessel surface. This inaccessibility is caused by the close proximity of magnetic coils, diagnostics and, for those vessels that are baked, the need to completely surround the vessel with a thermal insulation blanket. These obstructions reduce the effectiveness of the standard leak checking method of using a mass spectrometer and spraying a search gas such as helium on the vessel exterior. Even when the presence of helium is detected, its entry point into the vessel cannot always be pinpointed. This paper will describe a method of overcoming this problem. By slightly depressurizing the vessel, an influx of helium through the leak is created. The leak site can then be identified by personnel within the vessel using standard sniffing procedures. There are two conditions which make this method of leak checking practical. First, the vessel need only be depressurized 2 psi, thus allowing personnel inside to perform the sniffing operation. Second, the sniffing probe used (Leybold–Heraus ‘‘Quick Test’’) could detect a change in helium concentration as small as 100 ppb, which allows for faster scanning of the vessel inferior. Use of this technique to find an elusive 10−3 Torr⋅l/s leak in the Doublet III tokamak vacuum vessel will be presented.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

Mechanically certifying formula-based Noetherian induction reasoning