Soundness Research Articles

Vagueness as an epiphenomenon, and non-transitivity

ABSTRACT This paper deals with the linguistic phenomenon of vagueness. Based on certain observations regarding the intuitions and linguistic practices of the philosophically informed speaker, we make a series of assumptions concerning the nature and characteristics of the phenomenon. Vagueness is treated as an emerging phenomenon, caused, in essence, by the messy way in which linguistic communities reach classificatory equilibria. Any talk of ‘meaning’, ‘truth’, and such is treated as an indirect way of attempting to describe such equilibria, and it is in regard to these that ascriptions of truth are here relativised. Given such assumptions, we then proceed to formulate and study a semantic theory for languages that contain vague predicates. Finally, we describe two proof systems, one based on trees, and a natural deduction system in the style of Fitch, and outline the appropriate proofs of soundness and completeness.

On Logics and Semantics for Interpretability

AbstractWe study various properties of formalised relativised interpretability. In the central part of this thesis we study for different interpretability logics the following aspects: completeness for modal semantics, decidability and algorithmic complexity.In particular, we study two basic types of relational semantics for interpretability logics. One is the Veltman semantics, which we shall refer to as the regular or ordinary semantics; the other is called generalised Veltman semantics. In the recent years and especially during the writing of this thesis, generalised Veltman semantics was shown to be particularly well-suited as a relational semantics for interpretability logics. In particular, modal completeness results are easier to obtain in some cases; and decidability can be proven via filtration in all known cases. We prove various new and reprove some old completeness results with respect to the generalised semantics. We use the method of filtration to obtain the finite model property for various logics.Apart from results concerning semantics in its own right, we also apply methods from semantics to determine decidability (implied by the finite model property) and complexity of provability (and consistency) problems for certain interpretability logics.From the arithmetical standpoint, we explore three different series of interpretability principles. For two of them, for which arithmetical and modal soundness was already known, we give a new proof of arithmetical soundness. The third series results from our modal considerations. We prove it arithmetically sound and also characterise frame conditions w.r.t. ordinary Veltman semantics. We also prove results concerning the new series and generalised Veltman semantics.Abstract prepared by Luka Mikec.E-mail: luka.mikec1@gmail.comURL: http://hdl.handle.net/2445/177373

AN ESCAPE FROM VARDANYAN’S THEOREM

AbstractVardanyan’s Theorems [36, 37] state that $\mathsf {QPL}(\mathsf {PA})$ —the quantified provability logic of Peano Arithmetic—is $\Pi ^0_2$ complete, and in particular that this already holds when the language is restricted to a single unary predicate. Moreover, Visser and de Jonge [38] generalized this result to conclude that it is impossible to computably axiomatize the quantified provability logic of a wide class of theories. However, the proof of this fact cannot be performed in a strictly positive signature. The system $\mathsf {QRC_1}$ was previously introduced by the authors [1] as a candidate first-order provability logic. Here we generalize the previously available Kripke soundness and completeness proofs, obtaining constant domain completeness. Then we show that $\mathsf {QRC_1}$ is indeed complete with respect to arithmetical semantics. This is achieved via a Solovay-type construction applied to constant domain Kripke models. As corollaries, we see that $\mathsf {QRC_1}$ is the strictly positive fragment of $\mathsf {QGL}$ and a fragment of $\mathsf {QPL}(\mathsf {PA})$ .

SeCaV: A Sequent Calculus Verifier in Isabelle/HOL

We describe SeCaV, a sequent calculus verifier for first-order logic in Isabelle/HOL, and the SeCaV Unshortener, an online tool that expands succinct derivations into the full SeCaV syntax. We leverage the power of Isabelle/HOL as a proof checker for our SeCaV derivations. The interactive features of Isabelle/HOL make our system transparent. For instance, the user can simply click on a side condition to see its exact definition. Our formalized soundness and completeness proofs pertain exactly to the calculus as exposed to the user and not just to some model of our tool. Users can also write their derivations in the SeCaV Unshortener, which provides a lighter syntax, and expand them for later verification. We have used both tools in our teaching.

Two decades of automatic amortized resource analysis

AbstractThis article gives an overview of automatic amortized resource analysis (AARA), a technique for inferring symbolic resource bounds for programs at compile time. AARA has been introduced by Hofmann and Jost in 2003 as a type system for deriving linear worst-case bounds on the heap-space consumption of first-order functional programs with eager evaluation strategy. Since then AARA has been the subject of dozens of research articles, which extended the analysis to different resource metrics, other evaluation strategies, non-linear bounds, and additional language features. All these works preserved the defining characteristics of the original paper: local inference rules, which reduce bound inference to numeric (usually linear) optimization; a soundness proof with respect to an operational cost semantics; and the support of amortized analysis with the potential method.

Signatures of knowledge for Boolean circuits under standard assumptions

This paper constructs unbounded simulation sound proofs for boolean circuit satisfiability under standard assumptions with proof size O(n+d) bilinear group elements, where d is the depth and n is the input size of the circuit. Our technical contribution is to add unbounded simulation soundness to a recent NIZK of González and Ràfols (ASIACRYPT'19) with very small overhead. We give two different constructions: the first one is more efficient but not tight, and the second one is tight. Our new scheme can be used to construct Signatures of Knowledge based on standard assumptions that also can be composed universally with other cryptographic protocols/primitives.As an independent contribution we also detail a simple formula to encode Boolean circuits as Quadratic Arithmetic Programs.

Memory-limited model-based diagnosis

Various model-based diagnosis scenarios require the computation of the most preferred fault explanations. Existing algorithms that are sound (i.e., output only actual fault explanations) and complete (i.e., can return all explanations), however, require exponential space to achieve this task. As a remedy, and to enable successful diagnosis both on memory-restricted devices and for memory-intensive problem cases, we propose two novel diagnostic search algorithms which build upon tried and tested techniques from the heuristic search domain. The first method, dubbed Recursive Best-First Hitting Set Search (RBF-HS), is based on Korf's well-known Recursive Best-First Search (RBFS) algorithm. We show that RBF-HS can enumerate an arbitrary predefined finite number of fault explanations in best-first order within linear space bounds, without sacrificing the desirable soundness or completeness properties. The second algorithm, called Hybrid Best-First Hitting Set Search (HBF-HS), is a hybrid between RBF-HS and Reiter's seminal HS-Tree. The idea is to find a trade-off between runtime optimization and a restricted space consumption that does not exceed the available memory. Notably, both suggested algorithms are generally applicable to any model-based diagnosis problem, regardless of the used (monotonic) logical language to describe the diagnosed system and of the used reasoning mechanism.We conducted extensive experiments on real-world benchmarks from the knowledge-based systems field, a domain where the features soundness, completeness, the best-first property as well as a general applicability are pivotal and where Reiter's HS-Tree is the predominantly used diagnostic search. The evaluation reveals that, when computing fault explanations minimal-cardinality-first, RBF-HS compared to HS-Tree reduces memory requirements substantially in most cases by up to several orders of magnitude, while also saving runtime in more than a third of the cases. When computing fault explanations most-probable-first, RBF-HS compared to HS-Tree tends to trade memory savings more or less one-to-one for runtime overheads. Whenever runtime overheads were significant, using HBF-HS instead of RBF-HS reduced the runtime to values comparable with HS-Tree while keeping the used memory reasonably bounded.

Reasoning about “reasoning about reasoning”: semantics and contextual equivalence for probabilistic programs with nested queries and recursion

Metareasoning can be achieved in probabilistic programming languages (PPLs) using agent models that recursively nest inference queries inside inference queries. However, the semantics of this powerful, reflection-like language feature has defied an operational treatment, much less reasoning principles for contextual equivalence. We give formal semantics to a core PPL with continuous distributions, scoring, general recursion, and nested queries. Unlike prior work, the presence of nested queries and general recursion makes it impossible to stratify the definition of a sampling-based operational semantics and that of a measure-theoretic semantics—the two semantics must be defined mutually recursively. A key yet challenging property we establish is that probabilistic programs have well-defined meanings: limits exist for the step-indexed measures they induce. Beyond a semantics, we offer relational reasoning principles for probabilistic programs making nested queries. We construct a step-indexed, biorthogonal logical-relations model. A soundness theorem establishes that logical relatedness implies contextual equivalence. We demonstrate the usefulness of the reasoning principles by proving novel equivalences of practical relevance—in particular, game-playing and decisionmaking agents. We mechanize our technical developments leading to the soundness proof using the Coq proof assistant. Nested queries are an important yet theoretically underdeveloped linguistic feature in PPLs; we are first to give them semantics in the presence of general recursion and to provide them with sound reasoning principles for contextual equivalence.

Software model-checking as cyclic-proof search

This paper shows that a variety of software model-checking algorithms can be seen as proof-search strategies for a non-standard proof system, known as a cyclic proof system . Our use of the cyclic proof system as a logical foundation of software model checking enables us to compare different algorithms, to reconstruct well-known algorithms from a few simple principles, and to obtain soundness proofs of algorithms for free. Among others, we show the significance of a heuristics based on a notion that we call maximal conservativity ; this explains the cores of important algorithms such as property-directed reachability (PDR) and reveals a surprising connection to an efficient solver of games over infinite graphs that was not regarded as a kind of PDR.

О ТЕХНОЛОГИИ РАСЧЕТА СИСТЕМ ЖЕЛЕЗНОДОРОЖНОГО ТРАНСПОРТА

The paper considers principles of creating methods for calculation of railway stations and polygons. It shows that methods in existing instructions are theoretically based, in the essence, on logical soundness. The authors propose to consider a system approach in original statement as a scientific basis. A system is considered as a generally natural form of construction of organized matter which allows steadily functioning in environment with disorganization. An important property of the system is adaptability when structural elements enter a dynamic interaction. The authors state that in calculation models an element is assigned (consciously or not) a certain functional role that often doesn’t match a real role. Accordance of these roles is assumed as a main criterion of correctness of a calculation method. But the functional role is determined not only by properties of the element but also by a character of its interaction with other elements. That is why you can’t calculate a capacity of structural elements separately. The authors show on specific examples that it is possible to calculate the capacity only in a comprehensive model of the whole system. As a result, it is necessary to consider simulation modeling as a main method for calculation of systems of railway transport.

Formal categorical reasoning

In this paper, we present a category theory library developed in the proof assistant Coq. We discuss the design principles of the library in comparison with those existing out there. To explicitly demonstrate the utility of the library, we conclude with a case study in which a Coq formalized soundness proof of the intuitionistic propositional logic within a category theoretical settings is examined.

Where is The Limit of Questioning More? A Study of The Proper Use of The Concept of Critical Thinking

Critical Thinking is a sum of abilities that are more and more required today in this world of disinformation. Critical Thinking even started to become a philosophy of life where people must carefully evaluate the information they encounter. However, this statement of „think critically” has started to be used by entities who promote false theories, fake news and unsound arguments. For example, the motto of the controversed news outlet Russia Today is „Question more”. Another example is that on the social network of Instagram, the hashtag #criticalthinking is used in posts that promote the theory of the flat earth, that promote antivaxxing movement and so on. If we are to be put in an average person’s shoes, how would we know which critical thinking is the right one? How would we know who to believe? This paper’s aim is to emphasize the importance of critical thinking standards and criteria and to analyse them in order to make the difference between a proper use of this statement and an abusive and alleged one. Keywords: critical thinking, fake news, thinking standards, question more.

A Transient Semantics for Typed Racket

Mixed-typed languages enable programmers to link typed and untyped components in various ways. Some offer rich type systems to facilitate the smooth migration of untyped code to the typed world; others merely provide a convenient form of type Dynamic together with a conventional structural type system. Orthogonal to this dimension, Natural systems ensure the integrity of types with a sophisticated contract system, while Transient systems insert simple first-order checks at strategic places within typed code. Furthermore, each method of ensuring type integrity comes with its own blame-assignment strategy. Typed Racket has a rich migratory type system and enforces the types with a Natural semantics. Reticulated Python has a simple structural type system extended with Dynamic and enforces types with a Transient semantics. While Typed Racket satisfies the most stringent gradual-type soundness properties at a significant performance cost, Reticulated Python seems to limit the performance penalty to a tolerable degree and is nevertheless type sound. This comparison raises the question of whether Transient checking is applicable to and beneficial for a rich migratory type system. This paper reports on the surprising difficulties of adapting the Transient semantics of Reticulated Python to the rich migratory type system of Typed Racket. The resulting implementation, Shallow Typed Racket, is faster than the standard Deep Typed Racket but only when the Transient blame assignment strategy is disabled. For language designers, this report provides valuable hints on how to equip an existing compiler to support a Transient semantics. For theoreticians, the negative experience with Transient blame calls for a thorough investigation of this strategy.

Combining state- and event-based semantics to verify highly available applications

Replicated databases are attractive for managing the data of distributed applications. To achieve high availability, low latency, and high throughput for such applications, consistency has to be weakened. This comes at a price: it is harder to prove application correctness. In this paper, we describe a new and powerful specification and verification technique for highly available applications together with the tool Repliss that partially automates its use. The verification technique is based on existing event-based semantics for database replication and embeds the semantics into a classical state-based verification approach for application programs. As a central contribution, we carefully designed the specification and programming framework in such a way, that the complex concurrent semantics are reduced to sequential verification, which is much easier to handle and provide a better basis for automated tool support. We prove the soundness of this reduction and a restricted form of completeness. All these proofs and the underlying semantics of the system have been formalized in Isabelle/HOL. • Composable proof technique for working with weak consistency. • Expressible specifications with history invariants. • Semantics, technique, and soundness proof verified in Isabelle/HOL. • Implementation of a prototype verification tool to support the technique.

Differential logical relations, part II increments and derivatives

We study the deep relation existing between differential logical relations and incremental computing by showing how self-differences in the former precisely correspond to derivatives in the latter. The byproduct of such a relationship is twofold: on the one hand, we show how differential logical relations can be seen as a powerful meta-theoretical tool in the analysis of incremental computations, enabling an easy proof of soundness of differentiation. On the other hand, we generalize differential logical relations so as to be able to interpret full recursion, something not possible in the original system.

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

Checking the soundness of cyclic induction reasoning for first-order logic with inductive definitions (FOLID) is decidable but the standard checking method is based on an exponential complement operation for B\"uchi automata. Recently, we introduced a polynomial checking method whose most expensive steps recall the comparisons done with multiset path orderings. We describe the implementation of our method in the Cyclist prover. Referred to as E-Cyclist, it successfully checked all the proofs included in the original distribution of Cyclist. Heuristics have been devised to automatically define, from the analysis of the proof derivations, the trace-based ordering measures that guarantee the soundness property.

Contextual modal types for algebraic effects and handlers

Programming languages with algebraic effects often track the computations’ effects using type-and-effect systems. In this paper, we propose to view an algebraic effect theory of a computation as a variable context; consequently, we propose to track algebraic effects of a computation with contextual modal types . We develop ECMTT, a novel calculus which tracks algebraic effects by a contextualized variant of the modal □ (necessity) operator, that it inherits from Contextual Modal Type Theory (CMTT). Whereas type-and-effect systems add effect annotations on top of a prior programming language, the effect annotations in ECMTT are inherent to the language, as they are managed by programming constructs corresponding to the logical introduction and elimination forms for the □ modality. Thus, the type-and-effect system of ECMTT is actually just a type system. Our design obtains the properties of local soundness and completeness, and determines the operational semantics solely by β-reduction, as customary in other logic-based calculi. In this view, effect handlers arise naturally as a witness that one context (i.e., algebraic theory) can be reached from another, generalizing explicit substitutions from CMTT. To the best of our knowledge, ECMTT is the first system to relate algebraic effects to modal types. We also see it as a step towards providing a correspondence in the style of Curry and Howard that may transfer a number of results from the fields of modal logic and modal type theory to that of algebraic effects.

GhostCell: separating permissions from data in Rust

The Rust language offers a promising approach to safe systems programming based on the principle of aliasing XOR mutability : a value may be either aliased or mutable, but not both at the same time. However, to implement pointer-based data structures with internal sharing, such as graphs or doubly-linked lists, we need to be able to mutate aliased state. To support such data structures, Rust provides a number of APIs that offer so-called interior mutability : the ability to mutate data via method calls on a shared reference. Unfortunately, the existing APIs sacrifice flexibility, concurrent access, and/or performance, in exchange for safety. In this paper, we propose a new Rust API called GhostCell which avoids such sacrifices by separating permissions from data : it enables the user to safely synchronize access to a collection of data via a single permission. GhostCell repurposes an old trick from typed functional programming: branded types (as exemplified by Haskell’s ST monad), which combine phantom types and rank-2 polymorphism to simulate a lightweight form of state-dependent types. We have formally proven the soundness of GhostCell by adapting and extending RustBelt, a semantic soundness proof for a representative subset of Rust, mechanized in Coq.

A Characterisation of Open Bisimilarity using an Intuitionistic Modal Logic

Open bisimilarity is defined for open process terms in which free variables may appear. The insight is, in order to characterise open bisimilarity, we move to the setting of intuitionistic modal logics. The intuitionistic modal logic introduced, called $\mathcal{OM}$, is such that modalities are closed under substitutions, which induces a property known as intuitionistic hereditary. Intuitionistic hereditary reflects in logic the lazy instantiation of free variables performed when checking open bisimilarity. The soundness proof for open bisimilarity with respect to our intuitionistic modal logic is mechanised in Abella. The constructive content of the completeness proof provides an algorithm for generating distinguishing formulae, which we have implemented. We draw attention to the fact that there is a spectrum of bisimilarity congruences that can be characterised by intuitionistic modal logics.

Linear-Space Best-First Diagnosis Search

Various model-based diagnosis scenarios require the computation of the most preferred fault explanations. Existing algorithms that are sound (i.e., output only actual fault explanations) and complete (i.e., can return all explanations), however, require exponential space to achieve this task. As a remedy, to enable successful diagnosis on memory-restricted devices and for memory-intensive problem cases, we propose RBF-HS, a diagnostic search based on Korf’s seminal RBFS algorithm. RBF-HS can enumerate an arbitrary fixed number of fault explanations in best-first order within linear space bounds, without sacrificing the desirable soundness or completeness properties. Evaluations using real-world diagnosis cases show that RBF-HS, when used to compute minimum-cardinality fault explanations, in most cases saves substantial space (up to 98 %) while requiring only reasonably more or even less time than Reiter’s HS-Tree, one of the most widely used diagnostic algorithms with the same properties.