Conservative Extension Research Articles

Sequential rough set: a conservative extension of Pawlak’s classical rough set

Rough set theory is an important approach to deal with uncertainty in data mining. However, Pawlak’s classical rough set has low fault-tolerance on concept approximation based on knowledge granules, which may influence the classification accuracy in practical application. To address this problem, the present paper proposes a novel sequential rough-set model that is proved to be a conservative extension of Pawlak’s classical rough set. As a result, it effectively improves the fault-tolerance ability, classification accuracy and concept approximation accuracy of the latter without any additional assumption. Based on the properties and theoretical analysis of the proposed model, an algorithm is presented to automatically determine the sequential thresholds and compute the three regions for the given concept. Experiments on real data verify the validity of the algorithm, and also show the stable improvement on the two types of accuracy.

The Riḥlahs and Road Trips of Modern Iraqi Literature: Ṣafāʾ Khulūṣī’s Abū Nuwās fī Amrīkā

Abstract This article situates the 1955 novel Abū Nuwās fī Amrīkā by Ṣafāʾ Khulūṣī as an exemplar of and commentary on the dynamics of twentieth-century Iraqi cultural production. Drawing on Margaret Litvin’s application of “travel literature readings” to modern diasporic and expatriate literatures, it analyzes the novel’s protagonist—in the form of a reincarnated, radically reformed version of the Abbasid-era poet Abū Nuwās—as engaging in a “riḥlah-road-trip,” traversing the United States alongside an anonymous narrator. Rather than fashioning a conservative extension of the pre- and early modern riḥlah genre, in view of the author’s historical moment and works, this article argues that Khulūṣī parodies this genre by constructing critical distance and building intertextuality simultaneously. Khulūṣī draws on riḥlahs of the past, in which Iraq is represented as an advanced cosmopolitan destination, to speak to Iraq’s future as a differently globalized, modernizing state that grapples with its perceived temporal and cultural regress, much like the novel’s time-traveling protagonist.

A Case for First-Class Environments

Formalizations of programming languages typically adopt the substitution model from the lambda calculus. However, substitution creates notorious complications for reasoning and implementation. Furthermore, it is disconnected from practical implementations, which normally adopt environments and closures. In this paper we advocate for formalizing programming languages using a novel style of small-step environment-based semantics , which avoids substitution and is closer to implementations. We present a call-by-value statically typed calculus, called λ E , using our small-step environment semantics. With our alternative environment semantics programming language constructs for first-class environments arise naturally, without creating significant additional complexity. Therefore, λ E also adopts first-class environments, adding expressive power that is not available in conventional lambda calculi. λ E is a conservative extension of the call-by-value Simply Typed Lambda Calculus (STLC), and employs de Bruijn indices for its formalization, which fit naturally with the environment-based semantics. Reasoning about λ E is simple, and in many cases simpler than reasoning about the traditional STLC. We show an abstract machine that implements the semantics of λ E , and has an easy correctness proof. We also extend λ E with references. We show that λ E can model a simple form of first-class modules, and suggest using first-class environments as an alternative to objects for modelling capabilities. All technical results are formalized in the Coq proof assistant. In summary, our work shows that the small-step environment semantics that we adopt has three main and orthogonal benefits: 1) it simplifies the notorious binding problem in formalizations and proof assistants; 2) it is closer to implementations; and 3) additional expressive power is obtained from first-class environments almost for free.

Iris-MSWasm: Elucidating and Mechanising the Security Invariants of Memory-Safe WebAssembly

WebAssembly offers coarse-grained encapsulation guarantees via its module system, but does not support fine-grained sharing of its linear memory. MSWasm is a recent proposal which extends WebAssembly with fine-grained memory sharing via handles, a type of capability that guarantees spatial and temporal safety, and thus enables an expressive yet safe style of programming with flexible sharing. In this paper, we formally validate the pen-and-paper design of MSWasm. To do so, we first define MSWasmCert, a mechanisation of MSWasm that makes it a fully-defined, conservative extension of WebAssembly 1.0, including the module system. We then develop Iris-MSWasm, a foundational reasoning framework for MSWasm composed of a separation logic to reason about known code, and a logical relation to reason about unknown, potentially adversarial code. Iris-MSWasm thereby makes explicit a key aspect of the implicit universal contract of MSWasm: robust capability safety. We apply Iris-MSWasm to reason about key use cases of handles, in which the effect of calling an unknown function is bounded by robust capability safety. Iris-MSWasm thus works as a framework to prove complex security properties of MSWasm programs, and provides a foundation to evaluate the language-level guarantees of MSWasm.

Statically Contextualizing Large Language Models with Typed Holes

Large language models (LLMs) have reshaped the landscape of program synthesis. However, contemporary LLM-based code completion systems often hallucinate broken code because they lack appropriate code context, particularly when working with definitions that are neither in the training data nor near the cursor. This paper demonstrates that tighter integration with the type and binding structure of the programming language in use, as exposed by its language server, can help address this contextualization problem in a token-efficient manner. In short, we contend that AIs need IDEs, too! In particular, we integrate LLM code generation into the Hazel live program sketching environment. The Hazel Language Server is able to identify the type and typing context of the hole that the programmer is filling, with Hazel's total syntax and type error correction ensuring that a meaningful program sketch is available whenever the developer requests a completion. This allows the system to prompt the LLM with codebase-wide contextual information that is not lexically local to the cursor, nor necessarily in the same file, but that is likely to be semantically local to the developer's goal. Completions synthesized by the LLM are then iteratively refined via further dialog with the language server, which provides error localization and error messages. To evaluate these techniques, we introduce MVUBench, a dataset of model-view-update (MVU) web applications with accompanying unit tests that have been written from scratch to avoid data contamination, and that can easily be ported to new languages because they do not have large external library dependencies. These applications serve as challenge problems due to their extensive reliance on application-specific data structures. Through an ablation study, we examine the impact of contextualization with type definitions, function headers, and errors messages, individually and in combination. We find that contextualization with type definitions is particularly impactful. After introducing our ideas in the context of Hazel, a low-resource language, we duplicate our techniques and port MVUBench to TypeScript in order to validate the applicability of these methods to higher-resource mainstream languages. Finally, we outline ChatLSP, a conservative extension to the Language Server Protocol (LSP) that language servers can implement to expose capabilities that AI code completion systems of various designs can use to incorporate static context when generating prompts for an LLM.

Nonconservative extensions by propositional quantifiers and modal incompleteness

Abstract Propositional modal logics can be extended by propositional quantifiers, i.e. quantifiers binding proposition letters understood as variables. This paper investigates whether such an extension is always conservative. It is shown that the answer depends on the way in which propositional quantifiers are added. On a minimal approach, according to which propositional quantifiers only have to satisfy the classical principles of quantification, every classical modal logic has a conservative extension by propositional quantifiers. However, in the context of normal modal logics it is natural to require propositionally quantified extensions also to be closed under the rule of necessitation. It is shown that in this setting, there are normal modal logics whose propositionally quantified extensions are nonconservative. Nonconservativity is shown to be a special case of a number of model-theoretic notions of incompleteness. More tentatively, it is suggested that nonconservativity indicates incompleteness in a more substantial sense, concerning the intended target of capturing the logics of modalities.

Multi-level nonstandard analysis and the axiom of choice

Model-theoretic frameworks for Nonstandard Analysis depend on the existence of nonprincipal ultrafilters, a strong form of the Axiom of Choice (AC). Hrbacek and Katz, APAL Volume 72 formulate axiomatic nonstandard set theories SPOT and SCOT that are conservative extensions of respectively ZF and ZF + ADC (the Axiom of Dependent Choice), and in which a significant part of Nonstandard Analysis can be developed.The present paper extends these theories to theories with many levels of standardness, called respectively SPOTS and SCOTS. It shows that Jin's recent nonstandard proof of Szemeredi's Theorem can be carried out in SPOTS and that SCOTS is a conservative extension of ZF + ADC.

Denying Infinity: Pragmatism in Abraham Robinson’s Philosophy of Mathematics

Abraham Robinson is well-known as the inventor of nonstandard analysis, which uses nonstandard models to give the notions of infinitesimal and infinitely large magnitudes a precise interpretation. Less discussed, although subtle and original–if ultimately flawed–is Robinson's work in the philosophy of mathematics. The foundational position he inherited from David Hilbert undermines not only the use of nonstandard analysis, but also Robinson's considerable corpus of pre-logic contributions to the field in such diverse areas as differential equations and aeronautics. This tension emerges from Robinson's disbelief in the existence of infinite totalities (any mention of them is ‘literally meaningless’) and the fact that much of his work involves them. I argue that he treats infinitary avenues of mathematics as useful tools to avoid this difficulty, but that this is not successful to the extent that these tools must be justified by a conservative extension from finitary mathematics. While Robinson provides a compelling and unorthodox pragmatic justification for the role of formal systems in mathematical practice despite their apparent infinitary presuppositions, he deflates mainstream mathematics to a collection of games that occasionally produces meaningful results. This amounts to giving up on a commitment to reconciling his finitism with his mathematical practice.

On Combining Intuitionistic and S4 Modal Logic

We address the problem of combining intuitionistic and S4 modal logic in a non-collapsing way inspired by the recent works in combining intuitionistic and classical logic. The combined language includes the shared constructors of both logics namely conjunction, disjunction and falsum as well as the intuitionistic implication, the classical implication and the necessity modality. We present a Gentzen calculus for the combined logic defined over a Gentzen calculus for the host S4 modal logic. The semantics is provided by Kripke structures. The calculus is proved to be sound and complete with respect to this semantics. We also show that the combined logic is a conservative extension of each component. Finally we establish that the Gentzen calculus for the combined logic enjoys cut elimination.

Admissible extensions of subtheories of second order arithmetic

In this paper we study admissible extensions of several theories T of reverse mathematics. The idea is that in such an extension the structure M=(N,S,∈) of the natural numbers and collection of sets of natural numbers S has to obey the axioms of T while simultaneously one also has a set-theoretic world with transfinite levels erected on top of M governed by the axioms of Kripke-Platek set theory, KP.In some respects, the admissible extension of T can be viewed as a proof-theoretic analog of Barwise's admissible cover of an arbitrary model of set theory; see [2]. However, by contrast, the admissible extension of T is usually not a conservative extension of T. Owing to the interplay of T and KP, either theory's axioms may force new sets of naturals to exist which in turn may engender yet new sets of naturals on account of the axioms of the other.The paper discerns a general pattern though. It turns out that for many familiar theories T, the second order part of the admissible cover of T equates to T augmented by transfinite induction over all initial segments of the Bachmann-Howard ordinal. Technically, the paper uses a novel type of ordinal analysis, expanding that for KP to the higher set-theoretic universe while at the same time treating the world of subsets of N as an unanalyzed class-sized urelement structure.Among the systems of reverse mathematics, for which we determine the admissible extension, are Π11-CA0 and ATR0 as well as the theory of bar induction, BI.

On provability logics of Niebergall arithmetic

K. G. Niebergall suggested a simple example of a non-gödelean arithmetical theory $\mathrm{NA}$, in which a natural formalization of its consistency is derivable. In the present paper we consider the provability logic of $\mathrm{NA}$ with respect to Peano arithmetic. We describe the class of its finite Kripke frames and establish the corresponding completeness theorem. For a conservative extension of this logic in the language with an additional propositional constant, we obtain a finite axiomatization. We also consider the truth provability logic of $\mathrm{NA}$ and the provability logic of $\mathrm{NA}$ with respect to $\mathrm{NA}$ itself. We describe the classes of Kripke models with respect to which these logics are complete. We establish $\mathrm{PSpace}$-completeness of the derivability problem in these logics and describe their variable free fragments. We also prove that the provability logic of $\mathrm{NA}$ with respect to Peano arithmetic does not have the Craig interpolation property.

Axiomatization multisets: a comparative analysis

A multiset, unlike the classical set, allows for multiple instances of its elements. In this paper, we present a comparative analysis of theories on multisets. In particular, we examine, the first-order two-sorted multiset theory MST, and the single-sorted multiset theory MS that employs the same sort for multiplicities and the set they support. The logical strengths and significance of some axioms presented in these theories are investigated. The theory MST contains a copy of the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) but is independent of ZFC. The single-sorted multiset theory describes a stronger theory that mirrors the Zermelo-Fraenkel set theory (ZF) and is equiconsistent with ZF and antifoundation. The two-sorted multisettheory MST is a conservative extension of the classical set theory, making it a suitable theory to assume when dealing with studies that involve multisets.

Residuated Basic Logic

Residuated basic logic (RBL) is the logic of residuated basic algebras, which constitutes a conservative extension of basic propositional logic (BPL). The basic implication is a residual of a non-associative binary operator in RBL. The conservativity is shown by relational semantics. A Gentzen-style sequent calculus GRBL, which is an extension of the distributive full non-associative Lambek calculus, is established for residuated basic logic. The calculus GRBL admits the mix-elimination, subformula, and disjunction properties. Moreover, the class of all residuated basic algebras has the finite embeddability property. The consequence relation of GRBL is decidable.

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

In this paper, we rigorously prove the existence of type-level ordered pairs in Quine’s New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU + Inf + AC). The proof uses the cardinal squaring principle; more precisely, its instance for the (infinite) universe (VCSP), which is a theorem of NFU + Inf + AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost from the beginning. This axiomatic extension is NFU + Inf + AC + VCSP, which is equivalent to NFU + Inf + AC, but easier to reason about.

BENIGREEN: Blockchain-Based Energy-Efficient Privacy-Preserving Scheme for Green IoT

The energy conservative extension of Internet-of-Thing (IoT), green IoT, is a revolutionary approach in connecting people, processes, and things in an energy-efficient way. The existing research works in the domain of green IoTs forbid the use of decentralized management (e.g., blockchain) of data due to its intrinsic disadvantages of block mining, transaction incentives, and less throughput. However, the advantages of blockchains urge the development of new decentralized strategies utilizing the architecture of green IoTs. The infancy stage of the conjunction between green IoTs and blockchains, the need of decentralization in energy management in green IoTs motivate us for the present research. In this paper, we address the problems of decentralization, energy-conservation, and privacy simultaneously. We introduce the first blockchain-based privacy-preserving framework for green IoTs. We name this framework as "Blockchain-based ENergy-efficient and prIvacy-preserving data management scheme for GREEN-iot (BENIGREEN)" for smart cities. BENIGREEN uses weight metrics for energy-efficient cluster heads selection. The use of weight metrics is a novel contribution in the field of green IoTs. Further, we integrate a decentralized blockchain framework with an authentication scheme for secure transmission among Base Station (BS) and sensor nodes by employing registration, certification, and revocation phases. Consequently, BS allocates the collected information from cluster heads to decentralized blockchain and cloud storage. The BS eliminates all malicious nodes from the network by employing a certificate revocation process. We execute thorough experiments in terms of the operation time, throughput, average energy consumption, and computational latency. The comparative analysis with the state-of-the-art schemes show that BENIGREEN is efficient for IoT paradigm.

Dual counterpart intuitionistic logic

Abstract We introduce dual counterpart intuitionistic logic (or DCInt): a constructive logic that is a conservative extension of intuitionistic logic, a sublogic of bi-intuitionistic logic, has the logical duality property of classical logic, and also retains the modal character of its interpretation of the connective dual to intuitionistic implication. We define its Kripke semantics along with the corresponding notion of a bisimulation, and then prove that it has both the disjunction property and (its dual) the constructible falsity property. Also, for any class $ {\mathcal{C}}$ of Kripke frames from our semantics, we identify a condition such that $ {\mathcal{C}}$ will have the disjunction property if it satisfies the condition. This provides a method for generating extensions of DCInt that retain the disjunction property.

Some techniques for reasoning automatically on co-inductive data structures

Abstract Some techniques are proposed for reasoning on co-inductive structures. First, we devise a sound axiomatization of (conservative extensions) of such structures, thus reducing the problem of checking whether a formula admits a co-inductive model to a first-order satisfiability test. We devise a class of structures, called regularly co-inductive, for which the axiomatization is complete (for other co-inductive structures, the proposed axiomatization is sound, but not complete). Then, we propose proof calculi for reasoning on such structures. We first show that some of the axioms mentioned above can be omitted if the inference rules are able to handle rational terms. Furthermore, under some conditions, some other axioms may be replaced by an additional inference rule that computes the solutions of fixpoint equations. Finally, we show that a stronger completeness result can be established under some additional conditions on the signature.

Squeeziness for non-deterministic systems

Failed Error Propagation greatly reduces the effectiveness of Software Testing by masking faults present in the code. This situation happens when the System Under Test executes a faulty statement, the state of the system is affected by this fault, but the expected output is observed. Therefore, it is a must to assess its impact in the testing process. Squeeziness has been shown to be a useful measure to assess the likelihood of fault masking in deterministic systems. The main goal of this paper is to define a new Squeeziness notion that can be used in a scenario where we may have non-deterministic behaviours. The new notion should be a conservative extension of the previous one. In addition, it would be necessary to evaluate whether the new notion appropriately estimates the likelihood that a component of a system introduces Failed Error Propagation. We defined our black-box scenario where non-deterministic behaviours might appear. Next, we presented a new Squeeziness notion that can be used in this scenario. Finally, we carried out different experiments to evaluate the usefulness of our proposal as an appropriate estimation of the likelihood of Failed Error Propagation. We found a high correlation between our new Squeeziness notion and the likelihood of Failed Error Propagation in non-deterministic systems. We also found that the extra computation time with respect to the deterministic version of Squeeziness was negligible. Our new Squeeziness notion is a good measure to estimate the likelihood of Failed Error Propagation being introduced by a component of a system (potentially) showing non-deterministic behaviours. Since it is a conservative extension of the original notion and the extra computation time needed to compute it, with respect to the time needed to compute the former notion, is very small, we conclude that the new notion can be safely used to assess the likelihood of fault masking in deterministic systems.

A Gradual Probabilistic Lambda Calculus

Probabilistic programming languages have recently gained a lot of attention, in particular due to their applications in domains such as machine learning and differential privacy. To establish invariants of interest, many such languages include some form of static checking in the form of type systems. However, adopting such a type discipline can be cumbersome or overly conservative. Gradual typing addresses this problem by supporting a smooth transition between static and dynamic checking, and has been successfully applied for languages with different constructs and type abstractions. Nevertheless, its benefits have never been explored in the context of probabilistic languages. In this work, we present and formalize GPLC, a gradual source probabilistic lambda calculus. GPLC includes a binary probabilistic choice operator and allows programmers to gradually introduce/remove static type–and probability–annotations. The static semantics of GPLC heavily relies on the notion of probabilistic couplings, as required for defining several relations, such as consistency, precision, and consistent transitivity. The dynamic semantics of GPLC is given via elaboration to the target language TPLC, which features a distribution-based semantics interpreting programs as probability distributions over final values. Regarding the language metatheory, we establish that TPLC–and therefore also GPLC–is type safe and satisfies two of the so-called refined criteria for gradual languages, namely, that it is a conservative extension of a fully static variant and that it satisfies the gradual guarantee, behaving smoothly with respect to type precision.

Quantifying over information change with common knowledge

Public announcement logic (PAL) extends multi-agent epistemic logic with dynamic operators modelling the effects of public communication. Allowing quantification over public announcements lets us reason about the existence of an announcement that reaches a certain epistemic goal. Two notable examples of logics of quantified announcements are arbitrary public announcement logic (APAL) and group announcement logic (GAL). While the notion of common knowledge plays an important role in PAL, and in particular in characterisations of epistemic states that an agent or a group of agents might make come about by performing public announcements, extensions of APAL and GAL with common knowledge still haven’t been studied in detail. That is what we do in this paper. In particular, we consider both conservative extensions, where the semantics of the quantifiers is not changed, as well as extensions where the scope of quantification also includes common knowledge formulas. We compare the expressivity of these extensions relative to each other and other connected logics, and provide sound and complete axiomatisations. Finally, we show how the completeness results can be used for other logics with quantification over information change.