Higher-order Logic Research Articles

Formalising the Double-Pushout Approach to Graph Transformation

In this paper, we utilize Isabelle/HOL to develop a formal framework for the basic theory of double-pushout graph transformation. Our work includes defining essential concepts like graphs, morphisms, pushouts, and pullbacks, and demonstrating their properties. We establish the uniqueness of derivations, drawing upon Rosens 1975 research, and verify the Church-Rosser theorem using Ehrigs and Kreowskis 1976 proof, thereby demonstrating the effectiveness of our formalisation approach. The paper details our methodology in employing Isabelle/HOL, including key design decisions that shaped the current iteration. We explore the technical complexities involved in applying higher-order logic, aiming to give readers an insightful perspective into the engaging aspects of working with an Interactive Theorem Prover. This work emphasizes the increasing importance of formal verification tools in clarifying complex mathematical concepts.

Neural networks in closed-loop systems: Verification using interval arithmetic and formal prover

Machine Learning approaches have been successfully used for the creation of high-performance control components of cyber–physical systems, where the control dynamics result from the combination of many subsystems. However, these approaches may lack the trustworthiness required to guarantee their reliable application in a safety-critical context. In this paper, we propose a combination of interval arithmetic and theorem-proving verification techniques to analyze safety properties in closed-loop systems that embed neural network components. We show the application of the proposed approach to a model-predictive controller for autonomous driving comparing the neural network verification performance with other existing tools. The results show that open-loop neural network verification through interval arithmetic can outperform existing approaches proving properties with a smaller time overhead. Furthermore, we demonstrate the capability of combining the two approaches to construct a formal model of the network in higher-order logic of the controlled system in a closed-loop.

Translating meaning representations to behavioural interface specifications

Higher-order logic can be used for meaning representation in natural language processing to encode the semantic relationships in text. Alternatively, using a formal specification language for meaning representation is more precise for specifying programs and widely supported by automatic theorem provers, while deductive verification based on higher order logic is less common for mainstream programming languages. This paper addresses the research question of translating higher-order logic meaning representations generated from method-level code comments into a formal specification language that extends first-order logic. Doing so requires resolving possible ambiguities in determining the appropriate semantics for predicates. This is an open challenge in the path toward using natural language processing with formal methods. To address this, the paper proposes an approach and constructs a compiler for translating meaning representations, generated from Java programs with method-level comments, into Java Modelling Language. We evaluate the compiler on a set of representative benchmarks, including programs and specifications from the Java API, by generating Java Modelling Language specifications and statically checking them with a theorem prover. Results show that in 94% of the cases Java Modelling Language is accurately generated and in 97% of those cases it can be automatically checked with a state-of-the-art theorem prover.

Agile Logical Semantics for Natural Languages

This paper presents an agile method of logical semantics based on high-order Predicate Logic. An operator of predicate abstraction is introduced that provides a simple mechanism for logical aggregation of predicates and for logical typing. Monadic high-order logic is the natural environment in which predicate abstraction expresses the semantics of typical linguistic structures. Many examples of logical representations of natural language sentences are provided. Future extensions and possible applications in the interaction with chatbots are briefly discussed as well.

An encoding of abstract dialectical frameworks into higher-order logic

Abstract An approach for encoding abstract dialectical frameworks and their semantics into classical higher-order logic is presented. Important properties and semantic relationships are formally encoded and proven using the proof assistant Isabelle/HOL. This approach allows for the computer-assisted analysis of abstract dialectical frameworks using automated and interactive reasoning tools within a uniform logic environment. Exemplary applications include the formal analysis and verification of meta-theoretical properties, and the generation of interpretations and extensions under specific semantic constraints.

Research on the higher-order logic formalization of fractance element

Fractance element reflects the fractional order behavior of circuits, which can show the characteristics of the actual circuits. Higher-order logic theorem proving is based on the rigorous and correct mathematical theories. It becomes more and more important in the verifications of high-reliability systems. Fractance element is formalized using the proof of higher-order logic theorem in this paper. Firstly, the formalized model of fractional calculus which is based on Caputo definition is established in higher-order logic theorem proof tool. Then some properties of fractional calculus are proved, including the zero order property, the fractional differential of a constant and the consistency of fractional calculus and integer order calculus. Finally, fractance element and fractional differential circuit constituted by fractance element are formally analyzed. These formalizations demonstrate the effectiveness of the formal method in the analysis of fractance element.

Mathematical Modality: An Investigation in Higher-order Logic

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the ‘width’ of the set theoretic universe, such as Cantor’s continuum hypothesis. Within a higher-order framework I show that contingency about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the ‘Leibniz biconditionals’ stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions.

Ordinal type theory

ABSTRACT Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorising about properties, relations, and states of affairs – or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood as a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities.

Finitary Type Theories With and Without Contexts

We give a definition of finitary type theories that subsumes many examples of dependent type theories, such as variants of Martin–Löf type theory, simple type theories, first-order and higher-order logics, and homotopy type theory. We prove several general meta-theorems about finitary type theories: weakening, admissibility of substitution and instantiation of metavariables, derivability of presuppositions, uniqueness of typing, and inversion principles. We then give a second formulation of finitary type theories in which there are no explicit contexts. Instead, free variables are explicitly annotated with their types. We provide translations between finitary type theories with and without contexts, thereby showing that they have the same expressive power. The context-free type theory is implemented in the nucleus of the Andromeda 2 proof assistant.

Complete and Efficient Higher-Order Reasoning via Lambda-Superposition

Superposition is a highly successful proof calculus for reasoning about first-order logic with equality. We present λ-superposition, which extends superposition to higher-order logic. Its design goals include soundness, completeness, efficiency, and gracefulness with respect to standard first-order superposition. The calculus is implemented in two automatic theorem provers: E and Zipper position. These provers regularly win trophies at the CADE ATP System Competition, confirming the calculus's applicability. This paper is a summary of research that took place between 2017 and 2022.

Verification Column

Automated theorem proving can be seen as a logic-based approach for generating mathematical proofs mechanically. Important applications are hardware and software verification, general-purpose proof assistants and proof checking. Most state-of-the-art theorem provers rely on proof calculi for higher-order logic that incorporate superposition techniques or satisfiability modulo theories. The article by Bentkamp, Blanchette, Nummelin, Tourret and Waldmann provides an overview of the recently developed λ-superposition approach. It takes inspirations of classical superposition for first-order logic and extends it for higher-order constructs. This new approach paves the way for a new generation of automated theorem provers for powerful logics, still being impressively efficient.

Semantical investigations on non-classical logics with recovery operators: negation

Abstract We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to provide a semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant.

Combining Higher-Order Logic with Set Theory Formalizations

The Isabelle Higher-order Tarski–Grothendieck object logic includes in its foundations both higher-order logic and set theory, which allows importing the libraries of Isabelle/HOL and Isabelle/Mizar. The two libraries, however, define all the basic concepts independently, which means that the results in the two are disconnected. In this paper, we align significant parts of these two libraries, by defining isomorphisms between their concepts, including the real numbers and algebraic structures. The isomorphisms allow us to transport theorems between the foundations and use the results from the libraries simultaneously.

Robert Trueman’s Defence of Higher-Order Logic

The paper contains a review and a discussion of Robert Trueman's book Properties and Propositions: The Metaphysics of Higher-Order Logic, Cambridge University Press, 2021, pp. xii + 227. ISBN 978-1-108-81410-2. The discussion is focused on the consistency of Truema's language-based ontology and on its value in defending higher-order logic.

Log-Linear-Based Logic Mining with Multi-Discrete Hopfield Neural Network

Choosing the best attribute from a dataset is a crucial step in effective logic mining since it has the greatest impact on improving the performance of the induced logic. This can be achieved by removing any irrelevant attributes that could become a logical rule. Numerous strategies are available in the literature to address this issue. However, these approaches only consider low-order logical rules, which limit the logical connection in the clause. Even though some methods produce excellent performance metrics, incorporating optimal higher-order logical rules into logic mining is challenging due to the large number of attributes involved. Furthermore, suboptimal logical rules are trained on an ineffective discrete Hopfield neural network, which leads to suboptimal induced logic. In this paper, we propose higher-order logic mining incorporating a log-linear analysis during the pre-processing phase, the multi-unit 3-satisfiability-based reverse analysis with a log-linear approach. The proposed logic mining also integrates a multi-unit discrete Hopfield neural network to ensure that each 3-satisfiability logic is learned separately. In this context, our proposed logic mining employs three unique optimization layers to improve the final induced logic. Extensive experiments are conducted on 15 real-life datasets from various fields of study. The experimental results demonstrated that our proposed logic mining method outperforms state-of-the-art methods in terms of widely used performance metrics.

Formal Analysis of Trust and Reputation for Service Composition in IoT

The exponential growth in the number of smart devices connected to the Internet of Things (IoT) that are associated with various IoT-based smart applications and services, raises interoperability challenges. Service-oriented architecture for IoT (SOA-IoT) solutions has been introduced to deal with these interoperability challenges by integrating web services into sensor networks via IoT-optimized gateways to fill the gap between devices, networks, and access terminals. The main aim of service composition is to transform user requirements into a composite service execution. Different methods have been used to perform service composition, which has been classified as trust-based and non-trust-based. The existing studies in this field have reported that trust-based approaches outperform non-trust-based ones. Trust-based service composition approaches use the trust and reputation system as a brain to select appropriate service providers (SPs) for the service composition plan. The trust and reputation system computes each candidate SP's trust value and selects the SP with the highest trust value for the service composition plan. The trust system computes the trust value from the self-observation of the service requestor (SR) and other service consumers' (SCs) recommendations. Several experimental solutions have been proposed to deal with trust-based service composition in the IoT; however, a formal method for trust-based service composition in the IoT is lacking. In this study, we used the formal method for representing the components of trust-based service management in the IoT, by using higher-order logic (HOL) and verifying the different behaviors in the trust system and the trust value computation processes. Our findings showed that the presence of malicious nodes performing trust attacks leads to biased trust value computation, which results in inappropriate SP selection during the service composition. The formal analysis has given us a clear insight and complete understanding, which will assist in the development of a robust trust system.

On Exams with the Isabelle Proof Assistant

We present an approach for testing student learning outcomes in a course on automated reasoning using the Isabelle proof assistant. The approach allows us to test both general understanding of formal proofs in various logical proof systems and understanding of proofs in the higher-order logic of Isabelle/HOL in particular. The use of Isabelle enables almost automatic grading of large parts of the exam. We explain our approach through a number of example problems, and explain why we believe that each of the kinds of problems we have selected are adequate measures of our intended learning outcomes. Finally, we discuss our experiences using the approach for the exam of a course on automated reasoning and suggest potential future work.

Completeness in partial type theory

Abstract The present paper provides a completeness proof for a system of higher-order logic framed within partial type theory. The framework is a modification of Tichý’s extension of Church’s simple type theory, equipped with his innovative natural deduction system in sequent style. The system deals with both total and partial (multiargument) functions-as-mappings and also accommodates algorithmic computations arriving at various objects of the framework. The partiality of a function or a failure of a computation is not represented by a postulated null object such as the third truth value. The logical operators of the system are classical. Another welcome feature of this expressive system is that its consequence relation is monotonic.

Admissible Types-to-PERs Relativization in Higher-Order Logic

Relativizing statements in Higher-Order Logic (HOL) from types to sets is useful for improving productivity when working with HOL-based interactive theorem provers such as HOL4, HOL Light and Isabelle/HOL. This paper provides the first comprehensive definition and study of types-to-sets relativization in HOL, done in the more general form of types-to-PERs (partial equivalence relations). We prove that, for a large practical fragment of HOL which includes container types such as datatypes and codatatypes, types-to-PERs relativization is admissible, in that the provability of the original, type-based statement implies the provability of its relativized, PER-based counterpart. Our results also imply the admissibility of a previously proposed axiomatic extension of HOL with local type definitions. We have implemented types-to-PERs relativization as an Isabelle tool that performs relativization of HOL theorems on demand.

HFL(Z) Validity Checking for Automated Program Verification

We propose an automated method for checking the validity of a formula of HFL(Z), a higher-order logic with fixpoint operators and integers. Combined with Kobayashi et al.'s reduction from higher-order program verification to HFL(Z) validity checking, our method yields a fully automated, uniform verification method for arbitrary temporal properties of higher-order functional programs expressible in the modal mu-calculus, including termination, non-termination, fair termination, fair non-termination, and also branching-time properties. We have implemented our method and obtained promising experimental results.