Abstract This paper presents a fully verified interactive theorem prover for higher-order logic, more specifically: a fully verified clone of HOL Light. Our verification proof of this new system results in an end-to-end correctness theorem that guarantees the soundness of the entire system down to the machine code that executes at runtime. Our theorem states that every exported fact produced by this machine-code program is valid in higher-order logic. Our implementation consists of a read-eval-print loop (REPL) that executes the CakeML compiler internally. Throughout this work, we have strived to make the REPL of the new system provide a user experience as close to HOL Light’s as possible. To this end, we have, e.g., made the new system parse the same variant of OCaml syntax as HOL Light. All of the work described in this paper has been carried out in the HOL4 theorem prover.

As information plays an ever more central role across disciplines, the lack of a precise and reusable definition of state impedes comparison, measurement, and verification. Building on Objective Information Theory (OIT), this paper proposes a logic-based framework that defines the state of an object or system at a time point (or interval) as the semantic valuation of a set of well-formed formulas over a given domain and interpretation. Within first-order and higher-order logic—extended to infinitary logic when needed—we show how finite and broad classes of infinite structures can be characterized, drawing on core results from model theory. We then instantiate the framework in economics, sociology, computer science, and natural language, demonstrating that logic provides a unifying language for representing, reasoning about, and relating states across domains. Finally, we refine OIT by supplying a universal state representation that supports cross-domain exchange, measurement, and verification.

Higher-order logic HOL offers a very simple syntax and semantics for knowledge representation and reasoning in various particular domains, including in particular representing and reasoning about typed data structures. But its type system lacks advanced features where types may depend on terms. Dependent type theory offers such a rich type system, but has rather substantial conceptual differences to HOL, as well as comparatively poor proof automation support. We introduce a dependently-typed extension DHOL of HOL that retains the style and conceptual framework of HOL. Moreover, we build a translation from DHOL to HOL and implement it as a preprocessor to HOL theorem provers able to parse TPTP, thereby making all such provers able to run on DHOL problems.

This paper describes a formal theory of smooth vector fields, Lie groups and the Lie algebra of a Lie group in the theorem prover Isabelle. Lie groups are abstract structures that are composable, invertible and differentiable; they are useful in the study of continuous transformations in fields such as particle physics and robotics. The formalisation of this theory in an interactive theorem prover poses challenges beyond those encountered in textbook developments. We comment on representational choices we made to integrate involved concepts, such as smoothness of vector fields, with the simple type theory of higher-order logic (HOL) and existing material in Isabelle/HOL.

As AI systems increasingly operate with autonomy and adaptability, the traditional boundaries of moral responsibility in techno-social systems are being challenged. This paper explores the evolving discourse on the delegation of responsibilities to intelligent autonomous agents and the ethical implications of such practices. Synthesizing recent developments in AI ethics, including concepts of distributed responsibility and ethical AI by design, the paper proposes a functionalist perspective as a framework. This perspective views moral responsibility not as an individual trait but as a role within a socio-technical system, distributed among human and artificial agents. As an example of "AI ethical by design," we present Basti and Vitiello's implementation. They suggest that AI can act as artificial moral agents by learning ethical guidelines and using Deontic Higher-Order Logic to assess decisions ethically. Motivated by the possible speed and scale beyond human supervision and ethical implications, the paper argues for "AI ethical by design," while acknowledging the distributed, shared, and dynamic nature of responsibility. This functionalist approach offers a practical framework for navigating the complexities of AI ethics in a rapidly evolving technological landscape.

Abstract Notes on Kurt Gödel’s modal ontological argument and Dana Scott’s variant of it are presented. These remarks, supported by experimental studies with a proof assistant system for classical higher-order logic, implicitly answer some questions the authors have received over the last decade(s). In addition, some new insights resulting from the conducted experiments are reported.

We add an efficient function for computation to the kernels of higher-order logic interactive theorem provers. First, we develop and prove sound our approach for Candle. Candle is a port of HOL Light which has been proved sound with respect to the inference rules of its higher-order logic; we extend its implementation and soundness proof. Second, we replicate our now-verified implementation for HOL4 with only minor changes, and build additional automation for ease of use. The automation exists outside of the HOL4 kernel, and requires no additional trust. We exercise our new computation function and associated automation on the evaluation of the CakeML compiler backend within HOL4’s logic, demonstrating an order of magnitude speedup. This is an extended version of our previous conference paper [2], which described implementation and soundness proofs for Candle. Our HOL4 implementation and automation are new, as are the CakeML benchmarks.

Abstract This article describes an evaluation of Automated Theorem Proving (ATP) systems on problems taken from the Quantified Modal Logics Theorem Proving (QMLTP) library of first-order modal logic problems. Principally, the problems are translated to both typed first-order and higher-order logic (HOL) in the TPTP language using an embedding approach, and solved using first-order resp. HOL ATP systems and model finders. Additionally, the results from native modal logic ATP systems are considered, and compared with the results from the embedding approach. The findings are that the embedding process is reliable and successful when state-of-the-art ATP systems are used as backend reasoners, the first- and higher-order embeddings perform similarly, native modal logic ATP systems have comparable performance to classical systems using the embedding for proving theorems, native modal logic ATP systems are outperformed by the embedding approach for disproving conjectures and the embedding approach can cope with a wider range of modal logics than the native modal systems considered.

Logical reasoning in young children is difficult to ascertain experimentally even for single propositional operators. We present a novel argument that four- and five-year old children are capable of reasoning with complex representations containing multiple logical operators. The argument is based on an interaction between sentence interpretation and intonation. This interaction depends on the computation of logical inferences between the sentence uttered and possible alternative utterances containing proportional generalized quantifiers, and how adults arrive at different interpretations is well understood. The account that explains the interaction predicts that a specific intonation will disambiguate scopal interpretation in sentences with a negation and a universal quantifier, but not in sentences involving two quantifiers. We show that preschool children speaking German are sensitive to the interaction between logical scope of expressions and intonation in the same way as adult speakers. This result entails that preschool children can carry out logical reasoning within a higher order logic.

I take some initial steps toward a theory of real definition, drawing upon recent developments in higher-order logic. The resulting account allows for extremely fine-grained distinctions (it can distinguish between any relata that differ in their syntactic structure, while avoiding the Russell-Myhill problem). It is the first account that can consistently embrace three desirable logical principles that initially appear to be incompatible: the Identification Hypothesis (if F is, by definition, G, then there is a sense in which F is the same as G), Irreflexivity (there are no reflexive definitions), and Leibniz’s Law. Additionally, it possesses the resources needed to resolve the paradox of analysis.

Isabelle/DOF is an ontology framework on top of Isabelle/HOL. It allows for the formal development of ontologies and continuous conformity-checking of integrated documents, including the tracing of typed meta-data of documents. Isabelle/DOF deeply integrates into the Isabelle/HOL ecosystem, allowing to write documents containing (informal) text, executable code, (formal and semiformal) definitions, and proofs. Users of Isabelle/DOF can either use HOL or one of the many formal methods that have been embedded into Isabelle/HOL to express formal parts of their documents.In this paper, we extend Isabelle/DOF with annotations of ▪-terms, a pervasive data-structure underlying Isabelle to syntactically represent expressions and formulas. We achieve this by using Higher-order Logic (HOL) itself for query-expressions and data-constraints (ontological invariants) executed via code-generation and reflection. Moreover, we add support for parametric ontological classes, thus exploiting HOL's polymorphic type system.The benefits are: First, the HOL representation allows for flexible and efficient run-time checking of abstract properties of formal content under evolution. Second, it is possible to prove properties over generic ontological classes. We demonstrate these new features by a number of smaller ontologies from various domains and a case study using a substantial ontology for formal system development targeting certification according to CENELEC 50128.

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.

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.

Abstract We propose a stable model semantics for higher-order logic programs. Our semantics is developed using Approximation Fixpoint Theory (AFT), a powerful formalism that has successfully been used to give meaning to diverse non-monotonic formalisms. The proposed semantics generalizes the classical two-valued stable model semantics of Gelfond and Lifschitz as well as the three-valued one of Przymusinski, retaining their desirable properties. Due to the use of AFT, we also get for free alternative semantics for higher-order logic programs, namely supported model, Kripke-Kleene, and well-founded. Additionally, we define a broad class of stratified higher-order logic programs and demonstrate that they have a unique two-valued higher-order stable model which coincides with the well-founded semantics of such programs. We provide a number of examples in different application domains, which demonstrate that higher-order logic programming under the stable model semantics is a powerful and versatile formalism, which can potentially form the basis of novel ASP systems.

Translating meaning representations to behavioural interface specifications

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.

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.

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.

Properties and Propositions: The Metaphysics of Higher-Order Logic, by Robert Trueman

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.