In this paper we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalize the corresponding properties of the class of feasible functions. We also improve the Kapron - Cook result on mashine representation of basic feasible functionals. Our proofs are based on essential applications of logic. We introduce a weak fragment of second order arithmetic with second order variables ranging over functions from N into N which suitably characterizes basic feasible functionals, and show that it is a useful tool for investigating the properties of basic feasible functionals. In particular, we provide an example how one can extract feasible "programs" from mathematical proofs which use non-feasible functionals (like second order polynomials).
We present an illative system I_s of classical higher-order logic with subtyping and basic inductive types. The system I_s allows for direct definitions of partial and general recursive functions, and provides means for handling functions whose termination has not been proven. We give examples of how properties of some recursive functions may be established in our system. In a technical appendix to the paper we prove consistency of I_s. The proof is by model construction. We then use this construction to show conservativity of I_s over classical first-order logic. Conservativity over higher-or…
We present a simple resolution proof system for higher-order constrained Horn clauses (HoCHC) - a system of higher-order logic modulo theories - and prove its soundness and refutational completeness w.r.t. the standard semantics. As corollaries, we obtain the compactness theorem and semi-decidability of HoCHC for semi-decidable background theories, and we prove that HoCHC satisfies a canonical model property. Moreover a variant of the well-known translation from higher-order to 1st-order logic is shown to be sound and complete for HoCHC in standard semantics. We illustrate how to transfer deci…
We introduce a variation on Barthe et al.'s higher-order logic in which formulas are interpreted as predicates over open rather than closed objects. This way, concepts which have an intrinsically functional nature, like continuity, differentiability, or monotonicity, can be expressed and reasoned about in a very natural way, following the structure of the underlying program. We give open higher-order logic in distinct flavors, and in particular in its relational and local versions, the latter being tailored for situations in which properties hold only in part of the underlying function's domai…
The computer-mechanization of an ambitious explicit ethical theory, Gewirth's Principle of Generic Consistency, is used to showcase an approach for representing and reasoning with ethical theories exhibiting complex logical features like alethic and deontic modalities, indexicals, higher-order quantification, among others. Harnessing the high expressive power of Church's type theory as a meta-logic to semantically embed a combination of quantified non-classical logics, our work pushes existing boundaries in knowledge representation and reasoning. We demonstrate that intuitive encodings of comp…
TLA is a popular temporal logic for writing stuttering-invariant specifications of digital systems. However, TLA lacks higher-order features useful for specifying modern software written in higher-order programming languages. We use categorical techniques to recast a real-time semantics for TLA in terms of the actions of a group of time dilations, or "stutters," and an extension by a monoid incorporating delays, or "falters." Via the geometric morphism of the associated presheaf topoi induced by the inclusion of stutters into falters, we construct the first model of a higher-order TLA.
Abstraction logic is a new logic, serving as a foundation of mathematics. It combines features of both predicate logic and higher-order logic: abstraction logic can be viewed both as higher-order logic minus static types as well as predicate logic plus operators and variable binding. We argue that abstraction logic is the best foundational logic possible because it maximises both simplicity and practical expressivity. This argument is supported by the observation that abstraction logic has simpler terms and a simpler notion of proof than all other general logics. At the same time, abstraction …
Bisimulations and Modal Logics for Higher Dimensional Automata
Norwegian University of Science and Technology（挪威科技大学）；University of Edinburgh（爱丁堡大学）；University of New South Wales（新南威尔士大学）；University of Warsaw（华沙大学）
Most of the engineering and physical systems are generally characterized by differential and difference equations based on their continuous-time and discrete-time dynamics, respectively. Moreover, these dynamical models are analyzed using transform methods to prove various properties of these systems, such as, transfer function, frequency response and stability, and to find out solutions of the differential/difference equations. The conventional techniques for performing the transform methods based analysis have been unable to provide an accurate analysis of these systems. Therefore, higher-or…
Satisfiability modulo theories (SMT) solvers have throughout the years been able to cope with increasingly expressive formulas, from ground logics to full first-order logic modulo theories. Nevertheless, higher-order logic within SMT is still little explored. One main goal of the Matryoshka project, which started in March 2017, is to extend the reasoning capabilities of SMT solvers and other automatic provers beyond first-order logic. In this preliminary report, we report on an extension of the SMT-LIB language, the standard input format of SMT solvers, to handle higher-order constructs. We al…
We present an approach towards the deep, pluralistic logical analysis of argumentative discourse that benefits from the application of state-of-the-art automated reasoning technology for classical higher-order logic. Thanks to its expressivity this logic can adopt the status of a uniform \textit{lingua franca} allowing the encoding of both formalized arguments (their deep logical structure) and dialectical interactions (their attack and support relations). We illustrate this by analyzing an excerpt from an argumentative debate on climate engineering. Another, novel contribution concerns the de…
Synthetic Biology is an interdisciplinary field that utilizes well-established engineering principles, ranging from electrical, control and computer systems, for analyzing the biological systems, such as biological circuits, enzymes, pathways and controllers. Traditionally, these biological systems, i.e., the genetic circuits are analyzed using paper-and-pencil proofs and computer-based simulations techniques. However, these methods cannot provide accurate results due to their inherent limitations such as human error-proneness, round-off errors and the unverified algorithms present in the core…
Higher-order logic programming is an interesting extension of traditional logic programming that allows predicates to appear as arguments and variables to be used where predicates typically occur. Higher-order characteristics are indeed desirable but on the other hand they are also usually more expensive to support. In this paper we propose a program specialization technique based on partial evaluation that can be applied to a modest but useful class of higher-order logic programs and can transform them into first-order programs without introducing additional data structures. The resulting fir…
We present an approach for representing abstract argumentation frameworks based on an encoding into classical higher-order logic. This provides a uniform framework for computer-assisted assessment of abstract argumentation frameworks using interactive and automated reasoning tools. This enables the formal analysis and verification of meta-theoretical properties as well as the flexible generation of extensions and labellings with respect to well-known argumentation semantics.
This paper studies the logical properties of a very general class of infinite ranked trees, namely those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal mu-calculus, three main problems: model-checking, logical reflection (aka global model-checking, that asks for a finite description of the set of elements for which a formula holds) and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems we provide an effective solut…
Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, including security, privacy and machine learning. In this paper, we develop program logics for reasoning about adversarial computations in a higher-order setting. Our logics are built on top of a simply typed $λ$-calculus extended with a graded monad for probabilities and state. The grading is used to model and restrict the memory footprint and the co…
