Abstract We present deterministic model construction algorithms for sets of modal clauses saturated with respect to three refinements of the modal-layered resolution calculus implemented in the prover "Image missing". The model construction algorithms are inspired by the Bachmair-Ganzinger method for constructing a model for a set of ground first-order clauses saturated with respect to ordered resolution with selection. The challenge is that the inference rules of the modal-layered resolution calculus for modal operators are more restrictive than an adaptation of ordered resolution with selection for these would be. While these model construction algorithms provide an alternative means to proving completeness of the calculus, our main interest is the provision of a ‘certificate’ for satisfiable modal formulae that can be independently checked to assure a user that the result of "Image missing" is correct. This complements the existing provision of proofs for unsatisfiable modal formulae.

Abstract We are interested in widening the reasoning support for propositional modal logics in the so-called modal cube. The modal cube consists of extensions of the basic modal logic $$\textsf{K}_{}$$ K with an arbitrary combination of the modal axioms $$\textsf{B}$$ B , $$\textsf{D}$$ D , $$\textsf{T}$$ T , $$\textsf{4}$$ 4 and $$\textsf{5}$$ 5 . We revisit recently developed local reductions from all logics in the modal cube to a normal form comprising sets of clausal formulae with associated modal levels. We extend these reductions further to the basic modal logic $$\textsf{K}_{}$$ K , called definitional reductions. This enables any prover for $$\textsf{K}_{}$$ K to be used to solve the satisfiability problem for all logics in the modal cube. We also present alternative, axiomatic, reductions based on ideas originally proposed by Kracht, providing new theoretical results and improved bounds on the size of the reductions. We compare both sets of reductions combined with state-of-the-art provers for $$\textsf{K}_{}$$ K on a large set of parametric benchmarks for all logics in the modal cube. The results show that the provers perform better with reductions based on the clausal normal form than the axiomatic reductions.

Abstract We present novel reductions of extensions of the basic modal logic $${\textsf {K} }$$ K with axioms $$\textsf {B} $$ B , $$\textsf {D} $$ D , $$\textsf {T} $$ T , $$\textsf {4} $$ 4 and $$\textsf {5} $$ 5 to Separated Normal Form with Sets of Modal Levels $$\textsf {SNF} _{sml}$$ SNF sml . The reductions typically result in smaller formulae than the reductions by Kracht. The reductions to $$\textsf {SNF} _{sml}$$ SNF sml combined with a reduction to $$\textsf {SNF} _{ml}$$ SNF ml allow us to use the local reasoning of the prover $${\text {K}_{\text {S}}}{\text {P}}$$ K S P to determine the satisfiability of modal formulae in the considered logics. We show experimentally that the combination of our reductions with the prover $${\text {K}_{\text {S}}}{\text {P}}$$ K S P performs well when compared with a specialised resolution calculus for these logics, the built-in reductions of the first-order prover SPASS, and the higher-order logic prover LEO-III.

Abstract We present novel reductions of the propositional modal logics "Image missing" , "Image missing" , "Image missing" , "Image missing" and "Image missing" to Separated Normal Form with Sets of Modal Levels. The reductions result in smaller formulae than the well-known reductions by Kracht and allow us to use the local reasoning of the prover "Image missing" to determine the satisfiability of modal formulae in these logics. We show experimentally that the combination of our reductions with the prover "Image missing" performs well when compared with a specialised resolution calculus for these logics and with the b̆uilt-in reductions of the first-order prover SPASS.

Abstract Algorithms for the synchronisation of clocks across networks are both common and important within distributed systems. We here address not only the formal modelling of these algorithms, but also the formal verification of their behaviour. Of particular importance is the strong link between the very different levels of abstraction at which the algorithms may be verified. Our contribution is primarily the formalisation of this connection between individual models and population-based models, and the subsequent verification that is then possible. While the technique is applicable across a range of synchronisation algorithms, we particularly focus on the synchronisation of (biologically-inspired) pulse-coupled oscillators, a widely used approach in practical distributed systems. For this application domain, different levels of abstraction are crucial: models based on the behaviour of an individual process are able to capture the details of distinguished nodes in possibly heterogenous networks, where each node may exhibit different behaviour. On the other hand, collective models assume homogeneous sets of processes, and allow the behaviour of the network to be analysed at the global level. System-wide parameters may be easily adjusted, for example environmental factors inhibiting the reliability of the shared communication medium. This work provides a formal bridge across the “abstraction gap” separating the individual models and the population-based models for this important class of synchronisation algorithms.

Abstract We study translations from metric temporal logic (MTL) over the natural numbers to linear temporal logic (LTL). In particular, we present two approaches for translating from MTL to LTL which preserve the complexity of the satisfiability problem for MTL. In each of these approaches we consider the case where the mapping between states and time points is given by (i) a strict monotonic function and by (ii) a non-strict monotonic function (which allows multiple states to be mapped to the same time point). We use this logic to model examples from robotics, traffic management, and scheduling, discussing the effects of different modelling choices. Our translations allow us to utilise LTL solvers to solve satisfiability and we empirically compare the translations, showing in which cases one performs better than the other. We also define a branching-time version of the logic and provide translations into computation tree logic.

In this paper we describe the implementation of , a resolution-based prover for the basic multimodal logic $${\textsf {K}}_{n}^{}$$ Kn. The prover implements a resolution-based calculus for both local and global reasoning. The user can choose different normal forms, refinements of the basic resolution calculus, and strategies. We describe these options in detail and discuss their implications. We provide experiments comparing some of these options and comparing the prover with other provers for this logic.

Resolution-based provers for multimodal normal logics require pruning of the search space for a proof to ameliorate the inherent intractability of the satisfiability problem for such logics. We present a clausal modal-layered hyper-resolution calculus for the basic multimodal logic, which divides the clause set according to the modal level at which clauses occur to reduce the number of possible inferences. We show that the calculus is complete for the logics being considered. We also show that the calculus can be combined with other strategies. In particular, we discuss the completeness of combining modal layering with negative and ordered resolution and provide experimental results comparing the different refinements.

In this paper, we briefly describe an implementation of a hyper-resolution-based calculus for the propositional basic multimodal logic, Kn. The prover, KSP, is designed to support experimentation with different combinations of refinements for its basic calculus. The prover allows for both local and global reasoning. We present an experimental evaluation that compares KSP with a range of existing reasoners for Kn.

We present a resolution-based calculus for Coalition Logic CL, a non-normal modal logic used for reasoning about cooperative agency. We introduce a normal form and a set of inference rules to solve the satisfiability problem in CL. We also show that the calculus presented here is sound, complete, and terminating.

The branching-time temporal logic CTL is useful for specifying systems that change over time and involve quantification over possible futures. Here we present a resolution calculus for CTL that involves the translation of formulae to a normal form and the application of a number of resolution rules. We use indices in the normal form to represent particular paths and the application of the resolution rules is restricted dependent on an ordering and selection function to reduce the search space. We show that the translation preserves satisfiability, the calculus is sound, complete, and terminating, and consider the complexity of the calculus.

With tableaux-based reasoning approaches or model checking techniques for propositional linear-time temporal logics, PLTL, it is easily possible to construct counter examples for formulae that are not valid. In contrast, only the information that a formula is satisfiable is usually available in resolution-based inference systems. In this paper we present a resolution-based approach for constructing models for satisfiable PLTL formulae. Our approach is based on using the standard model construction for sets of propositional clauses saturated under ordered resolution in the different time points of a temporal model. The temporal model construction procedure is also designed in such a way that it can be easily implemented in existing theorem rovers for PLTL.

In this paper we present a translation principle, called the axiomatic translation , for reducing propositional modal logics with background theories, including triangular properties such as transitivity, Euclideanness and functionality, to decidable fragments of first-order logic. The goal of the axiomatic translation principle is to find simplified theories, which capture the inference problems in the original theory, but in a way that can be readily automated and is easier to deal with by existing (first-order) theorem provers than the standard translation. The principle of the axiomatic translation is conceptually very simple and can be almost completely automated. Soundness is automatic under reasonable assumptions, general decidability results can be stated and termination of ordered resolution is easily achieved. The non-trivial part of the approach is proving completeness. We prove results of completeness, decidability, model generation, the small model property and the interpolation property for a number of common and less common modal logics. We also present results of experiments with a number of first-order logic theorem provers which are very encouraging.

First-order temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order temporal logics has identified important enumerable and even decidable fragments. In this paper, we develop a clausal resolution method for the monodic fragment of first-order temporal logic over expanding domains. We first define a normal form for monodic formulae and then introduce novel resolution calculi that can be applied to formulae in this normal form. We state correctness and completeness results for the method. We illustrate the method on a comprehensive example. The method is based on classical first-order resolution and can, thus, be efficiently implemented.

This paper proposes two methods for realising automated reasoning about agent-based systems. The framework for modelling intelligent agent behaviour that we focus on is a core of KARO logic, an expressive combination of various modal logics including propositional dynamic logic, a modal logic of knowledge, a modal logic of wishes, and additional non-standard operators. The first method we present is based on a translation of core KARO logic to first-order logic combined with first-order resolution. The second method uses an embedding of core KARO logic into a combination of branching-time temporal logic CTL and multi-modal S5 plus a clausal resolution calculus for these combined logics. We discuss the advantages and shortcomings of each approach and suggest ways to extend each variant to cover more of the KARO framework.