Tableau System Research Articles

Complexity results for modal logic with recursion via translations and tableaux

This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the $\mu$-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce terminating and non-terminating tableau systems for the logics we study, based on Kozen's tableau for the $\mu$-calculus and the one of Fitting and Massacci for modal logic. Finally, we describe these tableaux with $\mu$-calculus formulas, thus reducing the satisfiability of each of these logics to the satisfiability of the $\mu$-calculus, resulting in a general 2EXP upper bound for satisfiability testing.

SAT Meets Tableaux for Linear Temporal Logic Satisfiability

Linear temporal logic (LTL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{LTL}\\,$$\\end{document}) and its variant interpreted on finite traces (LTLf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{LTL}_{\ extsf{f}\\,}$$\\end{document}) are among the most popular specification languages in the fields of formal verification, artificial intelligence, and others. In this paper, we focus on the satisfiability problem for LTL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{LTL}\\,$$\\end{document}and LTLf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{LTL}_{\ extsf{f}\\,}$$\\end{document}formulas, for which many techniques have been devised during the last decades. Among these are tableau systems, of which the most recent is Reynolds’ tree-shaped tableau. We provide a SAT-based algorithm for LTL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{LTL}\\,$$\\end{document}and LTLf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{LTL}_{\ extsf{f}\\,}$$\\end{document}satisfiability checking based on Reynolds’ tableau, proving its correctness and discussing experimental results obtained through its implementation in the BLACK satisfiability checker.

4DL: a four-valued dynamic logic and its proof theory

Abstract Transition systems are often used to describe the behaviour of software systems. If viewed as a graph then, at their most basic level, vertices correspond to the states of a program and each edge represents a transition between states via the (atomic) action labelled. In this setting, systems are thought to be consistent and at each state formulas are evaluated as either true or false. On the other hand, when a structure of this sort—e.g. a map where states represent locations, some local properties are known and labelled transitions represent information available about different routes—is built resorting to multiple sources of information, it is common to find inconsistent or incomplete information regarding what holds at each state, both at the level of propositional variables and transitions. This paper aims at bringing together Belnap’s four values, Dynamic Logic and hybrid machinery such as nominals and the satisfaction operator, so that reasoning is still possible in face of contradicting evidence. Proof-theory for this new logic is explored by means of a terminating, sound and complete tableau system.

Complexity through Translations for Modal Logic with Recursion

This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the mu-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce a terminating tableau system for the logics we study, based on Kozen's tableau for the mu-calculus, and the one of Fitting and Massacci for modal logic.

Tableaux for some deontic logics with the explicit permission operator

In this paper we present a tableau system for deontic logics with the operator of explicit permission. By means of this system the decidability of the considered logics can be proved. We will sketch how these logics are semantically defined by means of relating semantics and how they provide a simple solution to the free choice permission problem. In short, these logics employ relating implication and a certain propositional constant. These two are in turn used to define deontic operators similarly as in Andersonian-Kangerian reduction, which uses different intensional implications and constants.

Tableaux for Logics of Content Relationship and Set-Assignment Semantics

In the paper, we examine tableau systems for R. Epstein’s logics of content relationship: mathbf {D} (Dependence Logic), mathbf {DD} (Dual Dependence Logic), mathbf {Eq} (Logic of Equality of Content), mathbf {S} (Symmetric Relatedness Logic) and mathbf {R} (Nonsymmetric Relatedness Logic) (Epstein in The semantic foundations of Logic, Springer Science + Business Media, Dordrecht, (1990), cf. Epstein in Philos Stud 36:137–173, 1979, Epstein in Rep. Math. Logic 21:19–34, 1987, Klonowski in Logic Log Philos, accepted for publication, Krajewski in J Non Class Logic 8:7–33, 1991). The first tableau systems for those logics were defined by Carnielli (Rep Math Logic 21:35–46, 1987). However, his approach has some limitations, for example, it requires a proof of functional completeness and axiomatization. Notwithstanding the first two constraints, it does not include all Epstein logics, e.g., logic mathbf {Eq}. Unlike Carnielli’s approach, here we use set-assignment semantics to determine those logics. Since syntax and semantics of a given logic usually determine a minimal syntax and structure of a tableau system for the logic along with other properties, we propose a uniform tableau framework for the logics determined by set-assignment semantics. What distinguishes our tableau systems is that they combine the features of tableaux for propositional logics and syllogistic logics when the problem of content of propositions is analysed in tableau proofs. To denote the content of propositions in the proofs, we use generalised labels (explored in the syllogistic context in Jarmużek and Goré (In: Fitting (ed.) Landscapes in Logic, College Publications, London, accepted)).

Analytic Non-Labelled Proof-Systems for Hybrid Logic: Overview and a couple of striking facts

This paper is about non-labelled proof-systems for hybrid logic, that is, proofsystems where arbitrary formulas can occur, not just satisfaction statements. We give an overview of such proof-systems, focusing on analytic systems: Natural deduction systems, Gentzen sequent systems and tableau systems. We point out major results and we discuss a couple of striking facts, in particular that nonlabelled hybrid-logical natural deduction systems are analytic, but this is not proved in the usual way via step-by-step normalization of derivations.

Tableau for the logic ILP

A tableau system for a logic suitable for intuitionistic reasoning about probabilities is presented. Soundness and completeness of the system are proved. A decision procedure based on the tableau system is given.

Axiomatization of Some Basic and Modal Boolean Connexive Logics

Boolean connexive logic is an extension of Boolean logic that is closed under Modus Ponens and contains Aristotle’s and Boethius’ theses. According to these theses (i) a sentence cannot imply its negation and the negation of a sentence cannot imply the sentence; and (ii) if the antecedent implies the consequent, then the antecedent cannot imply the negation of the consequent and if the antecedent implies the negation of the consequent, then the antecedent cannot imply the consequent. Such a logic was first introduced by Jarmużek and Malinowski, by means of so-called relating semantics and tableau systems. Subsequently its modal extension was determined by means of the combination of possible-worlds semantics and relating semantics. In the following article we present axiomatic systems of some basic and modal Boolean connexive logics. Proofs of completeness will be carried out using canonical models defined with respect to maximal consistent sets.

Tableau Systems for Epistemic Positional Logics

The goal of the article is twofold. The first one is to provide logics based on positional semantics which will be suitable for the analysis of epistemic modalities such as ‘agent ... knows/beliefs that ...’. The second one is to define tableau systemsfor such logics. Firstly, we present the minimal positional logic MR. Then, we change the notion of formulas and semantics in order to consider iterations of the operator of realization and “free” classical formulas. After that, we move on to weaker logics in order to avoid the well known problem of logical omniscience. At the same time, we keep the positional counterparts of modal axioms (T), (4) and (5). For all of the considered logics we present sound and complete tableau systems.

Non-dual modal operators as a basis for 4-valued accessibility relations in Hybrid logic

The modal operators usually associated with the notions of possibility and necessity are classically duals. This paper aims to defy that duality in a paraconsistent environment, namely in a Belnapian Hybrid logic where both propositional variables and accessibility relations are four-valued. Hybrid logic, which is an extension of Modal logic, incorporates extra machinery such as nominals – for uniquely naming states – and a satisfaction operator – so that the formula under its scope is evaluated in the state whose name the satisfaction operator indicates.In classical Hybrid logic the semantics of negation, when it appears before compound formulas, is carried towards subformulas, meaning that eventual inconsistencies can be found at the level of nominals or propositional variables but appear unrelated to the accessibility relations. In this paper we allow inconsistencies in propositional variables and, by breaking the duality between modal operators, inconsistencies at the level of accessibility relations arise. We introduce a sound and complete tableau system and a decision procedure to check if a formula is a consequence of a set of formulas. Tableaux will be used to extract syntactic models for databases, which will then be compared using different inconsistency measures. We conclude with a discussion about bisimulation.

Quantified Temporal Alethic Boulesic Doxastic Logic

The paper develops a set of quantified temporal alethic boulesic doxastic systems. Every system in this set consists of five parts: a ‘quantified’ part, a temporal part, a modal (alethic) part, a boulesic part and a doxastic part. There are no systems in the literature that combine all of these branches of logic. Hence, all systems in this paper are new. Every system is defined both semantically and proof-theoretically. The semantic apparatus consists of a kind of T times W models, and the proof-theoretical apparatus of semantic tableaux. The ‘quantified part’ of the systems includes relational predicates and the identity symbol. The quantifiers are, in effect, a kind of possibilist quantifiers that vary over every object in the domain. The tableaux rules are classical. The alethic part contains two types of modal operators for absolute and historical necessity and possibility. According to ‘boulesic logic’ (the logic of the will), ‘willing’ (‘consenting’, ‘rejecting’, ‘indifference’ and ‘non-indifference’) is a kind of modal operator. Doxastic logic is the logic of beliefs; it treats ‘believing’ (and ‘conceiving’) as a kind of modal operator. I will explore some possible relationships between these different parts, and investigate some principles that include more than one type of logical expression. I will show that every tableau system in the paper is sound and complete with respect to its semantics. Finally, I consider an example of a valid argument and an example of an invalid sentence. I show how one can use semantic tableaux to establish validity and invalidity and read off countermodels. These examples illustrate the philosophical usefulness of the systems that are introduced in this paper.

Model Checking a Logic for True Concurrency

We study the model-checking problem for a logic for true concurrency, whose formulae predicate about events in computations and their causal dependencies. The logic, which represents the logical counterpart of history-preserving bisimilarity, is naturally interpreted over event structures or any formalism that can be given a causal semantics, like Petri nets. It includes least and greatest fixpoint operators and thus it can express properties of infinite computations. Since the event structure associated with a system is typically infinite (even if the system is finite state), already the decidability of model-checking is non-trivial. We first develop a local model-checking technique based on a tableau system, for which, over a class of event structures satisfying a suitable regularity condition, referred to as strong regularity, we prove termination, soundness, and completeness. The tableau system allows for a clean and intuitive proof of decidability, but a direct implementation of the procedure can be extremely inefficient. For easing the development of a more efficient model-checking technique, we move to an automata-theoretic framework. Given a formula and a strongly regular event structure, we show how to construct a parity tree automaton whose language is non-empty if and only if the event structure satisfies the formula. The automaton is usually infinite. We discuss how it can be quotiented to an equivalent finite automaton, where emptiness can be checked effectively. To show the applicability of the approach, we discuss how it instantiates to finite safe Petri nets, providing also a corresponding proof-of-concept model-checking tool.

One-pass and tree-shaped tableau systems for TPTL and TPTLb+Past

Linear Temporal Logic (LTL) is one of the most commonly used formalisms for representing and reasoning about temporal properties of computations. Its application domains range from formal verification to artificial intelligence. Many real-time extensions of LTL have been proposed over the years, including Timed Propositional Temporal Logic (TPTL), that makes it possible to constrain the temporal ordering of pairs of events as well as the exact time elapsed between them.The paper focuses on TPTL and BoundedTPTLwith Past (▪), a bounded variant of TPTL enriched with past operators, which has been recently introduced to formalise a meaningful class of timeline-based planning problems. ▪ allows one to refer to the past while keeping the computational complexity under control: in contrast to the full TPTLwith Past (TPTL+P), whose satisfiability problem is non-elementary, the satisfiability problem for ▪ is ▪-complete.The paper deals with the satisfiability problem for TPTL and ▪ by providing an original tableau system for each of them that suitably generalises Reynolds' one-pass and tree-shaped tableau for LTL. First, we show how to handle past operators, by devising a one-pass and tree-shaped tableau system for LTLwith Past (LTL+P). Then, we adapt it to TPTL and ▪, providing full proofs of the soundness and completeness of the resulting systems. In particular, completeness is proved by exploiting a novel model-theoretic argument that, compared to the one originally employed for the LTL system, provides a deeper understanding of the crucial role of the prune rule of the system.

Tableaux for essence and contingency

Abstract We offer tableaux systems for logics of essence and accident and logics of non-contingency, showing their soundness and completeness for Kripke semantics. We also show an interesting parallel between these logics based on the semantic insensitivity of the two non-normal operators by which these logics are expressed.

Non-distributive Relatives of ETL and NFL

In this paper we devise non-distributive relatives of Exactly true logic (ETL) by Pietz and Riveccio and its dual Non-falsity logic (NFL) by Shramko, Zaitsev and Belikov. We consider two pre-orders which are algebraic counterparts of the ETL’s and NFL’s entailment relations on the de Morgan lattice 4. We generalise these pre-orders and determine which distributive properties that hold on 4 are not forced by either of the pre-orders. We then construct relatives of ETL and NFL but lack such distributive properties. For these logics we also devise a truth table semantics which uses non-distributive lattice M3 as their lattice of truth values. We also provide analytic tableaux systems which work with sequents of the form $$\phi \vdash \chi $$. We then prove correctness and completeness results for these proof systems and provide a neat generalisation for non-distributive ETL- and NFL-like logics built over a certain family of non-distributive modular lattices.

A Four-Valued Dynamic Epistemic Logic

Epistemic logic is usually employed to model two aspects of a situation: the factual and the epistemic aspects. Truth, however, is not always attainable, and in many cases we are forced to reason only with whatever information is available to us. In this paper, we will explore a four-valued epistemic logic designed to deal with these situations, where agents have only knowledge about the available information (or evidence), which can be incomplete or conflicting, but not explicitly about facts. This layer of available information or evidence, which is the object of the agents’ knowledge, can be seen as a database. By adopting this sceptical posture in our semantics, we prepare the ground for logics where the notion of knowledge—or more appropriately, belief—is entirely based on evidence. The technical results include a set of reduction axioms for public announcements, correspondence proofs, and a complete tableau system. In summary, our contributions are twofold: on the one hand we present an intuition and possible application for many-valued modal logics, and on the other hand we develop a logic that models the dynamics of evidence in a simple and intuitively clear fashion.

Synthetic Tableaux with Unrestricted Cut for First-Order Theories

The method of synthetic tableaux is a cut-based tableau system with synthesizing rules introducing complex formulas. In this paper, we present the method of synthetic tableaux for Classical First-Order Logic, and we propose a strategy of extending the system to first-order theories axiomatized by universal axioms. The strategy was inspired by the works of Negri and von Plato. We illustrate the strategy with two examples: synthetic tableaux systems for identity and for partial order.

Deduction in Non-Fregean Propositional Logic SCI

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.

Alternating-time temporal logic ATL with finitely bounded semantics

We study a variant ATLFB of the alternating-time temporal logic ATL with a non-standard, ‘finitely bounded’ semantics (FBS). FBS was originally defined as a game-theoretic semantics where players must commit to time limits when attempting to verify eventuality (respectively, to falsify safety) formulae. It turns out that FBS has a natural corresponding compositional semantics that essentially evaluates formulae only on finite initial segments of paths and imposes uniform bounds on all plays for the fulfilment of eventualities. The resulting version ATLFB differs in some essential features from the standard ATL, as it no longer has the finite model property, though the two logics are equivalent on finite models.We develop two tableau systems for ATLFB. The first one deals with infinite sets of formulae and may run in a transfinite sequence of steps, whereas the second one deals only with finite sets of formulae in an extended language allowing explicit symbolic indication of time limits in formulae. We prove soundness and completeness of the infinitary tableau system and prove that it is equivalent to the finitary one. We also show that the finitary tableau system provides an exponential-time decision procedure for the satisfiability problem of ATLFB and thus establishes its EXPTIME-completeness. Furthermore, we present an infinitary axiomatization for ATLFB and prove its soundness and completeness.