Tableau System Research Articles

Boulesic-Doxastic Logic

In this paper, I will develop a set of boulesic-doxastic tableau systems and prove that they are sound and complete. Boulesic-doxastic logic consists of two main parts: a boulesic part and a doxastic part. By ‘boulesic logic’ I mean ‘the logic of the will’, and by ‘doxastic logic’ I mean ‘the logic of belief’. The first part deals with ‘boulesic’ concepts, expressions, sentences, arguments and theorems. I will concentrate on two types of boulesic expression: ‘individual x wants it to be the case that’ and ‘individual x accepts that it is the case that’. The second part deals with ‘doxastic’ concepts, expressions, sentences, arguments and theorems. I will concentrate on two types of doxastic expression: ‘individual x believes that’ and ‘it is imaginable to individual x that’. Boulesic-doxastic logic investigates how these concepts are related to each other. Boulesic logic is a new kind of logic. Doxastic logic has been around for a while, but the approach to this branch of logic in this paper is new. Each system is combined with modal logic with two kinds of modal operators for historical and absolute necessity and predicate logic with necessary identity and ‘possibilist’ quantifiers. I use a kind of possible world semantics to describe the systems semantically. I also sketch out how our basic language can be extended with propositional quantifiers. All the systems developed in this paper are new.

Deciding probabilistic simulation between probabilistic pushdown automata and finite-state systems

Checking whether a pushdown automaton is simulated – in the sense of a simulation pre-order – by a finite-state automaton is EXPTIME-complete. This paper shows that the same computational complexity is obtained in a probabilistic setting. That is, the problem of deciding whether a probabilistic pushdown automaton (pPDA) is simulated by a finite Markov decision process (MDP) is EXPTIME-complete. The considered pPDA contain both probabilistic and non-deterministic branching. The EXPTIME-membership is shown by combining a partition-refinement algorithm with a tableaux system that is inspired by Stirling's technique for bisimilarity checking of ordinary pushdown automata. The hardness is obtained by exploiting the EXPTIME-hardness for the non-probabilistic case. Moreover, our decision problem is in PTIME when both the number of states of the pPDA and the number of states in the MDP are fixed.

Doxastic logic: a new approach

ABSTRACTIn this paper, I develop a new set of doxastic logical systems and I show how they can be used to solve several well-known problems in doxastic logic, for example the so-called problem of logical omniscience. According to this puzzle, the notions of knowledge and belief that are used in ordinary epistemic and doxastic symbolic systems are too idealised. Hence, those systems cannot be used to model ordinary human or human-like agents' beliefs. At best, they can describe idealised individuals. The systems in this paper can be used to symbolise not only the doxastic states of perfectly rational individuals, but also the beliefs of finite humans (and human-like agents). Proof-theoretically, I will use a tableau technique. Every system is combined with predicate logic with necessary identity and ‘possibilist’ quantifiers and modal logic with two kinds of modal operators for relative and absolute necessity. The semantics is a possible world semantics. Finally, I prove that every tableau system in the paper is sound and complete with respect to its semantics.

One-Pass and Tree-Shaped Tableau Systems for TPTL and TPTLb+Past

In this paper, we propose a novel one-pass and tree-shaped tableau method for Timed Propositional Temporal Logic and for a bounded variant of its extension with past operators. Timed Propositional Temporal Logic (TPTL) is a real-time temporal logic, with an EXPSPACE-complete satisfiability problem, which has been successfully applied to the verification of real-time systems. In contrast to LTL, adding past operators to TPTL makes the satisfiability problem for the resulting logic (TPTL+P) non-elementary. In this paper, we devise a one-pass and tree-shaped tableau for both TPTL and bounded TPTL+P (TPTLb+P), a syntactic restriction introduced to encode timeline-based planning problems, which recovers the EXPSPACE-complete complexity. The tableau systems for TPTL and TPTLb+P are presented in a unified way, being very similar to each other, providing a common skeleton that is then specialised to each logic. In doing that, we characterise the semantics of TPTLb+P in terms of a purely syntactic fragment of TPTL+P, giving a translation that embeds the former into the latter. Soundness and completeness of the system are proved fully. In particular, we give a greatly simplified model-theoretic completeness proof, which sidesteps the complex combinatorial argument used by known proofs for the one-pass and tree-shaped tableau systems for LTL and LTL+P.

Labelled Tableau Systems for Some Subintuitionistic Logics

Labelled tableau systems are developed for subintuitionistic logics \(\mathbf {wK}_\sigma \), \(\mathbf {wKT}_\sigma \) and \(\mathbf {wK4}_\sigma \). These subintuitionistic logics are embedded into corresponding normal modal logics. Hintikka’s model systems are applied to prove the completeness of labelled tableau systems. The finite model property, decidability and disjunction property are obtained by labelled tableau method.

Deontic Paradoxes and Tableau System for Kalinowski’s Deontic Logic K1

In 1953, Jerzy Kalinowski published his paper on the logic of normative sentences. The paper is recognized as one of the first publications on the formal system of deontic logic. The aim of this paper is to present a tableau system for Kalinowski’s deontic logic and to discuss some of the topics related to the paradoxes of deontic logic.

Temporal Alethic Dyadic Deontic Logic and the Contrary-to-Duty Obligation Paradox

A contrary-to-duty obligation (sometimes called a reparational duty) is a conditional obligation where the condition is forbidden, e.g. “if you have hurt your friend, you should apologise”, “if he is guilty, he should confess”, and “if she will not keep her promise to you, she ought to call you”. It has proven very difficult to find plausible formalisations of such obligations in most deontic systems. In this paper, we will introduce and explore a set of temporal alethic dyadic deontic systems, i.e., systems that include temporal, alethic and dyadic deontic operators. We will then show how it is possible to use our formal apparatus to symbolise contrary-to-duty obligations and to solve the so-called contrary-to-duty (obligation) paradox, a problem well known in deontic logic. We will argue that this response to the puzzle has many attractive features. Semantic tableaux are used to characterise our systems proof theoretically and a kind of possible world semantics, inspired by the so-called T× W semantics, to characterise them semantically. Our models contain several different accessibility relations and a preference relation between possible worlds, which are used in the definitions of the truth conditions for the various operators. Soundness results are obtained for every tableau system and completeness results for a subclass of them.

Tautology Elimination, Cut Elimination, and S5

Tautology elimination rule was successfully applied in automated deduction and recently considered in the framework of sequent calculi where it is provably equivalent to cut rule. In this paper we focus on the advantages of proving admissibility of tautology elimination rule instead of cut for sequent calculi. It seems that one may find simpler proofs of admissibility for tautology elimination than for cut admissibility. Moreover, one may prove its admissibility for some calculi where constructive proofs of cut admissibility fail. As an illustration we present a cut-free sequent calculus for S5 based on tableau system of Fitting and prove admissibility of tautology elimination rule for it.

An Axiomatic System and a Tableau Calculus for STIT Imagination Logic

We formulate a Hilbert-style axiomatic system and a tableau calculus for the STIT-based logic of imagination recently proposed in Wansing (2015). Completeness of the axiom system is shown by the method of canonical models; completeness of the tableau system is also shown by using standard methods.

Two-Dimensional Tableaux

We present two-dimensional tableau systems for the actuality, fixedly, and up-arrow operators. All systems are proved sound and complete with respect to a two-dimensional semantics. In addition, a decision procedure for the actuality logics is discussed.

Soundness and Completeness Proofs by Coinductive Methods

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL's recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL's new coinductive specification language such as nesting through non-free types and mixed recursion---corecursion.

Reflecting on Social Influence in Networks

In many social contexts, social influence seems to be inescapable: the behavior of others influences us to modify ours, and vice-versa. However, social psychology is full of examples of phenomena where individuals experience a discrepancy between their public behavior and their private opinion. This raises two central questions. First, how does an individual reason about the behavior of others and their private opinions in situations of social influence? And second, what are the laws of the resulting information dynamics? In this paper, we address these questions by introducing a formal framework for representing reasoning about an individual's private opinions and public behavior under the dynamics of social influence in social networks. Moreover, we dig deeper into the involved information dynamics by modeling how individuals can learn about each other based on this reasoning. This compels us to introduce a new formal notion of reflective social influence. Finally, we initialize the work on proof theory and automated reasoning for our framework by introducing a sound and complete tableaux system for a fragment of our logic. Furthermore, this constitutes the first tableau system for the Facebook logic of J. Seligman, F. Liu, and P. Girard.

Tableau Systems for Deontic Action Logics Based on Finite Boolean Algebras, and Their Complexity

We introduce a family of tableau calculi for deontic action logics based on finite boolean algebras (or DAL for short), these logics provide deontic operators (e.g., obligation, permission, prohibition) which are applied to a finite number of actions (the vocabulary of the logic); furthermore, in these formalisms, actions can be combined by means of boolean operators, this provides an expressive algebra of actions. We define a tableau calculus for the basic logic and then we extend this calculus to cope with extant variations of this formalism; we prove the soundness and completeness of these proof systems. In addition, we investigate the computational complexity of the satisfiability problem for DAL and its extensions; we show this problem is NP-complete when the number of actions considered is fixed, and it is $${\Sigma^p_2}$$Σ2p-Hard (in Stockmeyer's polynomial hierarchy) when the number of actions is taken as an extra parameter. The tableau systems introduced here can be implemented in PSPACE, this seems reasonable taking into consideration the computational complexity of the logics.

A New Rule for LTL Tableaux

Propositional linear time temporal logic (LTL) is the standard temporal logic for computing applications and many reasoning techniques and tools have been developed for it. Tableaux for deciding satisfiability have existed since the 1980s. However, the tableaux for this logic do not look like traditional tree-shaped tableau systems and their processing is often quite complicated. In this paper, we introduce a novel style of tableau rule which supports a new simple traditional-style tree-shaped tableau for LTL. We prove that it is sound and complete. As well as being simple to understand, to introduce to students and to use, it is also simple to implement and is competitive against state of the art systems. It is particularly suitable for parallel implementations.

Grafting hypersequents onto nested sequents

We introduce a new Gentzen-style framework of grafted hypersequents that combines the formalism of nested sequents with that of hypersequents. To illustrate the potential of the framework, we present novel calculi for the modal logics $\mathsf{K5}$ and $\mathsf{KD5}$, as well as for extensions of the modal logics $\mathsf{K}$ and $\mathsf{KD}$ with the axiom for shift reflexivity. The latter of these extensions is also known as $\mathsf{SDL}^+$ in the context of deontic logic. All our calculi enjoy syntactic cut elimination and can be used in backwards proof search procedures of optimal complexity. The tableaufication of the calculi for $\mathsf{K5}$ and $\mathsf{KD5}$ yields simplified prefixed tableau calculi for these logic reminiscent of the simplified tableau system for $\mathsf{S5}$, which might be of independent interest.

Completeness and termination for a Seligman-style tableau system

Proof systems for hybrid logic typically use @-operators to access information hidden behind modalities; this labelling approach lies at the heart of the best known hybrid resolution, natural deduction, and tableau systems. But there is another approach, which we have come to believe is conceptually clearer. We call this Seligman-style inference, as it was rst introduced and explored by Jerry Seligman in natural deduction [31] and sequent calculus [32] in the 1990s. The purpose of this paper is to introduce a Seligman-style tableau system, to prove its completeness, and to show how it can be made to terminate. The most obvious feature of Seligman-style systems is that they work with arbitrary formulas, not just statements prexed by @-operators. They do so by introducing machinery for switching to other proof contexts. We capture this idea in the setting of tableaus by introducing a rule called GoTo which allows us to \jump to a named world on a tableau branch. We rst develop a Seligman-style tableau system for basic hybrid logic and prove its completeness. We then prove termination of a restricted version of the system without resorting to loop checking, and show that the restrictions do not eect completeness. Both completeness and termination results are proved by explicit translations that transform tableaus in a standard labelled system into Seligman-style tableaus and vice-versa.

Tableau reductions: Towards an optimal decision procedure for the modal necessity

We present a new prefixed tableau system TK for verification of validity in modal logic K. The system TK is deterministic, it uniquely generates exactly one proof tree for each clausal representation of formulas, and, moreover, it uses some syntactic reductions of prefixes. TK is defined in the original methodology of tableau systems, without any external technique such as backtracking, backjumping, etc. Since all the necessary bookkeeping is built into the rules, the system is not only a basis for a validity algorithm, but is itself a decision procedure. We present also a deterministic tableau decision procedure which is an extension of TK and can be used for the global assumptions problem.

Bivalent semantics, generalized compositionality and analytic classic-like tableaux for finite-valued logics

The paper is a contribution both to the theoretical foundations and to the actual construction of efficient automatizable proof procedures for non-classical logics. We focus here on the case of finite-valued logics, and exhibit: (i) a mechanism for producing a classic-like description of them in terms of an effective variety of bivalent semantics; (ii) a mechanism for extracting, from the bivalent semantics so obtained, uniform (classically-labeled) cut-free standard analytic tableaux with possibly branching invertible rules and paired with proof strategies designed to guarantee termination of the associated proof procedure; (iii) a mechanism to also provide, for the same logics, uniform cut-based tableau systems with linear rules. The latter tableau systems are shown to be adequate even when restricted to analytic cuts, and they are also shown to polynomially simulate truth-tables, a feature that is not enjoyed by the former standard type of tableau systems (not even in the 2-valued case). The results are based on useful generalizations of the notions of analyticity and compositionality, and illustrate a theory that applies to many other classes of non-classical logics.

Many-valued hybrid logic

In this paper we define a many-valued semantics for hybrid logic and we give a sound and complete tableau system which is proof theoretically well-behaved, in particular, it gives rise to a decision procedure for the logic. This shows that many-valued hybrid logics is a natural enterprise and opens up the way for future applications.

A logic for diffusion in social networks

This paper introduces a general logical framework for reasoning about diffusion processes within social networks. The new “Logic for Diffusion in Social Networks” is a dynamic extension of standard hybrid logic, allowing to model complex phenomena involving several properties of agents. We provide a complete axiomatization and a terminating and complete tableau system for this logic and show how to apply the framework to diffusion phenomena documented in social networks analysis.