Tableau System Research Articles

First-order belief and paraconsistency

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. A tableau system with labels, signs, and suffixes is defined, extending the basic language L QB by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since φ ∧ ¬φ does not “explode” with arbitrary syntactic consequences.

An implementation of a dual tableaux system for order-of-magnitude qualitative reasoning

Logic programming has been used as a natural framework to automate deduction in the logic of order-of-magnitude reasoning. Specifically, we introduce a Prolog implementation of the Rasiowa–Sikorski proof system associated with the relational translation Re(OM) of the multi-modal logic of order-of-magnitude qualitative reasoning OM.

Counterfactuals and semantic tableaux

The purpose of this paper is to develop a class of semantic tableau systems for some counterfactual logics. All in all I will discuss 1024 systems. Possible world semantics is used to interpret our formal languages. Soundness results are obtained for every tableau system and completeness results for a large subclass of these.

Relational approach for a logic for order of magnitude qualitative reasoning with negligibility, non-closeness and distance

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for orderof-magnitude reasoning and its dual tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object.

Dual Systems of Tableaux and Sequents for PLTL

On one hand, traditional tableau systems for temporal logic (TL) generate an auxiliary graph that must be checked and (possibly) pruned in a second phase of the refutation procedure. On the other hand, traditional sequent calculi for TL make use of a kind of inference rules (mainly, invariant-based rules or infinitary rules) that complicates their automatization. A remarkable consequence of using auxiliary graphs in the tableaux framework and invariants or infinitary rules in the sequents framework is that TL fails to carry out the classical correspondence between tableaux and sequents. In this paper, we first provide a tableau method TTM that does not require auxiliary graphs to decide whether a set of PLTL-formulas is satisfiable. This tableau method TTM is directly associated to a one-sided sequent calculus called TTC. Since TTM is free from all the structural rules that hinder the mechanization of deduction, e.g. weakening and contraction, then the resulting sequent calculus TTC is also free from this kind of structural rules. In particular, TTC is free of any kind of cut, including invariant-based cut. From the deduction system TTC, we obtain a two-sided sequent calculus GTC that preserves all these good freeness properties and is finitary, sound and complete for PTL. Therefore, we show that the classical correspondence between tableaux and sequent calculi can be extended to TL. Every deduction system is proved to be complete. In addition, we provide illustrative examples of deductions in the different systems.

Analysis of Authentication Protocols in Agent-Based Systems Using Labeled Tableaux

The study of multiagent systems (MASs) focuses on systems in which many intelligent agents interact with each other using communication protocols. For example, an authentication protocol is used to verify and authorize agents acting on behalf of users to protect restricted data and information. After authentication, two agents should be entitled to believe that they are communicating with each other and not with intruders. For specifying and reasoning about the security properties of authentication protocols, many researchers have proposed the use of belief logics. Since authentication protocols are designed to operate in dynamic environments, it is important to model the evolution of authentication systems through time in a systematic way. We advocate the systematic combinations of logics of beliefs and time for modeling and reasoning about evolving agent beliefs in MASs. In particular, we use a temporal belief logic called TML (+) for establishing trust theories for authentication systems and also propose a labeled tableau system for this logic. To illustrate the capabilities of TML (+), we present trust theories for several well-known authentication protocols, namely, the Lowe modified wide-mouthed frog protocol, the amended Needham-Schroeder symmetric key protocol, and Kerberos. We also show how to verify certain security properties of those protocols. With the logic TML (+) and its associated modal tableaux, we are able to reason about and verify authentication systems operating in dynamic environments.

Terminating Tableau Systems for Hybrid Logic with Difference and Converse

This paper contributes to the principled construction of tableau-based decision procedures for hybrid logic with global, difference, and converse modalities. We also consider reflexive and transitive relations. For converse-free formulas we present a terminating control that does not rely on the usual chain-based blocking scheme. Our tableau systems are based on a new model existence theorem.

How True It Is = Who Says It’s True

This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary (finite) Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the proper formalization, in this context, of natural language statements, particularly those involving negative knowledge and common knowledge. A case study is presented which brings these problems to the fore. None of the basic material presented here is new to this paper—all has appeared in several papers over many years, by the present author and by others. Much of the development in the literature is more general than here—we have confined things to the Boolean case for simplicity and clarity. Most proofs are omitted, but several of the examples are new. The main virtue of the present paper is its coherent presentation of a systematic point of view—identify the truth value of a formula with the set of those who say the formula is true.

Terminating Tableau Calculi for Hybrid Logics Extending K

This article builds on work by Bolander and Blackburn [Thomas Bolander and Patrick Blackburn. Termination for hybrid tableaus. Journal of Logic and Computation , 17(3):517–554, 2007] on terminating tableau systems for the minimal hybrid logic K . We provide (for the basic uni-modal hybrid language) terminating tableau systems for a number of non-transitive hybrid logics extending K , such as the logic of irreflexive frames, antisymmetric frames, and so on; these systems don't employ loop-checks. We also provide (for hybrid tense logic enriched with the universal modality) a terminating tableau calculus for the logic of transitive frames; this system makes use of loop-checks.

A new methodology for developing deduction methods

This paper explores the use of resolution as a meta-framework for developing various, different deduction calculi. In this work the focus is on developing deduction calculi for modal dynamic logics. Dynamic modal logics are PDL-like extended modal logics which are closely related to description logics. We show how tableau systems, modal resolution systems and Rasiowa-Sikorski systems can be developed and studied by using standard principles and methods of first-order theorem proving. The approach is based on the translation of reasoning problems in modal logic to first-order clausal form and using a suitable refinement of resolution to construct and mimic derivations of the desired proof method. The inference rules of the calculus can then be read off from the clausal form. We show how this approach can be used to generate new proof calculi and prove soundness, completeness and decidability results. This slightly unusual approach allows us to gain new insights and results for familiar and less familiar logics, for different proof methods, and compare them not only theoretically but also empirically in a uniform framework.

The Logic of Partitions: Introduction to the Dual of the Logic of Subsets

Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms. Modern categorical logic as well as the Kripke models of intuitionistic logic suggest that the interpretation of classical logic might be the logic of subsets of a given universe set. The propositional interpretation is isomorphic to the special case where the truth and falsity of propositions behave like the subsets of a one-element set. If classical logic is thus seen as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic.

In what sense is Kantian principle of contradiction non-classical?

On the ground of Kant’s reformulation of the principle of contradiction, a non-classical logic KC and its extension KC+ are constructed. In KC and KC+, ¬(φ ∧ ¬φ),φ → (¬φ → ψ), and φ ∨ ¬φ are not valid due to specific changes in the meaning of connectives and quantifiers, although there is the explosion of derivable consequences from {φ,¬φ} (the deduction theorem lacking). KC and KC+ are interpreted as fragments of an S5-based first-order modal logic M. The quantification in M is combined with a “subject abstraction” device, which excepts predicate letters from the scope of modal operators. Derivability is defined by an appropriate labeled tableau system rules. Informally, KC is mainly ontologically motivated (in contrast, for example, to Jaśkowski’s discussive logic), relativizing state of affairs with respect to conditions such as time.

Tableaux and Dual Tableaux: Transformation of Proofs

We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that the two systems are dual to each other. The duality is expressed in a formal way which enables us to define a transformation of proofs in one of the systems into the proofs of the other.

Analytic tableaux and interpolation

A tableau system for the predicate logic with countable conjunctions and disjunctions is presented and the completeness of the set of rules proved. These tableaux are used to prove a slightly more general form of the Malitz interpolation theorem.

Tableau-based Decision Procedures for Hybrid Logic

Hybrid logics are a principled generalization of both modal logics and description logics. It is well known that various hybrid logics without binders are decidable, but decision procedures are usually not based on tableau systems, a kind of formal proof procedure that lends itself to computer implementation. In this article, we give four different tableau-based decision procedures for a very expressive hybrid logic including the universal modality; three of the procedures are based on different tableau systems, and one procedure is based on a Gentzen system. The decision procedures make use of so-called loop-checks, which is a standard technique used in connection with tableau systems for other logics, namely, prefixed tableau systems for transitive modal logics as well as for certain description logics. The loop-checks used in our four decision procedures are similar, but the four proof systems on which the procedures are based constitute a spectrum of different systems: prefixed and internalized systems, tableau and Gentzen systems.

Tableau Systems for Some Paraconsistent Modal Logics

In this paper we present tableau-style proof theories for some modal extensions of two paraconsistent propositional logics: RM3, which allows for truth value gluts, and the weaker BN4, which also allows for truth value gaps. These proof theories are shown to be sound and complete with respect to their corresponding semantics. For comparison, we then present some Hilbert-style axiomatizations of these systems proposed by Lou Goble, and bring out some of the comparative advantages and disadvantages of these vis-à-vis our tableau systems.

Graded tableaux for Rational Pavelka Logic: Research Articles

In this article, a sound and complete tableau system for Rational Pavelka Logic (RPL) is introduced. Extended formulas are used as the counterpart of the graded formulas. In this calculus, if we want to show that the graded formula (x, r) is tableau provable (in the finite fuzzy theory F, respectively), we develop a tableau for the extended formula [r, x] (for the set of extended formulas l[r, x], [x1, a1],…, [xn, an] r, respectively). If this tableau closes we claim that (x, r) is tableau provable (in the fuzzy theory F, respectively). We claim also that x is valid at the degree equal to the l.u.b. that allows the closure of the tableaux. Our tableaux are a first step toward efficient procedures of automated deduction in narrow fuzzy logic with truth constants. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1273–1285, 2005.

Finite and Physical Modalities

The logic Kf of the modalities of finite, devised to capture the notion of 'there exists a finite number of accessible worlds such that . . . is true', was introduced and axiomatized by Fattorosi. In this paper we enrich the logical framework of Kf: we give consistency properties and a tableau system (which yields the decidability) explicitly designed for Kf, and we introduce a shorter and more natural axiomatization. Moreover, we show the strong and suggestive relationship between Kf and the much older logic of the physical modalities of Burks.

Graded tableaux for Rational Pavelka Logic

In this article, a sound and complete tableau system for Rational Pavelka Logic (RPL) is introduced. Extended formulas are used as the counterpart of the graded formulas. In this calculus, if we want to show that the graded formula (x, r) is tableau provable (in the finite fuzzy theory F, respectively), we develop a tableau for the extended formula [r, x] (for the set of extended formulas {[r, x], [x1, a1],…, [xn, an] }, respectively). If this tableau closes we claim that (x, r) is tableau provable (in the fuzzy theory F, respectively). We claim also that x is valid at the degree equal to the l.u.b. that allows the closure of the tableaux. Our tableaux are a first step toward efficient procedures of automated deduction in narrow fuzzy logic with truth constants. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1273–1285, 2005.

Analytical tableaux for da Costa's hierarchy of paraconsistent logics Cn, 1≤n

In this paper we present a new hierarchy of analytical tableaux systems TNDC n, 1≤n<ω, for da Costa's hierarchy of propositional paraconsistent logics Cn, 1≤n<ω. In our tableaux formulation, we introduce da Costa's “ball” operator “o”, the generalized operators “k” and “(k)”, for 1≤k, and the negations “~k”, for k≥1, as primitive operators, differently to what has been done in the literature, where these operators are usually defined operators. We prove a version of Cut Rule for the TNDC n, 1≤n<ω, and also prove that these systems are logically equivalent to the corresponding systems Cn, 1≤n<ω. The systems TNDC n constitute completely automated theorem proving systems for the systems of da Costa's hierarchy Cn, 1≤n<ω.