Tableau-based Decision Procedure Research Articles

Deciding the unguarded modal -calculus

The modal -calculus extends basic modal logic with second-order quantification in terms of arbitrarily nested fixpoint operators. Its satisfiability problem is EXPTIME-complete. Decision procedures for the modal -calculus are not easy to obtain though since the arbitrary nesting of fixpoint constructs requires some combinatorial arguments for showing the well-foundedness of least fixpoint unfoldings. The tableau-based decision procedures so far also make assumptions on the unfoldings of fixpoint formulas, e.g., explicitly require formulas to be in guarded normal form. In this paper we present a tableau calculus for deciding satisfiability of arbitrary, i.e., not necessarily guarded -calculus formulas. It is based on a new unfolding rule for greatest fixpoint formulas which allows unguarded formulas to be handled without an explicit transformation into guarded form, thus avoiding a (seemingly) exponential blow-up in formula size. We prove soundness and completeness of the calculus, and compare it empirically to using guarded transformation instead. The new unfolding rule can also be used to handle nested star operators in PDL formulas correctly.

Tableau-based decision procedure for the multiagent epistemic logic with all coalitional operators for common and distributed knowledge

We develop a conceptually clear, intuitive, and feasible de- cision procedure for testing satisability in the full multiagent epistemic logic CMAEL(CD) with operators for common and distributed knowl- edge for all coalitions of agents mentioned in the language. To that end, we introduce Hintikka structures for CMAEL(CD) and prove that sat- isability in such structures is equivalent to satisability in standard models. Using that result, we design an incremental tableau-building procedure that eventually constructs a satisfying Hintikka structure for every satisable input set of formulae of CMAEL(CD) and closes for every unsatisable input set of formulae.

Modal logics for reasoning about infinite unions and intersections of binary relations

We consider multi-modal logic (with countably infinite number of basic modalities) extended with additional modalities and corresponding to the union and intersection of all basic modalities. We present complete and sound axiomatic systems and polynomial-space terminating tableau-based decision procedures for the basic logic in this language, and its deterministic counterpart . We also show that admits filtration, which can be used independently of our tableaux to establish its decidability.

A Tableau Based Decision Procedure for an Expressive Fragment of Hybrid Logic with Binders, Converse and Global Modalities

In this paper we provide the first (as far as we know) direct calculus deciding satisfiability of formulae in negation normal form in the fragment of FHL (full hybrid logic with the binder, including the global and converse modalities), where no occurrence of a universal operator is in the scope of a binder. By means of a satisfiability preserving translation of formulae, the calculus can be turned into a satisfiability decision procedure for the fragment $\textsf{FHL}\setminus\Box \mathord\downarrow\Box$ , i.e. formulae in negation normal form where no occurrence of the binder is both in the scope of and contains in its scope a universal operator. The calculus is based on tableaux and termination is achieved by means of a form of anywhere blocking with indirect blocking. Direct blocking is a relation between nodes in a tableau branch, holding whenever the respective labels (formulae) are equal up to (a proper form of) nominal renaming. Indirect blocking is based on a partial order on the nodes of a tableau branch, which arranges them into a tree-like structure.

ExpTime Tableaux for Using Sound Global Caching

We present a simple ExpTime (complexity-optimal) tableau decision procedure based on and-or graphs with sound global caching for checking satisfiability of a concept w.r.t. a TBox in $\mathcal{ALC}$ . Our algorithm is easy to implement and provides a foundation for ExpTime (complexity-optimal) tableau-based decision procedures for many modal and description logics, to which various optimisation techniques can be applied.

Clausal Tableaux for Hybrid PDL

We present the first tableau-based decision procedure for PDL with nominals. The procedure is based on a prefix-free clausal tableau system designed as a basis for gracefully degrading reasoners. The clausal system factorizes reasoning into regular, propositional, and modal reasoning. This yields a modular decision procedure and pays off in transparent correctness proofs.

A tableau-based decision procedure for CTL*

AbstractWe present a sound, complete and implementable tableau method for deciding satisfiability of formulas in the propositional version of computation tree logic CTL*. This is the first such tableau. CTL* is an exceptionally important temporal logic with applications from hardware design to agent reasoning, but there is no easily automated reasoning approach to CTL*. The tableau here is a traditional tree-shaped or top-down style tableau, and affords the possibility of reasonably quick decisions on the satisfiability of medium-sized formulas and construction of small models for them. A straightforward subroutine is given for determining when looping allows successful branch termination, but much needed further development is left as future work. In particular, a more general repetition prevention mechanism is needed to speed up the task of tableau construction.

Dual tableau-based decision procedures for relational logics with restricted composition operator

We consider fragments of the relational logic RL(1) obtained by posing various constraints on the relational terms involving the operator of composition of relations. These fragments allow to express several non classical logics including modal and description logics. We show how relational dual tableaux can be employed to provide decision procedures for each of them.

Tableau-based decision procedures for logics of strategic ability in multiagent systems

We develop an incremental tableau-based decision procedure for the alternating-time temporal logic ATL and some of its variants. While running within the theoretically established complexity upper bound, we believe that our tableaux are practically more efficient in the average case than other decision procedures for ATL known so far. Besides, the ease of its adaptation to variants of ATL demonstrates the flexibility of the proposed procedure.

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.

Hybrid Tableaux for the Difference Modality

We present the first tableau-based decision procedure for basic hybrid logic with the difference modality. The decision procedure is gracefully degrading in that the less expressive constructs don't pay for the computationally expensive difference modality. The procedure can be specialized to reflexive and transitive frames. Key features of our approach are nominal elimination, pattern-based blocking, and expansion control.

Complete and Terminating Tableau for the Logic of Proper Subinterval Structures Over Dense Orderings

We introduce special pseudo-models for the interval logic of proper subintervals over dense linear orderings. We prove finite model property with respect to such pseudo-models, and using that result we develop a decision procedure based on a sound, complete, and terminating tableau for that logic. The case of proper subintervals is essentially more complicated than the case of strict subintervals, for which we developed a similar tableau-based decision procedure in a recent work.

An On-the-fly Tableau-based Decision Procedure for PDL-satisfiability

We present a tableau-based algorithm for deciding satisfiability for propositional dynamic logic ( PDL ) which builds a finite rooted tree with ancestor loops and passes extra information from children to parents to separate good loops from bad loops during backtracking. It is easy to implement, with potential for parallelisation, because it constructs a pseudo-model “on the fly” by exploring each tableau branch independently. But its worst-case behaviour is 2EXPTIME rather than EXPTIME. A prototype implementation in the TWB ( http://twb.rsise.anu.edu.au ) is available.

Tableaux for Logics of Subinterval Structures over Dense Orderings

In this article, we develop tableau-based decision procedures for the logics of subinterval structures over dense linear orderings. In particular, we consider the two difficult cases: the relation of strict subintervals (with both endpoints strictly inside the current interval) and the relation of proper subintervals (that can share one endpoint with the current interval). For each of these logics, we establish a small pseudo-model property and construct a sound, complete and terminating tableau that searches systematically for existence of such a pseudo-model satisfying the input formulas. Both constructions are non-trivial, but the latter is substantially more complicated because of the presence of beginning and ending subintervals which require special treatment. We prove PSPACE completeness for both procedures and implement them in the generic tableau-based theorem prover Lotrec.

Termination for Hybrid Tableaus

This article extends and improves work on tableau-based decision methods for hybrid logic by Bolander and Braüner. Their paper gives tableau-based decision procedures for basic hybrid logic (with unary modalities) and the basic logic extended with the global modality. All their proof procedures make use of loop-checks to ensure termination. Here we take a closer look at termination for hybrid tableaus. We cover both types of system used in hybrid logic: prefixed tableaus and internalized tableaus. We first treat prefixed tableaus. We prove a termination result for the basic language (with n-ary operators) that does not involve loop-checks. We then successively add the global modality and n-ary inverse modalities, show why various different types of loop-check are required in these cases, and then re-prove termination. Following this we consider internalized tableaus. At first sight, such systems seem to be more complex. However, we define a internalized system which terminates without loop-checks. It is simpler than previously known internalized systems (all of which require loop-checks to terminate) and simpler than our prefix systems (no non-local side conditions on rules are required).

An Optimal Decision Procedure for Right Propositional Neighborhood Logic

Propositional interval temporal logics are quite expressive temporal logics that allow one to naturally express statements that refer to time intervals. Unfortunately, most such logics turn out to be (highly) undecidable. In order to get decidability, severe syntactic or semantic restrictions have been imposed to interval-based temporal logics to reduce them to point-based ones. The problem of identifying expressive enough, yet decidable, new interval logics or fragments of existing ones that are genuinely interval-based is still largely unexplored. In this paper, we focus our attention on interval logics of temporal neighborhood. We address the decision problem for the future fragment of Neighborhood Logic (Right Propositional Neighborhood Logic, RPNL for short), and we positively solve it by showing that the satisfiability problem for RPNL over natural numbers is NEXPTIME-complete. Then, we develop a sound and complete tableau-based decision procedure, and we prove its optimality.

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.

Cut and Pay

In this paper we study families of resource aware logics that explore resource restriction on rules; in particular, we study the use of controlled cut-rule and introduce three families of parameterised logics that arise from different ways of controlling the use of cut. We start with a formulation of classical logic in which cut is non-eliminable and then impose restrictions on the use of cut. Three Cut-and-Pay families of logics are presented, and it is shown that each family provides an approximation process for full propositional classical logic when the control over the use of cut is progressively weakened. A sound and complete semantics is given for each component of each of the three families of approximated logics. One of these families is shown to possess the uniform substitution property, a new result for approximated reasoning. A tableau based decision procedure is presented for each element of the approximation families and the complexity of each decision procedure is studied. We show that there are families in which every component logic can be decided polynomially.

A Tableau-Based Decision Procedure for a Fragment of Set Theory with Iterated Membership

MLSS is a decidable fragment of set theory involving the predicates membership and set equality and the operators union, intersection, set difference, and singleton. In this paper we extend MLSS wi...

Decidability of [formula omitted] with complex role inclusion axioms

Motivated by medical terminology applications, we investigate the decidability of an expressive and prominent description logic (DL), SHIQ , extended with role inclusion axioms of the form R ○ S ⊑ ˙ T . It is well known that a naive such extension leads to undecidability, and thus we restrict our attention to axioms of the form R ○ S ⊑ ˙ R or S ○ R ⊑ ˙ R , which is the most important form of axioms in the applications that motivated this extension. Surprisingly, this extension is still undecidable. However, it turns out that by restricting our attention further to acyclic sets of such axioms, we regain decidability. We present a tableau-based decision procedure for this DL and report on its implementation, which promises to behave well in practice and provides important additional functionality in a medical terminology application.