# Why are Modal Logics so Robustly Decidable?

@inproceedings{Grdel1999WhyAM, title={Why are Modal Logics so Robustly Decidable?}, author={Erich Gr{\"a}del}, booktitle={Bull. EATCS}, year={1999} }

Modal logics are widely used in a number of areas in computer science, in particular for the speciication and veriication of hardware and software systems, for knowledge representation, in databases, and in artiicial intelligence. The most important reason for the successful applications of these logics is that they provide a good balance between expressive power and computational complexity. In 30] Vardi adressed the question to identify the main reasons for the robust decidabil-ity properties…

No Paper Link Available

## 100 Citations

### Why are combined modal logics so robustly undecidable ?

- Philosophy

One of the main reasons for the success of modal logics in computer science is their unusual robust decidability. Indeed, standard modal logics like polymodal K, S4, and S5, temporal logics like LTL…

### On the Complexity of Elementary Modal Logics

- Computer Science, MathematicsSTACS
- 2008

A very general classification for a wide class of relevant logics, showing that the satisfiability problem for each of these logics is either NP-complete or PSPACE-hard, and exhibit a simple classification criterion.

### Automata-Theoretic Decision Procedures for Information Logics

- Computer ScienceFundam. Informaticae
- 2002

An EXPTIME decision procedure based on the emptiness problem of Buchi automata on infinite trees for the very expressive information logic SIM designed for reasoning about information systems is defined.

### Automated reasoning techniques for hybrid logics

- Computer Science, Philosophy
- 2009

This thesis investigates satisfiability for hybrid logics using first-order resolution (via translations) and variations of a resolution calculus that operates directly on hybrid formulas and arrives at a formulation of modal semantics in terms of a novel type of models that are coinductively defined.

### Products of modal logics. Part 2: Relativised quantifiers in classical logic

- Philosophy, MathematicsLog. J. IGPL
- 2000

This paper identifies a new Square Fragment (SF) of the classical logic, where the basic predicates are binary and all quantiers are relativised, and shows the f.p.m.s. holds for products of modal logics in which some of the modalities are reflexive or serial.

### Binding Forms in First-Order Logic

- Computer ScienceCSL
- 2015

A hierarchy of four fragments focused on the Boolean combinations of these forms is described, showing that the less expressive one is already incomparable with several first-order limitations proposed in the literature, as the guarded and unary negation fragments.

### Complexity results and practical algorithms for logics in knowledge representation

- Computer ScienceArXiv
- 2001

A number of novel complexity results and practical algorithms for expressive DLs that provide different forms of counting quantifiers are established and it is shown that, in many cases, adding local counting in the form of qualifying number restrictions to DLs does not increase the complexity of the inference problems, even if binary coding of numbers in the input is assumed.

### Parameterized complexity of some problems in concurrency and verification[HBNI Th 39]

- Computer Science
- 2011

This thesis proposes to use the framework of parameterized complexity to study the impact of various restrictions on the complexity of problems related to some models and logics of concurrent systems.

### Advances in Modal Logic, Volume 4

- Philosophy, Computer Science
- 2001

By working in a hybrid logic setting, this work is able to develop a model-theoretic understanding of both assertional and terminological information in description logic, and shows how to use the connection between description and hybrid logics to transfer results on complexity and expressive power from one to the other.

### Guarded fixed point logics and the monadic theory of countable trees

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2002

## References

SHOWING 1-10 OF 31 REFERENCES

### Why is Modal Logic So Robustly Decidable?

- Philosophy, Computer ScienceDescriptive Complexity and Finite Models
- 1996

It is argued that the robust decidability of modal logic can be explained by the so-called tree- model property, and it is shown how the tree-model property leads to automata-based decision procedures.

### The complexity of tree automata and logics of programs

- Computer Science[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
- 1988

It is shown that for tree automata with m states and n pairs nonemptiness can be tested in time O((mn)/sup 3n/), even though the problem is in general NP-complete, and it follows that satisfiability for propositional dynamic logic with a repetition construct and for the propositional mu-calculus can be tests in deterministic single exponential time.

### A linear-time model-checking algorithm for the alternation-free modal mu-calculus

- Computer ScienceFormal Methods Syst. Des.
- 1993

A model-checking algorithm for a logic that permits propositions to be defined using greatest and least fixed points of mutually recursive systems of equations is developed, which improves on the best known algorithm for similar fixed-point logics.

### On the Restraining Power of Guards

- Mathematics, Computer ScienceJournal of Symbolic Logic
- 1999

It is proved that the satisfiability problems for the guarded fragment and the loosely guarded fragment of first-order logic are complete for deterministic double exponential time and that some natural, modest extensions of the guarded fragments are undecidable.

### Undecidability results on two-variable logics

- MathematicsArch. Math. Log.
- 1997

It is shown that going beyond L2 by adding any one of the following leads to an undecidable logic: very weak forms of recursion, such as transitive closure or monadic fixed-point operations.

### Decision Procedures and Expressiveness in the Temporal Logic of Branching Time

- Computer ScienceJ. Comput. Syst. Sci.
- 1985

It is established that CTL has the small property by showing that any satisfiable CTL formulae is satisfiable in a small finite model obtained from a small -&-ldquo;pseudo-model-&-rdquo%; resulting from the Fischer Ladner quotient construction.

### Model Checking and the Mu-calculus

- Computer ScienceDescriptive Complexity and Finite Models
- 1996

This work describes model checking algorithms and discusses their application on a particularly important type of temporal logic known as the Mu-calculus, which can provide an eecient and expressive tool for automatic veriication that a nite state system meets a correctness speciication formulated in temporal logic.

### On the Expressive Completeness of the Propositional mu-Calculus with Respect to Monadic Second Order Logic

- PhilosophyCONCUR
- 1996

It is shown that every formula of MSOL which does not distinguish between bisimilar models is equivalent to a formula of the propositional Μ-calculus, which implies that every logic over transition systems invariant under bisimulation and translatable into MSOL can be also translated into the Μ -calculus.

### An automata-theoretic approach to branching-time model checking

- Computer ScienceJACM
- 2000

It is shown that alternating tree automata are the key to a comprehensive automata-theoretic framework for branching temporal logics, and can be used to obtain optimal decision procedures and make it possible to derive optimal model-checking algorithms.