The MONA tool provides an implementation of the decision procedures for the logics WS1S and WS2S. It has been used for numerous applications, and it is remarkably efficient in practice, even though it faces a theoretically non-elementary worst-case complexity. The implementation has matured over a period of six years. Compared to the first naive version, the present tool is faster by several orders of magnitude. This speedup is obtained from many different contributions working on all levels of the compilation and execution of formulas. We present a selection of implementation "secrets" that have been discovered and tested over the years, including formula reductions, DAGification, guided tree automata, three-valued logic, eager minimization, BDD-based automata representations, and cache-conscious data structures. We describe these techniques and quantify their respective effects by experimenting with separate versions of the MONA tool that in turn omit each of them.

this paper we investigate this issue and consider model checking and satisfiability for all fragments of PLTL obtainable by restricting (1) the temporal connectives allowed, (2) the number of atomic propositions, and (3) the temporal height. Key Words: logic in computer science, computational complexity, verification, temporal logic, model checking 1.

Reproduction of all or part of this document is permitted on condition that it is unmodified, includes this copyright notice, and is distributed for free. The MONA tool is available under the GNU General Public License.

This chapter provides a modern overview of the field of hybrid logic. Hybrid logics are extensions of standard modal logics, involving symbols that name individual states in models. The first results that are nowadays considered as part of the field date back to the early work of Arthur

The monadic logics M2L-Str and WS1S have been successfully used for verification, although they are nonelementary decidable. Motivated by ideas from bounded model checking, we investigate procedures for bounded model construction for these logics. The problem is, given a formula and a bound k, does there exist a word model for of length k. We give a bounded model construction algorithm for M2L-Str that runs in a time exponential in k. For WS1S, we prove a negative result: bounded model construction is as hard as validity checking, i.e., it is nonelementary. From this, negative results for other monadic logics, such as S1S, follow. We present too preliminary tests using a SAT-based implementation of bounded model construction; for certain problem classes it can find counter-examples substantially faster than automata-based decision procedures.

We investigate the combination of the weak second-order monadic logic of one successor (WS1S) with higher-order logic (HOL). We show how these two logics can be combined, how theorem provers based on them can be safely integrated, and how the result can be used. In particular, we present an embedding of the semantics of WS1S in HOL that provides a basis for coupling the MONA system, a decision procedure for WS1S, with an implementation of HOL in the Isabelle system. Afterwards, we describe methods that reduce problems formalized in HOL to problems in the language of WS1S. We present applications to arithmetic reasoning and proving properties of parameterized sequential systems.

Logic offers the possibility of modeling and reasoning about hardware and software. But which logic? We propose monadic logics of strings and trees as good candidates for many kinds of discrete systems. These logics are natural, decidable, yet substantially more expressive, extensions of Boolean logic. We motivate their applicability through examples.

Accurate evaluation of delays of combinatorial circuits is crucial in circuit verification and design. In this paper we present a logical approach to timing analysis which allows us to compute exact stabilization bounds while proving the correctness of the boolean behavior.

This paper views modal logics as languages that are especially tailored for describing relational structures. After pointing out the connections between modal logics and other elds, it describes a natural extension to traditional modal languages: hybrid logics.

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