In saturation-based theorem provers, the reasoning process consists in constructing the closure of an axiom set under inferences.

In the last years, the development of automated theorem provers has been advancing in a so to speak Olympic spirit, following the motto "faster, higher, stronger"; and the Waldmeister system has been a part of that endeavour. We will survey the concepts underlying this prover, which implements Knuth-Bendix completion in its unfailing variant. The system architecture is based on a strict separation of active and passive facts, and is realized via speci cally tailored representations for each of the central data structures: indexing for the active facts, set-based compression for the passive facts, successor sets for the conjectures. In order to cope with large search spaces, specialized redundancy criteria are employed, and the empirically gained control knowledge is integrated to ease the use of the system. We conclude with a discussion of strengths and weaknesses, and a view of future prospects.

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In recent years, an important number of theoretical results concerning axiomatizability, proof systems (tableaux, natural deduction, etc.), interpolation, expressive power, complexity, etc. for hybrid logics has been obtained. The next natural step is to develop provers that can handle these languages. HyLoRes is a direct resolution prover for hybrid logics implementing a sound and complete algorithm for satisfiability of sentences in H (@;#). The most interesting distinguishing feature of HyLoRes is that it is not based on tableau algorithms but on (direct) resolution.

Introduction Hybrid languages are modal languages that allow direct reference to the elements of the model. Already the basic hybrid language (H(@)) which extends the basic modal language with the addition of nominals (i; j; k; : : :) and satis ability operators (@ i ; @ j ; @ k ; : : :), obviously increases the expressive power of the language as we can now explicitly check whether the actual point of evaluation is some speci c, named point in the model (w i), and whether a named point satis es a given formula (w @ i '). More interestingly, this extended expressive power permits also the de nition of very elegant decision algorithm, where nominals and @ play, inside the object language, the role of labels, or pre xes, which are usually needed during the construction of a proof in the modal setup (see, e.g. [8, 4]). And all these features we get with no increase in complexity (up to a polynomial): the complexity of the satis ability problem for H(@) is the same as for the ba

The architecture of the Waldmeister prover is based on a strict separation of active and passice facts...

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Mechanically checked, formal reasoning as envisioned by Leibniz has in the past few decades become a reality. Proof assistants are programs that can rigorously and mechanically check the details of logical arguments. Such formal reasoning has applications to nearly all areas of computer science, from formal proof in mathematics to specifying and verifying critical properties of hardware, software systems, and security protocols. Twelf is a proof assistant specialized for reasoning about deductive systems such as logics and programming languages. In its domain, it is one of the most powerful tools available. Significant Twelf developments cover many different application areas. For example, in the theory of programming languages, the formalization and proof of type safety for Standard ML represents the first machine verified proofs of important properties of a full-featured programming language. Proof carrying authentication frameworks based on Twelf allow for computer security based on formal proof. Proof carrying code and typed assembly languages also based on Twelf allow users to check properties of programs such as memory safety before they are executed. The combination of higher order abstract syntax for representing structures with variable binding, the Elf logic programming interpretation, and the M + 2 meta-logic for reasoning about

We investigate resolution calculi for hybrid logics with inference rules restricted via selection functions and orders. We start by providing a sound and refutationally complete resolution calculus for the hybrid logic H(