HABILITATIONSSCHRIFT Using Resolution as Decision Procedure

1 Introduction Logic made a major step in 1879, when Gottlob Frege published his 'Begriffss-chrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens', see [26]. In this paper, which marks the start of modern logic, Frege introduced a formal language, (which he called Begriffsschrift), in which mathematical statements can be formally stated, and also formally proven. (without any resort to intuition) The reasons for the development of mathematical logic around 1900 were the problems that arose when mathematics moved attention towards truly abstract notions, like for example general functions, general sequences and sets. For those abstract mathematical structures, intuitive reasoning turned out problematic. (For example, Frege started his research on logic when he tried to analyze the notion of sequence) In the beginning of the twentieth century, various systems had been developed as foundational instrument, see [26]. The largest work in this direction was the Principia Mathematica [21], in which properties of cardinals and ordinals were formalized. In later volumes, the integers, the rationals, and the reals were introduced, and some of their properties were formally proven. Once one has a formal calculus in which mathematical proofs can be verified, the next natural question is: Is there a mechanical method for finding those proofs? Unfortunately, this question was answered with 'no' around 1930. There do exist algorithms that are guaranteed to find a proof if there exists one, but there exists no algorithm that is guaranteed to report an answer in the case that no proof exists. But still one can try to find an algorithm that searches for a proof and that is guaranteed to find a proof if there exists one. This became particularly interesting when computers became generally available, in the second half of the twentieth century. For many applications, establishing that no proof exists is equally important as finding a proof, and an algorithm that can only stop on positive answers is not enough. Examples of such applications are question answering from a database, or anaphora resolution in natural language processing: A user asking for all flights from A to B also wants to receive an answer, when there is no such flight. In anaphora resolution, deduction can be used for deciding possible references of anaphora. Consider the discourses 'After the Titanic hit the iceberg, it sank', or 'After the Titanic hit the iceberg, it slowly melted'. In order to determine what 'it' refers to, one …