Pure and Applied Fixed−Point Logics

TLDR: It is shown that in the context of modal logic, inflationary fixed points are far more expressive than least fixed points, and already relatively simple logics such as the transitive closure logic lead to undecidable query languages on constraint databases.

Abstract: Fixed-point logics are logics with an explicit operator for forming fixed points of definable mappings. They are particularly well suited for modelling recursion in logical languages and consequently they have found applications in various areas of theoretical computer science such as database theory, finite model theory, and computer-aided verification. The topic of this thesis is the study of fixed-point logics with respect to their expressive power. Of particular interest are logics based on inflationary fixed points and their comparison to least fixed-point logics. The first part focuses on fixed-point extensions of first-order logic. In the main result we show that inflationary and least fixed-point logic – the extensions of first-order logic by least and inflationary fixed points – have the same expressive power on all structures, i.e. LFP = IFP. In the second part of this thesis, we study fixed-point extensions of modal logic. Such logics are widely used in the field of computer-aided verification. Again, the least fixed-point extension of modal logic, the modal μ-calculus, is of particular interest and is among the best studied logics in this area. The main contribution of the second part is the introduction and study of the corresponding inflationary fixed-point logic. Contrary to the case of first-order logic mentioned above, where least and inflationary fixed points lead to equivalent logics, it is shown that in the context of modal logic, inflationary fixed points are far more expressive than least fixed points. On the other hand, they are algorithmically far more complex. Besides the two main results, we study a variety of different fixed-point logics and develop methods to compare their expressive power. Finally, in the third part, we study fixed-point logics as query languages for constraint databases. It is shown that already relatively simple logics such as the transitive closure logic lead to undecidable query languages on constraint databases. Therefore we consider suitable restrictions of fixedpoint logics to obtain tractable query languages, i.e. languages with polynomial time evaluation. A detailed overview of the results presented in this thesis can be found in the second part of the introduction.