Forward looking logics and automata - Ley, C (2011)
This thesis is concerned with extending properties of regular word languages to richer structures. We consider intricate properties like the relationship between one-way and two-way temporal logics, minimization of automata, and the ability to effectively characterize logics. We investigate whether these properties can be extended to tree languages or word languages over an infinite alphabet. It is known that linear temporal logic (LTL) is as expressive as first-order logic over finite words [Kam68, GPSS80]. LTL is a unidirectional logic, that can only navigate forwards in a word, hence it is quite surprising that it can capture all of first-order logic. In fact, one of the main ideas of the proof of [GPSS80] is to show that the expressiveness of LTL is not increased if modalities for navigating backwards are added. It is also known that an extension of bidirectional LTL to ordered trees, called Conditional XPath, is first-order complete [Mar04]. We investigate whether the unidirectional fragment of Conditional XPath is also first-order complete. We show that this is not the case. In fact we show that there is a strict hierarchy of expressiveness consisting of languages that are all weaker than first-order logic. Unidirectional Conditional XPath is contained in the lowest level of this hierarchy. In the second part of the thesis we consider data word languages. That is, word languages over an infinite alphabet. We extend the theorem of Myhill and Nerode to a class of automata for data word languages, called deterministic finite memory automata (DMA). We give a characterization of the languages that are accepted by DMA, and also provide an algorithm for minimizing DMA. Finally we extend theorems of Büchi, Schützenberger, McNaughton, and Papert to data word languages. A theorem of Büchi states that a language is regular iff it can be defined in monadic second-order logic. Schützenberger, McNaughton, and Papert have provided an effective characterization of first-order logic, that is, an algorithm for deciding whether a regular language can be defined in first-order logic. We provide a counterpart of Büchi's theorem for data languages. More precisely we define a new logic and we show that it has the same expressiveness as non-deterministic finite memory automata. We then turn to a smaller class of data languages, those that are recognized by algebraic objects called orbit finite data monoids. We define a second new logic and show that it can define precisely the languages accepted by orbit finite data monoids. We provide an effective characterization of a first-order variant of this second logic, as well as of restrictions of first-order logic, such as its two variable fragment and local variants.