Good-for-Game QPTL: An Alternating Hodges Semantics

An extension of QPTL is considered where functional dependencies among the quantified variables can be restricted in such a way that their current values are independent of the future values of the other variables. This restriction is tightly connected to the notion of behavioral strategies in game-theory and allows the resulting logic to naturally express game-theoretic concepts. The fragment where only restricted quantifications are considered, called behavioral quantifications, can be decided, for both model checking and satisfiability, in 2ExpTime and is expressively equivalent to QPTL, though significantly less succinct

Logic and Games on Automatic Structures

The evaluation of a logical formula can be viewed as a game played by two opponents, one trying to show that the formula is true and the other trying to prove it false. This correspondence is exploited algorithmically to evaluate formulas of first and second-order logic on finite structures. We extend the game-based algorithmic approach to first-order logic on infinite structures that arise in computer science. Such structures are stored and manipulated by a computer, thus elements and relations must be represented in a finite way. We study a prominent class of finitely presentable structures, namely automatic structures. Automatic structures consist of elements represented by words over a finite alphabet. Relations within these structures are represented by synchronous automata that perform step-by-step transitions on tuples of symbols from the alphabet. A prominent example of an automatic structure is Presburger arithmetic, for which the natural way of writing numbers as sequences of digits and the standard column addition method constitute an automatic presentation. An important property of automatic structures is that first-order logic is decidable on these structures. To develop the correspondence between games and logic on automatic structures, we first look for suitable extensions of first-order logic that remain decidable. We study the notion of game quantification and extend the notions of open and closed game quantifiers, introduced in the model theory of infinitary logics, to a regular game quantifier. We show that this game quantifier preserves regularity on automatic presentations and investigate its expressive power. We identify the classes of structures on which first-order logic extended with this quantifier collapses to pure first-order logic. For better understanding of the classes where this is not the case, we introduce the notion of inductive automorphisms and show that they preserve relations defined using the game quantifier. Model-checking games for the above extension of first-order logic can be defined in a more natural way than for pure first-order logic. Towards this, we extend the classical two-player parity games to a multiplayer setting where two coalitions play against each other with a particular kind of hierarchical imperfect information about actions of the players. In contrast to the classical case, in hierarchical games it is essential that players alternate. We show that otherwise the problem which coalition wins a hierarchical game is undecidable. Nevertheless, there is an important aspect in which they are similar to classical games: hierarchical games are robust with respect to changes of the representation of winning condition. One reason for this robustness of hierarchical games is that only a finite memory is needed to reduce games with complex winning conditions to games with simpler ones. While this technique has been well-known for games with finitely many priorities, our approach is the first to extend it to games with infinitely many priorities on infinite arenas. We generalize the classical notion of latest appearance record to a new memory structure, which we show to be sufficient for winning Muller games with a finite or co-finite number of sets in the Muller condition, and additionally for a few other classes of games with infinitely many priorities. Certain classes of winning conditions on infinitely many priorities admit descriptions in terms of generalized Zielonka trees. We investigate such conditions and show that the correspondence between the number of branches of a Zielonka tree and the memory needed for strategies generalizes to the case of infinitely many priorities. A basic way of extending first-order logic is by adding generalized Lindström quantifiers. We address the following question: which generalized unary quantifiers can be added to first-order logic without introducing non-regular relations on automatic structures. We answer this question by giving a complete characterization of such quantifiers in terms of definability using cardinality and modulo counting quantifiers. We show that these quantifiers indeed preserve regularity on all automatic structures, including the non-injective omega-automatic ones. As a corollary we answer a question of Blumensath and prove that all countable omega-automatic structures are in fact finite-word automatic. Further, we study cardinality quantifiers on a large class of generalized-automatic structures. We use the composition method for monadic second-order logic to show that on such structures cardinality quantifiers collapse to first-order logic