How unprovable is Rabin's decidability theorem?

We study the strength of set-theoretic axioms needed to prove Rabin's theorem on the decidability of the MSO theory of the infinite binary tree. We first show that the complementation theorem for tree automata, which forms the technical core of typical proofs of Rabin's theorem, is equivalent over the moderately strong second-order arithmetic theory $\mathsf{ACA}_0$ to a determinacy principle implied by the positional determinacy of all parity games and implying the determinacy of all Gale-Stewart games given by boolean combinations of ${\bf \Sigma^0_2}$ sets. It follows that complementation for tree automata is provable from $\Pi^1_3$- but not $\Delta^1_3$-comprehension. We then use results due to MedSalem-Tanaka, M\"ollerfeld and Heinatsch-M\"ollerfeld to prove that over $\Pi^1_2$-comprehension, the complementation theorem for tree automata, decidability of the MSO theory of the infinite binary tree, positional determinacy of parity games and determinacy of $\mathrm{Bool}({\bf \Sigma^0_2})$ Gale-Stewart games are all equivalent. Moreover, these statements are equivalent to the $\Pi^1_3$-reflection principle for $\Pi^1_2$-comprehension. It follows in particular that Rabin's decidability theorem is not provable in $\Delta^1_3$-comprehension.Comment: 21 page

Should Sports Professionals Consider Their Adversary's Strategy? A Case Study of Match Play in Golf

This study explores strategic considerations in professional golf's Match Play format, challenging the conventional focus on individual performance. Leveraging PGA Tour data, we investigate the impact of factoring in an adversary's strategy. Our findings suggest that while slight strategy adjustments can be advantageous in specific scenarios, the overall benefit of considering an opponent's strategy remains modest. This confirms the common wisdom in golf, reinforcing the recommendation to adhere to optimal stroke-play strategies due to challenges in obtaining precise opponent statistics. We believe that the methodology employed here could offer valuable insights into whether opponents' performances should also be considered in other two-player or team sports, such as tennis, darts, soccer, volleyball, etc. We hope that this research will pave the way for new avenues of study in these areas

Solving parity games through fictitious play

The thesis aims to find an efficient algorithm for solving parity games. Parity games are graph-based, 0-sum, 2-person games with infinite plays. It is known that these games are determined: all nodes in these games are won by exactly one player. Solving parity games is equivalent to the model checking problem of modal mu-calculus; an efficient solution has important implications to program verification and controller synthesis. Although the decision problem of which player wins a given node is generally believed to be in PTIME, all known algorithms so far have been shown to run in (sub)exponential time. The design of existing algorithms either derives from the determinacy proof of parity games or from a purely graph theoretical perspective, using certain rank functions to iteratively search for an optimal solution. Since parity games are 2-person, 0-sum games, in this thesis I borrow ideas of game theory and investigate the viability of using fictitious play to solve them. Fictitious play is a method where two players choose strategies in strict alternation, and where these choices are “best responses” against the last k (so called bounded recall length) or against all strategies (unbounded recall length) of the other player chosen so far. I use this method to design an algorithm that can solve partity games and study its theoretical and experimental properties. For example, I prove that the basic algorithm solves fully connected games in polynomial time through a number of iterations that is bounded by a small constant. Although the proof is not extended to the general cases in the thesis, the basic algorithm performs demonstrably well against existing solvers in experiments over a large number and variety of games. In particular, the empirically obtained number of iterations that our basic algorithm requires appears to increase polynomially against the game sizes for all the games tested. Furthermore, the algorithm is conjectured to have a run time complexity bounded by O(n4 log2(n)) and I provide a discussion of strategy graphs and their emperically observed properties that motivates this conjecture. One caveat of fictitious play with bounded recall length is that the algorithm may fail to converge to the optimal solution due to the presence of nonoptimal strategy cycles of length greater than 2. In this thesis, I observe that in practice such cases account for less than 0.01% of the games tested. Different cycle resolution methods are explored in the thesis to address this. One particular method combines our basic algorithm and the discrete strategy solver together such that the resulting algorithm is guaranteed to terminate with the optimal solution. Also, this combined solver shares the runtime performance of fictitious play.Open Acces

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

Exponential Lower Bounds for Solving Infinitary Payoff Games and Linear Programs

Parity games form an intriguing family of infinitary payoff games whose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of mean and discounted payoff games, which in turn are very natural subclasses of turn-based stochastic payoff games. From a theoretical point of view, solving these games is one of the few problems that belong to the complexity class NP intersect coNP, and even more interestingly, solving has been shown to belong to UP intersect coUP, and also to PLS. It is a major open problem whether these game families can be solved in deterministic polynomial time. Policy iteration is one of the most important algorithmic schemes for solving infinitary payoff games. It is parameterized by an improvement rule that determines how to proceed in the iteration from one policy to the next. It is a major open problem whether there is an improvement rule that results in a polynomial time algorithm for solving one of the considered game classes. Linear programming is one of the most important computational problems studied by researchers in computer science, mathematics and operations research. Perhaps more articles and books are written about linear programming than on all other computational problems combined. The simplex and the dual-simplex algorithms are among the most widely used algorithms for solving linear programs in practice. Simplex algorithms for solving linear programs are closely related to policy iteration algorithms. Like policy iteration, the simplex algorithm is parameterized by a pivoting rule that describes how to proceed from one basic feasible solution in the linear program to the next. It is a major open problem whether there is a pivoting rule that results in a (strongly) polynomial time algorithm for solving linear programs. We contribute to both the policy iteration and the simplex algorithm by proving exponential lower bounds for several improvement resp. pivoting rules. For every considered improvement rule, we start by building 2-player parity games on which the respective policy iteration algorithm performs an exponential number of iterations. We then transform these 2-player games into 1-player Markov decision processes ii which correspond almost immediately to concrete linear programs on which the respective simplex algorithm requires the same number of iterations. Additionally, we show how to transfer the lower bound results to more expressive game classes like payoff and turn-based stochastic games. Particularly, we prove exponential lower bounds for the deterministic switch all and switch best improvement rules for solving games, for which no non-trivial lower bounds have been known since the introduction of Howard’s policy iteration algorithm in 1960. Moreover, we prove exponential lower bounds for the two most natural and most studied randomized pivoting rules suggested to date, namely the random facet and random edge rules for solving games and linear programs, for which no non-trivial lower bounds have been known for several decades. Furthermore, we prove an exponential lower bound for the switch half randomized improvement rule for solving games, which is considered to be the most important multi-switching randomized rule. Finally, we prove an exponential lower bound for the most natural and famous history-based pivoting rule due to Zadeh for solving games and linear programs, which has been an open problem for thirty years. Last but not least, we prove exponential lower bounds for two other classes of algorithms that solve parity games, namely for the model checking algorithm due to Stevens and Stirling and for the recursive algorithm by Zielonka

On the complexity of parity games

Parity games underlie the model checking problem for the modal μ-calculus, the complexity of which remains unresolved after more than two decades of intensive research. The community is split into those who believe this problem - which is known to be both in NP and coNP - has a polynomial-time solution (without the assumption that P=NP) and those who believe that it does not. (A third, pessimistic, faction believes that the answer to this question will remain unknown in their lifetime.)In this paper we explore the possibility of employing Bounded Arithmetic to resolve this question, motivated by the fact that problems which are both NP and coNP, and where the equivalence between their NP and coNP description can be formulated and proved within a certain fragment of Bounded Arithmetic, necessarily admit a polynomial-time solution. While the problem remains unresolved by this paper, we do proposed another approach, and at the very least provide a modest refinement to the complexity of parity games (and in turn the μ-calculus model checking problem): that they lie in the class of Polynomial Local Search problems. This result is based on a new proof of memoryless determinacy which can be formalised in Bounded Arithmetic.The approach we propose may offer a route to a polynomial-time solution. Alternatively, there may be scope in devising a reduction between the problem and some other problem which is hard with respect to PLS, thus making the discovery of a polynomial-time solution unlikely according to current wisdom

On the complexity of parity games

Parity games underlie the model checking problem for the modal µ-calculus, the complexity of which remains unresolved after more than two decades of intensive research. The community is split into those who believe this problem – which is known to be both in NP and coNP – has a polynomial-time solution (without the assumption that P = NP) and those who believe that it does not. (A third, pessimistic, faction believes that the answer to this question will remain unknown in their lifetime.) In this paper we explore the possibility of employing Bounded Arithmetic to resolve this question, motivated by the fact that problems which are both NP and coNP and which can be formulated within a certain fragment of Bounded Arithmetic necessarily admit a polynomial-time solution. While the problem remains unresolved by this paper, we do proposed another approach, and at the very least provide a modest refinement to the complexity of parity games (and in turn the µ-calculus model checking problem): that they lie in the class PLS of Polynomial Local Search problems. This result is based on a new proof of memoryless determinacy which can be formalised in Bounded Arithmetic. The approach we propose may offer a route to a polynomial-time solution. Alternatively, there may be scope in devising a reduction of the problem to some other problem which is hard with respect to PLS, thus making the discovery of a polynomial-time solution unlikely according to current wisdom. 1

On the complexity of parity games

Parity games underlie the model checking problem for the modal mu-calculus, the complexity of which remains unresolved after decades of intensive research. In this paper we explore the possibility of employing Bounded Arithmetic to resolve this question, motivated by the fact that problems which are both NP and co-NP (such as the problem of solving parity games), and where the equivalence between their NP and co-NP description can be formulated and proved within a certain fragment of Bounded Arithmetic, necessarily admit a polynomial-time solution. While the problem remains unresolved by this paper, we do proposed a novel approach, and provide a demonstration that the problem of solving parity games lies in the class PLS of Polynomial Local Search problems. This result is based on a new proof of memoryless determinacy which can be formalised in Bounded Arithmetic. The approach we propose may offer a route to a polynomial-time solution. Alternatively, there may be scope in devising a reduction between the problem and some other problem which is hard with respect to PLS, thus making the discovery of a polynomial-time solution unlikely according to current wisdom