Counting logics and games with counters

In verification and synthesis, properties or models of interest are oftenquantitative, and many quantitative aspects involve counting. For example, onemight be interested in the amount of memory required by a system, how manysimultaneous requests a system has to process along a run, or how many infixesbelonging to some regular language a word contains. In this thesis, we study twomodels of infinite games with counters as well as two quantitative countinglogics.The first game model we consider is that of counter parity games. In thesequantitative games, a finite set of counters is updated along the edges. Payoffsof finite plays are obtained via the counters, while payoffs of infinite playsare determined via a parity condition. The games are played by a maximizing anda minimizing player, and satisfying the parity condition is good for themaximizing player. We prove that the value of a counter parity game can becomputed, and give an algorithm to do so. The key step in the algorithm is tosolve the unboundedness problem for the value. This is done with the help of agame with imperfect recall, for which we show that finite-memory strategiessuffice for the player suffering from imperfect recall.In mean counter games, the second class of games studied in this thesis, thepayoff of infinite plays is obtained via a mean-payoff condition on a specialcounter. We prove that the unboundedness problem for the value can be solved ingames with only one counter. Furthermore, we show that the exact value can becomputed in one-counter single-player games.The first logic we study is Qμ[#MSO], a counting logic based on the quantitativeμ-calculus. This logic is designed specifically for transition systems where thestates are labeled with finite relational structures. Counting is introduced toQμ with the help of counting terms of monadic second-order logic, which areevaluated on the relational structures the states are labeled with. We prove,via a reduction to counter parity games, that the model-checking problem forQμ[#MSO] is decidable on a generalization of pushdown systems.We also consider a quantitative counting extension of monadic second-order logiccalled qcMSO. In this logic, we adapt the classical interpretations ofdisjunctions as maxima and conjunctions as minima and use counting atoms |X| tocount the sizes of sets. We investigate the connections between qcMSO and twoother extensions of MSO, namely costMSO and MSO+U. We prove that theqcWMSO-theory of the natural numbers with order is decidable via a reduction tothe model-checking problem for an extension of FO+RR on resource-automaticstructures. We also provide decomposition algorithms for qcMSO on the naturalnumbers with order and the infinite binary tree