The probabilistic (or quantitative) modal μ-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS’s). Two semantics have been studied for this logic, both assigning to every process state a value in the interval [0, 1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic games. The two semantics have been proved to coincide on all finite PLTS’s. A first contribution of the thesis is to extend this coincidence result to arbitrary PLTS’s. A shortcoming of the probabilistic μ-calculus is the lack of expressiveness required to encode other important temporal logics for PLTS’s such as Probabilistic Computation Tree Logic (PCTL). To address this limitation, we extend the logic with a new pair of operators: independent product and coproduct, and we show that the resulting logic can encode the qualitative fragment of PCTL. Moreover, a further extension of the logic, with the operation of truncated sum and its dual, is expressive enough to encode full PCTL. A major contribution of the thesis is the definition of appropriate game semantics for these extended probabilistic μ-calculi. This relies on the definition of a new class of games, called tree games, which generalize standard 2-player stochastic games. In tree games, a play can be split into concurrent subplays which continue their evolution independently. Surprisingly, this simple device supports the encoding of the whole class of imperfect-information games known as Blackwell games. Moreover, interesting open problems in game theory, such as qualitative determinacy for 2-player stochastic parity games, can be reformulated as determinacy problems for suitable classes of tree games. Our main technical result about tree games is a proof of determinacy for 2-player stochastic metaparity games, which is the class of tree games that we use to give game semantics to the extended probabilistic μ-calculi. In order to cope with measure-theoretic technicalities, the proof is carried out in ZFC set theory extended with Martin’s Axiom at the first uncountable cardinal (MAℵ1). The final result of the thesis shows that the game semantics of the extended logics coincides with the denotational semantics, for arbitrary PLTS’s. However, in contrast to the earlier coincidence result, which is proved in ZFC, the proof of coincidence for the extended calculi is once again carried out in ZFC +MAℵ1.

An interesting problem in proof theory is to find representations of proof that do not distinguish between proofs that are ‘morally’ the same. For many logics, the presentation of proofs in a traditional formalism, such as Gentzen’s sequent calculus, introduces artificial syntactic structure called ‘bureaucracy’; e.g., an arbitrary ordering of freely permutable inferences. A proof system that is free of bureaucracy is called canonical for a logic. In this dissertation two canonical proof systems are presented, for two logics: a notion of proof nets for additive linear logic with units, and ‘classical proof forests’, a graphical formalism for first-order classical logic. Additive linear logic (or sum–product logic) is the fragment of linear logic consisting of linear implication between formulae constructed only from atomic formulae and the additive connectives and units. Up to an equational theory over proofs, the logic describes categories in which finite products and coproducts occur freely. A notion of proof nets for additive linear logic is presented, providing canonical graphical representations of the categorical morphisms and constituting a tractable decision procedure for this equational theory. From existing proof nets for additive linear logic without units by Hughes and Van Glabbeek (modified to include the units naively), canonical proof nets are obtained by a simple graph rewriting algorithm called saturation. Main technical contributions are the substantial correctness proof of the saturation algorithm, and a correctness criterion for saturated nets. Classical proof forests are a canonical, graphical proof formalism for first-order classical logic. Related to Herbrand’s Theorem and backtracking games in the style of Coquand, the forests assign witnessing information to quantifiers in a structurally minimal way, reducing a first-order sentence to a decidable propositional one. A similar formalism ‘expansion tree proofs’ was presented by Miller, but not given a method of composition. The present treatment adds a notion of cut, and investigates the possibility of composing forests via cut-elimination. Cut-reduction steps take the form of a rewrite relation that arises from the structure of the forests in a natural way. Yet reductions are intricate, and initially not well-behaved: from perfectly ordinary cuts, reduction may reach unnaturally configured cuts that may not be reduced. Cutelimination is shown using a modified version of the rewrite relation, inspired by the game-theoretic interpretation of the forests, for which weak normalisation is shown, and strong normalisation is conjectured. In addition, by a more intricate argument, weak normalisation is also shown for the original reduction relation.