This work is dedicated to the public domain. The discrete strategy improvement algorithm for pari... more This work is dedicated to the public domain. The discrete strategy improvement algorithm for parity games and complexity measures for directed graphs
We expand the structure theory of finite Cayley graphs that avoid specific cyclic coset patterns.... more We expand the structure theory of finite Cayley graphs that avoid specific cyclic coset patterns. A focus lies on the exploration of duality in related structures and associated hypergraphs, especially applied to the local analysis of paths and cycles. We present several characterisations of local tree-likeness for these structures and show a close connection to α-acyclicity of hypergraphs.
2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
We investigate multi-agent epistemic modal logic with common knowledge modalities for groups of a... more We investigate multi-agent epistemic modal logic with common knowledge modalities for groups of agents and obtain van Benthem style model-theoretic characterisations, in terms of bisimulation invariance of classical first-order logic over the non-elementary classes of (finite or arbitrary) common knowledge Kripke frames. The fixpoint character of common knowledge modalities and the rôle that reachability and transitive closures play for the derived accessibility relations take our analysis beyond classical model-theoretic terrain and technically pose a novel challenge to the analysis of model-theoretic games. Over and above the more familiar locality-based techniques we exploit a specific structure theory for specially adapted Cayley groups: through the association of agents with sets of generators, all epistemic frames can be represented up to bisimilarity by suitable Cayley groups with specific acyclicity properties; these support a locality analysis at different levels of granularity as induced by distance measures w.r.t. various coalitions of agents.
We investigate multi-agent epistemic modal logic with common knowledge modalities for groups of a... more We investigate multi-agent epistemic modal logic with common knowledge modalities for groups of agents and obtain van Benthem style model-theoretic characterisations, in terms of bisimulation invariance of classical first-order logic over the non-elementary classes of (finite or arbitrary) common knowledge Kripke frames. The technical challenges posed by the reachability and transitive closure features of the derived accessibility relations are dealt with through passage to (finite) bisimilar coverings of epistemic frames by Cayley graphs of permutation groups whose generators are associated with the agents. Epistemic frame structure is here induced by an algebraic coset structure. Cayley structures with specific acyclicity properties support a locality analysis at different levels of granularity as induced by distance measures w.r.t. various coalitions of agents.
Van Benthem's theorem states that basic modal logic \ML is expressively equivalent to the bis... more Van Benthem's theorem states that basic modal logic \ML is expressively equivalent to the bisimulation-invariant fragment of first-order logic $\FO/{\sim}$; we write $\ML\equiv\FO/{\sim}$ for short. Hence, \ML can express every bisimulation-invariant first-order property and, moreover, \ML can be considered an effective syntax for the undecidable fragment $\FO/{\sim}$. Over the years, many variations of this theorem have been established. Rosen proved that $\ML\equiv\FO/{\sim}$ is still true when restricted to finite transition systems. Going beyond first-order logic, Janin and Walukiewicz showed that the bisimulation-invariant fragment of monadic second-order logic \MSO is precisely as expressive as the modal $\mu$-calculus~\Lmu, and several important fragments of~\Lmu have been characterised classically in a similar vein. However, whether $\Lmu\equiv\MSO/{\sim}$ is true over finite transition systems, remains an open problem. This thesis is concerned with modal common knowledg...
For some time the discrete strategy improvement algorithm due to Jurdziński and Vöge had been con... more For some time the discrete strategy improvement algorithm due to Jurdziński and Vöge had been considered as a candidate for solving parity games in polynomial time. However, it has recently been proved by Oliver Friedmann that the strategy improvement algorithm requires super-polynomially many iteration steps, for all popular local improvements rules, including switch-all (also with Fearnley's snare memorisation), switch-best, random-facet, random-edge, switch-half, least-recently-considered, and Zadeh's Pivoting rule. We analyse the examples provided by Friedmann in terms of complexity measures for directed graphs such as treewidth, DAG-width, Kelly-width, entanglement, directed pathwidth, and cliquewidth. It is known that for every class of parity games on which one of these parameters is bounded, the winning regions can be efficiently computed. It turns out that with respect to almost all of these measures, the complexity of Friedmann's counterexamples is bounded, and indeed in most cases by very small numbers. This analysis strengthens in some sense Friedmann's results and shows that the discrete strategy improvement algorithm is even more limited than one might have thought. Not only does it require super-polynomial running time in the general case, where the problem of polynomialtime solvability is open, it even has super-polynomial lower time bounds on natural classes of parity games on which efficient algorithms are known.
We expand the structural theory of \ca graphs that avoid specific cyclic coset patterns. We prese... more We expand the structural theory of \ca graphs that avoid specific cyclic coset patterns. We present several characterisations of tree-likeness for these structures and show a close connection to $\alpha$-acyclic hypergraphs. A focus lies on the behaviour of short paths of overlapping cosets in these \ca graphs, and their relation to short chordless paths in hypergraphs that are locally acyclic.
For some time the discrete strategy improvement algorithm due to Jurdziński and Vöge had been con... more For some time the discrete strategy improvement algorithm due to Jurdziński and Vöge had been considered as a candidate for solving parity games in polynomial time. However, it has recently been proved by Oliver Friedmann that the strategy improvement algorithm requires super-polynomially many iteration steps, for all popular local improvements rules, including switch-all (also with Fearnley's snare memorisation), switch-best, random-facet, random-edge, switch-half, least-recently-considered, and Zadeh's Pivoting rule.
This work is dedicated to the public domain. The discrete strategy improvement algorithm for pari... more This work is dedicated to the public domain. The discrete strategy improvement algorithm for parity games and complexity measures for directed graphs
We expand the structure theory of finite Cayley graphs that avoid specific cyclic coset patterns.... more We expand the structure theory of finite Cayley graphs that avoid specific cyclic coset patterns. A focus lies on the exploration of duality in related structures and associated hypergraphs, especially applied to the local analysis of paths and cycles. We present several characterisations of local tree-likeness for these structures and show a close connection to α-acyclicity of hypergraphs.
2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
We investigate multi-agent epistemic modal logic with common knowledge modalities for groups of a... more We investigate multi-agent epistemic modal logic with common knowledge modalities for groups of agents and obtain van Benthem style model-theoretic characterisations, in terms of bisimulation invariance of classical first-order logic over the non-elementary classes of (finite or arbitrary) common knowledge Kripke frames. The fixpoint character of common knowledge modalities and the rôle that reachability and transitive closures play for the derived accessibility relations take our analysis beyond classical model-theoretic terrain and technically pose a novel challenge to the analysis of model-theoretic games. Over and above the more familiar locality-based techniques we exploit a specific structure theory for specially adapted Cayley groups: through the association of agents with sets of generators, all epistemic frames can be represented up to bisimilarity by suitable Cayley groups with specific acyclicity properties; these support a locality analysis at different levels of granularity as induced by distance measures w.r.t. various coalitions of agents.
We investigate multi-agent epistemic modal logic with common knowledge modalities for groups of a... more We investigate multi-agent epistemic modal logic with common knowledge modalities for groups of agents and obtain van Benthem style model-theoretic characterisations, in terms of bisimulation invariance of classical first-order logic over the non-elementary classes of (finite or arbitrary) common knowledge Kripke frames. The technical challenges posed by the reachability and transitive closure features of the derived accessibility relations are dealt with through passage to (finite) bisimilar coverings of epistemic frames by Cayley graphs of permutation groups whose generators are associated with the agents. Epistemic frame structure is here induced by an algebraic coset structure. Cayley structures with specific acyclicity properties support a locality analysis at different levels of granularity as induced by distance measures w.r.t. various coalitions of agents.
Van Benthem's theorem states that basic modal logic \ML is expressively equivalent to the bis... more Van Benthem's theorem states that basic modal logic \ML is expressively equivalent to the bisimulation-invariant fragment of first-order logic $\FO/{\sim}$; we write $\ML\equiv\FO/{\sim}$ for short. Hence, \ML can express every bisimulation-invariant first-order property and, moreover, \ML can be considered an effective syntax for the undecidable fragment $\FO/{\sim}$. Over the years, many variations of this theorem have been established. Rosen proved that $\ML\equiv\FO/{\sim}$ is still true when restricted to finite transition systems. Going beyond first-order logic, Janin and Walukiewicz showed that the bisimulation-invariant fragment of monadic second-order logic \MSO is precisely as expressive as the modal $\mu$-calculus~\Lmu, and several important fragments of~\Lmu have been characterised classically in a similar vein. However, whether $\Lmu\equiv\MSO/{\sim}$ is true over finite transition systems, remains an open problem. This thesis is concerned with modal common knowledg...
For some time the discrete strategy improvement algorithm due to Jurdziński and Vöge had been con... more For some time the discrete strategy improvement algorithm due to Jurdziński and Vöge had been considered as a candidate for solving parity games in polynomial time. However, it has recently been proved by Oliver Friedmann that the strategy improvement algorithm requires super-polynomially many iteration steps, for all popular local improvements rules, including switch-all (also with Fearnley's snare memorisation), switch-best, random-facet, random-edge, switch-half, least-recently-considered, and Zadeh's Pivoting rule. We analyse the examples provided by Friedmann in terms of complexity measures for directed graphs such as treewidth, DAG-width, Kelly-width, entanglement, directed pathwidth, and cliquewidth. It is known that for every class of parity games on which one of these parameters is bounded, the winning regions can be efficiently computed. It turns out that with respect to almost all of these measures, the complexity of Friedmann's counterexamples is bounded, and indeed in most cases by very small numbers. This analysis strengthens in some sense Friedmann's results and shows that the discrete strategy improvement algorithm is even more limited than one might have thought. Not only does it require super-polynomial running time in the general case, where the problem of polynomialtime solvability is open, it even has super-polynomial lower time bounds on natural classes of parity games on which efficient algorithms are known.
We expand the structural theory of \ca graphs that avoid specific cyclic coset patterns. We prese... more We expand the structural theory of \ca graphs that avoid specific cyclic coset patterns. We present several characterisations of tree-likeness for these structures and show a close connection to $\alpha$-acyclic hypergraphs. A focus lies on the behaviour of short paths of overlapping cosets in these \ca graphs, and their relation to short chordless paths in hypergraphs that are locally acyclic.
For some time the discrete strategy improvement algorithm due to Jurdziński and Vöge had been con... more For some time the discrete strategy improvement algorithm due to Jurdziński and Vöge had been considered as a candidate for solving parity games in polynomial time. However, it has recently been proved by Oliver Friedmann that the strategy improvement algorithm requires super-polynomially many iteration steps, for all popular local improvements rules, including switch-all (also with Fearnley's snare memorisation), switch-best, random-facet, random-edge, switch-half, least-recently-considered, and Zadeh's Pivoting rule.
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Papers by Felix Canavoi