## Periodic Reporting for period 1 - ReACT (A Realizability Approach to Complexity Theory)

Reporting period: 2015-11-01 to 2017-10-31

Complexity theory lies at the intersection between mathematics and computer science, and studies the amount of resources needed to run a specific program (complexity of an algorithm) or solve a particular problem (complexity of a problem). The ReACT project aimed at building on recent work in realizability models for linear logic to provide new characterizations of existing complexity classes, in particular L (logarith- mic space) and P (polynomial time). The final goal was that these characterizations would enable researchers to attack long-standing open problems in complexity theory by using mathematical techniques, tools and invariants from the fields of operators algebras and dynamical systems. The complexity-through-realizability techniques developed by the ReACT project were expected to provide a unified framework for studying many computational paradigms and their associated computational complexity theory grounded on well-studied mathematical concepts. This would allow comparison of complexity classes defined from different computational paradigms (e.g. sequential and quantum computation), as well as establish a theory of complexity for computational paradigms currently lacking such a theory (e.g. concurrent processes).

The ReACT project had two objectives. The first objective aimed at establishing this new approach to complexity as an emerging and promising field of study on the basis that it captures, generalizes and extends the techniques developed by previous approaches such as Implicit Computational Compelxity (ICC). The second objective was more exploratory and its goal was to investigate how the new methods and techniques derived from the mathematical foundations of Interaction Graphs models can be used to address open problems in complexity, namely problems related to the question of classifying complexity classes, i.e. deciding if two classes are equal or not.

The ReACT project had two objectives. The first objective aimed at establishing this new approach to complexity as an emerging and promising field of study on the basis that it captures, generalizes and extends the techniques developed by previous approaches such as Implicit Computational Compelxity (ICC). The second objective was more exploratory and its goal was to investigate how the new methods and techniques derived from the mathematical foundations of Interaction Graphs models can be used to address open problems in complexity, namely problems related to the question of classifying complexity classes, i.e. deciding if two classes are equal or not.

The ReACT project was quite successful in establishing Complexity-through-Realisability techniques as a solid and promising new approach to computational complexity.

Concerning the first objective of the project, the expected characterisations of standard complexity classes were obtained, and it was shown how the method improves on standard techniques from Implicit Computational Complexity. Moreover, it was shown how the method applies to non-sequential models of computation, providing in particular sound mathematical models of probabilistic computation and of parallel random access machines.

Concerning the second objective a number of connections were discovered between standard tools and invariants of dynamical systems and the theory of Interaction Graphs, which underlies the Complexity-through-Realisability techniques. Building on this, it was possible to obtain a characterisation of a sufficient condition for two complexity classes to be equal. Some progress was also made concerning the question of showing that two classes are not equal, a major open problem in mathematics and computer science. Indeed we showed, using Complexity-through-Realisability methods, how several standard lower bounds results in complexity can be understood in terms of topological entropy, an invariant for dynamical systems.

Lastly, a number of additional results were obtained within the ReACT project through the collaboration with the supervisor and other researchers at the hosting institution. In particular it was shown that techniques related to Interaction Graphs could lead to methods in static analysis and optimisation of programs, which could be implemented within the LLVM compiler. Some theoretical results within the theory of dynamical systems were also obtained.

Overall, the project has lead to five publications in peer-reviewed venues (published or accepted), most of them in top-ranked venues, two tools/software, and seven papers currently submitted or in preparation.

Concerning the first objective of the project, the expected characterisations of standard complexity classes were obtained, and it was shown how the method improves on standard techniques from Implicit Computational Complexity. Moreover, it was shown how the method applies to non-sequential models of computation, providing in particular sound mathematical models of probabilistic computation and of parallel random access machines.

Concerning the second objective a number of connections were discovered between standard tools and invariants of dynamical systems and the theory of Interaction Graphs, which underlies the Complexity-through-Realisability techniques. Building on this, it was possible to obtain a characterisation of a sufficient condition for two complexity classes to be equal. Some progress was also made concerning the question of showing that two classes are not equal, a major open problem in mathematics and computer science. Indeed we showed, using Complexity-through-Realisability methods, how several standard lower bounds results in complexity can be understood in terms of topological entropy, an invariant for dynamical systems.

Lastly, a number of additional results were obtained within the ReACT project through the collaboration with the supervisor and other researchers at the hosting institution. In particular it was shown that techniques related to Interaction Graphs could lead to methods in static analysis and optimisation of programs, which could be implemented within the LLVM compiler. Some theoretical results within the theory of dynamical systems were also obtained.

Overall, the project has lead to five publications in peer-reviewed venues (published or accepted), most of them in top-ranked venues, two tools/software, and seven papers currently submitted or in preparation.

The ReACT project lead to significant progress on the researcher's Complexity-through-Realisability approach, establishing the latter as a strong and credible new approach to separation problems. As the current state-of-the-art on separation problems is a documented lack of methods for tackling open problems, the results obtained within the project have the potential to impact durably and substantially to the field of computational complexity.

More precisely, it was shown how the use of Interaction Graphs models allows for a concrete and feasible uniform approach to complexity theory, one that encompasses many different computational paradigms (sequential, probabilistic, parallel). This improves on standard approaches which are more a mixed collection of methods than a uniform and homogeneous theory. Moreover the project gathered evidence on how this point of view provides new insights on the classification problem for complexity classes, for instance showing how invariants for topological dynamical systems such as entropy plays a crucial role in several proofs of lower bounds. This sheds a new light on standard methods and provides concrete evidence of the role played by dynamics in computational complexity, opening the way for the use of previously unavailable methods and invariants to be used in complexity theory.

More precisely, it was shown how the use of Interaction Graphs models allows for a concrete and feasible uniform approach to complexity theory, one that encompasses many different computational paradigms (sequential, probabilistic, parallel). This improves on standard approaches which are more a mixed collection of methods than a uniform and homogeneous theory. Moreover the project gathered evidence on how this point of view provides new insights on the classification problem for complexity classes, for instance showing how invariants for topological dynamical systems such as entropy plays a crucial role in several proofs of lower bounds. This sheds a new light on standard methods and provides concrete evidence of the role played by dynamics in computational complexity, opening the way for the use of previously unavailable methods and invariants to be used in complexity theory.