Periodic Reporting for period 1 - PIGGY (Reasoning using tangles on parity games)
Berichtszeitraum: 2020-04-01 bis 2022-03-31
The main problem addressed by the project is that of solving parity games in polynomial time. Parity games are one of the conceptually simplest problems that are expected to have a polynomial-time algorithm, but for which no such algorithm has been found. Primary objective of the project was to use tangle attractors, that is, attractor-based reasoning extended by knowledge of tangles, that is, regions of a parity game that are known to be won by one of the two players, as the basic building block of a polynomial-time parity game algorithm. The conclusion is that no such algorithm has been found. Different approaches have been tried resulting in several counterexamples (of games that require more than polynomial time) which are insightful. These counterexamples may be used to find a general framework for classes of algorithms that cannot be solved in polynomial time. Several modifications to tangle learning algorithms remain open at this time.
Another objective was towards symbolic representation of tangles. I have made significant progress here using a different approach than originally envisioned in the project, and have a semisymbolic approach that translates a symbolic parity game to an explicit parity game. While efficient symbolic representation of a set of tangles remains an open problem and likely requires as many symbolic representations as there are tangles, I have found that the expected improvement from using symbolic algorithms can already be reached by using the semisymbolic approach.
Three other objectives were to use tangles to improve solving large games, using tangles in neighboring graph problems, and to develop tangles into a format suitable for the general public, such as board games that rely on tangle-similar concepts. These objectives have not been reached for various reasons, including limitations due to COVID19.
The project is important for society in two broader contexts. First, understanding theoretical limits of computing is fundamental in computer science. The most famous problem is the P vs NP problem. If parity games are proven to be unsolvable in polynomial time, then this solves the P vs NP problem. If parity games can be shown to be solvable in polynomial time, then this solves one of the pieces of the puzzle, namely both the parity game problem and the related model checking problem of the mu calculus, which have been open for 40 years.
Secondly, parity games play a central role for verification and synthesis problems of reactive controllers. Thus, improved algorithms for parity games and a better understanding of the complexity of synthesis problems and structures helps towards more practical application of controller synthesis in the future, that is, reliable controllers that are correct by construction.
Another objective was towards symbolic representation of tangles. I have made significant progress here using a different approach than originally envisioned in the project, and have a semisymbolic approach that translates a symbolic parity game to an explicit parity game. While efficient symbolic representation of a set of tangles remains an open problem and likely requires as many symbolic representations as there are tangles, I have found that the expected improvement from using symbolic algorithms can already be reached by using the semisymbolic approach.
Three other objectives were to use tangles to improve solving large games, using tangles in neighboring graph problems, and to develop tangles into a format suitable for the general public, such as board games that rely on tangle-similar concepts. These objectives have not been reached for various reasons, including limitations due to COVID19.
The project is important for society in two broader contexts. First, understanding theoretical limits of computing is fundamental in computer science. The most famous problem is the P vs NP problem. If parity games are proven to be unsolvable in polynomial time, then this solves the P vs NP problem. If parity games can be shown to be solvable in polynomial time, then this solves one of the pieces of the puzzle, namely both the parity game problem and the related model checking problem of the mu calculus, which have been open for 40 years.
Secondly, parity games play a central role for verification and synthesis problems of reactive controllers. Thus, improved algorithms for parity games and a better understanding of the complexity of synthesis problems and structures helps towards more practical application of controller synthesis in the future, that is, reliable controllers that are correct by construction.
For Objective 1, finding a polynomial-time algorithm, I have found mostly “negative” results. While tangle attractors remain a versatile tool to invent creative algorithms, my attempts during the fellowship yielded exponential counterexamples. These counterexamples appear to have common principles behind them, which have potential for a single common framework. This remains future work.
I have also mostly reached Objective 4: To encode tangles using symbolic methods such as binary decision diagrams for large practical games, which enables verification and synthesis of implicitly defined systems without enumerating all system states, which is essential to avoid exponential blowups. I have found that direct application of binary decision diagrams to solve a parity game encoded using binary decision diagrams is significantly slower than solving an explicit parity game derived from the encoded symbolic parity game. Thus, the solution I found is to encode a parity game symbolically using binary decision diagrams, and then derive from this symbolic representation a potentially much smaller explicit parity game, which can then be solved using traditional methods and/or tangle learning. This has been implemented in a program called Knor, which is a tangible result of the fellowship and which provides plenty of room for future improvements through research.
Concrete results are several papers related to WP1 concerning different algorithms for solving parity games that in the end had a negative conclusion, that is, proofs by example that they do not run in polynomial time. Furthermore, I presented a paper together with Remco Abraham and Salomon Sickert at SYNT 2021 describing an almost fully symbolic tool chain for linear temporal logic (LTL) synthesis – this is not published unfortunately as SYNT is a workshop without published proceedings. See also https://workshops.inf.ed.ac.uk/SYNT2021/program.html. This work is tied to a deliverable Otus, an implementation that is integrated with the Owl toolset by Salomon Sickert et al.
Another concrete result is the computer program Knor, which implements part of the LTL synthesis workflow using a variety of symbolic techniques.
I produced a course for Computer Science students on parity games algorithms; the materials are publicly available.
I have also mostly reached Objective 4: To encode tangles using symbolic methods such as binary decision diagrams for large practical games, which enables verification and synthesis of implicitly defined systems without enumerating all system states, which is essential to avoid exponential blowups. I have found that direct application of binary decision diagrams to solve a parity game encoded using binary decision diagrams is significantly slower than solving an explicit parity game derived from the encoded symbolic parity game. Thus, the solution I found is to encode a parity game symbolically using binary decision diagrams, and then derive from this symbolic representation a potentially much smaller explicit parity game, which can then be solved using traditional methods and/or tangle learning. This has been implemented in a program called Knor, which is a tangible result of the fellowship and which provides plenty of room for future improvements through research.
Concrete results are several papers related to WP1 concerning different algorithms for solving parity games that in the end had a negative conclusion, that is, proofs by example that they do not run in polynomial time. Furthermore, I presented a paper together with Remco Abraham and Salomon Sickert at SYNT 2021 describing an almost fully symbolic tool chain for linear temporal logic (LTL) synthesis – this is not published unfortunately as SYNT is a workshop without published proceedings. See also https://workshops.inf.ed.ac.uk/SYNT2021/program.html. This work is tied to a deliverable Otus, an implementation that is integrated with the Owl toolset by Salomon Sickert et al.
Another concrete result is the computer program Knor, which implements part of the LTL synthesis workflow using a variety of symbolic techniques.
I produced a course for Computer Science students on parity games algorithms; the materials are publicly available.
I have found several new algorithms that solve parity games using tangle attractors. These algorithms are flexible and offer new perspectives for finding a future polynomial-time algorithm. However, the variations of the algorithms that have been studied during the fellowship have resulted in example parity game families for which these algorithms have exponential runtime. The game family constructions share common principles, from which future research could derive a common framework for hard parity games for a variety of algorithms. This may result in new fundamental understanding of why parity games are difficult to solve, and may lead to an understanding of why they are maybe not solvable in polynomial time. Potential impacts here concern the fundamental problem in computer science of the P vs NP problem, which has been unsolved for many decades.
Apart from this, clear progress is made on symbolic linear temporal logic (LTL) synthesis in the form of the computer programs Otus and Knor, both are prototypes of symbolic LTL synthesis using completely different approaches, although they share extensive usage of symbolic methods using binary decision diagrams. Knor in particular focused on the "parity automaton/game" part of the typical LTL synthesis toolchain and is significantly faster than other LTL synthesis tools in the international SYNTCOMP competition, winning first place for speed and solving most of the inputs in the PGAME track. Future research (after the fellowship) will focus on the efficiency of the generated controllers, that is, the size of these controllers. Wider implications of this strand of the research are reactive controllers that are correct-by-construction with respect to LTL specifications.
Apart from this, clear progress is made on symbolic linear temporal logic (LTL) synthesis in the form of the computer programs Otus and Knor, both are prototypes of symbolic LTL synthesis using completely different approaches, although they share extensive usage of symbolic methods using binary decision diagrams. Knor in particular focused on the "parity automaton/game" part of the typical LTL synthesis toolchain and is significantly faster than other LTL synthesis tools in the international SYNTCOMP competition, winning first place for speed and solving most of the inputs in the PGAME track. Future research (after the fellowship) will focus on the efficiency of the generated controllers, that is, the size of these controllers. Wider implications of this strand of the research are reactive controllers that are correct-by-construction with respect to LTL specifications.