- 1. Introduction.
- Logic and Games on Automatic Structures - Playing with Quantifiers and Decompositions;
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We present a new method for proving liveness and termination properties for fair concurrent programs, which does not rely on finding a ranking function orx on computing the transitive closure of the transition relation. The set of states from which termination or some liveness property is guaranteed is computed by a backwards reachability analysis.
A central tools is a check for a certain commutativity property.
The method is not complete. Its power can be extended by repeated applications, and by extending the transition relation as in abstraction. It can also be seen as a complement to other methods for proving termination, in that it transforms a termination problem into a simpler one with a larger set of terminated states. We show the usefulness of our method by applying it to existing programs from the literature.
We have also implemented it in the framework of Regular Model Checking, and used it to automatically verify non-starvation for infinite-state and parameterized algorithms. In an average payoff game two players, Min and Max, construct an infinite path by taking turns to move a token along an edge of a graph in which every vertex has an integer payoff. Player Min wants to minimize the limit of average payoffs along the path and player Max wants to maximize it.
This talk complements existing literature on average payoff games on finite graphs by formulating optimality equations whose solutions yield optimal strategies. It also introduces a natural strategy improvement algorithm for solving optimality equations. The algorithm generalizes classical policy iteration algorithms for Markov decision processes and yields an alternative proof of positional determinacy for average payoff games.
Natural generalizations of optimality equations for infinite state average payoff games will be also discussed. Game quantification is an expressive concept and has been studied in model theory and descriptive set theory, especially in relation to infinitary logics. Automatic structures on the other hand play an important role in computer science, especially in program verification. We extend first-order logic on structures on words by allowing to use an infinite string of alternating quantifiers on letters of a word, the game quantifier. This extended logic is decidable and preserves regularity on automatic structures, but can be undecidable on other structures even with decidable first-order theory.
We show that in the presence of game quantifier any relation that allows to distinguish successors is enough to define all regular relations and therefore the game quantifier is strictly more expressive than first-order logic in such cases.
Conversely, if there is an automorphism of atomic relations that swaps some successors then we prove that it can be extended to any relations definable with game quantifier. After investigating it's expressiveness, we use game quantification to introduce a new type of combinatorial games with multiple players and imperfect information exchanged with respect to a hierarchical constraint.
It is shown that these games on finite arenas exactly capture the logic with game quantifier when players alternate their moves but are undecidable and not necessarily determined in the other case. We study channel systems whose behaviour sending and receiving messages via unbounded FIFO channels must follow given timing constraints specifying the execution speeds of the local components. The goal is to study the borderline between decidable and undecidable classes of channel systems in the timed setting.
Note that in the untimed setting, these systems are no more expressive than finite state machines. The capability of synchronizing on time makes it substantially more difficult to verify channel systems. In this talk we present two different model checking games for Fixpoint Logic with Chop - an extension of the modal mu-calculus with a sequential composition operator. We concentrate on explaining the winning conditions in these games, but also survey their known complexity results and applications.
This is a joint work with Wojciech Rytter, Warsaw University. In the past decade game semantics has emerged as a successful paradigm in semantics of programming languages. Its precision and concreteness have recently inspired a new approach to program analysis based on automata-theoretic representations of game models.
In this talk we give an overview of this programme, focussing on relationships between game semantics and various classes of automata. We present a new algorithm actually two algorithms, but they have much in common for solving parity games. The algorithm is based on the notion of spine, a structural way of capturing the possible winning sets and counter-strategies.
The definition of spine and the algorithms were inspired by the strategy improvement algorithm, but there are important differences. Second, in our algorithm we do not perform arbitrary improvement steps. Instead we try to get rid of winning cycles by hopefully temporarily making the associated measure worse.
Third, we tried to give an algorithm which is symmetric, i. This is joint work with Colin Stirling. Numerous tools have been developed to translate LTL formulas into automata and check them against a system model. Other tools, like SMV, specify the automaton symbolically, rather then explicitly. We can test the efficiency and the quality of these translations by testing for satisfiability of the LTL formulas.go to link
Highlights of Logic, Games and Automata
We will present benchmark data comparing ten such tools, examine their shortcomings, and consider the question: Can we do better? Two variants of pebble tree-walking automata are considered that were introduced in the literature. It is shown that for each number of pebbles, the two models have the same expressive power both in the deterministic case and in the nondeterministic case. Furthermore, nondeterministic resp. Moreover, there is a regular tree language that is not recognized by any tree-walking automaton with pebbles. Be the first to write a review. Add to Wishlist. Ships in 15 business days.
Link Either by signing into your account or linking your membership details before your order is placed. Description Table of Contents Product Details Click on the cover image above to read some pages of this book! This is Service Design Doing. In Stock. Life 3. Novacene The Coming Age of Hyperintelligence. Alan Turing: The Enigma. Automatic structures on the other hand appear very often in computer science, especially in program verification. We extend first-order logic on structures on words by allowing to use an infinite string of alternating quantifiers on letters of a word, the game quantifier.
This extended logic is decidable and preserves regularity on automatic structures, but can be undecidable on other structures even with decidable first-order theory.
Logic, automata, algebra and games
We show that in the presence of game quantifier any relation that allows to distinguish successors is enough to define all regular relations and therefore the game quantifier is strictly more expressive than first-order logic in such cases. Conversely, if there is an automorphism of atomic relations that swaps some successors then we prove that it can be extended to any relations definable with game quantifier.
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After investigating it's expressiveness, we use game quantification to introduce a new type of combinatorial games with multiple players and imperfect information exchanged with respect to a hierarchical constraint. It is shown that these games on finite arenas exactly capture the logic with game quantifier when players alternate their moves but are undecidable and not necessarily determined in the other case.
In this way we define the first model checking games with finite arenas that can be used for model checking first-order logic on automatic structures. Weak monadic second-order logic is a very expressive logic which has been used successfully for verification of software and hardware systems. Moreover, structures of bounded clique width have a decidable weak MSO theory which allows to Moreover, structures of bounded clique width have a decidable weak MSO theory which allows to specify and verify detailed correctness properties.
However, one obstacle to a wide application of WMSO verification techniques in practice is that the prevalent model-checking algorithms are based on tree automata, and hence require to encode the structure of interest in the binary tree which is often a cause of inefficiency. We present a new algorithm for model-checking weak MSO on inductive structures, a certain kind of structures of bounded clique width, together with a proof of its decidability which follows from Shelah's composition method. Our algorithm directly manipulates formulas and checks them on the structure of interest without the need to encode it in the binary tree.
In fact, the model-checking problem is reduced to solving a finite reachability game. In addition to the new algorithm, we also show that our method can be extended to obtain decidability of weak MSO extended with the unbounding quantifier on the binary tree, which was open before. Directed Graphs of Entanglement Two. Entanglement is a complexity measure for directed graphs that was used to show that the variable hierarchy of the propositional modal mu-calculus is strict. While graphs of entanglement zero and one are indeed very simple, some graphs of While graphs of entanglement zero and one are indeed very simple, some graphs of entanglement two already contain interesting nesting of cycles.
This motivates our study of the class of graphs of entanglement two, as these are both simple in a sense and already complex enough for modelling certain structured systems. Undirected graphs of entanglement two were already studied by Belkhir and Santocanale and a structural decomposition for such graphs was given. We study the general case of directed graphs of entanglement two and prove that they can be decomposed as well, in a way similar to the known decompositions for tree-width, DAG-width and Kelly-width. Moreover, we show that all graphs of entanglement two have both DAG-width and Kelly-width three.
Since there exist both graphs with DAG-width three and graphs with Kelly-width three, but with arbitrary high entanglement, this confirms that graphs of entanglement two are a very basic class of graphs with cycles intertwined in an interesting way. Synthesis for Structure Rewriting Systems. The description of a single state of a modelled system is often complex in practice, but few procedures for synthesis address this problem in depth. We study systems in which a state is described by an arbitrary finite structure, and We study systems in which a state is described by an arbitrary finite structure, and changes of the state are represented by structure rewriting rules, a generalisation of term and graph rewriting.
Both the environment and the controller are allowed to change the structure in this way, and the question we ask is how a strategy for the controller that ensures a given property can be synthesised. We focus on one particular class of structure rewriting rules, namely on separated structure rewriting, a limited syntactic class of rules.
To counter this restrictiveness, we allow the property to be ensured by the controller to be specified in a very expressive logic: a combination of monadic second-order logic evaluated on states and the modal mu-calculus for the temporal evolution of the whole system. We show that for the considered class of rules and this logic, it can be decided whether the controller has a strategy ensuring a given property, and in such case a finite-memory strategy can be synthesised.
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Additionally, we prove that the same holds if the property is given by a monadic second-order formula to be evaluated on the limit of the evolution of the system. Information Tracking in Games on Graphs. When seeking to coordinate in a game with imperfect information, it is often relevant for a player to know what other players know.
Keeping track of the information acquired in a play of infinite duration may, however, lead to infinite Keeping track of the information acquired in a play of infinite duration may, however, lead to infinite hierarchies of higher-order knowledge. We present a construction that makes explicit which higher-order knowledge is relevant in a game and allows us to describe a class of games that admit coordinated winning strategies with finite memory.
GDL inherits from Datalog the GDL inherits from Datalog the use of Horn clauses as rules and recursion, but it too requires stratification and does not allow to use quantifiers.