Universal Graphs and Good for Games Automata: New Tools for Infinite Duration Games

Several problems in planning and reactive synthesis can be reduced to the analysis of two-player quantitative graph games. Optimization is one form of analysis. We argue that in many cases it may be better to replace the optimization problem with the satisficing problem, where instead of searching for optimal solutions, the goal is to search for solutions that adhere to a given threshold bound.This work defines and investigates the satisficing problem on a two-player graph game with the discounted-sum cost model. We show that while the satisficing problem can be solved using numerical methods just like the optimization problem, this approach does not render compelling benefits over optimization. When the discount factor is, however, an integer, we present another approach to satisficing, which is purely based on automata methods. We show that this approach is algorithmically more performant – both theoretically and empirically – and demonstrates the broader applicability of satisficing over optimization.

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“…Proof. The first two points use a standard synchronized product argument on the following formal reduction [15]:…”

Section: Satisficing Via Safety and Reachability Gamesmentioning

confidence: 99%

On Satisficing in Quantitative Games

Bansal

Chatterjee

Vardi

2021

Tools and Algorithms for the Construction and Analysis of Systems

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“…time, Calude et al [9] essentially show how to compute this particular fixpoint in quasipolynomial time, that is, in time 2 O((log n) c ) for some constant c. Subsequently, it has been shown [13,14,28] that universal graphs (that is, even graphs into which every even graph of a certain size embeds by a graph morphism) can be used to transform parity games to equivalent safety games obtained by pairing the original game with a universal graph; the size of these safety games is determined by the size of the employed universal graphs and it has been shown [13,14] that there are universal graphs of quasipolynomial size. This yields a uniform algorithm for solving parity games to which all currently known quasipolynomial algorithms for parity games have been shown to instantiate using appropriately defined universal graphs [13,14].…”

Section: In Recent Breakthrough Work On the Solution Of Parity Games mentioning

confidence: 99%

“…This method has also been described as pairing separating automata with safety games [14]. It has been shown [13,14] that there are exponentially sized universal graphs (essentially yielding the basis for e.g. the fixpoint iteration algorithm [8] or the small progress measures algorithm [27]) and quasipolynomially sized universal graphs (corresponding, e.g., to the succinct progress measure algorithm [28], or to the recent quasipolynomial variant of Zielonka's algorithm [38]).…”

Section: In Recent Breakthrough Work On the Solution Of Parity Games mentioning

confidence: 99%

“…Leaves in universal trees are identified by navigation paths, that is, sequences of branching directions, so that the leaves are linearly ordered by the lexicographic order ≤ on navigation paths (which orders leafs from the left to the right). As described in [13], one can obtain a universal graph (T, K) over T in which transitions (q, i, q ) ∈ K for odd i (the crucial case) move to the left, that is, q is a leaf that is to the left of q in the universal tree (so that q < q), ensuring universality. As it turns out, the lexicographic ordering on T is a simulation order.…”

Section: A Progress Measure Algorithmmentioning

confidence: 99%

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Quasipolynomial Computation of Nested Fixpoints

Hausmann

Schröder

2021

Tools and Algorithms for the Construction and Analysis of Systems

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It is well-known that the winning region of a parity game with n nodes and k priorities can be computed as a k-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires $$\mathcal {O}(n^{\frac{k}{2}})$$ O ( n k 2 ) iterations of the function. Calude et al.’s recent quasipolynomial-time parity game solving algorithm essentially shows how to compute the same fixpoint in only quasipolynomially many iterations by reducing parity games to quasipolynomially sized safety games. Universal graphs have been used to modularize this transformation of parity games to equivalent safety games that are obtained by combining the original game with a universal graph. We show that this approach naturally generalizes to the computation of solutions of systems of any fixpoint equations over finite lattices; hence, the solution of fixpoint equation systems can be computed by quasipolynomially many iterations of the equations. We present applications to modal fixpoint logics and games beyond relational semantics. For instance, the model checking problems for the energy $$\mu $$ μ -calculus, finite latticed $$\mu $$ μ -calculi, and the graded and the (two-valued) probabilistic $$\mu $$ μ -calculus – with numbers coded in binary – can be solved via nested fixpoints of functions that differ substantially from the function for parity games but still can be computed in quasipolynomial time; our result hence implies that model checking for these $$\mu $$ μ -calculi is in $$\textsc {QP}$$ QP . Moreover, we improve the exponent in known exponential bounds on satisfiability checking.

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“…time, Calude et al [9] essentially show how to compute this particular fixpoint in quasipolynomial time, that is, in time 2 O((log n) c ) for some constant c. Subsequently, it has been shown [16,17,34] that universal graphs (that is, even graphs into which every even graph of a certain size embeds by a graph morphism) can be used to transform parity games to equivalent safety games obtained by pairing the original game with a universal graph; the size of these safety games is determined by the size of the employed universal graphs and it has been shown [16,17] that there are universal graphs of quasipolynomial size. This yields a uniform algorithm for solving parity games to which all currently known quasipolynomial algorithms for parity games have been shown to instantiate using appropriately defined universal graphs [16,17].…”

Section: In Recent Breakthrough Work On the Solution Of Parity Games In Quasipolynomialmentioning

confidence: 99%

Quasipolynomial Computation of Nested Fixpoints

Hausmann¹,

Schröder²

2021

Preprint

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It is well-known that the winning region of a parity game with n nodes and k priorities can be computed as a k-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires O(n k 2 ) iterations of the function. Calude et al.'s recent quasipolynomial-time parity game solving algorithm essentially shows how to compute the same fixpoint in only quasipolynomially many iterations by reducing parity games to quasipolynomially sized safety games. Universal graphs have been used to modularize this transformation of parity games to equivalent safety games that are obtained by combining the original game with a universal graph. We show that this approach naturally generalizes to the computation of solutions of systems of any fixpoint equations over finite lattices; hence, the solution of fixpoint equation systems can be computed by quasipolynomially many iterations of the equations. We present applications to modal fixpoint logics and games beyond relational semantics. For instance, the model checking problems for the energy µ-calculus, finite latticed µ-calculi, and the graded and the (two-valued) probabilistic µ-calculus -with numbers coded in binary -can be solved via nested fixpoints of functions that differ substantially from the function for parity games but still can be computed in quasipolynomial time; our result hence implies that model checking for these µ-calculi is in QP. Moreover, we improve the exponent in known exponential bounds on satisfiability checking.

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Universal Graphs and Good for Games Automata: New Tools for Infinite Duration Games

Cited by 20 publications

References 14 publications

On Satisficing in Quantitative Games

On Satisficing in Quantitative Games

Quasipolynomial Computation of Nested Fixpoints

Quasipolynomial Computation of Nested Fixpoints

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