Note on winning positions on pushdown games with ω-regular conditions

2008 23rd Annual IEEE Symposium on Logic in Computer Science

Hague

Meyer

et al. 2008

Self Cite

In this paper we consider parity games defined by higher-order pushdown automata. These automata generalise pushdown automata by the use of higher-order stacks, which are nested "stack of stacks" structures. Representing higher-order stacks as well-bracketed words in the usual way, we show that the winning regions of these games are regular sets of words. Moreover a finite automaton recognising this region can be effectively computed.A novelty of our work are abstract pushdown processes which can be seen as (ordinary) pushdown automata but with an infinite stack alphabet. We use the device to give a uniform presentation of our results.From our main result on winning regions of parity games we derive a solution to the Modal Mu-Calculus Global Model-Checking Problem for higher-order pushdown graphs as well as for ranked trees generated by higher-order safe recursion schemes.

Section: Remarkmentioning

confidence: 99%

“…In [6,32], the winning region is shown to be regular when a configuration (q, w) is represented by the word qw. Note that this result can easily be derived from the results by Vardi [36].…”

Section: Introductionmentioning

confidence: 99%

Winning Regions of Higher-Order Pushdown Games

2008 23rd Annual IEEE Symposium on Logic in Computer Science

Hague

Meyer

et al. 2008

Self Cite

“…The proof is by induction on the order, and the induction step can be divided in three sub-steps (for order-1, the result is a classical one [18]). Assume one starts with an n-CPDA parity game G (using colours {0, .…”

Section: Theoremmentioning

confidence: 99%

Recursion Schemes and Logical Reflection

Broadbent

2010 25th Annual IEEE Symposium on Logic in Computer Science

Ong

et al. 2010

Self Cite

Let R be a class of generators of node-labelled infinite trees, and L be a logical language for describing correctness properties of these trees. Given R ∈ R and ϕ ∈ L, we say that Rϕ is a ϕ-reflection of R just if (i) R and Rϕ generate the same underlying tree, and (ii) suppose a node u of the tree [[R]] generated by R has label f , then the label of the node u of] is the computation tree of a program R, we may regard Rϕ as a transform of R that can internally observe its behaviour against a specification ϕ. We say that R is (constructively) reflective w.r.t. L just if there is an algorithm that transforms a given pair (R, ϕ) to Rϕ. In this paper, we prove that higher-order recursion schemes are reflective w.r.t. both modal µ-calculus and monadic second order (MSO) logic. To obtain this result, we give the first characterisation of the winning regions of parity games over the transition graphs of collapsible pushdown automata (CPDA): they are regular sets defined by a new class of automata. (Order-n recursion schemes are equi-expressive with order-n CPDA for generating trees.) As a corollary, we show that these schemes are closed under the operation of MSO-interpretation followed by tree unfoldingà la Caucal.

“…Cachat [5] and Serre [14] have independently generalised Walukiewicz's algorithm to compute the winning regions of these games. That is, the set of all positions in the game where a given player can force a win.…”

Section: Introductionmentioning

confidence: 99%

Regular Strategies in Pushdown Reachability Games

Lecture Notes in Computer Science

Hague

2014

Abstract. We show that positional winning strategies in pushdown reachability games can be implemented by deterministic finite state automata of exponential size. Such automata read the stack and control state of a given pushdown configuration and output the set of winning moves playable from that position. This result can originally be attributed to Kupferman, Piterman and Vardi using an approach based on two-way tree automata. We present a more direct approach that builds upon the popular saturation technique. Saturation for analysing pushdown systems has been successfully implemented by Moped and WALi. Thus, our approach has the potential for practical applications to controller-synthesis problems.