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Defining Winning Strategies in Fixed-Point Logic

Felix Canavoi, Erich Gradel, Simon Lessenich, Wied Pakusa
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
A closely related, yet different, facet of this problem concerns the definability of winning regions and winning strategies in logical systems such as monadic second-order logic, least fixed-point logic  ...  We study definability questions for positional winning strategies in infinite games on graphs.  ...  We infer that not only the winning regions in reachability games are definable in fixed-point logic, but also the optimal winning strategies.  ... 
doi:10.1109/lics.2015.42 dblp:conf/lics/CanavoiGLP15 fatcat:3cndjmbxujgelmx63y7piyakqq

Backtracking games and inflationary fixed points

Anuj Dawar, Erich Grädel, Stephan Kreutzer
2006 Theoretical Computer Science  
The corresponding increase in expressiveness allows us to use backtracking games as model checking games for inflationary fixed-point logics such as IFP or MIC.  ...  As a consequence, winning strategies become more complex objects and computationally harder.  ...  Inflationary fixed point logic.  ... 
doi:10.1016/j.tcs.2005.10.030 fatcat:5jyecmxhmbgnvh2ys3nn546ipu

Backtracking Games and Inflationary Fixed Points [chapter]

Anuj Dawar, Erich Grädel, Stephan Kreutzer
2004 Lecture Notes in Computer Science  
The corresponding increase in expressiveness allows us to use backtracking games as model checking games for inflationary fixed-point logics such as IFP or MIC.  ...  As a consequence, winning strategies become more complex objects and computationally harder.  ...  Inflationary fixed point logic.  ... 
doi:10.1007/978-3-540-27836-8_37 fatcat:nfbe6j4qebhf5jtphd5gtljfda

Games for Inclusion Logic and Fixed-Point Logic [chapter]

Erich Grädel
2016 Dependence Logic  
One of the most intriguing results on logics of dependence and independence is the tight connection between inclusion logic and the least fixed-point logic LFP.  ...  We study the model-checking games for inclusion logic and for posGFP, the fragment of LFP that uses only (non-negated) greatest fixed points.  ...  Least fixed-point logic Least fixed-point logic, denoted LFP, extends first order logic by least and greatest fixed points of definable relational operators.  ... 
doi:10.1007/978-3-319-31803-5_5 fatcat:tn7wxvmcmfb5hfd673ycczeqnq

Semiring Provenance for Büchi Games: Strategy Analysis with Absorptive Polynomials

Erich Grädel, Niels Lücking, Matthias Naaf
2021 Electronic Proceedings in Theoretical Computer Science  
Evaluating the fixed-point formula that defines the winning region in a given game in an appropriate semiring of polynomials provides not only the Boolean information on who wins, but also tells us how  ...  This is well-understood for reachability games, where the winning region is definable as a least fixed point.  ...  well-defined semantics of fixed-point logics.  ... 
doi:10.4204/eptcs.346.5 fatcat:ud3sn6q2ebdnfkvdpvhbm7uubu

Modular Games for Coalgebraic Fixed Point Logics

Corina Cîrstea, Mehrnoosh Sadrzadeh
2008 Electronical Notes in Theoretical Computer Science  
The logics obtained from syntax constructors are originally boolean, but in order to ensure that fixed points have a well-defined semantics, we leave out negation from these languages.  ...  For instance, we obtain the fixed points of Dynamic Epistemic Logic [2] via the coalgebraic semantics for this logic described in [5] .  ...  winning strategy in (c, φ). 2 As a consequence of Theorem 4.2, we obtain a result about the implicit negation of a fixed point formula.  ... 
doi:10.1016/j.entcs.2008.05.020 fatcat:d27lk7nsp5es3esonyd2j2aflm

A Game Semantics for Proof Search: Preliminary Results

Dale Miller, Alexis Saurin
2006 Electronical Notes in Theoretical Computer Science  
The syntactic variable A denotes atomic formulas: that is, a formula with a predicate (a non-logical constant) as its head: the formulas ⊥ and and t = s are not atomic formulas.  ...  The positive translation of fix is the least fixed point operation µ; negative translation of fix is the greatest fixed point operation ν.  ...  Extending for recursion Extend expressions with the fixed point constructors {fix n } n≥0 . In (fix n λP λx 1 . . . λx n .M ) the bound variable P is an n-ary recursive function.  ... 
doi:10.1016/j.entcs.2005.11.072 fatcat:my7zdcym3bekrf56ta37d7hyfu

Semiring Provenance for Büchi Games: Strategy Analysis with Absorptive Polynomials [article]

Erich Grädel, Niels Lücking, Matthias Naaf
2021 arXiv   pre-print
Evaluating the fixed-point formula that defines the winning region in a given game in an appropriate semiring of polynomials provides not only the Boolean information on who wins, but also tells us how  ...  This is well-understood for reachability games, where the winning region is definable as a least fixed point.  ...  The Formula It is well known that the winning region (of Player 0) in a Büchi game is definable in fixed-point logic.  ... 
arXiv:2106.12892v1 fatcat:y2tlngpa3fatbkup6wc5vvk7ja

Decisions, Actions, and Games: A Logical Perspective [chapter]

Johan van Benthem
2008 Lecture Notes in Computer Science  
In recent years, standard 'static' logics describing information states of agents have been generalized to dynamic logics describing actions and events that produce information, revise beliefs, or change  ...  Next, we introduce dynamic logics, and see what they add in scenarios with information update and belief revision where given games can change as new information arrives.  ...  This fixed point can still be defined in propositional dynamic logic, using the formula < ((turn i )? ; E ) ∪ (turn j )?  ... 
doi:10.1007/978-3-540-92701-3_1 fatcat:q7n5gsqmtjcm7biwz2fpy5ijrm

In Praise of Strategies [chapter]

Johan van Benthem
2012 Lecture Notes in Computer Science  
of winning strategies and the like in fixed finite games.  ...  This note high-lights one major theme in my lecture notes Logic in Games (van Benthem 1999 -2002 : the need for explicit logics that define agents' strategies, as the drivers of interaction in games.  ...  Thus, propositional dynamic logic does a reasonable job in defining explicit strategies in simple extensive games.  ... 
doi:10.1007/978-3-642-29326-9_6 fatcat:3atr6272gbfzhel5of5pnj5swe

Static Analysis of Parity Games: Alternating Reachability Under Parity [chapter]

Michael Huth, Jim Huan-Pu Kuo, Nir Piterman
2015 Lecture Notes in Computer Science  
nodes in its greatest fixed point are won by said player in the parity game.  ...  We prove the determinacy of these games and use this determinacy to define, for each player, a monotone fixed point over an ordered domain of height linear in the size of the parity game such that all  ...  finite game graph G (the setting of our chapter), it is proved that least fixed-point logic can define the winning regions of G iff these winning regions are computable in polynomial time.  ... 
doi:10.1007/978-3-319-27810-0_8 fatcat:co2qjkdsunhnrjzc32bklgutcy

A double arity hierarchy theorem for transitive closure logic

Martin Grohe, Lauri Hella
1996 Archive for Mathematical Logic  
In this paper we prove that the k-ary fragment of transitive closure logic is not contained in the extension of the (k − 1)-ary fragment of partial fixed point logic by all (2k − 1)-ary generalized quantifiers  ...  As a consequence, the arity hierarchies of all the familiar forms of fixed point logic are strict simultaneously with respect to the arity of the induction predicates and the arity of generalized quantifiers  ...  For example, the arity of a fixed point operator is the arity of the relation defined by the fixed point.  ... 
doi:10.1007/bf01268616 fatcat:jzilabwfofcytdi33oab7uvm5i

Reasoning about Strategies [chapter]

Johan van Benthem
2013 Lecture Notes in Computer Science  
In this little piece, I discuss one theme in the overlap of our interests, namely, logical systems for reasoning with strategies -in gentle exploratory mode. 1  ...  Samson Abramsky has placed landmarks in the world of logic and games that I have long admired.  ...  For one, while it does have a natural definition in the first-order fixed-point logic LFP(FO), it does not seem to have an obvious program definition in the above PDL terms.  ... 
doi:10.1007/978-3-642-38164-5_23 fatcat:gjqtj4xwbjcxxdk46e6m3slzna

Fixed-Point Logics and Solitaire Games

Dietmar Berwanger, Erich Gr�del
2004 Theory of Computing Systems  
The model-checking games associated with fixed-point logics are parity games, and it is currently not known whether the strategy problem for parity games can be solved in polynomial time.  ...  On finite structures (but not on infinite ones), Solitaire-LFP is equivalent to transitive closure logic. We also consider the solitaire fragment of guarded fixed-point logics.  ...  Least Fixed-Point Logic Leastfixed-point logic, denoted LFP, extends first-order logic by least and greatest fixed points of definable relational operators.  ... 
doi:10.1007/s00224-004-1147-5 fatcat:o7bj3ruwkzgxxdqqnsdcboehpm

OUP accepted manuscript

2019 Journal of Logic and Computation  
However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy.  ...  We propose a new version of formula size game for modal logic.  ...  A new challenge in defining such a game is that if S uses a fixed point ηX (η ∈ {µ, ν}) as the logical operator in his move, and later uses the corresponding variable X, then in the next round, the game  ... 
doi:10.1093/logcom/exz025 fatcat:dtlk5dn7fbdxbgtinich5ubbsi
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