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Defining Winning Strategies in Fixed-Point Logic

2015
*
2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
*

A closely related, yet different, facet of this problem concerns the

doi:10.1109/lics.2015.42
dblp:conf/lics/CanavoiGLP15
fatcat:3cndjmbxujgelmx63y7piyakqq
*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*. ...##
###
Backtracking games and inflationary fixed points

2006
*
Theoretical Computer Science
*

The corresponding increase

doi:10.1016/j.tcs.2005.10.030
fatcat:5jyecmxhmbgnvh2ys3nn546ipu
*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*. ...##
###
Backtracking Games and Inflationary Fixed Points
[chapter]

2004
*
Lecture Notes in Computer Science
*

The corresponding increase

doi:10.1007/978-3-540-27836-8_37
fatcat:nfbe6j4qebhf5jtphd5gtljfda
*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*. ...##
###
Games for Inclusion Logic and Fixed-Point Logic
[chapter]

2016
*
Dependence Logic
*

One of the most intriguing results on

doi:10.1007/978-3-319-31803-5_5
fatcat:tn7wxvmcmfb5hfd673ycczeqnq
*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. ...##
###
Semiring Provenance for Büchi Games: Strategy Analysis with Absorptive Polynomials

2021
*
Electronic Proceedings in Theoretical Computer Science
*

Evaluating the

doi:10.4204/eptcs.346.5
fatcat:ud3sn6q2ebdnfkvdpvhbm7uubu
*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*. ...##
###
Modular Games for Coalgebraic Fixed Point Logics

2008
*
Electronical Notes in Theoretical Computer Science
*

The

doi:10.1016/j.entcs.2008.05.020
fatcat:d27lk7nsp5es3esonyd2j2aflm
*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. ...##
###
A Game Semantics for Proof Search: Preliminary Results

2006
*
Electronical Notes in Theoretical Computer Science
*

The syntactic variable A denotes atomic formulas: that is, a formula with a predicate (a non-

doi:10.1016/j.entcs.2005.11.072
fatcat:my7zdcym3bekrf56ta37d7hyfu
*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. ...##
###
Semiring Provenance for Büchi Games: Strategy Analysis with Absorptive Polynomials
[article]

2021
*
arXiv
*
pre-print

Evaluating the

arXiv:2106.12892v1
fatcat:y2tlngpa3fatbkup6wc5vvk7ja
*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*. ...##
###
Decisions, Actions, and Games: A Logical Perspective
[chapter]

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 )? ...

##
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In Praise of Strategies
[chapter]

2012
*
Lecture Notes in Computer Science
*

of

doi:10.1007/978-3-642-29326-9_6
fatcat:3atr6272gbfzhel5of5pnj5swe
*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. ...##
###
Static Analysis of Parity Games: Alternating Reachability Under Parity
[chapter]

2015
*
Lecture Notes in Computer Science
*

nodes

doi:10.1007/978-3-319-27810-0_8
fatcat:co2qjkdsunhnrjzc32bklgutcy
*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. ...##
###
A double arity hierarchy theorem for transitive closure logic

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*. ...

##
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Reasoning about Strategies
[chapter]

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. ...

##
###
Fixed-Point Logics and Solitaire Games

2004
*
Theory of Computing Systems
*

The model-checking games associated with

doi:10.1007/s00224-004-1147-5
fatcat:o7bj3ruwkzgxxdqqnsdcboehpm
*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. ...##
###
OUP accepted manuscript

2019
*
Journal of Logic and Computation
*

However, due to a crucial difference

doi:10.1093/logcom/exz025
fatcat:dtlk5dn7fbdxbgtinich5ubbsi
*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 ...
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