TheAlternating-Time ExplicitStrategies Joint work with Lutz Schröder and Dirk Pattinson by Merlin Göttlinger 7th January 2020 µ -CalculusWithDisjunctive
Introduction - 2/49 Introduction
Introduction - 3/49 consisting of determining the set of moves (or actions ) available to for each agent j and each state q , a natural number k q Structure a finite set Q of states , 1 Alur, Henzinger, and Kupferman ‘Alternating-Time Temporal Logic’ (2002) set A of atoms set N of agents , Given a Definition (Concurrent game structure 1 ) a concurrent game structure (CGS) is a tuple ( Q , v , k , f ) for each q ∈ Q , a set v ( q ) ⊆ A of propositions true at q , j ≥ 1 agent j at state q to be [ k q j ] , for each q ∈ Q a transition function f q : [ k q N ] → Q
Introduction - 4/49 Given a set A of (propositional) atoms , a set V of variables , a finite set N of agents , fixpoints, respectively. 2 Alur, Henzinger, and Kupferman ‘Alternating-Time Temporal Logic’ (2002) Alternating-time µ -Calculus Definition (The alternating-time µ -calculus (AMC) 2 ) formulae φ, ψ are then given by the grammar φ, ψ ::= p | x | ⊤ | ⊥ | φ ∧ ψ | φ ∨ ψ | [ C ] φ | � C � φ | µ x . φ | ν x . φ where x ∈ V , p ∈ A , and C ⊆ N , i.e. a coalition . We generally write ¯ C = N \ C . As usual, µ and ν take least and greatest
Introduction - 5/49 AMC cont. over history-dependent joint strategies for coalition A . Embeds into the AMC as fixpoints given that memory-less strategies suffice. 3 Alur, Henzinger, and Kupferman ‘Alternating-Time Temporal Logic’ (2002) Alternating-time temporal logic (ATL) 3 φ, ψ ::= p | ⊥ | ¬ φ | φ ∧ ψ | � � A � � x x ::= � φ | � φ | φ U ψ � � A � � builds paths from the current state by quantifying
Introduction - 6/49 Limited ATL and even the AMC could be more expressive. “No matter what the other network actors do, Alice and Bob can collaborate to exchange keys via Server S provided that S adheres to the protocol”. Restricting some agents’ choices of action is quite natural in human reasoning. This requires moves or strategies to be part of the syntax.
Introduction - 7/49 at the ATL path quantifiers. 7 Walther, Hoek, and Wooldridge ‘Alternating-time Temporal Logic with Logics’ (2013) 6 Herzig, Lorini, and Walther ‘Reasoning about Actions Meets Strategic (2006) 5 Ågotnes ‘Action and Knowledge in Alternating-Time Temporal Logic’ 4 Hoek, Jamroga, and Wooldridge ‘A Logic for Strategic Reasoning’ (2005) Explicit Strategies’ (2007) Strategic Reasoning I the current world at the ATL path quantifiers. at the ATL path quantifiers. dynamic modality. Counterfactual ATL (CATL) 4 has action commitment as a ATL with actions (ATL-A) 5 has action restriction for one step ATL with explicit actions (ATLEA) 6 has action commitment for ATL with explicit strategies (ATLES) 7 has strategy commitment
Introduction - 8/49 1 1 1 0 1 Strategic Reasoning II 0 0 1 1 0 0 B Despite syntactically capable of reasoning about ATLEA M Due to an error in their axiomatization the commitment is only A checking results only cover the memory-less case. history-dependent strategies their satisfiability and model for the next step. ATLES ∗ AM �→ ∗ BM �→ ∗ A �→ ρ = ∗ B �→ p , ¬ q ¬ p , q ¬ p , ¬ q ∗ M �→
Introduction - 9/49 Even More Expressiveness! Notice that all those extensions build on ATL rather than on the full AMC. Idea Treating the logics in coalgebraic modal logic produces multiple other benefits). The simplified treatment via the one-step logic enables easy further extensions. Let us first look at the extended logic we are dealing with. results for the full alternating µ -calculus (as well as
The AMC With Disjunctive Explicit Strategies - 10/49 TheAMCWithDisjunctive ExplicitStrategies
The AMC With Disjunctive Explicit Strategies - 11/49 The AMCDES We allow for disjunctive commitments: “Johnson has a strategy to enforce Brexit and stay in power in the process, provided that Labour opts to either support the Brexit deal or to proceed with new elections”. We include full support for least and greatest fixpoint operators, with associated gains in expressivity analogous to the extension from ATL to the AMC.
The AMC With Disjunctive Explicit Strategies - 12/49 The AMCDES explicit strategies (AMCDES)) Given atoms A , variables V , and agents N as in CGSs, formulae are then given by the grammar disjunctive explicit strategy , for some coalition D , disjoint strategies (AMCES) is the fragment of the AMCDES obtained by disallowing strategy disjunction. Definition (The alternating-time µ -calculus with disjunctive a set M j of explicit strategies for each agent j , φ, ψ ::= p | x | ⊤ | ⊥ | φ ∧ ψ | φ ∨ ψ | [ C , O ] φ | � C , O � φ | µ x . φ | ν x . φ where x ∈ V , p ∈ A , and C ⊆ N , i.e. a coalition . Moreover, O ⊆ � j ∈ D M j is a set of joint explicit strategies, called a from C , that we denote by Ag ( O ) . The AMC with explicit
The AMC With Disjunctive Explicit Strategies - 13/49 Extent Empty agents AMCDES and even the AMCES subsumes both AMC and ATL. Singleton strategy When Ag ( O ) = ∅ , [ C , O ] corresponds to [ C ] in the AMC. So the When O is a singleton i.e. a non-disjunctive strategy, [ C , O ] corresponds to the ATLES � � C � � ρ � for a memory-less strategy ρ ∈ O and the other path formulae are expressible via fixpoints.
The AMC With Disjunctive Explicit Strategies - 14/49 CGSES Definition (CGSES) A concurrent game structure with explicit strategies (CGSES) for agents N , atoms A , and explicit strategies M j for j ∈ N is a tuple ( Q , v , k , f , M , i ) consisting of a CGS ( Q , v , k , f ) for N , A , for each q ∈ Q a strategy interpretation i q : � j ∈ N ( M j → [ k q j ]) .
The AMC With Disjunctive Explicit Strategies - 15/49 S S CGSES Semantics of the AMCDES S S The semantics of the AMCDES is then defined by assigning to each formula φ an extension � φ � σ S ⊆ Q , which depends on a CGSES S = ( Q , v , k , f , M , i ) and valuation σ : V → P ( Q ) : � p � σ � x � σ S = { q ∈ Q | p ∈ v ( q ) } S = σ ( x ) � ⊤ � σ � ⊥ � σ S = Q S = ∅ � φ ∧ ψ � σ S = � φ � σ S ∩ � ψ � σ � φ ∨ ψ � σ S = � φ � σ S ∪ � ψ � σ � [ C , O ] φ � σ S = { q ∈ Q | ∃ m C ∈ [ k q C ] . ∀ m N ∈ [ k q N ] . m C ⊑ m N ∧ m N | Ag ( O ) ∈ i q [ O ] ⇒ f ( m ) ∈ � φ � σ S } S = � { B ⊆ Q | � φ ( x ) � σ [ x �→ B ] � µ x . φ ( x ) � σ ⊆ B } S = � { B ⊆ Q | B ⊆ � φ ( x ) � σ [ x �→ B ] � ν x . φ ( x ) � σ }
The AMC With Disjunctive Explicit Strategies - 16/49 Coalition vs Opposition Disjunction In the disjunctive case the semantics vary depending on Restricts the choices of the coalition. whether Ag ( O ) is made a part of C or not. Ag ( O ) ⊆ C ⇒ Disjunction at the ∃ -level. ⇒ Can be encoded as disjunction over boxes. Ag ( O ) ⊆ N \ C The choice happens at the ∀ -level. ⇒ Can not be equivalently encoded. ⇒ We opted for this kind of disjunction as part of the syntax.
Preliminaries: Coalgebraic Logic - 17/49 Preliminaries: CoalgebraicLogic
Preliminaries: Coalgebraic Logic - 18/49 Coalgebraic Modal Logic 8 Cîrstea, Kurz, Pattinson, Schröder, and Venema ‘Modal Logics are CGSESs are generated by the following functor: CGSES functor such systems as set-functor F . Parameterizes the semantics of logics over the type of A uniform framework for modal and temporal logics Coalgebraic’ (2011) interpreted over state-based systems 8 . F -coalgebras ( W , γ ) represent systems where W are the states and γ : W → FW the transition map . G ES = { (( k j ) j ∈ N , f , i ) | � 1 ≤ ( k j ) ∈ N N , f : [ k N ] → W , i : ( M j → [ k j ]) } j ∈ N
Preliminaries: Coalgebraic Logic - 19/49 Coalgebraic Modal Logic modal operators with assigned finite arities. atoms are encoded as nullary modalities. given by the grammar 9 Cîrstea, Kupke, and Pattinson ‘EXPTIME Tableaux for the Coalgebraic The syntax is parameterized over a set Λ of (next-step) We require that for every ♥ ∈ Λ there is a dual operator ¯ ♥ ∈ Λ The coalgebraic µ -calculus 9 over Λ then has formulae φ, ψ φ, ψ ::= ⊤ | ⊥ | x | φ ∧ ψ | φ ∨ ψ | ♥ φ | µ x . φ | ν x . φ where x ranges over a reservoir of fixpoint variables , and ♥ over Λ . µ -Calculus’ (2011)
Preliminaries: Coalgebraic Logic - 20/49 Coalgebraic Modal Logic ones plus A modal operator ♥ ∈ Λ is interpreted by assigning to it a predicate lifting � ♥ � . � ♥ � W for any set W assigns to each subset Y ⊆ W a subset � ♥ � W ( Y ) ⊆ FW . Given an F -coalgebra C = ( W , γ ) and a valuation σ : Fix → P W the extension � φ � σ C ⊆ W of a formula φ are then the standard � ♥ φ � σ C = γ − 1 [ � ♥ � W ( � φ � σ C )] .
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