TheAlternating-Time ExplicitStrategies Joint work with Lutz Schrder - - PowerPoint PPT Presentation

TheAlternating-Time µ-CalculusWithDisjunctive ExplicitStrategies

Joint work with Lutz Schröder and Dirk Pattinson by

Merlin Göttlinger

7th January 2020

Introduction - 2/49

Introduction

Introduction - 3/49

Structure

Definition (Concurrent game structure 1)

Given a set N of agents, set A of atoms a concurrent game structure (CGS) is a tuple (Q, v, k, f) consisting of a finite set Q of states, for each q ∈ Q, a set v(q) ⊆ A of propositions true at q, for each agent j and each state q, a natural number kq

j ≥ 1

determining the set of moves (or actions) available to agent j at state q to be [kq

j ],

for each q ∈ Q a transition function fq : [kq

N] → Q

1Alur, Henzinger, and Kupferman ‘Alternating-Time Temporal Logic’ (2002)

Introduction - 4/49

Alternating-time µ-Calculus

Definition (The alternating-time µ-calculus (AMC) 2)

Given a set A of (propositional) atoms, a set V of variables, a finite set N of agents, formulae φ, ψ are then given by the grammar φ, ψ ::= p | x | ⊤ | ⊥ | φ ∧ ψ | φ ∨ ψ | [C]φ | Cφ | µx. φ | νx. 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 fixpoints, respectively.

2Alur, Henzinger, and Kupferman ‘Alternating-Time Temporal Logic’ (2002)

Introduction - 5/49

AMC cont.

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

• ver history-dependent joint strategies for coalition A.

Embeds into the AMC as fixpoints given that memory-less strategies suffice.

3Alur, Henzinger, and Kupferman ‘Alternating-Time Temporal Logic’ (2002)

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

Strategic Reasoning I

Counterfactual ATL (CATL) 4 has action commitment as a dynamic modality. ATL with actions (ATL-A) 5 has action restriction for one step at the ATL path quantifiers. ATL with explicit actions (ATLEA) 6 has action commitment for the current world at the ATL path quantifiers. ATL with explicit strategies (ATLES) 7 has strategy commitment at the ATL path quantifiers.

4Hoek, Jamroga, and Wooldridge ‘A Logic for Strategic Reasoning’ (2005) 5Ågotnes ‘Action and Knowledge in Alternating-Time Temporal Logic’

(2006)

6Herzig, Lorini, and Walther ‘Reasoning about Actions Meets Strategic

Logics’ (2013)

7Walther, Hoek, and Wooldridge ‘Alternating-time Temporal Logic with

Explicit Strategies’ (2007)

Introduction - 8/49

Strategic Reasoning II

ATLEA

Due to an error in their axiomatization the commitment is only for the next step.

ATLES

Despite syntactically capable of reasoning about history-dependent strategies their satisfiability and model checking results only cover the memory-less case. A p, ¬q M ¬p, ¬q B ¬p, q 1 1 1 ρ =                ∗AM → 1 ∗BM → ∗A → 1 ∗B → 1 ∗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 results for the full alternating µ-calculus (as well as 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.

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

• perators, with associated gains in expressivity analogous

to the extension from ATL to the AMC.

The AMC With Disjunctive Explicit Strategies - 12/49

The AMCDES

Definition (The alternating-time µ-calculus with disjunctive explicit strategies (AMCDES))

Given atoms A, variables V, and agents N as in CGSs, a set Mj of explicit strategies for each agent j, formulae are then given by the grammar φ, ψ ::= p | x | ⊤ | ⊥ | φ∧ψ | φ∨ψ | [C, O]φ | C, Oφ | µx. φ | νx. φ where x ∈ V, p ∈ A, and C ⊆ N, i.e. a coalition. Moreover, O ⊆

j∈D Mj is a set of joint explicit strategies, called a

disjunctive explicit strategy, for some coalition D, disjoint from C, that we denote by Ag(O). The AMC with explicit strategies (AMCES) is the fragment of the AMCDES obtained by disallowing strategy disjunction.

The AMC With Disjunctive Explicit Strategies - 13/49

Extent

Empty agents

When Ag(O) = ∅, [C, O] corresponds to [C] in the AMC. So the AMCDES and even the AMCES subsumes both AMC and ATL.

Singleton strategy

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 Mj 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 iq :

j∈N(Mj → [kq j ]).

The AMC With Disjunctive Explicit Strategies - 15/49

CGSES Semantics of the AMCDES

The semantics of the AMCDES is then defined by assigning to each formula φ an extension φσ

S ⊆ Q, which depends on a

CGSES S = (Q, v, k, f, M, i) and valuation σ : V → P(Q): pσ

S = {q ∈ Q | p ∈ v(q)}

xσ

S = σ(x)

⊤σ

S = Q

⊥σ

S = ∅

φ ∧ ψσ

S = φσ S ∩ ψσ S

φ ∨ ψσ

S = φσ S ∪ ψσ S

[C, O] φσ

S = {q ∈ Q | ∃mC ∈ [kq C]. ∀mN ∈ [kq N].

mC ⊑ mN ∧ mN|Ag(O) ∈ iq[O] ⇒ f(m) ∈ φσ

S}

µx. φ(x)σ

S = {B ⊆ Q | φ(x)σ[x→B] S

⊆ B} νx. φ(x)σ

S = {B ⊆ Q | B ⊆ φ(x)σ[x→B] S

}

The AMC With Disjunctive Explicit Strategies - 16/49

Coalition vs Opposition Disjunction

In the disjunctive case the semantics vary depending on whether Ag(O) is made a part of C or not.

Ag(O) ⊆ C

Restricts the choices of the coalition. ⇒ 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

A uniform framework for modal and temporal logics interpreted over state-based systems 8. Parameterizes the semantics of logics over the type of such systems as set-functor F. F-coalgebras (W, γ) represent systems where W are the states and γ : W → FW the transition map.

CGSES functor

CGSESs are generated by the following functor: GES = {((kj)j∈N, f, i) | 1 ≤ (kj) ∈ NN, f : [kN] → W, i :

• j∈N

(Mj → [kj])}

8Cîrstea, Kurz, Pattinson, Schröder, and Venema ‘Modal Logics are

Coalgebraic’ (2011)

Preliminaries: Coalgebraic Logic - 19/49

Coalgebraic Modal Logic

The syntax is parameterized over a set Λ of (next-step) modal operators with assigned finite arities. atoms are encoded as nullary modalities. We require that for every ♥ ∈ Λ there is a dual operator ¯ ♥ ∈ Λ The coalgebraic µ-calculus 9 over Λ then has formulae φ, ψ given by the grammar φ, ψ ::= ⊤ | ⊥ | x | φ ∧ ψ | φ ∨ ψ | ♥φ | µx. φ | νx. φ where x ranges over a reservoir of fixpoint variables, and ♥

• ver Λ.

9Cîrstea, Kupke, and Pattinson ‘EXPTIME Tableaux for the Coalgebraic

µ-Calculus’ (2011)

Preliminaries: Coalgebraic Logic - 20/49

Coalgebraic Modal Logic

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 → PW the extension φσ

C ⊆ W of a formula φ are then the standard

• nes plus

♥φσ

C = γ−1[♥W(φσ C)].

Preliminaries: Coalgebraic Logic - 21/49

Tableau Soundness & Completeness

Definition

The rule ∆/Γ1 | · · · | Γn is one-step tableau sound if Γiτ = ∅ for some i whenever ∆τ = ∅.

Definition

Let R be a set of one-step tableau rules, closed under injective renaming of variables. Then R is one-step tableau complete if the following condition holds: For all W, τ : PV → P(W), and Θ ⊆ Λ(V), whenever for each rule ∆/Γ1 | · · · | Γn ∈ R such that ∆ ⊆ Θ, we have Γiτ = ∅ for some i, then Θτ = ∅.

Preliminaries: Coalgebraic Logic - 22/49

To obtain complexity results, rule sets formally need to be tractable in a suitable sense

10 11.

Theorem (Satisfiability checking)

If a coalgebraic µ-calculus admits a tractable one-step tableau complete set of one-step tableau sound rules, then its satisfiability problem is in ExpTime.

10Schröder and Pattinson ‘PSPACE bounds for rank-1 modal logics’ (2009) 11Cîrstea, Kupke, and Pattinson ‘EXPTIME Tableaux for the Coalgebraic

µ-Calculus’ (2011)

AMCDES Model Checking - 23/49

AMCDESModelChecking

AMCDES Model Checking - 24/49

Strategic Model Checking

Variants of model checking 12

fixed: interpretation of explicit strategies is considered part of the model.

• pen: interpretation has to be constructed during model

checking. fixed ATLES model checking with memory-less strategies is P-complete.

• pen model checking is NP-complete with the upper bound

from guessing the strategies.

12Walther, Hoek, and Wooldridge ‘Alternating-time Temporal Logic with

Explicit Strategies’ (2007)

AMCDES Model Checking - 25/49

Coalgebraic Model Checking

Given a finite set X, a modality [C, O], and t ∈ GESX:

One-step model checking 13

Given Y ⊆ X: If t ∈ [C, O]X(Y) can be decided in P then the full AMCDES model checking is in NP ∩ coNP. If we can design a one-step satisfaction arena having polynomially many inner nodes in the size of [C, O] and t, then the full AMCDES model checking is in QP.

13Hausmann and Schröder ‘Game-Based Local Model Checking for the

Coalgebraic mu-Calculus’ (2019)

AMCDES Model Checking - 26/49

One-step satisfaction

Lemma

The one-step satisfaction problem for the AMCDES is in P.

Proof.

Let W be a set, w = ((kj)j∈N, f, i) ∈ GES(W) and U ⊆ W. Show that w ∈ [C, O](U) can be checked in P. for mC ← [kC] do x := ⊤; for o ← O, m¯

C ← [kN\C\Ag(O)] do

if f(mc, m¯

C, i[o]) /

∈ U then x := ⊥; end if x then return ⊤; end return ⊥ Complexity: |[kC]| × |[k¯

C]| × |O| = O(size(w) × size([C, O]))

AMCDES Model Checking - 27/49

One-step arena

Lemma

AMCDES has small one-step satisfaction arenas.

Proof (adapted from Hausmann and Schröder 14).

Let W be a set, a modality [C, O], and w = ((kj)j∈N, f, i) ∈ GES(W). The one-step satisfaction arena A[C,O],w = (V[C,O],w, E[C,O],w) is constructed as follows. An initial node ([C, O], w) belonging to Eloise, inner nodes I[C,O],w := [kC] belonging to Abelard, and moves E[C,O],w := x →

• I[C,O],w if x = ([C, O], w)

{f(x, m¯

C, i) | m¯ C ∈ [kN\C], i ∈ i[O]}

. Inner nodes: |[kC]| = O(size(w)).

14Hausmann and Schröder ‘Game-Based Local Model Checking for the

Coalgebraic mu-Calculus’ (2019)

Set Valued Resolution - 28/49

SetValuedResolution

Set Valued Resolution - 29/49

Set Valued Resolution

Set-valued propositional resolution

Adaptation of the standard propositional resolution method. Formulas interpreted over Boolean algebras of the form P(W)Y. Literals of the form A(y) for y ∈ Y and A ⊆ W. A function f : Y → W satisfies A(y) if f(y) ∈ A. Set-valued resolution rule SRΓ, A(y) B(y), ∆ Γ, (A ∩ B)(y), ∆ A clause is blatantly inconsistent if all its literals are of the form ∅(y).

AMCDES Satisfiability - 30/49

AMCDESSatisfiability

AMCDES Satisfiability - 31/49

Order-2 Moves

Definition (Order-2 moves)

We use a special kind of moves al called order-2 moves of length k, made up of a move symbol a from some finite set Const, and an index l, which is an element of the order-2 abelian group Fk

2.

Indices can be seen as length-k bit vectors, with addition being component-wise XOR We write idx(al) = l for the index of al, and for a joint move mC

• f a coalition C, idx(mC) is the sum over all indices of the

components moves of mC.

AMCDES Satisfiability - 32/49

AMCDES SAT

Satisfiability checking in the AMCDES is in ExpTime. Containment implies completeness because already computation tree logic (CTL) satisfiability is ExpTime-hard).

• 1. Give one-step rules
• 2. Show soundness of the rules
• 3. Use set-valued resolution for the completeness proof

Definition (Predicate liftings for the AMCDES)

The modality [C, O] can be lifted over the functor GES [C, O]W(Y) = {(f, (kj)j∈N, i) ∈ GESW | ∃mC ∈ [kC] ∀m¯

C ∈ [kN\(C∪Ag(O))] :

• n∈O

f(mC, m¯

C, i[n]) ∈ Y}

AMCDES Satisfiability - 33/49

AMCDES Rules I

(DES0) [A1, PG1]a1, . . . [Aα, PGα]aα, C1, rH1c1, . . . , Cβ, rHβcβ a1, . . . , aα, c1, . . . , cβ where

• 1. for each j, k, Aj ∩ Ak = ∅
• 2. for each j, Cj ∪ Hj = N
• 3. no explicit strategies for agents in α

j=1 Aj

• 4. there is a joint explicit strategy l for the agents

α

j=1 Gj ∪ β j=1 Hj such that for j = 1, . . . , β, rHj ⊑ l and for

each j = 1, . . . , α there exists p ∈ PGj such that p ⊑ l

AMCDES Satisfiability - 34/49

AMCDES Rules II

(DES1) [A1, PG1]a1, . . . , [Aα, PGα]aα, B, QJb, C1, rH1c1, . . . , Cβ, rHβcβ (aj)j∈Aq, b, (cj)j∈Bq | . . . for each q ∈ QJ where Aq ⊆ {1, . . . , α}, Bq ⊆ {1, . . . , β} for each q 1.–3. as in (DES0)

• 4. for each j, Aj ⊆ B
• 5. B ∪ J ⊇

j Gj ∪ j Hj

• 6. rHj|Hj∩J = q|Hj∩J for each q ∈ QJ, j ∈ Bq
• 7. exists a joint explicit strategy l for B ∩ (α

j=1 Gj ∪ β j=1 Hj)

such that rHj|Hj∩B = l|Hj∩B for each q ∈ QJ, j ∈ Bq, and moreover for each j ∈ Aq there exists p ∈ PGj such that p|Gj∩J = q|Gj∩J and p|Gj∩B = l|Gj∩B

AMCDES Satisfiability - 35/49

AMCDES Completeness I

Theorem (One-step tableau completeness)

The rules (DES0) and (DES1) are one-step tableau complete w.r.t. a restricted version of AMCDES where grand coalition diamonds cannot have disjunctive strategies. In particular, the rule instances where strategies are not disjunctive are one-step tableau complete w.r.t. AMCES.

AMCDES Satisfiability - 36/49

AMCDES Completeness II

Assumptions

a P(W)-valuation τ, a finite set Γ = {[A1, PG1]a1, . . . , [Aα, PGα]aα, C1, RH1c1, . . . , Cβ, RHβcβ} where RHj not being a singleton implies Cj ∪ Hj N. Every instance of (DES0) or (DES1) applicable to Γ has a non-empty conclusion under τ.

Claim

We can satisfy Γ under τ (i.e. find an element of Γτ) using suitable order-2 moves of length k = ⌈log2(β) + 1⌉.

AMCDES Satisfiability - 37/49

AMCDES Completeness III

Construct Const = {ej | j = 1, . . . , α} ∪ {sj | j = 1, . . . , β} ∪ {rj | j = 1, . . . , β, r ∈ RHj} ∪ {pj | j = 1, . . . , α, p ∈ PGj}, where we restrict indices on ej, rj, and pj to be 0.

Translate Γ into the clause set φ

Cj, RHjcj generates dj = {τ(cj)(mCj, v Cj∪Hj, rj) | r ∈ RHj} where either Cj ∪ Hj = N or idx(mCj) + idx(v Cj∪Hj) = uj. [Aj, PGj]aj generates bp

j = {τ(aj)(eAj, v Aj∪Gj, pj)}

where p ∈ PGj.

AMCDES Satisfiability - 38/49

AMCDES Completeness IV

Clauses dj may now have more than one literal, corresponding to strategy disjunction. Thanks to the restrictions such clauses do not resolve among each other. ⇒ A proof of a blatantly inconsistent clause will involve either

• ne or zero instances of a clause dj where Cj ∪ Hj = N; we

will refer to resolution proofs of the first type as type 1 and to proofs of the second kind as type 0

AMCDES Satisfiability - 39/49

AMCDES Completeness V

Type 0 — Rule (DES0): Let mN be a joint move for N let G be the union of all Gj such that i[pj] ⊑ mN for some p ∈ PGj let H be the union of all Hj such that Cj ∪ Hj = N and i[rj] ⊑ mN for the unique r ∈ RHj and put l = mN|G∪H. Observe the following. bp

j , bq k then Aj ∩ Ak = ∅, as otherwise eAj and eAk would overlap.

bp

j then no other clause in φ with joint move mN can mention

explicit strategies for agents in Aj, as otherwise eAj

• verlaps with an explicit strategy.

By construction of l, (DES0) is applicable to a subset of Γ having non-empty conclusion extension under τ i.e. the resolution proof does not produce a blatantly inconsistent clause.

AMCDES Satisfiability - 40/49

AMCDES Completeness VI

Type 1 — Rule (DES1): successively resolve suitable singleton clauses of the form either bp

j or dk with Ck ∪ Hk = N into literals of the single

clause of the form dj0 for Cj0 ∪ Hj0 = N. To match the notation of rule (DES1), we rename Cj0, RHj0 into B, QJb. we denote the restriction of mN to B ∪ J as mB • q. Aq = {j ∈ {1, . . . , α} | p ∈ Pj, i[p] ⊑ mB • q} Bq = {j ∈ {1, . . . , β} | r ∈ RHj, i[r] ⊑ mB • q} G =

q∈QJ,j∈Aq Gj, H = q∈QJ,j∈Bq Hj, l = mB • q|B∩(G∪H).

AMCDES Satisfiability - 41/49

AMCDES Completeness VII

1.–3. as in the type 0 case and via the above renaming

• 4. If a clause of the form bp

j with joint move (mB, v B∪J, q) is

in φ, then Aj ⊆ B, as otherwise eAj overlaps with v B∪J or q.

• 5. Since interpretations of explicit strategies can not overlap

with sj0, we have B ∪ J ⊇ G ∪ H.

• 6. Follows from the fact that i[rHj] ⊑ mB • q for j ∈ Bq by

definition of Bq.

• 7. Similarly, l fits the conditions by the definitions of Aq and

Bq because l ⊑ mB • q.

Conclusion - 42/49

Conclusion

Conclusion - 43/49

COOL!

We extended the coalgebraic logic reasoner COOL to handle AMCES formulas. Strategies are written enclosed in () as

strategyname_agent e.g. (a_1) .

So [1, 2, 3, (a, b)]p for a ∈ M1, b ∈ M2 can be translated into

[{1 2 3},(a_1 b_2)]P or simply [{3},(a_1 b_2)]P .

To run the reasoner execute e.g. coalg.native sat CLN <<< ’[{1,2},(a_1)] False’

• r coalg.native sat CLN <<< ’µ x. [{1,2},(a_1)] x’ .

Conclusion - 44/49

Summary

Introduced the alternating-time µ-calculus with disjunctive explicit strategies (AMCDES), which extends ATLES with fixpoint operators and disjunction over explicit strategies

• f opposing agents.

Employed methods from coalgebraic logic to show fixed model checking in QP as well as in NP ∩ coNP, and in NP in the open case. Satisfiability checking in ExpTime under the restriction that grand diamonds do not contain strategy disjunction.

Conclusion - 45/49

Future Work

Axiomatization for full AMCDES i.e. where the grand coalition diamonds can have disjunctive strategies as well. N \ x, {1, 2}xφ ∧ N \ x, {1, 2, 3}x¬φ ∧ N \ x, {3}xφ Excluded strategies i.e. “C can enforce φ as long as D don’t execute strategy O” (suggested in Herzig, Lorini, and Walther 15). Investigate if history dependent strategies can be encoded into memoryless strategies via the construct of Alfaro, Henzinger, and Majumdar 16.

15Herzig, Lorini, and Walther ‘Reasoning about Actions Meets Strategic

Logics’ (2013)

16Alfaro, Henzinger, and Majumdar ‘From Verification to Control: Dynamic

Programs for Omega-regular Objectives’ (2001)

Conclusion - 46/49

References I

• R. Alur, T. A. Henzinger, and O. Kupferman,

‘Alternating-time temporal logic,’ J. ACM, vol. 49,

• pp. 672–713, 2002. doi: 10.1145/585265.585270.
• W. van der Hoek, W. Jamroga, and M. Wooldridge, ‘A logic

for strategic reasoning,’ Proceedings of the Fourth International Joint Conference on Autonomous Agents and Multiagent Systems - AAMAS ’05, 2005. doi: 10.1145/1082473.1082497.

• T. Ågotnes, ‘Action and knowledge in alternating-time

temporal logic,’ Synthese, vol. 149, no. 2, pp. 375–407,

• Mar. 2006, issn: 1573-0964. doi:

10.1007/s11229-005-3875-8.

Conclusion - 47/49

References II

• A. Herzig, E. Lorini, and D. Walther, ‘Reasoning about

actions meets strategic logics,’ Lecture Notes in Computer Science, pp. 162–175, 2013, issn: 1611-3349. doi: 10.1007/978-3-642-40948-6_13.

• D. Walther, W. van der Hoek, and M. Wooldridge,

‘Alternating-time temporal logic with explicit strategies,’ in Theoretical Aspects of Rationality and Knowledge, TARK 2007, D. Samet, Ed., ACM Press, 2007, pp. 269–278. doi: 10.1145/1324249.1324285.

• C. Cîrstea, A. Kurz, D. Pattinson, L. Schröder, and
• Y. Venema, ‘Modal logics are coalgebraic,’ Comput. J.,
• vol. 54, no. 1, pp. 31–41, 2011. doi:

10.1093/comjnl/bxp004.

Conclusion - 48/49

References III

• C. Cîrstea, C. Kupke, and D. Pattinson, ‘EXPTIME tableaux

for the coalgebraic µ-calculus,’ Log. Methods Comput. Sci., vol. 7, no. 3, 2011. doi: 10.2168/LMCS-7(3:3)2011.

• L. Schröder and D. Pattinson, ‘PSPACE bounds for rank-1

modal logics,’ ACM Trans. Comput. Log., vol. 10, no. 2, 13:1–13:33, 2009. doi: 10.1145/1462179.1462185.

• D. Hausmann and L. Schröder, ‘Game-based local model

checking for the coalgebraic mu-calculus,’ in Concurrency Theory, CONCUR 2019, ser. LIPIcs, vol. 140, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Aug. 2019, 35:1–35:16, isbn: 978-3-95977-121-4. doi: 10.4230/LIPIcs.CONCUR.2019.35.

Conclusion - 49/49

References IV

• L. de Alfaro, T. A. Henzinger, and R. Majumdar, ‘From

verification to control: Dynamic programs for

• mega-regular objectives,’ Proceedings 16th Annual IEEE

Symposium on Logic in Computer Science, 2001. doi: 10.1109/lics.2001.932504.