This paper proposes a logical framework for representing and reasoning about imperfect information games. We first extend the game description language (GDL) with the standard epistemic operators and provide it with a semantics based on the epistemic state transition model. We then demonstrate how to use the language to represent the rules of an imperfect information game and formalize its epistemic properties. We also show how to use the framework to reason about player's own as well as other players' knowledge during game playing. Finally we prove that the model-checking problem of the framework is in ∆ p 2 , which is the lowest among the existing similar frameworks, even though its lower bound is Θ p 2 . These results indicate that the framework makes a good balance between expressive power and computational efficiency.

Introduction

General Game Playing (GGP) is concerned with creating intelligent agents that understand the rules of arbitrary new games and learn to play these games without human intervention [START_REF] Genesereth | [END_REF]. Representing and reasoning about games is a core technique in GGP. A formal game description language (GDL) has been introduced as an official language for GGP since 2005. GDL is defined as a high-level, machine-processable language for representing the rules of arbitrary games [Love et al., 2006]. Originally designed for perfect information games, GDL has recently been extended to GDL-II so as to incorporate imperfect information games, such as Poker, Backgammon [Thielscher, 2010].

Playing games with imperfect information poses an intricate reasoning challenge for players since imperfect information requires a player to use the rules of a game to infer legal actions, draw conclusions from her own knowledge about the current game state and about knowledge of other agents. However, as a purely descriptive language, GDL-II is only a tool for describing the rules of an imperfect information game but does not provide a facility for reasoning about how a player infers unveiled information based on the rules [START_REF] Schiffel | Stephan Schiffel and Michael Thielscher. Reasoning about general games described in GDL-II[END_REF][START_REF] Schiffel | Stephan Schiffel and Michael Thielscher. Representing and reasoning about the rules of general games with imperfect information[END_REF]. Indeed, some information is essential for players to proceed with a game.

For example, players should always know their own available actions in non-terminal states and know their results in terminal states. Such epistemic properties of a game are normally implied by the game rules and thus need reasoning facilities to infer and verify them. Unfortunately, GDL-II (or GDL) is not designed for this purpose. To handle this issue, a few approaches have been proposed, mostly embedding GDL-II into a logical system, such as Situation Calculus or Alternating-time Temporal Epistemic Logic (ATEL), to use their reasoning facilities [START_REF] Schiffel | Stephan Schiffel and Michael Thielscher. Reasoning about general games described in GDL-II[END_REF][START_REF] Ruan | The epistemic logic behind the game description language[END_REF][START_REF] Ruan | [END_REF][START_REF] Huang | [END_REF]. As long as the targeting logics are expressive enough to interpret any GDL description, it is possible to use the inference mechanisms of these logics for reasoning about GDL-II games. However, a highly expressive logic may incur high complexity for reasoning tasks. For instance, [START_REF] Ruan | [END_REF] propose an adaption of ATEL to verify epistemic properties of GDL-II games, and show that the model-checking problem in that setting is 2EXPTIME-hard. Such high computational complexity may not be what we want.

This paper aims to propose a different approach to deal with this problem. We introduce a logical framework, called EGDL, equipped with a language for representing imperfect information games and a semantical model that can be used for reasoning about game information and players' epistemic status. More importantly, we develop a model-checking algorithm for EGDL and show that the complexity of the model-checking problem for the logic can be significantly reduced to ∆ p 2 . There are two major reasons that help us to reduce the complexity. Firstly, our language is a conservative extension of GDL with the standard epistemic operators [Fagin et al., 2003]. We take a cautious way of doing that without introducing the until operator or coalition operators. Secondly, we provide an imperfect recall semantics for knowledge. Other cases could be considered; nevertheless, the addition of perfect recall to GDL-II renders the model-checking problem of ATEL undecidable in general [START_REF] Ruan | [END_REF]. Also, in many applications, especially when modeling extremely large games, imperfect recall may provide considerably empirical and practical advantages [START_REF] Piccione | [END_REF]Waugh et al., 2009;Busard et al., 2015]. Despite of a moderate expressive power, we demonstrate with a running example that our language is enough for expressing game rules, formalizing essential epis-Epistemic GDL: A Logic for Representing and Reasoning about Imperfect Information Games Guifei Jiang, 1,2 Dongmo Zhang, 1 Laurent Perrussel, 2 and Heng Zhang 3 temic properties and specifying the interactions of knowledge and actions. In this sense, EGDL makes a good balance between expressive power and computational efficiency.

The rest of this paper is structured as follows: Section 2 establishes the syntax and the semantics of EGDL. Section 3 demonstrates its expressiveness and reasoning facility. Section 4 investigates the model-checking problem of EGDL. Finally, we conclude with a discussion of related work and future work.

The Framework

All games we consider in this paper are assumed to be played in multi-agent environments. A game signature S is a triple (N, A, Φ), where

• N = {1, 2, • • • , k} is a non-empty finite set of agents, • A = i∈N A i , where A i consists of a non-empty finite set of actions for agent i s.t. A i ∩ A j = ∅ if i = j, and • Φ = {p, q, • • • } is a finite set of propositional atoms for
specifying individual features of a game state.

Hereafter we will fix a game signature S and all concepts will be based on the same game signature unless otherwise specified.

Epistemic State Transition Models

We begin by introducing the semantical structures used to model synchronous games with imperfect information. By synchronous we mean that all players move simultaneously.

In particular, turn-based asynchronous games are modelled by only allowing the "noop" action for a player, when it is not her turn.

Definition 1. An epistemic state transition (ET) model M is a tuple (W, w, T, {R i } i∈N , L, U, g, V ), where • W is a non-empty set of possible states.

• w ∈ W , representing the initial state.

• T ⊆ W , representing the set of terminal states.

• R i ⊆ W × W is an equivalence relation for agent i, indicating the states that are indistinguishable for i. • L ⊆ W × A is a legality relation, describing the legal actions at each state.

• U : W × D → W is an update function, specifying state
transitions, where D is the set of joint actions i∈N A i .

• g : N → 2 W is a goal function, specifying the winning states for each agent.

• V : W → 2 Φ is a standard valuation function. For d ∈ D, let d(i) denote the i-th component of d.
That is, agent i's action in the joint action d. For convenience, let L(w) = {a ∈ A | (w, a) ∈ L} denote the set of all legal actions at state w. We now define the set of all possible ways in which a game can develop.

Definition 2. Let M = (W, w, T, {R i } i∈N , L, U, g, V ) be an ET-model. A path δ is a sequence of states and joint actions w

d1 → • • • de
→ w e such that for all 1 ≤ j ≤ e and any i ∈ N , 1. d j (i) ∈ L(w j-1 ) (that is, any action that is taken must be legal), 2. w j = U (w j-1 , d j ) (state transition), and 3. {w, • • • , w e-1 } ∩ T = ∅ (that is, only the last state may be terminal).

A path δ is complete if w e ∈ T . Let P(M ) denote the set of all complete paths in M . When M is fixed, we simply write P. Given δ ∈ P, the states on δ are called reachable states. Let δ[j] denote the j-th reachable state of δ and θ i (δ, j) denote the action of agent i taken at stage j of δ. The length of δ, written |δ|, is defined as the number of actions.

The following definition, by extending equivalence relations over states to complete paths, characterizes precisely what an agent with imperfect recall and perfect reasoning can in principle know at a specific stage of a game. Definition 3. Two complete paths δ, δ ′ ∈ P are imperfect recall (also called memoryless) equivalent for agent i at stage j ∈ N, written δ

≈ j i δ ′ , iff δ[j]R i δ ′ [j].
That is, imperfect recall requires an agent to be only aware of the present state but forget everything that happened. This is similar to the notion of imperfect recall in ATL [Schobbens, 2004]. It should be noted that the paper focuses on imperfect recall; nevertheless, the framework is flexible to define ATL state-based perfect recall [START_REF] Jamroga | Wojciech Jamroga and Wiebe van der Hoek. Agents that know how to play[END_REF] and GDL-II perfect recall [Thielscher, 2010].

To illustrate our framework, we use as our running example a variant of the Tic-Tac-Toe, called Krieg-Tictactoe in [START_REF] Schiffel | Stephan Schiffel and Michael Thielscher. Reasoning about general games described in GDL-II[END_REF].

Example. Krieg-Tictactoe is played by two players, cross x and naught o, who take turns marking grids in a 3 × 3 board. Different from standard Tic-Tac-Toe, each player can see her own marks, but not those of her opponent, just like the chess variant Kriegspiel [Pritchard, 1994]. Players are informed of the turn-taking. The game ends if the board is completely filled or one player wins by having completed a horizontal, vertical or diagonal line of three with her own symbol.

To represent Krieg-Tictactoe in terms of the ET-model, we first describe the game signature, written S KT , as follows: let

N KT = {x, o}, A i KT = {a i j,k | 1 ≤ j, k ≤ 3} ∪ {noop i }, and Φ KT = {p i j,k , tried(a i j,k ), turn(i) | i ∈ {x, o} and 1 ≤ j, k ≤ 3}
, where a i j,k denotes the action that player i fills grid (j, k) with her symbol, noop i denotes that player i does action noop, p i j,k represents the fact that grid (j, k) is filled with player i's symbol, tried(a i j,k ) represents the fact that player i has tried to fill grid (j, k) before, and turn(i) says that it is player i's turn now.

We next specify the ET-model for this game, written M KT , as follows: let

W KT = {(t x , t o , x 1,1 , • • • , x 3,3 ) : t x , t o ∈ {0, 1} & x 1,1 , • • • , x 3,3 ∈ { , ⊠, , ⊗,
⊙}} be the set of possible states, where t x , t o specify the turn taking and x i,j represents the fact that grid (i, j) is occupied by the cross and not tried by the nought ⊠, occupied by the nought and not tried by the cross , occupied by the nought and tried by the cross ⊗, occupied by the cross and tried by the nought ⊙, or empty . The initial state w is (1, 0, , • • • , ).

For any two states

w = (t x , t o , x 1,1 , • • • , x 3,3 ) and w ′ = (t ′ x , t ′ o , x ′ 1,1 , • • • , x ′ 3,3 ) in W KT , wR x w ′ iff (1) t i = t ′ i for any i ∈ N KT , (2) x j,k ∈ {⊠, ⊙} iff x ′ j,k ∈ {⊠, ⊙} for any 1 ≤ j, k ≤ 3, and (3) x j,k = ⊗ iff x ′ j,k = ⊗ for any 1 ≤ j, k ≤ 3. The equivalence relation for o is de- fined in a similar way. Let V KT be a valuation such that for each state w = (t x , t o , x 1,1 , • • • , x 3,3 ) ∈ W KT , V KT (w) = {turn(i) : t i = 1} ∪ {p x j,k : x j,k ∈ {⊠, ⊙}} ∪ {p o j,k : x j,k ∈ { , ⊗}} ∪ {tried(a x j,k ) : x j,k = ⊗} ∪ {tried(a o j,k ) : x j,k = ⊙}.
Moreover, we assume that each player takes the same action at stages of all her indistinguishable complete paths, i.e., θ i (δ, j) = θ i (δ ′ , j) whenever δ ≈ j i δ ′ . Due to the space limit, we refrain from explicitly listing the legality relation, the update function, and the terminal and goal states for the agents as this is possible but considerably lengthy. However, the syntactic descriptions of the game given in the following section detail them in a more compact way.

The Language

Let us now introduce an epistemic extension of a variant of GDL [Zhang and Thielscher, 2015b] to represent games with imperfect information, and further provide a semantics for the language based on the epistemic state transition model. We call this framework EGDL for short. Definition 4. The language, denoted by L, consists of

• the finite set Φ of propositional atoms;

• pre-defined propositions: initial, terminal, wins(•), legal(•) and does(•); • logical connectives ¬ and ∧; • temporal operator ; • epistemic operators: K and C.

A formula ϕ in L is defined by the following BNF:

ϕ ::= p |initial |terminal | legal(a i ) | wins(i) | does(a i ) | ¬ϕ | ϕ ∧ ψ | ϕ | K i ϕ | Cϕ where p ∈ Φ, i ∈ N and a i ∈ A i .
Other connectives ∨, →, ↔, ⊤, ⊥ are defined by ¬ and ∧ in a standard way. Intuitively, initial and terminal specify the initial state and the terminal states of a game, respectively; does(a i ) asserts that agent i takes action a at the current state; legal(a i ) asserts that action a is available to agent i at the current state; and wins(i) asserts that agent i wins at the current state. The formula ϕ means "ϕ holds in the next state". All these components are inherited from GDL. The epistemic operators K and C are taken from the Modal Epistemic Logic [Fagin et al., 2003]. The formula K i ϕ is read as "agent i knows ϕ", and Cϕ as "ϕ is common knowledge among all the agents in N ".

We use the following abbreviations in the rest of the paper:

Kiϕ = def ¬Ki¬ϕ Cϕ = def ¬C¬ϕ Eϕ = def i∈N Kiϕ
where K i and C are the dual operators of K i and C, respectively. K i ϕ says "ϕ is compatible with agent i's knowledge" and it is similar to Cϕ. Eϕ says "every agent in N knows ϕ".

Let us illustrate the intuition of the language with our running example. Example (continued) The rules of Krieg-Tictactoe are specified by EGDL in Figure 1.

1. initial ↔ turn(x) ∧ ¬turn(o) ∧ 3 j,k=1 (¬(p x j,k ∨ p o j,k ) ∧¬(tried(a x j,k ) ∨ tried(a o j,k ))) 2. wins(i) ↔ ( 3 j=1 2 l=0 p i j,1+l ) ∨ ( 3 k=1 2 l=0 p i 1+l,k ) ∨ ( 2 l=0 p i 1+l,1+l ) ∨ ( 2 l=0 p i 1+l,3-l ) 3. teminal ↔ wins(x) ∨ wins(o) ∨ 3 j,k=1 (p x j,k ∨ p o j,k ) 4. turn(i) → ¬turn(i) ∧ turn(-i) 5. legal(noop i ) ↔ turn(-i), where -i represents i's opponent 6. legal(a i j,k ) ↔ turn(i) ∧ ¬p i j,k ∧ ¬tried(a i j,k ) 7. p i j,k ↔ terminal ∨ p i j,k ∨ (does(a i j,k ) ∧ ¬(p x j,k ∨ p o j,k )) 8. tried(a i j,k ) ↔ terminal ∨ tried(a i j,k ) ∨ (does(a i j,k ) ∧ p -i j,k ) 9. does(a i j,k ) → Ki(does(a i j,k )) 10. initial → Einitial 11. (turn(i) → Eturn(i)) ∧ (¬turn(i) → E¬turn(i)) 12. (p i j,k → Kip i j,k ) ∧ (¬p i j,k → Ki¬p i j,k ) 13. (tried(a i j,k ) → Kitried(a i j,k ))∧(¬tried(a i j,k ) → Ki¬tried(a i j,k ))
Figure 1: An EGDL description of Krieg-Tictactoe.

The initial state, each player's winning states, the terminal states and the turn-taking are given by rules 1-4.

The preconditions of each action are specified by Rule 5 and Rule 6. The player who has the turn can fill any grid s.t. (i) it is not filled by herself, and (ii) she has never tried to fill the grid before. The other player can only do action noop.

Rules 7 and 8 are the combination of the frame axioms and the effect axioms [Reiter, 1991]. Rule 7 states that a grid is marked with a player's symbol in the next state if the player takes the corresponding action at the current state, or the grid has been filled by herself, or the game ends. Similarly, Rule 8 says that an action is tried by a player in the next state if the action is ineffective while still taken by the player at the current state, or it has been tried before, or the game ends.

The others are the epistemic rules. Rule 9 states each player knows which action she is taking. Rule 10 and Rule 11 says both players know the initial state and the turn-taking, respectively. Rule 12 says that each player knows which grid is filled or not by her own symbol. Similarly, Rule 13 states that each player knows which grid is tried or not by herself.

Let Σ KT be the set of rules 1-13. It should be noted that rules 11-13 together specify the epistemic relations for each player: two states are indistinguishable for a player if their configurations of the game board are the same in her view.

Semantics

The semantics of the language is based on the epistemic state transition model. Definition 5. Let M be an ET-model. Given a complete path δ in M , a stage j of δ and a formula ϕ ∈ L, we say ϕ is true at j of δ under M , denoted by M, δ, j |= ϕ, according to the following definition:

M, δ, j |= p iff p ∈ V (δ[j]) M, δ, j |= ¬ϕ iff M, δ, j |= ϕ M, δ, j |= ϕ1 ∧ ϕ2 iff M, δ, j |= ϕ1 and M, δ, j |= ϕ2 M, δ, j |= initial iff δ[j] = w M, δ, j |= terminal iff δ[j] ∈ T M, δ, j |= wins(i) iff δ[j] ∈ g(i) M, δ, j |= legal(a i ) iff a i ∈ L(δ[j]) M, δ, j |= does(a i ) iff θi(δ, j) = a i M, δ, j |= ϕ iff if j < |δ|, then M, δ, j + 1 |= ϕ M, δ, j |= Kiϕ iff for any δ ′ ∈ P with δ ≈ j i δ ′ , M, δ ′ , j |= ϕ M, δ, j |= Cϕ iff for any δ ′ ∈ P with δ ≈ j N δ ′ , M, δ ′ , j |= ϕ where ≈ j N is its transitive closure of i∈N ≈ j i . A formula ϕ is globally true in an ET-model M , written M |= ϕ, if M, δ, j |= ϕ for any δ ∈ P and any 0 ≤ j ≤ |δ|. A formula ϕ is valid, written |= ϕ, if M |= ϕ for any ET- model M . A formula ϕ is called true at a state w in M , written M, w |= ϕ, if it is
true for all complete paths going through w, i.e., M, δ, j |= ϕ for any δ ∈ P and any j ≥ 0 with δ[j] = w. Finally, let Σ be a set of formulas in L, then M is a model of Σ if M |= ϕ for all ϕ ∈ Σ.

We now show that EGDL provides a sound description for Krieg-Tictactoe. Proposition 1. M KT is a model of Σ KT .

It follows that these game rules are common knowledge among two players, which is just what we expect. Corollary 1. M KT |= Cϕ for all ϕ ∈ Σ KT .

Epistemic and Strategic Reasoning

In this section, we demonstrate the expressive power and flexibility of EGDL by showing how it allows us to specify epistemic properties and reason about agents' knowledge.

Epistemic Properties

The introduction of imperfect information raises new epistemic properties of a game. For instance, to make a game playable, each player should always know her own legal actions in the course of the game. This property as well as some other well-known properties can be naturally formulated by EGDL as follows: given i ∈ N and a

i ∈ A i , (1) initial → C initial (2) legal(a i ) → Ki(legal(a i )) (3) does(a i ) → Ki(does(a i )) (4) wins(i) → Ki(wins(i)) (5) terminal → C terminal
Formulas (1) and ( 5) express that the initial state, the terminal states are common knowledge, respectively. Formula (2) says that each agent knows her own legal actions. In ATEL, this is a required semantic property yet with no syntactic expression [ Ågotnes, 2006]. Formula (3) asserts that each agent is aware of her own actions. This is called the "uniform" property of actions (strategies) also with no syntactic counterpart in ATEL [ [START_REF] Van Der | [END_REF][START_REF] Jamroga | Wojciech Jamroga and Wiebe van der Hoek. Agents that know how to play[END_REF]. Finally, formula (4) specifies that each agent should know her winning result. Moreover, the above epistemic properties are precisely characterised by indistinguishable complete paths as follows:

Proposition 2. Let M be an ET-model. Then

1. M |= initial → C initial iff for all δ, δ ′ ∈ P and any j ∈ N, if δ ≈ j N δ ′ , then (δ[j] = w iff δ ′ [j] = w). 2. M |= legal(a i ) → Ki(legal(a i )) iff for all δ, δ ′ ∈ P and any j ∈ N, if δ ≈ j i δ ′ , then (a i ∈ L(δ[j]) iff a i ∈ L(δ ′ [j])). 3. M |= does(a i ) → Ki(does(a i )) iff for all δ, δ ′ ∈ P and any j ∈ N, if δ ≈ j i δ ′ , then (θi(δ, j) = a i iff θi(δ ′ , j) = a i ). 4. M |= wins(i) → Ki(wins(i)) iff for all δ, δ ′ ∈ P and any j ∈ N, if δ ≈ j i δ ′ , then (δ[j] ∈ g(i) iff δ ′ [j] ∈ g(i)). 5. M |= terminal → C terminal iff for all δ, δ ′ ∈ P and any j ∈ N, if δ ≈ j N δ ′ , then (δ[j] ∈ T iff δ ′ [j] ∈ T ).
Obviously, not all games with imperfect information satisfy these epistemic properties. For instance, property ( 5 for player o. Yet the left one is a terminal state while the right one is not. In fact, according to the game rules, Krieg-Tictactoe satisfies all the other properties.

Observation 1. Formulas ( 1)-( 4) are globally true in M KT .

It should be noted that each EGDL-formula may be interpreted as a property of a game. Typically, globally true formulas describe properties for a particular game, such as the rules for Krieg-Tictactoe, while valid formulas specify general properties of a class of games and thus can be used to classify games. For instance, different from Krieg-style board games, most card games have the property (5).

Strategic Reasoning

Let us now show how to use EGDL to reason about agents' knowledge and actions based on the game rules. In the context of imperfect information, epistemic reasoning is closely related to strategic reasoning. To start with, the following proposition shows that EGDL is suitable for reasoning about players' knowledge as it is a conservative extension of the standard Epistemic Modal Logic S5 C n [Fagin et al., 2003]. Proposition 3. Given an EGDL-formula ϕ without involving the operator and the pre-defined propositions, ϕ is valid in EGDL iff it is valid in S5 C n . This result indicates that EGDL is sufficient to provide a static characterization of agents' knowledge at a certain stage. For instance, with S5 C n , we can derive the following formulas from the rules of Krieg-Tictactoe. Observation 2.

1. MKT |= turn(i) → Cturn(i) 2. MKT |= Ki(K-ip -i j,k ∨ K-i¬p -i j,k ) 3. MKT |= Kitried(a i j,k ) → Kip -i j,k
Clause 1 says the turn-taking is common knowledge. Clause 2 says a player knows the opponent knows whether or not a grid is filled by herself. The last one says if a player knows she has tried an action, then she knows the corresponding grid has been filled by the opponent. The last two properties are important when players gather information. Furthermore, with the full expressive power of the language, we can use EGDL to specify agents' knowledge of particular game features and reason about how agent's knowledge changes as a game progresses. Observation 3.

1. MKT |= Kip i j,k → Ki p i j,k 2. MKT |= Ki tried(a i j,k ) → Kitried(a i j,k ) 3. MKT |= initial → C( 3 j,k=1 legal(a x j,k ) ∧ legal(noop o )) 4. MKT |= does(a i j,k ) → Ki(p i j,k ∨ tried(a i j,k ))
Intuitively, clause 1 says that if a player knows a grid has been filled by herself, then she still knows this fact holds at the next state. Clause 2 says that a player is able to remember the grid she has tried to fill before. Clause 3 says that at the initial state the legal actions are common knowledge among two players, and Clause 4 expresses that if a player takes an action now, then at the next state she will know either the corresponding grid has been filled by her symbol or she has tried that action. Most importantly, the interactions of actions and knowledge can be naturally formulated using EGDL. Specifically, they interact in three different ways:

(i) Knowledge is necessary for an agent to perform an action, which may be formulated by does(a i ) → K i ϕ. For instance, in Krieg-Tictactoe, with partial observation, a player might take an ineffective action by trying to fill a grid which has been filled by the opponent. Then we say a player i takes a good action a i j,k , written good(a i j,k ), if it is effective. It follows that, to take a good action, a player needs to know the grid she attempts to fill is empty. Formally,

M KT |= good(a i j,k ) → K i (¬(p x j,k ∨ p o j,k )).
(ii) Performing an action may increase an agent's knowledge, which may be specified by does(a i ) → K i ϕ. For example, if a player takes an ineffective action, then she would know the corresponding grid has been filled by the other player. Consider the following complete path

δ = w a x 2,2 ,noop o → w 1 noop x ,a o 2,2 → w 2 a x 1,1 ,noop o → w 3 • • • At stage 2
after player o tries to fill grid (2,2), by Rule 7 and Rule 13, she knows that the grid has been filled by player x.

Thus, M KT , δ, 1 |= does(a o 2,2 ) → K o (tried(a o 2,2 ) ∧ p x 2,2
). (iii) An agent makes her choice of actions based on her knowledge, which may be captured by K i ϕ → does(a i ). Let us consider the following two basic actions:

attack i = def Ki(does(a i j,k ) ∧ wins(i)) → does(a i j,k ) block i = def Ki (does(a -i j,k ) ∧ wins(-i)) → does(a i j,k )
Intuitively, attack says if a player knows that filling a grid leads to win, then she should fill that grid. Instead, block says if a player knows her opponent makes to win by filling a grid in the next state, then the player must fill that grid at the current state to avoid an immediate loss.

Model Checking

In this section, we investigate the complexity of the modelchecking problem for EGDL and develop a model-checking algorithm for EGDL.

The model checking problem for EGDL, denoted by EGDL-MC, is the following: Given an EGDL-formula ϕ, an ET-model M , a path δ of M and a stage j on δ, determine whether M, δ, j |= ϕ or not. In principle, two variants of EGDL-MC can be defined as follows: Given an ET-model M , a state w of M and an EGDL-formula ϕ, determine whether M, w |= ϕ and determine whether M |= ϕ. It should be noted that all the bounds presented in this section remain true for these variants. Proofs are similar to those of EGDL-MC, or can be obtained by simple reductions to/from EGDL-MC.

Let us first consider the upper bound of the complexity for model-checking. Our goal is to show the following bound. Theorem 1. EGDL-MC is in ∆ p 2 . To prove this upper bound, according to the definition of ∆ p 2 , we need to prove that there is a polynomial-time deterministic Turing machine M with an NP-oracle such that M solves the model-checking problem for EGDL. To show the existence, let us start with a simple property of EGDL.

Let ϕ be an EGDL-formula, and M = (G, V ) be an ETmodel over S. Take ψ to be any subformula of ϕ of the form ⊕ϑ, where ⊕ is either C or K i for some i ∈ N . We introduce a fresh propositional atom p ψ for ψ. Let M ψ be the ET-model (G, V ψ ) where V ψ is a valuation function defined by

V ψ (w) := V (w) ∪ {p ψ } if M, w |= ψ; V (w) otherwise
for each state w of M . Let ϕ ψ denote the formula obtained from ϕ by replacing ψ by p ψ . Then, by the definition of semantics for EGDL, the following property is clearly true. Lemma 1. For every path δ of M and every stage j on δ, it holds that M, δ, j |= ϕ iff M ψ , δ, j |= ϕ ψ . Thus, by applying the above lemma, the epistemic operators can be eliminated from the formula in a recursive way. For the EGDL-formulas without epistemic operators, we can show that its model-checking problem is tractable. Lemma 2. The following problem is in PTIME: Given an ET-model M , a path δ of M , a stage j on δ and an EGDLformula ϕ without involving any epistemic operators, determine whether M, δ, j |= ϕ or not.

Due to the space limit, we omit the proof here. Roughly speaking, in the context of model checking, the operator can be simply regarded as a standard modal operator. Note that model-checking for the basic modal logic is in PTIME.

To construct the model M ψ , the truth value of ψ under M and a given state is needed to be evaluated. To simplify the question, let us first consider a simple case as follows. Lemma 3. The following problem is in NP: Given an ETmodel M , a state w of M and an EGDL-formula ⊕ϕ where ⊕ ∈ { C} ∪ { Ki : i ∈ N } and ϕ does not involve any epistemic operators, determine whether M, w |= ⊕ϕ or not.

This lemma holds due to the observation that, given any path δ of M and any j ≥ 0, we have M, δ, j |= ϕ if, and only if, M, δ[j, j + k], 0 |= ϕ, where k is the number of occurrences of in ϕ, and δ[j, j + k] denotes the segment of δ starting from position j with length k. With it, one can design a nondeterministic Turing machine which first guesses a path of length ≤ k, and then check the truth value of ϕ under this path. By Lemma 2, the later can be done in PTIME.

To eliminate all the epistemic operators, it remains to consider the formulas with nested epistemic operator. With this complexity result for the non-nested case, we are now able to design an algorithm for the general case. Roughly speaking, the idea is to carry out the elimination of epistemic operator in a bottom-up way. As we can see in Algorithm 1, such an idea is implemented in the algorithm elimeop. It's easy to check that the algorithm can be implemented in a polynomial-time deterministic Turing machine, but with an NP-oracle. Here the procedure of checking N 0 , w |= ⊕¬ψ 0 is used as the NP-oracle. By Lemma 3, the checking is in NP. In addition, Algorithm elimeop visits each subformula of ϕ at most once, and that the number of subformulas of ϕ is not greater than the size of ϕ. These assure that the Turing machine will terminate in a polynomial number of stages.

With this algorithm, we can then devise an algorithm mc for the model-checking of EGDL such that, given any proper M, δ, j and ϕ as input, mc works as follows:

• Firstly, mc call the algorithm elimeop(M, ϕ), and let (M 0 , ϕ 0 ) be the results of this call. • Next, mc check whether M 0 , δ, j |= ϕ 0 or not, and return "true" if it holds, "false" otherwise. By Lemma 1 and the definition of algorithm elimeop, it is not difficult to verify that M, δ, j |= ϕ if, and only if, M 0 , δ, j |= ϕ 0 . This assures the soundness of the algorithm mc. On the other hand, by the previous analysis, the first stage can be implemented in a polynomial-time determinitic Turing machine with an NP-oracle; by Lemma 2, the second stage can be done in PTIME. Thus, the algorithm mc can be implemented in a polynomial-time determinitic Turing machine with an NP-oracle, which proves Theorem 1.

Next we identify a lower bound of the complexity of model-checking for EGDL.

Theorem 2. EGDL-MC is Θ p 2 -hard. Due to the space limit, we skip the proof here. The basic idea is to reduce the validity problem for Carnap's modal logic C to the model-checking problem of EGDL. The former has been proved to be Θ p 2 -complete [Gottlob, 1995]. It should be noted that the lower bound shows that the algorithm mc is nearly optimal, since ∆ p 2 and Θ p 2 are very close together, which both lie in the second level of the polynomial hierarchy.

Conclusion

We have presented a logical framework for representing and reasoning about imperfect information games with imperfect recall players. The framework allows us to represent game rules, formalize epistemic properties, specify the interactions of knowledge and actions as well as reason about agents' knowledge during game playing. We have also investigated the model-checking problem for the logic. These results indicate that we have made a reasonable compromise between expressive power and computational efficiency.

Most of the related work has been discussed in the Introduction. Besides that, the following is also worth mentioning. [START_REF] Ruan | The epistemic logic behind the game description language[END_REF] study the epistemic structure and expressiveness of GDL-II in terms of epistemic modal logic. They mainly provide a static characterization of players' knowledge at a certain stage without involving the temporal dimension. [START_REF] Haufe | Automated verification of epistemic properties for general game playing[END_REF] develop an automated reasoning method to deal with epistemic properties for GDL-II. Different from ours, their method is restricted to positive-epistemic formulas. Our underlying language is from [Zhang and Thielscher, 2015b]. It was originally proposed for reasoning about strategies of asynchronous games with perfect information, while we investigate its epistemic extension for representing and reasoning about synchronous games with imperfect information.

Directions of future research are manifold. As we have mentioned, besides imperfect recall, the framework is flexible enough to specify other memory types. To obtain a complete picture of the relation between perfect or imperfect information, and perfect or imperfect recall, we plan to study properties of these memory types; We also want to investigate the satisfiability problem and the axiomatization of EGDL based on the current literature [Zhang and Thielscher, 2015a;Halpern et al., 2004].

Figure 2 :

 2 Figure 2: The indistinguishable states for o

Acknowledgments

We are grateful to Michael Thielscher for his valuable suggestions, and special thanks are due to four anonymous ref-erees for their insightful comments. This research was partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02.


Lucas Eichenberger
email: lucas.eichenberger@univ-lorraine.fr

Michel François

Daniel Malterre

Romain Sibille

Nicolas Pietro

Maria Casati

Lucie Nataf

François Baudelet

Thomas Mazet

N P M Casati

Pressure-induced Yb valence transition and magnetism in YbMn 6 Ge 6-x Sn x

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

In condensed-matter physics, Yb-or Ce-based intermetallics are relevant candidates to study quantum critical effects such as non-Fermi-liquid behavior or unconventional superconductivity [1,[START_REF] Coleman | Handbook of Magnetism and Advanced Magnetic Materials[END_REF]. These exotic phases have been mainly investigated in materials where the rare-earth-metal element is alloyed with nonmagnetic elements [1,[START_REF] Morosan | [END_REF] and are often limited to very low temperature and trivalent rare earth metals [START_REF] Coleman | Handbook of Magnetism and Advanced Magnetic Materials[END_REF]4]. In most cases, the physics of such systems can be understood based on the Doniach picture [5], where Kondo screening competes with Ruderman-Kittel-Kasuya-Yoshida (RKKY) exchange interactions. Their relative magnitude can be varied using nonthermal control parameters (chemical substitution, pressure, magnetic field) to cross the rare-earthmetal magnetic instability where quantum critical effects may manifest [1,[START_REF] Coleman | Handbook of Magnetism and Advanced Magnetic Materials[END_REF]. Other heavy fermion materials deviate from this behavior and are termed unconventional [6][7][8][9] with, in some cases, quantum critical effects completely detached from the rare-earth-metal magnetic instability [10][11][12][13]. In all these materials, Yb or Ce are alloyed with nonmagnetic elements and, for the sake of simplicity, we hereafter term them standard materials.

In YbMn 6 Ge 6-x Sn x [14][15][16][17][18][19], the situation is rather different since the Mn sublattice, which orders magnetically near room temperature or above, interacts with intermediate valent Yb. The Mn(3d )-Yb(4 f ) exchange interaction is substantially stronger than 4 f -4 f RKKY interactions [20,21]. This results in unusually high magnetic ordering temperatures of intermediate valent Yb (up to T Yb ≈ 125 K) [17] and a shift of the magnetic instability toward strongly hybridized Yb (Yb valence υ ≈ 2.77 at the critical composition x c ≈ 5.23) [18]. This kind of heavy fermion systems also shows signatures of quantum criticality near x c , with an enhancement of the effective mass of the charge carriers [19] and of the Yb valence and 4 f magnetism [18]. This manifests by a marked peak in their composition dependence, in agreement with theories on unconventional quantum criticality which predict divergent valence and magnetic susceptibilities at the quantum critical point [22,23].

In YbMn 6 Ge 6-x Sn x , the hybridization between 4 f and conduction electrons, thus the Yb valence υ, is altered by chemical pressure using the isoelectronic substitution of Sn for Ge. The Yb valence ranges from υ ≈ 3 for x = 0.0 down to υ ≈ 2.6 for x = 6.0 [18]. Chemical or external pressure variation may modify physical properties through various mechanisms as the contraction of interatomic distances (in turn changing the magnetic exchange interactions and/or rare-earth-metal valence) [7,[24][25][26] or modifications of the electronic structure (increase of the bandwidth and depression of the electronic density of states at the Fermi level) [27,28]. In standard Yb-based heavy fermion materials, pressure increase promotes trivalent Yb and therefore Yb magnetism [7,[24][25][26], due to the decrease of the 4 f -conduction electron hybridization, which alters the balance between Kondo effect (T K ) and the RKKY interaction (T RKKY ) [START_REF] Thompson | Handbook on the Physics and Chemistry of Rare Earths[END_REF]. Ce-based materials behave oppositely, with an increase of the hybridization and loss of magnetism upon pressure increase.

In the present work, we investigate the influence of external pressure change on the crystal structure, Yb valence, and magnetism in YbMn 6 Ge 6-x Sn x , using x-ray diffraction, Yb L 3 edge x-ray absorption near edge spectroscopy (XANES), and x-ray magnetic circular dichroism (XMCD). We compare the effect of physical pressure to chemical pressure. This work is organized as follows: Section II provides the experimental details. The results are presented in Sec. III and discussed in Sec. IV before a short conclusion in Sec. V.

II. EXPERIMENTAL DETAILS

The polycrystalline YbMn 6 Ge 6-x Sn x samples investigated here (and the two related Lu-based samples) were prepared from high-purity elements as described in Refs. [14,17]. These two papers also report on the crystal and magnetic properties determined by x-ray diffraction, DC magnetization, and neutron diffraction measurements. Due to limited beam time, only a few samples were investigate with x-ray diffraction experiments under external pressure. The experiments were carried out up to ≈ 10 GPa at room temperature using the synchrotron radiation delivered by the beam line MS-X04SA (λ = 0.62 Å) at the Swiss Light Source (SLS, Paul Scherrer Institute, Villigen, Switzerland) [START_REF] Gozzo | [END_REF]. External pressure was applied using a diamond anvil cell with silicon oil as the pressure-transmitting medium. The pressure was evaluated from the variation of the quartz unit-cell parameters [31]. The x-ray diffraction patterns were analyzed by Le Bail refinements using the FULLPROF software [32].

The XANES and XMCD spectra at the Yb L 3 edge were recorded using the ODE beam line of the SOLEIL synchrotron (Saint-Aubin, France) [33]. The dichroism effect is given by the difference between the x-ray absorption intensities recorded under an applied field of 1.3 T aligned parallel and antiparallel to the beam while keeping constant the beam helicity. The experiments were carried out in the 5-300 K temperature range and under external pressure up to ≈ 50 GPa thanks to a diamond anvil cell with silicon oil as transmitting medium. The shift of the R1 fluorescence line of a ruby [34] was used to measure the pressure. Due to occasional failures of the high-pressure cell, some compositions were investigated in a more reduced pressure range.

III. EXPERIMENTAL RESULTS

A. X-ray diffraction under pressure X-ray diffraction experiments under pressure were carried out on five YbMn 6 Ge 6-x Sn x compositions (x = 0.00, 3.80, 4.25, 4.40, 5.80) as well as on two related Lu-based compounds (namely, LuMn 6 Ge 1.8 Sn 4.2 and LuMn 6 Sn 6 ), the latter two with a stable 4 f shell rare-earth-metal element serving as reference materials. Le Bail refinement of the data indicates that all compounds keep the HfFe 6 Ge 6 -type structure (P6/mmm), with one site for Mn, one site for Yb, and three sites for the p elements [14]. The cell volume contracts upon external pressure increase without structural transition up to the maximal applied pressure used of 10 GPa. Depending on composition, the volume contraction at 10 GPa lies between 7 and 10% of the ambient pressure unit cell volume V 0 [see Fig. 1(a)]. The contraction is slightly anisotropic since the ratio c/a increases with pressure [see inset of Fig. 1(a)]. By contrast, the ratio c/a decreases upon chemical pressure increase (i.e., upon decreasing the Sn content) [35]. This suggests that chemical and hydrostatic pressure variations do not have strictly the same effect. In YbRh 2 Si 2 , the ratio c/a also evolves dissimilarly depending on whether the contraction is due to hydrostatic pressure increase or to chemical pressure increase from Co for Rh substitution [7].

The bulk modulus K 0 , which characterizes the compression behavior under hydrostatic pressure, was extracted from the data by fitting a third-order Birch-Murnahgan equation of state [36]:

P = 3 2 K 0 V 0 V 7 3 - V 0 V 5 3 × 1 + 3 4 K 0 -4 V 0 V 2 3 -1 ,
where V 0 , K 0 , and K 0 are the ambient pressure volume, bulk modulus, and its pressure derivative, respectively. As an example, Fig. 1(b) shows the fit for x = 4.25.

In both the Yb and Lu series, K 0 decreases upon increasing the Sn content, indicating that compressing the alloys becomes easier [Fig. 1(c)]. This is most likely due to weaker chemical bonds with Sn than with Ge in such intermetallics [37,38]. In addition, our results reveal higher compressibility of compounds with intermediate valent Yb alloys [18] compared with the Lu-based compounds of similar tin content. This difference likely originates from the Yb 4 f 14 (spd ) 2 → 4 f 13 (spd ) 2 pressure induced gradual transition [25], absent in the stable 4 f shell Lu-based counterparts. The compressibility gap between the Yb and Lu series increases with the Sn content due to the concomitant reduction of the Yb valence [18].

B. XANES

XANES measurements have been performed at 5 K on several YbMn 6 Ge 6-x Sn x alloys (3.80 x 5.80) and also at 300 K for x = 4.95. The L 3 -edge XANES spectra of YbMn 6 Ge 0.2 Sn 5.8 recorded at 5 K upon increasing hydrostatic pressure are shown in Fig. 2

(a). A typical intermediate valent

Yb L 3 edge XANES spectrum consists of two structures: a main peak at 8.948 keV and a shoulder at 8.941 keV, which correspond to transitions from the initial state to the final states of mainly 2p 5 4 f 13 (trivalent) and 2p 5 4 f 14 (divalent) character, respectively. As expected [7,[24][25][26], increasing external pressure yields a spectral weight transfer from the Yb 2+ component toward the Yb 3+ one. The same is true for all compositions. The Yb valence υ is obtained by fitting the divalent and trivalent components as described in Refs. [18,39] Above 10 GPa, Yb reaches the fully trivalent state in almost all compounds, except in the richest Sn compound (x = 5.80) which remains slightly below trivalency at the highest applied pressure (υ max ∼ 2.98 at P ∼50 GPa). This points will be further discussed in Sec. IV.

Three kinds of behavior can be distinguished depending on the Sn content. For ( ∂υ ∂P ≈ 0GPa -1 ), which extends with the Sn content. Beyond this plateau, the Yb valence strongly increases with pressure ( ∂υ ∂P ≈ 0.03 GPa -1 ), as does the intermediate composition x = 4.95 at low pressure, before slowly reaching (5.23

x 5.30) or approaching (x = 5.80) the fully trivalent state above 10 GPa.

We now approximately evaluate the chemical pressure, to compare its influence with that of hydrostatic pressure, by combining the ambient pressure crystallographic data (see Refs. [14,17]) and data from x-ray diffraction under pressure (cf. Sec. III A). We proceed as follows. The richest Sn alloys (x = 5.8) with the larger cell volume is chosen as the zero of chemical pressure. We use the experimental υ(P) curve of x = 5.8 to locate the ambient pressure volumes of the other compounds with x < 5.8. For each composition, the corresponding pressure is taken as the chemical pressure [Fig. 3(a)]. For instance, for x = 4.65 the chemical pressure is estimated to be P chem = 4.1 GPa. were obtained by shifting of P chem those shown in Fig. 2(b). Most of the data points are then scattered around an S-shaped curve. This indicates that external and chemical pressure are almost equivalent. The main differences are observed for the Sn-rich compounds (x 5.23) for which the plateaus occur for unlike Yb valences. In addition, as mentioned above, the richest Sn alloys x = 5.8 does not reach the fully trivalent state at high pressure. This suggests that, in the Sn-rich region, hydrostatic pressure and isoelectronic Ge/Sn substitution are not strictly equivalent.

C. XMCD

The pressure-dependent Yb L 3 edge XMCD measurements were used to probe the local magnetism of Yb at 5 K. This is a qualitative study since sum rules do not apply to the L 3 edge of rare-earth metals [40]. A complete description of the XMCD spectra in YbMn 6 Ge 6-x Sn x can be found in Ref. [18]. The weak negative peak at E ≈8.944 keV is due to the hybridization of the 5d states of Yb with the self-polarized 3d states of Mn. The large positive peak at E ≈ 8.948 keV is ascribed to local exchange interaction with the 4 f spin. Thus, Yb 4 f magnetism yields the quadrupolar (E2) and the large dipolar contributions (E1) while the weak negative peak at E≈ 8.944 keV remains present and almost constant as long as the Mn sublattice is ordered. Thereafter, we only follow the pressure dependence of the main E1 contribution at 5 K (the evolution of the E2 contribution being very similar). The pressure dependence of the XMCD spectra of x = 3.8 and 5.8 is shown in Figs. 4(a) and 4(b), respectively. The pressure dependence of E1 is plotted in Fig. 4(c) for all investigated compositions.

External pressure induces an increase of the Yb valence (cf. Sec. III B) and thus strengthens Yb 4 f magnetism. This is evidenced by the strong increase of the E1 contribution under moderate external pressure (P < 10 GPa) [Fig. 4(c)]. For the Sn-rich alloys (x = 5.30 and 5.80), within which Yb is nonmagnetic at ambient pressure [18], the E1 and E2 contributions become nonzero beyond a composition-dependent critical pressure, proving the onset of 4 f magnetism. After the initial growth, the XMCD signal intensity keeps almost constant up to the highest pressure of the measurements (≈ 50 GPa) except for the three poorer Sn alloys (x = 3.80, 4.25, and 4.95) for which a sharp decrease of the XMCD signal is observed beyond a composition-dependent pressure P lim . This indicates that, although Yb is fully trivalent, it loses its magnetic character under high enough hydrostatic pressure.

IV. DISCUSSION

Our study shows that increasing the hydrostatic pressure or the chemical pressure through isoelectronic Ge for Sn substitution has globally similar effect: It yields an increase of the Yb valence and promotes Yb 4 f magnetism. However, besides the dissimilar influence of external and chemical pressure on the ratio c/a, some other differences manifest in the Sn-rich alloys (x 5.23).

When x 5.23, the Yb valence is found insensitive to low applied pressure (< 2 GPa) whereas the ambient pressure Yb valence evolves gradually with composition 5.23 x 5.80 [18], although in both cases the volume contraction is of comparable magnitude [a few percent; see Fig. 3(a)]. We can identify two possible origins for this discrepancy. First, the chemical pressure acts locally; the Ge for Sn substitution takes place on the metalloid 2d site [17] which is coplanar of Yb and in its first coordination shell. As a result, weak changes in the Ge/Sn content yields significant modification of the interatomic distances between Yb and its first neighbors. By contrast, hydrostatic pressure acts on the whole lattice, making it less influential on local interatomic distances around Yb. Additional x-ray diffraction experiments performed to allow Rietveld refinements should help to compare the effects of chemical and external pressure by following interatomic distances modifications. The second possible origin for the differences between chemical and physical pressure behaviors invokes electronic effects. Though Ge and Sn are isovalent, the dissimilarity between the Ge 4s4p and the less bound Sn 5s5p valence states could yield different electronic structures near the Fermi level that influence the Yb valence. For instance, in the magnetocaloric Gd 5 (Si x Ge 1-x ) 4 alloys, the influence of Ge for Si substitution arises predominantly from an electronic effect rather than from volume variation [41].

A difference in the electronic structure near the Fermi level might also explain the singular high-pressure behavior of x = 5.8-the sole alloy for which Yb does not reach trivalency and even tends to reduce under high hydrostatic pressure. Several examples reported in the literature [27,42,43] show that a saturation or a decrease of the Yb valence may be observed despite the concomitant reduction of the lattice parameters. This unusual decrease of the Yb valence with pressure cannot be explained with conventional cf hybridization. It might arise from the competition between two ways of coupling f -electrons and the lattice, as explained in Ref. [44], resulting in a pressure-induced enhancement of the Kondo effect due to an increase of both the hybridization strength and the width of the conduction band [43,44].

A last remark about the pressure dependence of the Yb valence involves intermediate compositions, in particular x = 4.95, whose Yb valence shows a strong sensitivity to low applied pressure variation [Fig. 3(a)], thus to weak interatomic distance changes (≈1%) comparable to those due to thermal contraction. As suggested in Refs. [15,16], this strong sensitivity could participate in the unusual increase of the Yb valence observed upon cooling for intermediate YbMn 6 Ge 6-x Sn x compositions (ca. 4 < x < 5), in addition to the effect due to the increasing exchange field.

At ambient pressure, the alloy with x = 5.23 (υ ≈ 2.77) is the richest Sn alloy whose low-temperature XMCD spectrum shows trace of 4 f magnetism [18]. It is thus located close to the Yb magnetic instability and the associated quantum criticality, while Yb is nonmagnetic in richer Sn alloys. Increasing the external pressure on the two alloys with initially nonmagnetic Yb (x = 5.30 and 5.80) allows stabilizing 4 f magnetism [see Figs. 4(a) and 4(b)], as observed in many standard Yb heavy fermion materials close to trivalency [24,45]. The low-pressure (P < 4 GPa) variation of the Yb valence and XMCD E1 contribution are shown together in Fig. 5 for these two alloys (x = 5.30 and 5.80). Both the Yb valence υ and the E1 contribution evolve similarly. We first note that the lower pressure point for which 4 f magnetism is detected in the XMCD spectra corresponds to a Yb valence of υ ≈ 2.76, almost identical to that reported at ambient pressure, but considerably lower than the valence at which Yb magnetism may appear in standard materials where Yb is alloyed with nonmagnetic elements. We further observe that the XMCD E1 contribution remains zero as long as the Yb valence keeps independent of pressure, and 4 f magnetism appears when the Yb valence starts to grow (i.e., when there is a marked slope change in the υ(P) curves), that is near ≈1 GPa and 2 GPa for, respectively, x = 5.30 and 5.80. Slope changes in the pressure dependence of the Yb valence-but generally corresponding to a slope reduction-are often considered as a marker of quantum criticality in standard heavy fermion materials [45][46][47][48].

In previous works [18,19], we identified the peak near x c ≈ 5.23 (whose base width is estimated to be x ≈ 0.05 from the data in Ref. [18]) in the composition dependence of the Yb valence, XMCD intensity, and electronic specific heat coefficient as signs of quantum criticality. We do not observe 3(a), the peak width is estimated to be P ≈ 0.2 GPa and located near P ≈ 2 GPa for x = 5.80. However, most of the pressure-dependent measurements have been performed before we realized the existence of that divergence, such that the pressure steps we used (at least ≈0.5 to 1 GPa) were too large and the peak might have been missed [see the shaded area in Fig. 5(b)]. Further, we used silicon oil as the pressuretransmitting medium, resulting in pressure gradients across the sample chamber (≈0.25 GPa for 2 GPa < P < 10 GPa) [50] that may hinder the detection of the divergence.

As noticed in Sec. III C, the E1 XMCD intensity dramatically decreases at high pressure in the three richest Ge investigated alloys (x = 3.80, 4.25, and 4.95) despite Yb is then fully trivalent [Fig. 4(c)]. The pressure at which the XMCD signal starts to fall off (P lim ) reduces upon increasing the Ge content, that is upon reducing the lattice parameters. Since this disappearance of 4 f Yb magnetism seems unrelated to Yb valence fluctuations, we propose it may come from the collapse of the Mn magnetic moment, in agreement with the almost flat XMCD signal of the x = 5.8 composition at ≈ 50 GPa [see Fig. 4(a)]. Since the unusually high temperature magnetic ordering of Yb is thought to be due to the strong Mn-Yb exchange interaction [15][16][17][18], the collapse of the Mn magnetism yields in turn the destabilization of the Yb sublattice magnetic order. The breakdown of Mn magnetism can be understood based on simple arguments in the framework of the Stoner's model [49]. At sufficiently high pressure [above P lim in Fig. 4(c)], the Mn-Mn distances are considerably reduced. The electronic bandwidth widens such that the density of states at the Fermi level becomes too low for the Stoner criterion to be satisfied, and in turn Mn loses its magnetic moment. Since the Mn-Mn distances shorten upon Ge for Sn substitution, P lim tends to decrease with the Ge content while the lost of Yb magnetism is not observed in the Sn-rich compositions.

V. CONCLUSION

We have investigated the pressure dependence of the crystal structure, Yb valence and Yb 4 f magnetism in YbMn 6 Ge 6-x Sn x using, respectively, x-ray diffraction, XANES, and XMCD measurements at the Yb L 3 edge. Our results show that external or chemical pressure have overall the same influence, as often observed in heavy fermion materials: A pressure increase promotes Yb trivalency and strengthens Yb magnetism. Some differences were observed, mainly in the Sn-rich region, that we tentatively ascribed to the local nature of chemical pressure changes and/or to different binding and spatial extent of the valence electrons of isoelectronic Ge and Sn.

Once plotted as a function of total pressure (= external + chemical), the Yb valence follows an S-shaped curve. At low external pressure, for alloys with initially nonmagnetic Yb, the Yb valence is insensitive to external pressure change, while chemical pressure variation yields a gradual change of the valence throughout the series. Upon further total pressure increase, the Yb valence changes quickly with pressure before slowing down when trivalency is approached. The strong slope increase at low pressure corresponds to the Yb magnetic instability, i.e., to the appearance of Yb magnetism. This instability occurs at nearly the same valence (υ ≈ 2.76) as for changes of the Sn content at ambient pressure. The pressureinduced magnetic instability might be connected to quantum critical effects, but we did not observe the singularity in the Yb valence and XMCD signal previously found at ambient pressure. This is likely due to the sparse pressure points available due to limited beam-time availability. We proposed that the fall of the XMCD signal at elevated pressure observed in the Ge-rich alloys might arise from the collapse of Mn magnetism due to 3d-band broadening.

Further experiments may be helpful for a better understanding of these atypical series and to verify some of our assumptions. For instance, x-ray data allowing Rietveld refinements to detect difference in interatomic distances variation upon changing chemical or external pressure as well as pressure-dependent XMCD measurements at the Mn K edge to verify the collapse of Mn magnetism.

FIG. 1 .

 1 FIG. 1. (a) Pressure variation of the normalized unit cell volume V /V 0 (V 0 is the ambient pressure unit cell volume) in YbMn 6 Ge 6-x Sn x and LuMn 6 Ge 6-x Sn x at 300 K. The insert shows c/a variation for Yb-based alloys. (b) Fit to third-order Birch-Murnahgan's equation of state of YbMn 6 Ge 1.75 Sn 4.25 . (c) Composition dependence of the bulk modulus K 0 in YbMn 6 Ge 6-x Sn x and LuMn 6 Ge 6-x Sn x .

  Figure 2(b) presents the pressure dependence of the Yb valence for the whole YbMn 6 Ge 6-x Sn x series.

  x 4.65, Yb is nearly trivalent at ambient pressure (υ > 2.95) and Yb valence increases slowly and continuously with pressure ( ∂υ ∂P ≈ 0.005 GPa -1 ) until saturation. For an intermediate composition (x = 4.95) the Yb valence increases sharply from the lowest pressures ( ∂υ ∂P ≈ 0.02 GPa -1 ) to reach trivalency above 8 GPa. For this composition only, the pressure dependence was also measured at room temperature and is almost identical to that recorded at low temperature [open blue diamonds in Fig. 2(b)]. Finally, in the Sn-rich compounds (x 5.23), Yb is farther from trivalency at ambient pressure (υ < 2.80). The υ(P) curves starts with a plateau-like behavior at low pressure

FIG. 2 .

 2 FIG. 2. (a) Yb L 3 XANES spectra of YbMn 6 Ge 0.2 Sn 5.8 at 5 K for different applied pressures. (b) Pressure dependence of the Yb valence at 5 K of YbMn 6 Ge 6-x Sn x extracted from the XANES spectra. The inset shows an enlargement of the low-pressure region (P < 10 GPa). Room-temperature data are also shown for x = 4.95 (open blue diamonds).

Figure 3 (

 3 FIG. 3. (a) Estimation of the initial chemical pressure P chem in YbMn 6 Ge 6-x Sn x . The pressure variation of the unit cell volume of the Sn-rich reference alloy x = 5.80 is taken from Fig. 1(a).Right scale shows the normalized unit cell volume, V 0 being the room-temperature lattice volume of x = 5.80. The room-temperature lattice volumes are also shown for all the measured compositions (data from Refs.[14,17]). (b) Yb valence at 5 K as a function of total pressure (P external + P chem ) of YbMn 6 Ge 6-x Sn x , inset showing an enlargement of the low-pressure region (P <10 GPa).

FIG. 4 .

 4 FIG. 4. (a) Yb L 3 XMCD spectra of YbMn 6 Ge 2.20 Sn 3.80 and (b) YbMn 6 Ge 0.20 Sn 5.80 at 5 K for different applied pressures. (c) Pressure dependence of the dipolar XMCD signal (E1) at 5 K in YbMn 6 Ge 6-x Sn x . P lim indicates the pressure at which the E1 signal starts to strongly decrease.

205146- 5 FIG. 5 .

 55 FIG. 5. Pressure dependence (P < 4 GPa) of the Yb valence (black squares, left scale) and of the Yb E1 XMCD contribution (red triangles, right scale) at 5 K for (a) YbMn 6 Ge 0.70 Sn 5.30 and (b) YbMn 6 Ge 0.20 Sn 5.80 . The shaded area shows the pressure domain where a peak in the pressure dependence of the Yb valence and XMCD signal is expected for x = 5.8 [18] (see text). such a peak in the pressure dependence of the Yb valence and E1 XMCD [Figs. 2(b), 4(c), and 5]. Based on the data in Fig.3(a), the peak width is estimated to be P ≈ 0.2 GPa and located near P ≈ 2 GPa for x = 5.80. However, most of the pressure-dependent measurements have been performed before we realized the existence of that divergence, such that the pressure steps we used (at least ≈0.5 to 1 GPa) were too large and the peak might have been missed [see the shaded area in Fig.5(b)]. Further, we used silicon oil as the pressuretransmitting medium, resulting in pressure gradients across the sample chamber (≈0.25 GPa for 2 GPa < P < 10 GPa)[50] that may hinder the detection of the divergence.As noticed in Sec. III C, the E1 XMCD intensity dramatically decreases at high pressure in the three richest Ge investigated alloys (x = 3.80, 4.25, and 4.95) despite Yb is then fully trivalent [Fig.4(c)]. The pressure at which the XMCD signal starts to fall off (P lim ) reduces upon increasing the Ge content, that is upon reducing the lattice parameters. Since this disappearance of 4 f Yb magnetism seems unrelated to Yb valence fluctuations, we propose it may come from the collapse of the Mn magnetic moment, in agreement with the almost flat XMCD signal of the x = 5.8 composition at ≈ 50 GPa [see Fig.4(a)]. Since the unusually high temperature magnetic ordering of Yb is thought to be due to the strong Mn-Yb exchange interaction[15][16][17][18], the collapse of the Mn magnetism yields in turn the destabilization of the Yb sublattice magnetic order. The breakdown of Mn

ACKNOWLEDGMENTS

We acknowledge the Paul Scherrer Institute, Villigen, Switzerland (Exp. No. 20131006) and the French Synchrotron facility SOLEIL, Saint-Aubin, France (Exps. No. 20130845, No. 20140775, and No. 20170707) for the allocated beam time.