Jérémy Turi
email: jeremy.turi@laas.fr

Arthur Bit-Monnot
email: abitmonnot@laas.fr

Guidance of a Refinement-based Acting Engine with a Hierarchical Temporal Planner

Endowing robotic platforms with advanced deliberation capabilities to support autonomous and intelligent behavior has been a key objective of the scientific community for several decades. Where many acting systems propose to first perform offline planning and then execute the plan, we propose here the OMPAS system, based on hierarchical operational models, which proposes to leverage planning as a guidance to an otherwise reactive system. For these hierarchical operational models, this paper proposes both a Lisp-based language to define them, and its automatic analysis to generate predictive models of the system's behavior that can be leveraged by a hierarchical planner. The generation of predictive models allows to have a unified language for execution and planning, while having a rich structure to program operational models.

Introduction

In robotics, deliberation has always been a key component of autonomous systems. Endowing them with advanced deliberation capabilities allows them to be deployed in complex environments, in which they will be able to make wise decisions based on knowledge about the world collected via perception systems, the mission they have to carry out, and their own capabilities, while being able to cope with unexpected events.

Acting systems such as the Procedural Reasoning System (PRS) [START_REF] Ingrand | PRS: A High Level Supervision and Control Language for Autonomous Mobile Robots[END_REF] and the Refinement Acting Engine (RAE) [START_REF] Ghallab | Automated Planning and Acting[END_REF] take advantage of a hierarchical representation of the agent skills, which allows reasoning about high-level goals, and the methods for achieving them by performing lower level tasks.

The agent skills can be represented as an acting domain A ∆ , a tuple (A, T, M t ) where A is the set actions (low-level skills), T the set of task (high-level skills), and M t the set of methods to perform tasks in T , each method representing a different way to perform an abstract task t ∈ T . In acting systems, a method is an executable procedure that we call an operational model.

As skills of an agent are defined in A ∆ , the acting engine will use this knowledge to achieve an abstract task t ∈ T by selecting a method m during the task refinement process. The refinement strategy of reactive systems is to execute an arbitrary method in a trial and repair manner until the mission succeeds. But such greedy approach has several shortcomings, as a suboptimal choice of method, can provoke inefficiencies, unnecessary failures or even lead the system into deadlocks. Such problems may be avoided by endowing the system with the capability of evaluating the long-term impact of a method. With such lookahead capabilities, the acting system is able to choose the method that will maximize its long-term success. One lookahead technique is planning, that exploits a model of the agent capabilities to modify its environment, to generate a sequence of commands called a plan to solve a goal.

As noted by [START_REF] Ghallab | Automated Planning and Acting[END_REF], planning and acting require different models. Planners exploit a descriptive model of the environment that makes explicit the dynamics of the environment (the actor's "know what"). On the other hand, the acting part typically relies on operational models (the actor's "know how"), which are defined using general purpose languages, such that existing rich programming tools can be used to define the agent behavior (including, e.g., error handling, conditions, loops). This discrepancy in representations can be the source of many integration problems, in particular when the descriptive model diverges from the observed evolution of the environment and the agent's capabilities.

In this paper, we focus on an integrated planning and acting engine that is centered on a hierarchical operational model, from which a descriptive model can be extracted for planning purposes. We propose to exploit planning to make an informed choice when selecting a procedure or method for a given task. To do that we propose a new implementation of RAE named Operational Model Planning and Acting System (OMPAS), together with a Lisp dialect based on the Scheme [START_REF] Moretti | The λ-scheme[END_REF]) variant to define operational models for RAE systems. Unlike previous implementations of RAE, this custom language enables the automated analysis of the operational model to extract planning domains from acting domains. We can then rely on existing planners (and in particular an extension of LCP (Bit-Monnot 2018) for hierarchical planning) to select the best method to achieve each task faced by the system.

Related Work

The deliberation community proposed several supervision approaches for acting systems. In the late 90s, PRS (In-grand et al. 1996) was proposed as a goal oriented acting system. The skills are represented in a hierarchical fashion, which eases the definition of the problem and permits to divide the resolution into smaller and easier problems. Like in PRS, RAE [START_REF] Ghallab | Automated Planning and Acting[END_REF] needs to refine high-level objectives into a set of lower-level objectives. While PRS proposes a goal-oriented representation, RAE uses a hierarchical task representation closer to a Hierarchical Task Network (HTN) [START_REF] Georgievski | An overview of hierarchical task network planning[END_REF] representation, which eases the use of planning techniques to guide the refinement process.

To maximize long-term efficiency and success rate of the execution platform, many acting systems work jointly with automated planners. A first approach is to first call a planner, wait for a plan, and then execute it. Propice-Plan (Despouys [START_REF] Despouys | Propice-Plan: Toward a Unified Framework for Planning and Execution[END_REF], proposed an architecture extending PRS with both (i) planning capabilities for long-term choices and (ii) an anytime anticipation module (based on simulated execution) which returns an estimation of the best method with an anytime algorithm. The addition of an anytime module makes the system more reactive, and less limited by the planner whose computation time largely dominates the one of other processes. IxTeT-eXeC [START_REF] Lemai | IxTeT-eXeC: planning, plan repair and execution control with time and resource management[END_REF]) is yet another approach where acting and planning are interleaved, the planning process being continuously updated with task status and state updates to generate a valid plan.

Extensions to RAE have been proposed to endow the system with lookahead capabilities similar to Propice-Plan to guide the choice on the method to achieve the goal, taking into account both the current state of the system, and possible refinements of subtasks. RAEPlan [START_REF] Patra | Acting and Planning Using Operational Models[END_REF]) is an anytime planning system using MCTS techniques to sample results of low-level commands in each method, which gives two metrics to sort methods: their cost and efficiency. It has been extended with UPOM [START_REF] Patra | Integrating Acting, Planning, and Learning in Hierarchical Operational Models[END_REF], taking into account the nondeterministic result of commands to guide the exploration of the search tree, and using learning techniques to speed up search and produce useful heuristics. Several techniques have been proposed along UPOM (i) Learnπ maps a tuple (τ, ξ) of a task and a state to a method, and is used when no time is allowed by the acting system to find a method, (ii) Learnπ i returns the best instantiation of arbitrary parameters for a method in a given state ξ and LearnH gives heuristic for the branching in UPOM. While they do speed up search and increase the quality of the chosen methods, no guarantee is given on the long-term validity of the plan, and on the possibility to reach the high-level goal. In this paper we propose to address those issues by guiding the refinement process thanks to planning.

Recent works by [START_REF] Bansod | A Constraint-Based Encoding for Domain-Independent Temporal Planning[END_REF] propose to extend the Run-Lazy-Lookahead algorithm first described by [START_REF] Ghallab | Automated Planning and Acting[END_REF] with Run-Lazy-Refineahead. While the former executes a plan π as far as possible, calling Lookahead again only when π ends or a plan simulator says that π will no longer work properly, Run-Lazy-Refineahead proposes to extend the former one with a hierarchical representation of the domain. The algorithm can be linked to any online HTN planner that provides the solution as a refined task network, and that provides control over its backtrack-ing features. This extension takes into account the whole refinement while checking the consistency of the plan, and thus fix some limitations of Run-Lazy-Lookahead. [START_REF] Bansod | A Constraint-Based Encoding for Domain-Independent Temporal Planning[END_REF] put forward the integration of RAE with IPy-Hop, an HTN Planner extending the previous iteration of Python planners Pyhop and GTPyHop [START_REF] Nau | GTPyhop: A hierarchical goal+ task planner implemented in Python[END_REF]. The use of Python to define operational models is justified as an easier approach to define an acting domain, and makes it more accessible. However, it limits the automatic analysis of operational models, and the extraction of descriptive models. Therefore, the definition of the planning domain relies on dedicated descriptive languages such as PDDL [START_REF] Fox | PDDL2.1: An Extension to PDDL for Expressing Temporal Planning Domains[END_REF] and ANML [START_REF] Smith | The ANML language[END_REF].

System Overview

OMPAS is an acting system based on RAE. Skills of an agent are modelled as hierarchical operational models, where an operational model is defined as a procedure, encoded in a general purpose language, that can be executed by the acting engine. The hierarchical combination of operational models allows the definition of abstract tasks, that can be refined by several methods. We define an acting domain A ∆ (A, T, M t ) as the structure representing the skills of an agent, where A is the set of actions also called primitive tasks, achieving low-level goals, T the set of abstract tasks, achieving high-level goals, and M t the set of methods, where each method m ∈ M t is a possible refinement of a task t ∈ T .

In this section, we present the overall architecture of OM-PAS as well as the main refinement process. Finally, we propose a specific operational language based on the Scheme variant of Lisp: Scheme OMPAS (SOMPAS).

Architecture of the system

OMPAS is a complete system that embeds the acting engine and a monitoring system, presented in Figure 1. The acting engine is connected to a platform, that is in charge of executing primitive commands and gathering the perceived state of the environment. In addition to update of the world, the platform communicates the current status of previously sent commands {Pending, Running, Failure, Success}. The monitor is an external process that can send tasks to the system or retrieve information about its internal status. The monitor can either be a program, or a human operator through the command line.

Refinement process

When facing a task t ∈ T the role of the acting engine is to choose one applicable method m ∈ M t that will achieve the high-level objective of t. The system typically has the choice between several methods, each representing a particular procedure that might be used to achieve it. Initially, the refinement engine should select an arbitrary method m ∈ M t such that m is applicable in the current state, which is verified thanks to its preconditions that define the set of states in which the execution of m is possible. When a method m is chosen, its body is executed and should result either in a success (i.e. achievement of t) or a failure. The body of a method might contain arbitrary logic (if conditions, loops, . . . ), computations as well as the requirement to execute subtasks in T or actions in A. If m succeeds, the task t is considered successful. However, if m fails the refinement system resorts to a retry procedure where it selects another (not previously tried) applicable method m ∈ M t and executes it. This procedure continues until either (i) a method succeeds, in which case the task is successful or (ii) all applicable methods have been tried unsuccessfully in which case the task is considered failed.

This task execution procedure is the same regardless of whether the execution was requested by an external process (t is a top level task) or as a result of executing a method of which t is a subtask.

Choices and programs

In the previous definition of the refinement process, it should be obvious that the algorithm will have to make choices, primarily regarding the choice of the method to execute. Ideally we would like the first selected method to be successful as the contrary will typically result in the controlled agent acting in useless, counterproductive or even dangerous manner. As the choice of the method is arbitrary, the refinement engine is free to use any heuristic to guide his choice. In this paper, we are interested in guiding this choice by the analysis of the impact of a method on the state evolution. A key requirement for this is the ability to inspect the body of methods to predict the behavior they will induce in the controlled agent.

Acting language

As methods can be thought of as programs, their definitions are made through an acting language. In this work we propose to use an expression oriented language based on Scheme. An expression can be either an atom (boolean, number, symbol or procedure), or a list of expressions. When the first element of a list is a function symbol, it can be evaluated as the application of the function with the rest of the list as parameters. As the language is expression oriented, we consider body of methods as a single expression. The execution of such a language is based on the recursive evaluation of the root expression by an Eval function (Algorithm 1). Eval takes as input an expression and returns its evaluation, resulting from the recursive computation of all subexpressions. As everything is defined as a symbol, an evaluation environment env is used to bind symbols to values (themselves expressions). The core of the implementation is based on the operators define (binds a symbol to an expression in env), begin (evaluates expressions sequentially and returns the result of the last one), if (evaluates expression conditionally), quote (returns the expression without evaluation) and lambda (creates a user-defined procedure). The language being purely functional, we consider all computations as pure (i.e. the result of a function application depends only on its parameters and has no side effects), except for a few identified acting primitives.

Acting primitives On top of Scheme, the acting language proposes several primitives offering ways to program parallelism, concurrency, and synchronization in the body of method. To keep the presentation focused, we only present here a restricted set of acting primitives that interest us in the present matter. Note that the following primitives are considered as impure.

• exec-task takes as argument a task label and a list of instantiated parameters. When it is an action, the engine sends a request to the platform to execute it. In the case of an abstract task, the refinement engine is called to find a suitable method to execute.

• read-state takes as argument a state-variable and returns its value at the time of evaluation.

Domain definition

Several facilities are provided to define an acting domain using the same language. We illustrate them with the specification of the well-known gripper domain. A robot robby should transport balls in different rooms. Robby is equipped with two grippers left and right, each gripper can carry one ball. The domain contains 3 primitive actions: (move ?from ?to), (pick ?ball ?room ?gripper) and (drop ?ball ?room ?gripper). The abstract task (pickand-drop ?b ?r) transports the ball ?b to room ?r. It can be refined by two methods m-noop, when it is already in the target room, and m-pick-and-drop for other cases. The state is represented via 3 state-functions: (i) at-robby : room returns the position of robby, (ii) at : ball → room returns the position of a ball or no place if robby is carrying the ball, (iii) carry : gripper → ball returns the ball that a gripper is holding or no ball if the gripper is empty. The following list of functions defines the acting domain.

• def-state-function takes as parameter the label of the statefunction, a list of parameters with their annotated types, and a typed result.

( d e fs t a t ef u n c t i o n c a r r y ' ( ? g g r i p p e r ) ' ( ? b b a l l ) )

• def-action takes as parameter the action label and a list of typed parameters.

( d e f -a c t i o n d r o p ' ( ? o b j b a l l ) ' ( ? room room ) ' ( ? g r i p p e r g r i p p e r ) )

• def-task takes as parameter the task label and a list of typed parameters.

( d e f -t a s k p i c k -and-d r o p ' ( ? b a l l b a l l ) ' ( ? room room ) )

• def-method takes the following list of parameters: :task the label of the task the method refines.

:params the list of typed parameters of the method. It contains the same list of parameters as the task, and more if necessary.

:pre-conditions a list of expressions that returns true if a method is applicable, an error otherwise.

:body the operational model of the method. In this section, we presented OMPAS, a supervision system using its own acting language to define operational models. The following section describes our exploitation of this language to extract descriptive models in the form of chronicles and the of a constraint-based planner to generate a plan.

Intermediate Representation of a Program

For our purpose of using planning as guidance of the refinement process in RAE, we propose to extract high-level descriptive models from the body of methods. The extraction process requires an intermediate representation that is more structured and amenable to automated analysis. In this section, we propose an interpretation of an expression (e.g. the body of a method) into a Single Static Assignment (SSA) form. A program in SSA has a structure like the following one:

. . .

t 1 : r 1 ← prim-expr 1 t 2 : r 2 ← prim-expr 2 t 3 : r 3 ← prim-expr 3 . . .
where each t i uniquely identifies a line, r i is a label associated to the value computed by a primitive expression that is only allowed to refer to previous labels. This intermediate representation defines all steps in the evaluation of an expression in terms of the low-level capabilities of the interpreter.

To derive this representation, we start from a single line,

t : r ← body
where body, a non-primitive expression, is the body of the method. Each line containing a non-primitive expression is systematically replaced by one or multiple lines based on a set of rules reflecting how each non-primitive expression should be interpreted.

For any SSA statement of the form, t : r ← expr we define below the rules for expanding the line based on the form of expr.

expr is an atom that evaluates to the value v. The corresponding SSA representation is the following:

t : r ← cst(v)
expr is a list (f e 1 ...e n ) In SOMPAS this corresponds to the application of the function f to the e i parameters (where each e i might be an arbitrary expression). Following the definition of the Eval function, it is expanded to:

t 0 : r 0 ← f t 1 : r 1 ← e 1 . . . t n : r n ← e n t n+1 : r ← apply(r 0 , r 1 , . . . , r n )
Note that after this expansion, other expansions will be triggered to, e.g., refine the computation e 1 into primitive expression or specialize the last line (function application) into a primitive expression depending on the nature of r 0 .

expr matches (def ine var val) The operator defines a name for the value val and returns nil. It is translated as

t 1 : r 1 ← val t 2 : r ← nil
and all subsequent uses of var name are replaced by r 1 .

expr matches (begin e 1 ...e n ) The operator begin evaluates sequentially a list of expressions, and returns the result of the last expression. It is translated as

t 1 : r 1 ← e 1 . . . t n : r ← e n
where the original label r is given the value of the last expression e n .

expr matches apply(r 0 , r 1 , . . . , r n ) where r 0 is a user defined function f with parameters x 1 . . . x n . In this case we replace the expression by the body of the function f where each parameter x i has been substituted by the corresponding r i value.

t : r ← body(f )[x i /r i ]
expr matches apply(r 0 , r 1 , . . . , r n ) where r 0 is the readstate primitive: t : r ← read-state(r 1 , . . . , r n ) expr matches apply(r 0 , r 1 , . . . , r n ) where r 0 is a task symbol. In the acting engine, a task is either a primitive task, or an abstract task that is refined into a method, and the task ends when the method ends. t : r ← exec-task(r 0 , r 1 , . . . , r n ) expr matches (if cond a b) The operator if first evaluates the expression cond that returns a boolean atom. If cond is true, a is evaluated and is the result of the expression, otherwise b will be evaluated. This is a form of conditional evaluation of the a and b expressions for which we rely on the existing semantics of tasks and methods. Hence, it is translated as, first an evaluation of the condition, and then the invocation of a newly created task τ that is given as parameter the result of cond.

t 1 : r 1 ← cond t 2 : r ← exec-task(τ, r 1 , . . . )
The synthetic task τ must be associated with two methods: one with precondition r 1 and body a and one with precondition ¬r 1 and body b. Note that for completeness, the synthetic task τ should also be given as parameter any variable that appears in the body of either a or b.

expr matches (quote e) The operator quote avoids evaluating an expression. The expression is therefore considered as an atom, and we obtain t : r ← cst(e) Termination If no more rules are applicable, then the program has been transformed into SSA, meaning that all expressions are in primitive form. In our case it means that each statement corresponds to one of the primitives: cst, read-state, exec-task or apply (with the restriction that the first parameter of apply must be a built-in function as all user-defined ones have been expanded).

A partial example of the conversion from a method into SSA form is given in Figure 2.

Translation into a Planning Model

Given a method m with a body in SSA form, we want to translate it into a form that is amenable for planning using a hierarchical temporal planner. For this, we first discuss how causes of failures can be identified when constructing the SSA form and then propose a way to encode the method into hierarchical chronicles, a rich temporal model for primitive and abstract tasks [START_REF] Godet | Chronicles for Representing Hierarchical Planning Problems with Time[END_REF].

Excluding error cases

The objective of planning is to find a choice of methods that would not result in any failure, i.e., that the final return value r is not err (r = err). This imposes a first requirement on the output of the program that can be propagated to other intermediate values.

Notably, an if-expression (if cond a b) is successful, if and only if the chosen branch is successful. If a branch cannot succeed then it must never be taken. For instance, if a is the err symbol, one can infer that cond should always be false in order to lead to a successful execution. Two notable cases where this happens is in the do and check constructs that are syntactic sugar around if expression where one branch returns an error. If a (do e 1 . . . e n ) is required to succeed, then all intermediate expressions are required to succeed as well. If a (check c) is required to succeed, then c is required to be true.

During the translation into SSA, this identification of necessarily successful outcomes and associated boolean constraints on conditions is done mechanically at each expansion, leading to a set of requirements on intermediate values.

Methods as chronicles

Having this SSA representation, it becomes quite natural to express them as predictive models adapted for planning and in particular into the formalism of hierarchical chronicles of [START_REF] Godet | Chronicles for Representing Hierarchical Planning Problems with Time[END_REF].

Each chronicle is associated to the task achieved by the method it represents. For a given method, a chronicle will contain a temporal variable for each of the t i timepoints and a variable for each of the r i intermediate results. In addition, a chronicle allows the definition of conditions that we exploit to represent the read-state primitive and subtasks that we exploit to represent the exec-task primitive. The other primitives will be translated as constraints, restricting the set of valid values for the variables.

For a given method, the corresponding chronicle is constructed by iterating through the SSA form and applying for each line the corresponding rule below.

Constant t i : r i ← cst(v) The constraint r i = v is added to the chronicle. Read state t i : r i ← read-state(sv, p 1 , . . . , p n ) The condition [t i , t i ] sv(p 1 , . . . , p n ) = r i is added to the chronicle. Subtask t i : r i ← exec-task(τ, p 1 , . . . , p n )
The subtask [t i , t i ] τ (p 1 , . . . , p n ) is added to the chronicle, together with variable t i and the constraint t i ≤ t i ≤ t i+1 . Note that in a hierarchical planner, all subtasks of a method must be successfully decomposed and do not provide any result value. The above translation is thus only valid when the r i result was identified as necessarily successful by the error propagation. Function application t i : r i ← apply(f, p 1 , . . . , p n ) A new variable r i is added to the chronicle, together with the constraint r i = f (p 1 , . . . , p n ). Note that by construction, f is a built-in function so it is assumed that the solver is able to handle it, possibly with semantic attachment.

Success and ordering constraints

We rely on the error analysis, to enforce that the plan is failure free if deemed valid by an automated planner. In particular, if a condition was detected as requiring a particular value, we enforce this as an additional constraint on the corresponding variable. We also enforce the order in the different statements, that is t i ≤ t i+1 for any two subsequent timepoints in the SSA.

Post processing

With this automated process, we would like to obtain a chronicle such as the one of Figure 3. However, our automated conversion of a method into a chronicle has several shortcomings: redundant constraints can be present, variables might be duplicated and some timepoints might not be used in expressions of the chronicle. We propose several post-processing steps in order to simplify the resulting variables : {s, e, ?ball, ?gripper, r1, r2, r3, r4, t1, t2} constraints

: {r1 = r2, r3 = r4, s ≤ t1 ≤ t2 ≤ e} conditions : {[s, s] at(?ball) = r1 [s, s] at-robby() = r2 [t2, t2] at-robby() = r3 [t2, t2] at(?ball) = r4} subtasks : {[s, t1] move(r2, r1)
[t2, e] pick(?ball, r4, ?gripper)} chronicle, and speed up planning by removing unnecessary variables and constraints.

Variable binding

The program analyzes the set of constraints, and for each equality constraint (e 1 = e 2 ), such that e 1 and e 2 are atoms, binds the atoms following some rules depending on the kind of the atoms. Three kinds of atoms are distinguishable: (i) constant values, (ii) parameter variables of the method and (iii) local variables, which are generated during the conversion into the SSA form (as timepoints and results). We use an algorithm inspired by the union-find algorithm [START_REF] Charguéraud | Verifying the Correctness and Amortized Complexity of a Union-Find Implementation in Separation Logic with Time Credits[END_REF] to bind two atoms a 1 and a 2 , such that binding a 2 to a 1 means that (i) a 1 replaces a 2 everywhere it appears in the program and (ii) the allowed values of a 1 are restricted to only contain allowed values of a 2 (i.e. their domains are intersected). 1 Note that as a consequence of an unsatisfiable constraint, this domain restriction might result in a variable with no allowed value (which is also a contradiction). Given two atoms a 1 and a 2 required to be equal, we propose to bind a 2 to a 1 in the following cases:

• a 1 is a constant and a 2 a variable (parameter or local)

• a 1 is a parameter and a 2 a local

• a 1 and a 2 are of the same kind. Note that this last case is symmetric with respect to a 1 and a 2 , so either a 1 or a 2 could be substituted.

After binding, we remove the constraint from the chronicle. Simplification of temporal constraints During the conversion process, some unnecessary constraints with timepoints not involved in any expression of the chronicle are still in the set of constraints. We call those timepoints ghost timepoints, and propose an algorithm to check if such timepoints can be removed without modifying the properties of the chronicle. For this purpose we use Point Algebra (PA) as defined by [START_REF] Gerevini | Incremental Qualitative Temporal Reasoning: Algorithms for the Point Algebra and the ORD-Horn Class[END_REF], to first check path consistency of the Temporal Network resulting from constraints of the chronicle, and then shrink the set of constraints by removing ghost timepoints.

We define p : (timepoint, timepoint) → {<, >, =, ≤ , ≥, =, , ⊥} as the PA relation between two timepoints. Any ghost timepoint t can be certainly removed from the chronicle in the following cases:

• t has at most one relation. Any relation it is implied in can also be removed from the chronicle. • t has two relations with timepoints t 1 and t 2 , with p 1 = p(t 1 , t ), p 2 = p(t , t 2 ), and p 3 = p(t 1 , t 2 ) the respective paths between t 1 and t , t and t 2 , and t 1 and t 2 in the network. We define p 3 = p 1 • p 2 as the path between t 1 and t 2 through t and state that p 3 can be replaced by p 3 = p 3 ∩ p 3 the resulting path constrained by both p 3 and p 3 as path consistency has been checked beforehand.

Relations p 1 and p 2 are then removed from the chronicle, and p 3 is overwritten by p 3 .

Here is an example where t 2 and t 3 can be removed from the chronicle.

variables : {t 1 , t 2 , t 3 , t 4 , r 1 } → {t 1 , t 4 , r 1 } constraints : {t 1 ≤ t 2 ≤ t 3 < t 4 , → {t 1 < t 4 } t 1 ≤ t 4 } subtasks : {[t 1 , t 4 ](t r 1 )} → {[t 1 , t 4 ](t r 1 )}

Merge conditions

To avoid duplications of read-state of the same state-variables at the same instant, the program merges conditions c 1 and c 2 , such that:

c 1 :[s 1 , e 1 ] sf (a 1 , ..., a n ) = x c 2 :[s 2 , e 2 ] sf (b 1 , ..., b n ) = y If s 1 = s 2 ∧ e 1 = e 2 ∧ a 1 = b 1 ∧ ... ∧ a n = b n ,

Planning-based Method Selection

Planning-based lookahead while Acting

Having now a hierarchical predictive model of available methods, we would like to exploit it to guide the acting system in the selection of the method to apply when facing a task t to execute, using a hierarchical planner. This is done through the PLANSELECTMETHOD of Algorithm 2. The most important part of the algorithm is the invocation of the planner (line 7) that, based on the current state ξ, will attempt to find a plan that refines the task t. This plan is then analyzed (line 8) to identify the method m that is used to refine t. If this succeeds and the method is one of the allowed methods in M , then the method is returned to be attempted by the acting engine (line 10). Otherwise, the system might resort to an arbitrary heuristic to select one of the allowed methods (line 12). The latter case might notably occur if the translation into chronicles fails (e.g. due to unsupported language features in the translation) or if the planner fails to find a plan within its allocated time.

Note that if t is a subtask of a method that was selected based on a plan π, then t will also appear in this earlier plan. When this occurs it is possible to check whether the previous plan π is still valid and if this is the case, avoid calling the planner to compute a new one (lines 5-6).

Algorithm 2 Plan-based method selection for a task τ .

1: function PLANSELECTMETHOD(τ )

2: return arbitrary m ∈ M

m ← ∅ 3: ξ ← GET-STATE 4: M ← APPLICABLE(τ , ξ) \ TRIED(τ ) 5: π ← GET-PARENT-PLAN(τ ) 6: if π = ∅ or not IS-VALID(π) then 7: π ← CALL-PLANNER(τ , ξ) 8: m ← GET-METHOD-FROM-PLAN(π, τ ) 9: if m = ∅ ∧ m ∈ M then

Generation of the planning problem

In order to implement the CALLPLANNER function, we need a set of chronicles in order to represent (i) the initial state and objectives, (ii) the available actions and (iii) the available methods. While we have already covered how to generate a chronicle template for each method, we now turn our attention to the generation of the initial chronicle and of the action models.

Initial state generation

The initial chronicle C 0 is instantiated with the current state ξ, such that for each state variable sv(p 1 , ..., p n ) with value v of the system, we define an initial effect [0, 0] sv(p 1 , ..., p n ) ← v. The task t to be refined is defined as a subtask of the initial chronicle.

Assuming we want to select a method for a task (pick-anddrop b1 kitchen) in a state where robby is in the living-room and the ball is currently in the bedroom, we would obtain a state like the following effects : {[0, 0] at(b1) ← bedroom, [0, 0] at-robby() ← living-room, . . . } subtasks : {[0, t] pick-and-drop(b1, kitchen)}

In nominal functioning, OMPAS will take a snapshot of the state returned by the platform. We can also use OMPAS as a simulator, in which the execution is simulated by the evaluation of the operational models of actions.

Primitive Action Model To state the planning problem, we suppose in this work that models of actions are available and can be converted into chronicles. In practice, we exploit a PDDL-like [START_REF] Fox | PDDL2.1: An Extension to PDDL for Expressing Temporal Planning Domains[END_REF] representation of an action, with :params a set of typed parameters, :pre-conditions a set of pre-conditions, :effects a set of effects. The above def-action-model macro expresses this as an operational model that can be converted into a chronicle (using the same procedure as for methods). Note the presence of the assert keyword that sets the value of a state variable. It is only allowed in primitive actions and will (i) be mapped to a new set-state primitive in SSA and (ii) results in an effect in the chronicle describing the action.

Preliminary Results

We experimented the present integration with Aries 2 (a hierarchical extension of the LCP planner) on a selection of domains. One of them is gripper-door in which several robots must displace balls in rooms that are connected by doors. An abstract task t move is used for the displacement of a robot in a target room, which associated methods are m noop that is applicable if the robot is already at the goal, and m recursive that is applicable if the robot is not at the goal position, the method performs the action move to an arbitrary location that is connected to the current position of the robot, and call again t move. As in gripper, the state is defined by at, and carry; at-rob : robot → room gives the position of a robot, and connected : (room, room) → bool states if it is possible to pass from a room to another. Note that in some problems, some doors can be passed in one direction only, possibly leading robots into loops or dead ends.

In this work we compare (1) the reactive system, (2) the system using Aries, and (3) the system using the optimal version of Aries (Aries-opt) in terms of total number of primitive actions and total refinement time to complete the task.

The plots in Figure 4 present the mean results on a series of runs for each of the 10 problems. For each problem we ran in simulation 30 times in reactive mode, 10 times with Aries, and once with Aries-opt. For small problems, both the total refinement time and the number of executed actions are equivalent between different techniques. However, as the complexity of the problem increases with the addition of rooms and connections, both Aries and Aries-opt outperform the purely reactive system on the total refinement time and the number of actions. This is explained by the structure of the problems, which makes the reactive system perform useless move actions, as it has no heuristic on the distance he is from the final destination, while planning systems finds a path in the topology of rooms. We can note that Aries and Aries-opt find very quickly solutions, with the planning cost compensated by the absence of failures (which in turn reduce the burden of selecting a recovery method).

Conclusion and Future works

The present work proposed a successful integration of a reactive system capable of generating temporal planning problems from the automatic analysis of operational models. While a limited number of primitives are currently analyzed, our preliminary results on selected acting domains show promising results on the capacity of the system to produce a totally refined plan to advise the supervisor on the method to choose. Even if those results are on simple acting domains, the approach is general enough to adapt to larger and more complex acting domains, and will be tested in the continuity of this work.

We aim to extend our system to support further programming patterns, such as loops, and translate it as a planning representation, to tackle a wider set of problems, and enrich its capacity to use planning on more acting domains. One limitation of the current work is that the translation requires the total success of subtasks, and does not take into account the capability of an operational model to handle failures locally. Following iterations of the system will address this problem by taking into account the result of the execution of a task in the chronicle representation.

Figure 1 :

 1 Figure 1: A global view on the architecture of OMPAS.

  ( d e f -method m1 ' ( ( : t a s k p i c k -and-d r o p ) ( : p a r a m s ( ? b a l l b a l l ) ( ? room room ) ( ? g r i p p e r g r i p p e r ) ( ? d e p a r t u r e room ) ) ( : p r e -c o n d i t i o n s ( = ( c a r r y ? g r i p p e r ) n o b a l l ) ( = ? d e p a r t u r e ( a t -r o b b y ) ) ) ( : body ( do ( i f ( ! = ( a t ? b a l l ) ( a t -r o b b y ) )( move ? d e p a r t u r e ( a t ? b a l l ) ) ) ( p i c k ? b a l l ? d e p a r t u r e ? g r i p p e r ) ( move ? d e p a r t u r e ? room ) ( d r o p ? b a l l ? room ? g r i p p e r ) ) ) ) )• Types and constants: A domain can define new types and constants. As the engine does not support undefined values for state-variable yet, we define constants no ball to represent the absence of ball, and no place for the representation of the lack of information about the exact position of a ball.

(

  d e f -t y p e s room g r i p p e r b a l l ) ( d e f -c o n s t a n t s ' ( l e f t r i g h t g r i p p e r ) ' ( n o p l a c e room ) ' ( n o b a l l b a l l ) )

  a t -r o b b y ) ( a t ? b a l l ) ) ( c h e c k ( = ( a t -r o b b y ) ( a t ? b a l l ) ) ) ( p i c k ? b a l l ( a t ? b a l l ) ? g r i p p e r ) ) ) (a) The body of a method that first move to the location of a ball, check if the action succeeded, and then pick the ball. t1 : r1 ← cst(move) t2 : r2 ← cst(at-robby) t3 : r3 ← read-state(r2) t4 : r4 ← cst(at) t5 : r5 ← cst(value(?ball)) t6 : r6 ← read-state(r4, r5) t7 : r7 ← exec-task(r1, r3, r6) (b) Part of the SSA form.

Figure 2 :

 2 Figure 2: An example of conversion of the body of a method into SSA form.

Figure 3 :

 3 Figure 3: The chronicle resulting from the conversion of the method of figure 2a (after post processing).

  then x and y are bound, and c 2 is removed. Here is an example of use case and the resulting chronicle.variables : {s, ?b1, x, y, r1, r2} → {s, ?b1, x, y, r1} constraints: {r1 = x, r2 = y} → {r1 = x, r1 = y} conditions : {[s] at(?b1) = r1 → {[s] at(?b1) = r1} [s] at(?b1) = r2}Simplify constraints We propose to simplify constraints of the form (true = c) where c is a constraint and replace it directly by c, which gives (true = c) → c. Remove useless variables During the translation into the SSA form, many temporal and result variables are created. The set of variables is analyzed to remove from the chronicle any local variable l which is neither in the set of constraints, conditions, effects nor subtasks.

(

  d e f -a c t i o n -model move ' ( ( : p a r a m s ( ? from room ) ( ? t o room ) ) ( : p r e -c o n d i t i o n s ( = ( a t -r o b b y ) ? from ) ( ! = ? from ? t o ) ) ( : e f f e c t s ( a s s e r t ' a t -r o b b y ? t o ) ) ) )

  2 https://github.com/plaans/aries Mean number of executed actions in function of the number of rooms defined in the problem. Note that the number of actions is strictly the same using either Aries or Aries-opt. Mean of the total refinement time (in seconds) in function of the number of rooms defined in the problem.

Figure 4 :

 4 Figure 4: Comparison of the number of executed actions and the total refinement time for several sizes of problems in the gripper-door domain in simulation.

Algorithm 1

 1 Overview of the recursive evaluation of expressions in Lisp function EVAL(expr, env) if expr is an atom then return VALUE(expr) else if expr is a list then f ← EVAL(expr[0], env)

	args ← []
	for i do in [1..|expr|]
	arg ← EVAL(expr[i], env)
	PUSH(args, arg)
	return APPLY(f, args)

To make this fully general, note that a constant can be interpreted as a variable with a single allowed value.


Lea Gimeno

Clemence Queffelec

Errol Blart

Yann Pellegrin
email: yann.pellegrin@univ-nantes.fr

Copper(I) bis(diimine) complexes with high photo-oxidation power: reductive quenching of the excited state with benzimidazoline sacrificial donor

Keywords: reductive quenching, copper(I) complexes, photochemistry, benzimidazoline, dipyrridophenazine spacer

The reductive quenching of photo-excited photosensitizers is a very efficient way to achieve challenging reduction reactions. In this process, the excited photosensitizer is reduced by a sacrificial electron donor. This mechanism is rarely observed with copper(I)-bis(diimine) complexes which are nevertheless acknowledged as very promising photosensitizers. This is due to the fact that they are very poor photo-oxidants, and prove unable to react with common donors once promoted in their excited state. In this contribution, we evidence the rare reductive quenching cycle with two specially designed copper(I) complexes. These complexes exhibit improved photo-oxidation power, thanks to an optimized coordination sphere made of strongly π-accepting ligands. Reductive quenching of the excited state of the latter complexes with a classical benzimidazoline sacrificial donor is monitored and reduced complexes are accumulated during prolonged photolysis. Trials to utilize the photo-generated reductive power are presented.

Introduction

Heavy metal based complexes are very useful photosensitizers because they are stable, feature long lived excited states, luminescence in the visible, and versatile redox properties in their excited state. However, heavy metals are rare and thus expensive, and are potentially noxious. A lot of efforts are now devoted to developing Earth abundant photosensitizers. [START_REF] Hockin | Photoredox catalysts based on earth-abundant metal complexes[END_REF] Among them, copper(I) complexes have been abundantly used in the fields of lighting devices, 2 photovoltaics, 3 solar fuel synthesis 4 and organic photochemistry. 5 There are several copper(I) species which can act as photosensitizers, among them copper(I) bis(diimine) complexes (where diimines are for example phenanthroline ligands bearing bulky groups R in α of the nitrogen atoms). Those complexes are particularly interesting because they are strongly colored in the visible (broad metal to ligand charge transfer (MLCT) spreading between 400 and 550 to 600 nm, depending on diimine ligand), they exhibit luminescence in the near infrared and a significantly long lifetime of the excited state (from ca. 70 ns for [Cu(dmp)2] + to 3.26 µs for [Cu(dtbp)2] + , [START_REF] Green | Photophysical Characteristics and Reactivity of Bis(2,9-di-tert-butyl-1,10-phenanthroline)copper(I)[END_REF] where dmp and dtbp respectfully stand for 2,9-dimethyl-1,10-phenanthroline and 2,9-di-tertbutyl-1,10-phenanthroline).

As can be observed, the overall properties of copper(I)-bis(diimine) complexes are strongly impacted by the diimine ligand and in particular by the nature of the bulky group R. Mostly, the diimine ligand is a phenanthroline derivative. It is mandatory to tether such bulky group in positions 2 and 9 of phenanthroline in order to observe long-lived, exploitable excited state properties. Indeed, upon excitation in the MLCT, an electron shifts from copper centered d orbitals to ligand centered π* empty orbitals. Thus, the copper complex transiently changes from a tetrahedral copper(I) complex to a square planar copper(II) complex before returning to the ground state. This translates into a photoinduced flattening of the coordination sphere of the complex, and opens several deactivation channels leading to the premature quenching of the excited state. Preventing such flattening by grafting bulky groups R in α of the nitrogen atoms improves the lifetime of the excited state. [START_REF] Armaroli | Photochemistry and photophysics of coordination compounds: copper[END_REF] Many complexes with different R groups on ligand L have been isolated and studied in the literature and a few of them have been used as photosensitizers to drive photochemical reactions. In most cases, the excited state of the complexes was oxidatively quenched by an electron acceptor (either a catalyst or an organic reagent). Indeed, homoleptic copper(I) complexes are very potent reductants in their excited state, even outcompeting famous [Ru(bpy)3] 2+ . Interestingly, there has been many less reports where the excited state of a homoleptic copper(I) complex [CuL2] + * is reductively quenched by an electron donor. 8 This is explained by the very weak oxidative power of [CuL2] + *, a common feature of all traditional homoleptic copper(I) complexes. 8a,8b,9 Yet, implementing the reductive quenching of [CuL2] + * would be an asset in the field of photochemistry. Indeed, reduction of photo-excited complex [CuL2] + * by an electron donor in solution leads to [Cu(L)(L •-)] 0 which can be a very potent reductant (e.g. -1.65 V vs. SCE for L = 2,9-diisopropyl-1,10-phenanthroline, designated as dipp in what follows). [START_REF] Garakyaraghi | Cuprous Phenanthroline MLCT Chromophore Featuring Synthetically Tailored Photophysics[END_REF] In addition, [Cu(L)(L •-)] 0 accumulates in the mixture during irradiation (if the electron donor is irreversibly oxidized) thus facilitating collisional reactions with substrates or catalysts.

Aiming at circumventing the low oxidation potential of [CuL2] + *, attempts to increase E([CuL2] + */ [Cu(L)(L •-)] 0 ) have been made. From the Rehm and Weller equation: 00 , and E 00 is the energy of the excited state). E* allows to estimate the photo-oxidation power. One strategy to increase E* consists in increasing E 00 , 8c another in shifting Ered to less negative potential. 8b,11 The LUMO of homoleptic copper(I) complexes [CuL2] + is located on the phenanthroline ligands L; therefore tethering electron withdrawing groups 8b,12 of fusing aromatic cycles onto L 11 leads to a stabilization of the LUMO and a shift of Ered([CuL2] + ) towards less cathodic potentials. This strategy was successfully implemented by Cunningham et al. with complex [Cu(TPTAP)2] + (Figure 1a, TPTAP = 2,3,[START_REF] Green | Photophysical Characteristics and Reactivity of Bis(2,9-di-tert-butyl-1,10-phenanthroline)copper(I)[END_REF]4,5, In this case, Ered([Cu(TPTAP)2] + ) = ca. -0.9 V vs. SCE which is considerably less cathodic than Ered([Cu(dipp)2] + ) = -1.65 V vs. SCE for example. Using equation 1, a spectacularly high oxidation potential of the excited state of [Cu(TPTAP)2] + of approximately 0.8 vs. SCE was estimated. This value is comparable to E* for [Ru(bpy)3] 2+ and significantly higher than E* for most traditional homoleptic copper(I) complexes (for example, E* = 0.3 V vs. SCE for [Cu(dipp)2] + ). Conclusively, RQ was observed when [Cu(TPTAP)2] + was illuminated in presence of diversely substituted ferrocene derivatives Fc' as electron donors. However, Fc' + /Fc' are reversible couples. Therefore the reduced species [Cu I (TPTAP •-)(TPTAP)] 0 quickly recombined with oxidized ferrocene Fc' + and could not be accumulated in the photolyzed mixture. The aim of the present work is to design copper(I) complexes with high E* (at least 0.8 V vs. SCE) by tuning their reduction potential, and to observe the productive reductive quenching of their excited state by a sacrificial donor, namely a species which is irreversibly oxidized by the photoexcited complexes. Inspired by Cunningham's work, we prepared two copper(I) complexes bearing the electron depleted ligand 3, 6-di-nbutyl-dipyrido[3,2-a:2',3'-c]phenazine (L A , Figure 1b) whose synthesis was reported earlier by us. [START_REF] Soulis | New luminescent copper(I) complexes with extended π-conjugation[END_REF] Ligand L A is a derivative of famous dppz, 13 (dppz = dipyrido[3,2-a:2′,3′-c]phenazine ) functionalized in α of the chelating nitrogen atoms by n-butyl chains to promote the excited state properties of the corresponding copper(I) complex. The reduction potentials of such ligands and associated coordination complexes are known to be shifted towards less negative potentials compared to plain phenanthroline. This guarantees that E*([Cu(L A )2] + ) will be more positive and that RQ with a given sacrificial donor will be encouraged.

E* = Ered + E 00 (equation 1) (where E* = E([CuL2] + */ [Cu(L)(L •-)] 0 ), Ered = E([CuL2] + / [Cu(L)(L •-)] 0 ) + E
We investigated two complexes: heteroleptic [Cu(L A )(L B )] + and homoleptic [Cu(L A )2] + (Cu1 and Cu2 respectively, Figure 1c) bearing respectively one and two ligands L A . Heteroleptic copper(bis-diimine) complexes are usually not kinetically stable because the coordination sphere around Cu(I) is very labile. Here, we managed nevertheless to isolate stable Cu1 using Schmittel's strategy, also known as HETPHEN concept (HETPHEN = HETeroleptic PHENanthroline complexes). 14 This concept relies on the use of very bulky ligands such as 2,9-dimesityl-1,10-phenanthroline (L B ). L B bears two mesityl groups in positions 2 and 9 and the important steric bulk around the coordinating cavity prevents the formation of homoleptic [Cu(L B )2] + , leading rather to [Cu(L B )(Solvent)n] + . The latter can readily react with a less encumbered ligand such as L A leading to Cu1.

Concerning the choice of the sacrificial donor, we focused on benzimidazolines such as BIH [START_REF] Zhu | Hydrogen Atom, Proton, and Electron Transfer Driving Forces of Various Five-Membered Heterocyclic Organic Hydrides and Their Reaction Intermediates in Acetonitrile[END_REF] (Figure 1d, here BIH = 1,3-Dimethyl-2-(4'-hydroxyphenyl)-2,3-dihydro-1H-benzimidazole). These organic molecules are well-known reductants, 16 their oxidation potential can be easily tuned 8d,15 and they have been abundantly used as sacrificial donors in photochemistry. 4f,4i,17 Importantly, they are irreversibly oxidized into stable benzimidazolium species in presence of a base like triethylamine (TEA) following the steps described in Figure S1, as commonly observed for benzimidazoline donors when oxidized in basic medium. [START_REF] Zhu | Hydrogen Atom, Proton, and Electron Transfer Driving Forces of Various Five-Membered Heterocyclic Organic Hydrides and Their Reaction Intermediates in Acetonitrile[END_REF] We report in this contribution the synthesis and characterization of Cu1 and Cu2 in THF. We evidence the occurrence of the reductive quenching of the excited states of Cu1 and Cu2 in presence of BIH and the accumulation of their reduced states during photolysis.

Materials and Methods

General: chemicals were purchased from Sigma-Aldrich or Alfa Aesar and used as received.

Thin-layer chromatography (TLC) was performed on aluminium sheets precoated with Merck 5735 Kieselgel 60F254. Column chromatography was carried out either with Merck 5735 Kieselgel 60F (0.040-0.063 mm mesh). 1 H spectra were recorded on an AVANCE 300 UltraShield BRUKER. Chemical shifts for 1 H NMR spectra are referenced relative to residual protium in the deuterated solvent ( = 7.26 ppm for CDCl3). NMR spectra were recorded at room temperature, chemical shifts are written in ppm and coupling constants in Hz. High-resolution mass (HR-MS) spectra were obtained either by electrospray ionization coupled with high resolution ion trap orbitrap (LTQ-Orbitrap, ThermoFisher Scientific,) or by MALDI-TOF-TOF (Autoflex III, Bruker), working in ion-positive or ion-negative mode. Electrochemical measurements were made under an argon atmosphere in CH2Cl2 with 0.1 M Bu4NPF6. Cyclic voltammetry experiments were performed by using an Autolab PGSTAT 302N potentiostat/galvanostat. A standard three-electrode electrochemical cell was used. Potentials were referenced to a saturated calomel electrode (SCE) as internal reference. All potentials are quoted relative to SCE. The working electrode was a glassy carbon disk and the auxiliary electrode was a Pt wire. In all the experiments the scan rate was 100 mV . s -1 . UV-visible absorption spectra were recorded on an Analytik Jena Specord 210 Plus double beam spectrophotometer, using 1 cm path length cells. Emission spectra were recorded on a Fluoromax-4 Horiba Scientific spectrofluorimeter (1 cm quartz cells).

Luminescence decays were recorded with a DeltaFlex Time Correlated Single Photon Counting (TCSPC) system (Horiba Scientific) equipped with a DeltaDiode laser DD-440L (440 ± 9 nm, 60 ps pulse width at 20 MHz) and PPD-850 detector (250-850 nm) on degassed dichloromethane and THF solutions.

Photolysis procedures: a classical 4 mL-UV-Vis quartz cuvette (1 cm path length) was charged with a small stirring bar, ca. 0.35 mg of copper complex (ca. 0.35 µmol) and 3 mL of analytical grade THF. The mixture was degassed by argon bubbling for 15 min, after which a Teflon cap was fit on the cuvette, thus protecting it from ambient air. Then ca. 20 mg BIH (80 µmol) was quickly added and the solution was further degassed for 1 mn. The obtained solution was then irradiated under one blue LED light for a few minutes. Every min, the irradiation was stopped and a UV-Vis spectrum was taken. The distance from light source irradiation to the cuvette was 0.3 cm without the use of any filters.

For photoreduction of 4-bromobenzophenone (Bz-Br): an oven-dried vial was charged with BIH (0.2 mmol) and 4-bromobenzophenone (0.10 mmol). The vial was sealed with a rubber septum and purged with Argon. Then tetrahydrofuran (THF, 3 mL) and freshly distilled triethylamine (TEA) (400 μL, 2.9 mmol) were added and the mixture was degassed for 10 min. Cu2 (5 mg, 0.005 mmol) was next added and the mixture was degassed again for 5 min. Then the reaction mixture was stirred under one blue LED (purchased from Farnell, 450 nm, 25 lumens, 75 mW) irradiation for 3 h. The distance from light source irradiation to the vial was 0.3 cm without the use of any filters. The conversion was then estimated using gas chromatography. Those measurements were performed on a Hewlett Packard HP 5 (l = 30 m, i.d.= 0.32 mm, film thickness = 0.25 mm), equipped with a FID detector (N2 as carrier gas (1.6 mL.min -1 ); temperature program: 80 °C for 1 min then 15 °C.min -1 to 320 °C). Bz-Br was purchased from Sigma-Aldrich.

Synthesis: ligands L A, 11 , L B, 18 and complex Cu2 11 were prepared as previously reported.

Complex Cu1: L B (23.4 mg, 56 µmol) is dissolved in dichloromethane (10 mL). The mixture is thoroughly degassed by argon bubbling and Cu(CH3CN)4PF6 is added quickly (21 mg, 56 µmol). The colourless solution turns yellow. After 5 min, a degassed solution of L A (20 mg, 51 µmol, 10 mL dichloromethane) is syringed in the mixture over 5 min. The deep orange mixture is left at room temperature for 30 min. The clear solution is then evaporated under reduced pressure. The orange residue is purified by thin layer chromatography on silica (with a gradient of petroleum ether/dichloromethane (100% petroleum ether to 100% dichloromethane). The orange fraction is collected and re-precipitated in toluene, affording a pure sample of Cu1.

Yield: 35 mg (68%). [START_REF] Hockin | Photoredox catalysts based on earth-abundant metal complexes[END_REF] 1, 161.1, 154.9, 149.8, 148.1, 142.7, 141.3, 139.6, 138.2, 138.1, 137.3, 136.4, 134.8, 134.2, 131.5, 129.7, 128.5, 128.3, 127.5, 127.4, 126.5, 124.2, 39.2, 30.3, 22.6, 20.7, 20.1, 13.

Results

Optical Properties

Cu1 and Cu2 exhibit the usual properties of all copper(I) complexes and are summarized in Table 1. The visible part of the absorption spectrum is dominated by a broad metal to ligand charge transfer transition (MLCT) with a maximum absorption wavelength λabs of 455 nm in THF for both Cu1 and Cu2 (Figure 2). The extinction coefficient for the MLCT of Cu1 (4500 L.mol -1 .cm -1 ) is roughly half the one of Cu2 (9200 L.mol -1 .cm -1 ). This is due the different dipole moments of Cu1 and Cu2. Indeed, L A is a diimine ligand with an extended π-conjugated system increasing the dipole moment of the complex and thus raising the probability of the MLCT dipolar transition. Since there are two ligands L A in Cu2 versus only one in Cu1, the dipole moment of Cu2 is larger than the one of Cu1, thus rationalizing the different extinction coefficients of the two complexes. Conversely, the mesityl groups on L B contribute to decrease the dipole moment, [START_REF] Phifer | The basis of aryl substituent effects on charge-transfer absorption intensities[END_REF] and likely contribute to the overall weak extinction coefficient of the MLCT of Cu1. The set of sharp transitions at 360 and 380 nm on both spectra corresponds to a complex mixture of phenazine centered transitions and high energy MLCT transitions, as previously demonstrated by quantum calculations on Cu2. 11 Upon excitation in the MLCT, a weak and broad emission band can be observed in the near infrared at ca. 730 and 720 nm for Cu1 and Cu2 respectively (Figure S3). The energies of the lowest lying excited singlet state for Cu1 and Cu2 were determined from the emission spectra using the tangent method [START_REF] Garakyaraghi | Cuprous Phenanthroline MLCT Chromophore Featuring Synthetically Tailored Photophysics[END_REF][START_REF] Arnold | Photochemistry: An Introduction[END_REF] and are comparable for both complexes. The emission quantum yields are weak (< 0.1%) and the emission lifetimes are rather short (Table 1). This observation was made before on other copper(I) complexes studied in THF. [START_REF] Garakyaraghi | Cuprous Phenanthroline MLCT Chromophore Featuring Synthetically Tailored Photophysics[END_REF] Such behavior is due to different very efficient deactivation channels competing with the radiative decay. First of all, the use of THF favors exciplex quenching. 21 This process consists in the nucleophilic attack of the excited state of the complex by Lewis bases. Here, THF acts as a Lewis base through the non-bonding doublets on the oxygen atom and deteriorates the emission properties of Cu1 and Cu2. The deleterious impact of THF on the emission properties is demonstrated by comparing the steady state emission spectra and emission decays recorded in dichloromethane and THF (Table 1 andFigures S4 to S7): indeed, the luminescence intensities and emission lifetimes are both smaller in THF compared to dichloromethane for Cu1 and Cu2 (for absorption matched solutions in both solvents). Interestingly, Cu2 seems to be much more affected than Cu1 by the solvent. The emission lifetime of Cu2 is indeed weaker than in dichloromethane, while the emission of Cu1 is a lot less affected. This could be due to the mesityl groups in L B , better shielding Cu1 vs. nucleophilic attacks from THF. Finally, internal conversion from populated excited states to lower lying dark states is possible too as previously observed on Cu2. [START_REF] Soulis | New luminescent copper(I) complexes with extended π-conjugation[END_REF][START_REF] Véry | Photophysical Properties of Ruthenium(II) Polypyridyl DNA Intercalators: Effects of the Molecular Surroundings Investigated by Theory[END_REF] All potentials given vs. saturated calomel electrode (SCE). E1/2 = (E p,a + E p,c )/2; ΔE = |E p,a -E p,c | where E p,a and E p,c are the anodic and cathodic peak potentials for a given electrochemical wave.

Solutions were purged from oxygen by argon bubbling. E 00 determined by the tangent method (Figure S2). E* calculated with equation 1, using square voltammetry data. a In THF. b In dichloromethane.

c Determined by square wave voltammetry.

Electrochemical Properties

The electrochemical behavior of Cu1 and Cu2 was analyzed by cyclic and square wave voltammetries (Figures S8-S11). The voltammograms are strongly distorted when recorded in THF vs. dichloromethane (see for example the cyclovoltammogram of the same sample of Cu2 recorded in THF and in dichloromethane, Figures S9 andS12). For example, the oxidation wave assigned to the Cu II /Cu I couple appears curiously irreversible, although it is well known that the oxidation of bis(imine) copper(I) complexes is pseudo reversible in dichloromethane and acetonitrile. [START_REF] Eggleston | Steric Effects in the Ground and Excited States of Cu(NN)2+ Systems[END_REF] Increasing the sweep rate did not affect the shape of the signal. We thus characterized the electrochemical behavior of Cu1 and Cu2 in dichloromethane.

In the anodic part of the voltammogram, the reversible wave associated to the Cu II /Cu I couple can be monitored at 1.02 and 1.12 V vs. SCE for Cu1 and Cu2 respectively. These values are in accordance with previously published work on copper(I) complexes where the metal ion is coordinated by two (identical or not) phenanthroline ligands. 8b,10,18,23-24 The slightly higher oxidation potential of Cu2 vs. Cu1 is likely due to the stronger π-accepting character of L A vs. L B . In the cathodic part, well-defined ligand-centered reduction waves are monitored for Cu1 and Cu2 at -1.28 and -1.21

V vs. SCE. Those reduction waves can be safely assigned to a L A centered reduction step for both For Cu1, the reduction and oxidation waves have similar sizes as expected for two one-electron electrochemical processes. But for Cu2, the reduction wave is anomalously large (ΔE = 230 mV) and is approximately twice larger than the oxidation wave (whereas both oxidation and reduction processes feature on average the same current intensities for Cu1). This is due to the fact that Cu2 bears two ligands L A coordinated to one copper(I) center, and that both ligands are reduced at similar potentials. As a matter of fact, square wave voltammograms in dichloromethane (Figure S11) in the cathodic region show that two peaks have merged in a broad deformed one. The first reduction process (Ered 1 , Table 1) appears as a shoulder at -1.17 V vs. SCE and the second (Ered 2 ) at -1.21 V vs. SCE, corresponding to the formation of [Cu(L A•-)(L A )] 0 and [Cu(L A•-)2] -, respectively. Consistently, the area under the cathodic peak is twice as large as the area under the anodic peak. The fact that both ligands L A in the same copper complex can be reduced at similar potentials shows that there is a limited electronic communication between them. Importantly, this means that the driving forces which must be wielded to reach mono-reduced Cu1 -and bis-reduced Cu2 2-are very similar, paving the way towards charge photo-accumulation upon excitation.

Photolysis Experiments

With all the collected electrochemical and optical data, the oxidation and reduction potential of Cu1 and Cu2 in their excited states can be estimated using the Rehm and Weller equation (Table 1).

Both Cu1 and Cu2 can a priori be reduced by BIH (E(BIH + /BIH) = 0.28 V vs. SCE) 17a upon excitation, since both feature respectable driving forces of -0.40 eV and -0.58 eV respectively for this reductive quenching process.

We thus performed preliminary photochemical trials where degassed solutions of Cu1 and Cu2

in THF are irradiated under 450 nm LED light in presence of electron donors, and we monitored the evolution of the photolyzed medium by UV-Vis spectroscopy.

No spectral evolution was noticed when solutions of Cu1 or Cu2 were irradiated in presence of classical sacrificial electron donor triethylamine (E(TEA + /TEA) ≈ 0.7 V vs. SCE) [START_REF] Chow | Nonaromatic aminium radicals[END_REF] despite an encouraging driving force of -0.16 eV for Cu2. This means that no reductive quenching step is taking place in these conditions. More important driving forces are required, probably to overcome the large reorganization energy that is supposed to take place upon reductive quenching.

When the sacrificial donor is BIH in presence of TEA, photolyzed solutions darken very quickly after the beginning of the irradiation. As mentioned above, TEA is essential to ensure that BIH is irreversibly oxidized and that charge recombination is thus limited (Figure S1). However, precipitation of a fine powder during irradiation prevented to record consistent UV-Vis spectra because of strong light scattering. The powder was identified as BI + by NMR. The precipitation of BI + and the darkening of the solutions are however qualitative pieces of evidence that reductive quenching is taking place.

We attempted to reproduce the same photolysis experiment in absence of TEA with BIH only.

Likewise, photolyzed solutions darken very quickly after the beginning of the irradiation. The coloration persisted for a sufficiently long time to allow recording absorption spectra of the photolyzed mixtures (Figure 3). After only one minute of irradiation at 450 nm, a broad absorption band with a maximum around 605 nm rises for both Cu1 and Cu2. This band is typical of reduced phenazine spacers (Figure S13 and comments) [START_REF] Fees | Electronic structure of the "molecular light switch" bis(bipyridine)dipyrido[3,2-a:2',3'-c]phenazineruthenium(2+). Cyclic voltammetric, UV/visible and EPR/ENDOR study of multiply reduced complexes and ligands[END_REF] and can be safely assigned to [Cu(L A•-)(L B )] 0 (Cu1 -) and [Cu(L A•-)(L A )] 0 (Cu2 -). The formation of the reduced forms of Cu1 and Cu2 upon irradiation in presence of BIH conclusively points to a reductive quenching of the excited states of both complexes by the benzimidazoline sacrificial donor. When irradiation is stopped, the solution returns to its initial orange color, although a significant loss of MLCT was observed pointing to some instability of the complexes during irradiation. This instability could be due to the well-known lability of the coordination sphere of the complexes. [START_REF] Hebbe-Viton | Chiral spiro Cu(I) complexes. Supramolecular stereocontrol and isomerisation dynamics by the use of TRISPHAT anions[END_REF] As mentioned above, Cu2 can be twice reduced at close potentials. Interestingly, it has been proven several times that the phenazine spacer was electronically decoupled from the dipyridyl chelate. 28 Thus, [Cu(L A•-)(L A )] 0 could be formally considered as a [Cu(L)(L A )] + moiety (where L is a sterically challenged phen ligand like e.g. 2,9-di-nbutyl-1,10-phenanthroline) linked to an isolated reduced phenazine spacer. As such, the reduced complex Cu2 -could in principle be excited a second time, and be involved in a second reductive quenching with BIH leading to doubly reduced Cu2 2-. If we assume that E 00 remains similar for Cu2 and Cu2 -, we can apply equation 1:

E*(Cu2 -) = Ered 2 + E 00 = ca. 0.8 V vs. SCE.

Reductive quenching of (Cu2 -)* by BIH would thus be thermodynamically allowed. However, photolyzed mixtures of Cu1 (where such photo-accumulation is not thermodynamically possible) and Cu2 show virtually identical spectral features and prolonged irradiation did not result in any further spectral change for Cu1 nor Cu2. Thus, in our conditions, it is very unlikely that a double photoinduced reduction of Cu2 take place. The excited state of Cu2 -is probably very quickly quenched by energy transfer towards low lying (L A•-)-centered states.

Photo-Induced Reduction Trials

We then attempted to utilize the implemented reductive quenching of the designed copper(I) complexes in presence of BIH, to perform the reductive dehalogenation of 4-bromobenzophenone (Bz-Br). This was recently successfully used as a benchmark reaction to test the efficacy of reductive quenching with copper 5b,8d and iridium complexes (scheme 1). 17f In particular, we demonstrated the efficient reduction of Bz-Br into Bz-H (benzophenone) when a mixture of [Cu(dipp)2] + , Bz-Br and a chemically engineered benzimidazoline derivative BI2H (Figure S14) is irradiated in the visible. (BI2H •+ ) is quickly deprotonated in presence of TEA leading to BI • (BI2 • ) which is a strong reductant (ca. -1.6 V vs. SCE). [START_REF] Zhu | Hydrogen Atom, Proton, and Electron Transfer Driving Forces of Various Five-Membered Heterocyclic Organic Hydrides and Their Reaction Intermediates in Acetonitrile[END_REF] Summarily, the reductive quenching of the excited state of Cu2 by BIH (BI2H) leads to two reductants with only one photon. In the present case, BI • is apparently not sufficiently reductive to react with Bz-Br (since there is no conversion for the photochemical reaction). BI2 • is slightly more reductive than BI • thanks to the additional methoxy groups tethered to the structure (Figure S14), which could account for the slight improvement of the conversion when BI2H is used as sacrificial donor.

Moreover, those results have an important fallout on our previous work. As mentioned above, we recently demonstrated that irradiating a mixture of Bz-Br, [Cu(dipp)2] + and BI2H lead to the efficient photo-reduction of Bz-Br into Bz-H (93% yield in 30 mn irradiation). However, we were unable to determine whether Bz-Br was reduced by photoproduced [Cu(dipp)(dipp • )] 0 or BI2 • . In the present work, we demonstrate that BI2 • (and incidentally BI • too) is unable to efficiently reduce Bz-Br. Stronger reductants are necessary to reduce such challenging substrates, such as homoleptic copper(I) complexes with more negative reduction potentials than E(Cu2/Cu2 -).

Given the results obtained with Cu2, we did not perform further trials with Cu1 which shows very similar properties.

Conclusions

We reported the synthesis and characterization of two copper(I) complexes Cu1 and Cu2 showing enhanced photo-oxidative power. This task was realized by using electron poor, phenazine-fused ligands, which allowed to significantly shift the reduction potential of both complexes to less negative values and subsequently increased the reduction potential of the excited state of each complex E(Cu1*/Cu1 -) and E(Cu2*/Cu2 -). Cu1 and Cu2 were therefore able to oxidize a common benzimidazoline sacrificial reductant BIH under blue light irradiation. In other words, Cu1 and Cu2 were involved in a reductive quenching cycle, a very rare photochemical process with copper(I)bis(diimine) complexes. The challenging photo-induced reduction of an aryl bromide via reductive quenching was then attempted but proved unsuccessful because reduced complexes are insufficiently reductive to react with such substrates.

Supplementary Materials:

Figure 1 .

 1 Figure 1. a) structure of complex [Cu(TPTAP)2] + . 8b b) Structure of ligands L A and L B . c) Structures of complexes Cu1 and Cu2. d) Structure of BIH (BIH = 1,3-Dimethyl-2-(4'-hydroxyphenyl)-2,3dihydro-1H-benzimidazole).

  H NMR (300 MHz, CDCl3, 25 °C): δ = 9.64-9.61 (2H, d, J = 8.4 Hz), 8.78-8.75 (2H, d, J = 8.1 Hz), 8.46-8.43 (2H, dd, J = 6.6 Hz, J ' = 3.3 Hz), 8.29 (2H, s), 8.05-8.01 (2H, dd, J = 6.6 Hz, J ' = 3.3 Hz), 7.81-7.78 (2H, d, J = 8.4 Hz), 7.71-7.68 (2H, d, J = 8.4 Hz), 6.18 (4H, s), 2.41-2.36 (4H, m), 1.73 (6H, s), 1.60 (12H, s + broad water peak), 1.23-1.19 (4H, m), 0.90-0.82 (4H, m), 0.72-0.67 (6H, m) ppm. NMR spectrum provided in Figure S2. 13 C NMR (75 MHz, CDCl3, 25°C): δ = 163.

  9 ppm. HRMS (MALDI) m/z: [M+H] + calculated for C56H54CuN6: 873.3706; found: 873.3696. Δ = 1.1 ppm.

Figure 2 .

 2 Figure 2. UV-Vis absorption spectra of Cu1 (red line) and Cu2 (black line) in THF.

Cu1

  and Cu2, leading to a species which can be formally written [Cu(L A•-)(L B )] 0 . The reduction of copper(I) complexes bearing classical bulky phenanthroline ligands like neocuproine or 2,9-diisopropyl-1,10-phenanthroline, are usually reduced at much more negative potential, ruling out the possibility that the reduction wave of Cu1 could correspond to L B•-formation.

Figure 3 .

 3 Figure 3. (a) UV-Vis spectra of a photolyzed mixture of Cu1 and BIH in THF; (b) UV-Vis spectra of a photolyzed mixture of Cu2 and BIH in THF. Plain lines: UV-Vis spectra before photolysis. Dashed lines: UV-Vis spectra after photolysis. Dotted lines: UV-Vis spectra of BIH alone in THF at the same concentration (25 mM).

Scheme 1 .

 1 Scheme 1. Reductive photo-dehalogenation of Bz-Br via reductive quenching of Cu2's excited state in presence of BIH.

Figure S1 :

 S1 Series of electron transfers following one electron oxidation of benzimidazolines. Figure S2: NMR spectra of Cu1. Figure S3: emission spectra of complex Cu1 (red line) and Cu2 (black line) in THF, Figures S3-S5: Steady state emission spectra in THF. Figures S6-S7: luminescence decays. Figures S7: Luminescence decays of solutions of Cu2 recorded by TC-SPC in degassed dichloromethane (grey line) and THF (black line). Figures S8-S12: Cyclovoltammograms and square wave voltammograms of Cu1 and Cu2. Figure S13: spectroelectrochemistry.

Figure S14 :

 S14 Figure S14: structure of BI2H.

Table 1 .

 1 Optical and electrochemical parameters for Cu1 and Cu2.

		λabs / nm (ε / M -1 .cm -1 ) a λem / nm a τem / ns E 00 / eV a	Cyclic voltammetry E1/2,ox / V b (ΔE / mV) E1/2,red / V b (ΔE / mV)	Square wave voltammetry Eox / V b,c Ered 1 / V b,c Ered 2 / V b,c	E* / V vs. SCE
	Cu1	455 (4500)	730	47.7 a (53.2) b	1.98	1.04 (100)	-1.23 (120)	1.04	-1.30		0.68
	Cu2	455 (9200)	720	14.0 a (55.5) b,11	2.03	1.14 (120)	-1.16 (230)	1.12	-1.17 (sh)	-1.21	0.86

Funding: this research was funded by ANR, grant number ANR-16-CE07-0012-01.

Conflicts of Interest: the authors declare no conflict of interest.

Author information: lea.gimeno@univ-nantes.fr; clemence.queffelec@univ-nantes.fr; Errol.blart@univ-nantes.fr; yann.pellegrin@univ-nantes.fr.