 Good morning everyone. I am Anirvan Mojumdar. I am a PhD student in LSB France and in reign in France, Sunder Nathalie and Patricia. So this work is on concurrent parameterized games. So these games are quite different games than we had this morning. So I will quickly recall what games are in these settings and I will move to our model. So when we talk about games, we generally talk about games on finite graphs. So we have a finite graph and in turn based to player games, what happens is that so we have two players player and player 1 and player 2. So we have two types of vertices in the graph, circle vertices and square vertices. So circle vertices are controlled by player 1 and square vertices are controlled by player 2. So how does the game proceed? So it starts from some initial vertex for example this one and so this is player 1's vertex, player 1 will choose to go to one of these two vertices. So maybe he will choose this one then this is player 2's vertex. So he will choose to go to the next one and next one and so on. So this is called a play and a winning objective is given for player 1 as a set of plays and we check if there exist winning strategies for the players. So it is well known that we have positional winning strategies for well known winning conditions for example reachability safety. So by positional winning strategies we mean from some particular vertex player strategy does not depend on the past history. Then people define something called concurrent to player games. What are these games? So here we have only one type of vertices. So do not confuse these circles with the circles in the previous slide. Here we have two players and they choose actions from AB. So in the graph the transitions are labeled with two later words and players choose actions simultaneously. And how does the game proceed? So next vertex is determined by the chosen actions of the player. For example if player 1 chooses A and player 2 chooses B since AB is in this transition. So game will proceed from V0 to V2. So the main difference is from the previous model is that here players choose the actions simultaneously where in the previous one we call that was turn based. So their player used to choose one by one. So if we want to find player 1 strategy here it is well known that we can convert this one to a turn based game where this circle vertex, well again this circle is not same as this circle. So this circle vertex will correspond to player 1's vertex and we can show that player 1 has a winning strategy in this concurrent game if and only if circle player has a winning strategy in this turn based game. So this is well known and one can generalize this thing to K player concurrent game in the sense that now the transitions have levels K length words and there are K players. So game proceeds according so they choose actions simultaneously and again game proceeds accordingly. For example if first one chooses A and the rest of all choose B then since AB power K minus 1 is in this transition it will go from V0 to V2. And now I will move to concurrent parameterized game. So one natural question one can ask in this concurrent game model is that what if we do not fix the number of players. So what is the parameter here is number of players. So again we have some actions AB. So if we want to general extend the previous example in this setting what happens is that we say if everyone so we consider player 1 versus rest of the world which we call environment. So player 1 does not know how many players are playing. So we say that whatever our player is playing if all of them choose the same letter then the game goes here otherwise game goes here. So this is just a toy example I will explain this in the next slide soon. So just before this so this is just a toy example so our main model is like this. So the transitions are leveled with regular languages and based on the choices the word belongs to which language depending on that the game will proceed to the next vertex. Not necessarily language can be there can be intersecting I mean not necessarily designed yes. So I will explain in the next slide. So and we say that player 1 wins if he can win against every number of opponents and every choice of the opponents. So formally how does the game proceed? First environment chooses the number of players which will be fixed toward the game but player 1 does not know good. Then player 1 will choose some action A1 and other players will also choose some actions and this forms a word and then environment chooses the vertex such that this word belongs to for example if this word belongs to L1 then environment will choose V1. So if there is some non-determinism environment will break the non-determinism environment has the power to break the non-determinism and again from Vi we again so player 1 again choose and others also choose so the game continues and we say that player 1 wins if he wins against every choice of others and for every number of opponents. So here is a small example which also answers the question that a spoiler is that positional strategies will not suffice. For example here at the first vertex player 1 only has one choice is that A. So what does it say if there is one opponent whatever he chooses the game goes from V1 to V2 if there is two or more opponents the game goes from V1 to V3 but player 1 does not know how many opponents he have he has here. So at V4 if player 1 has two choices now A and B but if he chooses A and there is only one opponent then go to V5 otherwise go to V6 and vice versa for B and objective for player 1 is to reach the yellow state. So here player 1 actually has winning strategy is that so when he comes to V4 looking at the history he can infer that if the history is V1 V2 V4 then there is only one opponent otherwise if the history was V1 V3 V4 then he had only he had two or more opponents and he can play accordingly A or B to reach this state. But there is no winning strategy no positional winning strategy because at V4 only playing A might lose only playing B also might lose. Now a simplification is that in this particular example we saw that players opponents choices actually do not matter. So we replace this sigma to be like number of opponents so similar number of opponents here and one observation is that for general case also if we are interested in looking at player 1's winning strategy then only number of opponents matter. So we replace this transitions by so first component is player 1's choice and the second component is the set of opponents. So yeah this is our new simplified game and it is already known that if L is regular so we assume that L is regular in that case a set of lengths of all overs is semi-linear we want to investigate this model for semi-linear sets but we do it for three sub cases is that intervals union of intervals and semi-linear sets so because we will have different complexities and yeah complexities are the main results here. So since the game has changed I will quickly tell you what the game is now so here environment chooses the number of opponents and player 1 then choose some action sigma and then environment chooses VI such that K belongs to that set. So now environment only breaks the non-determinism such that this number of player belongs to that set. So here opponents now do not have any actions to play and game will proceed to VI and so on and we say that P1 wins if he wins for every K. So now we want to solve this game we will reduce this game to a standard two player game suppose this is the game arena so here we have if only one opponent is there then goes to V2 if more than two then goes to V3. So this will reduce to two player game with some knowledge of player 1 so here if this is the initial vertex V1 is the initial vertex then here player 1 knows that he has he can have any number of opponent so he stores that like one infinity and this is an environment vertex if the game goes here then player 1 can infer that he had only one opponent so we mark this state as V2, 1 and similarly V2, 2 infinity. In the general case what we do is that it is quite similar for example so if in the knowledge game player 1 had knowledge K and V1 and if the game takes this transition then at this node he has this information that he is in V2 and the knowledge is K intersection S1 and it is yeah so this is just the previous example reducing to turn based game here we can see that there are two different paths for one opponent and two or more opponents and here player 1 actually has a winning strategy as we saw here and it can be shown that player 1 has a winning strategy if an early circle player has a winning strategy there. So this immediately gives the decidability result because this game is finite why we only take intersections over the sets given in the input so we cannot make like infinite number of sets. So this immediately gives the decidability but next I will talk about complexity. So briefly concurrent games were defined previously and what we do is that we say well a natural extension can be if we do not know the number of players and we saw the strategies, more or less strategies are not enough and it is enough to restrict to only number of opponents and we saw the decidability result. So here are the complexity results so as I said like the complexities are different for different sub cases intervals is easy it is P time and for unions of intervals it is actually interesting that for deterministic it is NP complete and for non-deterministic it is P space complete and for general semi-linear sets like for general language languages so it is P space complete and these two proofs are similar and yeah for unions of intervals and intervals the complexities are on number of end points given in the input and here also it is on the number of semi-linear sets given in the input it is not like it does not depend on the encoding of the input like how do you encode binary or something it does not depend on that. So the easiest case is intervals why is it polynomial because if in the game arena you are given only intervals then you can only construct quadratic many intervals out of that so this gives the polynomial size knowledge game and we know that turn-based games are solvable in polynomial time well for each you do not know for right and for so that gives us polynomial time solvable for this game. Now I will move to semi-linear sets so in general for semi-linear sets this game this knowledge game construction can be exponential size but we can see in the next slide that next slide that we can have a polynomial space algorithm. The polynomial space P space algorithm so we are only solving for iterative goes in two steps I will try to briefly give the idea so we have the knowledge game which is pretty big which is exponential but somehow we want to restrict the knowledge game. So suppose in the knowledge game we are in vertex v and knowledge k and suppose this is the knowledge game so dotted line means there might be more successors or something. So we want to somehow restrict this knowledge game how we stop at any vertex where the knowledge strictly decreases so for example here the knowledge is still k so we explore but here the knowledge strictly less so we do not explore we stop here. So what happens is that in this small game every vertex has knowledge k or exactly one with small so it will have a polynomial size so it is it gives a polynomial size game this restriction and this can be solvable in polynomial time so this gives a structure something like this maybe we start from v0 and n and it can go here here here so here the knowledge is k1 here the knowledge is k2 or something so each is of size polynomial. Now the idea is to solve this polynomial size games and reuse the same space so this is the step 2 applying DFS we want to so if v0 is the initial vertex and k0 is the initial knowledge then we want to check the winning status of this vertex for player 1. So what we do we expand this vertex to the restriction of the game so to the restriction and suppose somehow by DFS we already know the winning status of player 1 here and here so now we forget this subtree and we explore the red ones so we explore the red ones again suppose this one is done and we explore the red ones so like this branching is polynomial because this game is polynomial and also at a time we are only saving polynomial height tree so this at one time we are only saving polynomial size tree. Now how do we tag we not lose a vertex in this tree well if this vertex is target for example here we say that yes this is reachable by player 1 so we say this is winning and in other case so for example if here everything is known what we do is that we solve this small game for player 1 and if player 1 has a winning strategy in this small game then we tag this vertex as winning so for example if 4 of these we already know the status then we solve this small game and accordingly we tag this one and then forget the subtree and then again the again we explore the next vertex so this way we can actually explore using DFS this tree and also we can show that this tree so in this initial node the tree has tag winning if and only player 1 has a winning strategy in the knowledge game. So this gives a p space upper bound and this proof also works for also holds for unions of intervals yeah this is the same proof basically but for unions of intervals for deterministic case we can show that it is NP because so deterministic means this is basically a partition of N so yeah this is always a partition so player 1 actually can guess a strategy which will be of polynomial size why because at every step this will be a partition of N using only the end points given in the input so this width is polynomial and also this height is polynomial so this tree size is polynomial yeah and I am not showing the hardness proofs but we can we have hardness proofs also from reduction by reduction from set but I am not going to show that so now we are trying to work on a related model which we call which we may call parametric synthesis is like here also we do not know the number of players so the transitions are again leveled with regular languages now this is not player 1 versus everyone this is like a coalition game so everyone is playing together and want to win the game for example if here the transition is A star B then they reach the winning state so it means if the last player play B then they reach the winning state but they do not know how many players are there but in this particular example what they can do is that player I play B at ith round so for example player 1 first play B if he is the last player he goes to everyone goes to the winning state if he is not other ones will play A so they will look until the last player comes and plays a B and goes to the winning state so here yeah we assume that players somehow know their id but they do not know how many players are there so yeah it is not quite formalized but yeah this is kind of future direction we want to pursue so to conclude as we saw we generalize the two player concurrent games and we consider the scenario player 1 versus everyone else we saw that only memoryless strategies are not enough we proved decidedly by knowledge game construction which is finite and we saw that p-space completeness for general case but better bounds for other simpler cases that is it thank you so the well so the knowledge game construction is just reducing it to player game and we can check bushi for the knowledge game constructions so decidability is clear enough but we are not sure about complexity what happens I mean there ok yeah so yeah we don't know I mean yeah in this model we say that it's fixed throughout the game but we don't know about dynamic number of players