Bisimulation and Modal Logic in Distributed Computing

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1 Bisimulation and Modal Logic in Distributed Computing Tuomo Lempiäinen Distributed Algorithms group, Department of Computer Science, Aalto University (joint work with Lauri Hella, Matti Järvisalo, Antti Kuusisto, Juhana Laurinharju, Kerkko Luosto, Jukka Suomela and Jonni Virtema) Computational Logic Day 06 December 8, Aalto University / 5

2 Publications Brief overview of two papers: Hella, Järvisalo, Kuusisto, Laurinharju, Lempiäinen, Luosto, Suomela and Virtema: Weak models of distributed computing, with connections to modal logic PODC 0, Distributed Computing 05 Lempiäinen: Ability to count messages is worth Θ( ) rounds in distributed computing LICS 06 / 5

3 The model of computation A simple finite undirected graph, whose each node is a deterministic state machine that runs the same algorithm, can communicate with its neighbours, produces a local output. / 5

4 The model of computation A simple finite undirected graph, whose each node is a deterministic state machine that runs the same algorithm, can communicate with its neighbours, produces a local output. Anonymous nodes a weak model of computation. / 5

5 Communication in synchronous rounds In every round, each node v sends messages to its neighbours, a v b receives messages from its neighbours, updates its state. / 5

6 Communication in synchronous rounds In every round, each node v sends messages to its neighbours, c v d receives messages from its neighbours, updates its state. / 5

7 Communication in synchronous rounds v In every round, each node v sends messages to its neighbours, receives messages from its neighbours, updates its state. / 5

8 Communication in synchronous rounds v In every round, each node v sends messages to its neighbours, receives messages from its neighbours, updates its state. Eventually, each node halts and announces its own local output. / 5

9 Focus on communication, not computation The running time of an algorithm is the number of communications rounds. = n = 6 The running time may depend on two parameters: the maximum degree of the graph,, the number of nodes, n. 5 / 5

10 Graph problems We study graph problems where the problem instance is the communication graph G = (V, E), a solution is a mapping S : V Y from nodes to local outputs. 6 / 5

11 Graph problems We study graph problems where the problem instance is the communication graph G = (V, E), a solution is a mapping S : V Y from nodes to local outputs. Often the solution is an encoding of a subset of vertices or edges of the graph. Y = {0, } One typical example is the minimum vertex cover. 6 / 5

12 PODC 0: seven variants of the model Options for sending messages: a port number for each neighbour (V), a b v c Node v sends a vector (a, c, b). 7 / 5

13 PODC 0: seven variants of the model Options for sending messages: a port number for each neighbour (V), broadcast the same message to all neighbours (B). a a v a Node v broadcasts message a. 7 / 5

14 PODC 0: seven variants of the model Options for sending messages: a port number for each neighbour (V), broadcast the same message to all neighbours (B). Options for receiving messages: a port number for each neighbour (V), a a v b Node v receives a vector (a, b, a). 7 / 5

15 PODC 0: seven variants of the model Options for sending messages: a port number for each neighbour (V), broadcast the same message to all neighbours (B). Options for receiving messages: a port number for each neighbour (V), receive a multiset of messages (M), a a v b Node v receives a multiset {a, a, b}. 7 / 5

16 PODC 0: seven variants of the model Options for sending messages: a port number for each neighbour (V), broadcast the same message to all neighbours (B). Options for receiving messages: a port number for each neighbour (V), receive a multiset of messages (M), receive a set of messages (S). a a v b Node v receives a set {a, b}. 7 / 5

17 PODC 0: seven variants of the model Options for sending messages: a port number for each neighbour (V), broadcast the same message to all neighbours (B). Options for receiving messages: a port number for each neighbour (V), receive a multiset of messages (M), receive a set of messages (S). We can require the outgoing and incoming port numbers to be consistent the port-numbering model (VV c ). 7 / 5

18 PODC 0: a hierarchy of complexity classes VV c VV VB MV MB SV SB 8 / 5

19 PODC 0: a hierarchy of complexity classes VV c VV c VV VB VV VB = = MV MB MV MB = SV SB SV SB Theorem SB MB = VB SV = MV = VV VV c. 8 / 5

20 PODC 0: connections to modal logic The constant-time variant of each of the seven complexity classes can be characterised by a modal logic such that there is a canonical one-to-one correspondence between algorithms and modal formulas. 9 / 5

21 PODC 0: connections to modal logic The constant-time variant of each of the seven complexity classes can be characterised by a modal logic such that there is a canonical one-to-one correspondence between algorithms and modal formulas. Example: graded modal logic (GML), ϕ := q n (ϕ ϕ) ϕ ϕ, k ϕ, where q n are proposition symbols and k N. G, v = q n iff degree(v) = n, G, v = k ϕ iff {w V : (v, w) E and G, w = ϕ} k. 9 / 5

22 PODC 0: connections to modal logic The constant-time variant of each of the seven complexity classes can be characterised by a modal logic such that there is a canonical one-to-one correspondence between algorithms and modal formulas. Example: graded modal logic (GML), ϕ := q n (ϕ ϕ) ϕ ϕ, k ϕ, where q n are proposition symbols and k N. G, v = q n iff degree(v) = n, G, v = k ϕ iff {w V : (v, w) E and G, w = ϕ} k. GML corresponds to the complexity class MB (receive a multiset, send by broadcasting). 9 / 5

23 PODC 0: connections to modal logic In each variant of modal logic, one can characterise definability by a variant of bisimulation. A nonempty relation Z V V is a graded bisimulation between G = (V, E, τ) and G = (V, E, τ ) if the following conditions hold. If (v, v ) Z, then v τ(q n ) iff v τ (q n ) for each q n. If (v, v ) Z and X E(v), then there is a set X E (v ) such that X = X and for each w X there is a w X with (w, w ) Z. If (v, v ) Z and X E (v ), then there is a set X E(v) such that X = X and for each w X there is a w X with (w, w ) Z. We use bisimulation to derive the separation results between the complexity classes. 0 / 5

24 The relationship of MV and SV The simulation results used to show the equivalence of complexity classes do not increase the running time, except for one: Theorem (PODC 0) Assume that there is an MV-algorithm A that solves a problem Π in time T. Then there is an SV-algorithm B that solves Π in time T +. Is this result tight? / 5

25 LICS 06: the simulation overhead is tight Theorem For each there is a port-numbered graph G with nodes u, v, w such that when executing any SV-algorithm A in G, u receives identical messages from its neighbours v and w in rounds,,...,. We can also separate the models by a graph problem: Theorem There is a graph problem Π that can be solved in one round by an MV-algorithm but that requires at least rounds for all, when solved by an SV-algorithm. / 5

26 Example: separating SV and MV u v Output if there is an even number of neighbours of even degree, 0 otherwise. / 5

27 Generalisation: graph G (here = ) / 5. The blue nodes are bisimilar up to the distance.

28 Conclusion VV c We defined seven complexity classes and characterised the containment relations. VV = VB = Each constant-time class corresponds to a variant of modal logic. MV = SV MB SB Only in one case there is overhead in simulating a stronger model by a weaker one, and that overhead is unavoidable. 5 / 5

29 Conclusion VV c We defined seven complexity classes and characterised the containment relations. VV = VB = Each constant-time class corresponds to a variant of modal logic. MV = SV MB SB Only in one case there is overhead in simulating a stronger model by a weaker one, and that overhead is unavoidable. Thanks! Questions? 5 / 5

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