Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Computing the simulation preorder of a given Kripke structure (i.e., a directed graph with  n  labeled vertices) has crucial applications in model checking of temporal logic. It amounts to solving a specific two-players reachability game, called simulation game.  offer the first conditional lor bounds for this problem, and  relate its complexity (for computation, verification, and certification) to some variants of  n  ×  n  matrix multiplication.  show that any  O ( n α  )-time algorithm for simulation games, even restricting to acyclic games/structures, can be used to compute  n  ×  n  boolean matrix multiplication (BMM) in  O ( n α  ) time. In the acyclic case,  match this bound by presenting the first subcubic algorithm, based on fast BMM, and running in  n   ω + o (1)  time (where  ω   O ( n  2 )-size certificates, that can be verified in truly subcubic time by means of matrix multiplication. In the acyclic case,  O ( n  2 ) time is sufficient, employing standard (+, ×)-matrix product verification. In the cyclic case, a min-edge witness matrix multiplication (EWMM) is used, i.e., a matrix multiplication on the semi-ring (max, ×) where one matrix contains only 0's and 1's, which is computable in truly subcubic  n  (3+ ω )/2+ o (1)  time. Finally,  show a reduction from EWMM to cyclic simulation games which implies a separation beten the cyclic and the acyclic cases, unless EWMM can be verified in  n   ω + o (1)  time.
['Massimo Cairo', 'Romeo Rizzi']
I am writing a research paper on a idea The complexity of simulation and matrix multiplication
[{"from": "human", "value": "I am writing a research paper on a idea The complexity of simulation and matrix multiplication"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Computing the simulation preorder of a given Kripke structure (i.e., a directed graph with  n  labeled vertices) has crucial applications in model checking of temporal logic. It amounts to solving a specific two-players reachability game, called simulation game.  offer the first conditional lor bounds for this problem, and  relate its complexity (for computation, verification, and certification) to some variants of  n  \u00d7  n  matrix multiplication.  show that any  O ( n \u03b1  )-time algorithm for simulation games, even restricting to acyclic games/structures, can be used to compute  n  \u00d7  n  boolean matrix multiplication (BMM) in  O ( n \u03b1  ) time. In the acyclic case,  match this bound by presenting the first subcubic algorithm, based on fast BMM, and running in  n   \u03c9 + o (1)  time (where  \u03c9   O ( n  2 )-size certificates, that can be verified in truly subcubic time by means of matrix multiplication. In the acyclic case,  O ( n  2 ) time is sufficient, employing standard (+, \u00d7)-matrix product verification. In the cyclic case, a min-edge witness matrix multiplication (EWMM) is used, i.e., a matrix multiplication on the semi-ring (max, \u00d7) where one matrix contains only 0's and 1's, which is computable in truly subcubic  n  (3+ \u03c9 )/2+ o (1)  time. Finally,  show a reduction from EWMM to cyclic simulation games which implies a separation beten the cyclic and the acyclic cases, unless EWMM can be verified in  n   \u03c9 + o (1)  time."}]