Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Lecture 10: Space Complexity III Arijit Bishnu 27.03.2010 Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). The class NPSPACE is an analog of the class NP. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). The class NPSPACE is an analog of the class NP. Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). The class NPSPACE is an analog of the class NP. Definition L = SPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). The class NPSPACE is an analog of the class NP. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Is the following in L? Check whether the following language is in L? A = f0k 1k j k ≥ 0g Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Examples Is the following in L? Check whether the following language is in L? EVEN = fx j x has an even number of 1sg: Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Examples Is the following in L? Check whether the following language is in L? EVEN = fx j x has an even number of 1sg: Is the following in L? Check whether the following language is in L? A = f0k 1k j k ≥ 0g The idea for this definition was that the nondeterministic choices of the NDTM that result in acceptance can be viewed as a polynomial sized certificate that x 2 L and vice versa. A logspace NDTM, according to the concept of configuration graph that we studied, can generate a certificate that is polynomially long but it does not have the space to store it. Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes A Certificate Based Definition of NL Recall the Definition of the Class NP A language L ⊆ f0; 1g∗ is in NP if there exists a polynomial p : N ! N and a polynomial-time TM M such that for every x 2 f0; 1g∗, x 2 L () 9u 2 f0; 1gp(jxj) such that M(x; u) = 1 If x 2 L and u 2 f0; 1gp(jxj) satisfy M(x; u) = 1, then we call u a certificate for x (w.r.t. language L and machine M). A logspace NDTM, according to the concept of configuration graph that we studied, can generate a certificate that is polynomially long but it does not have the space to store it. Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes A Certificate Based Definition of NL Recall the Definition of the Class NP A language L ⊆ f0; 1g∗ is in NP if there exists a polynomial p : N ! N and a polynomial-time TM M such that for every x 2 f0; 1g∗, x 2 L () 9u 2 f0; 1gp(jxj) such that M(x; u) = 1 If x 2 L and u 2 f0; 1gp(jxj) satisfy M(x; u) = 1, then we call u a certificate for x (w.r.t. language L and machine M). The idea for this definition was that the nondeterministic choices of the NDTM that result in acceptance can be viewed as a polynomial sized certificate that x 2 L and vice versa. Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes A Certificate Based Definition of NL Recall the Definition of the Class NP A language L ⊆ f0; 1g∗ is in NP if there exists a polynomial p : N ! N and a polynomial-time TM M such that for every x 2 f0; 1g∗, x 2 L () 9u 2 f0; 1gp(jxj) such that M(x; u) = 1 If x 2 L and u 2 f0; 1gp(jxj) satisfy M(x; u) = 1, then we call u a certificate for x (w.r.t. language L and machine M). The idea for this definition was that the nondeterministic choices of the NDTM that result in acceptance can be viewed as a polynomial sized certificate that x 2 L and vice versa. A logspace NDTM, according to the concept of configuration graph that we studied, can generate a certificate that is polynomially long but it does not have the space to store it. At each step, the machine's head on this tape can either stay in place or move right. Thus, the tape is a read once tape. Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes A Certificate Based Definition of NL continued... So, we assume a separate read-only tape for the logspace machine. We call this tape to be the certificate tape. Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes A Certificate Based Definition of NL continued... So, we assume a separate read-only tape for the logspace machine. We call this tape to be the certificate tape. At each step, the machine's head on this tape can either stay in place or move right. Thus, the tape is a read once tape. Definition: Class NL A language L ⊆ f0; 1g∗ is in NL if there exists a polynomial p : N ! N and a deterministic TM M using at most O(log jxj) space on its read/write tape for every input x and M has a certificate tape, such that for every x 2 f0; 1g∗, x 2 L () 9u 2 f0; 1gp(jxj) such that M(x; u) = 1 M(x; u) denotes the output of M where x is placed on its input tape and u is placed on its certificate tape. Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes The Certificate Based Definition of NL Definition: Class NL A language L ⊆ f0; 1g∗ is in NL if there exists a polynomial p : N ! N and a deterministic TM M using at most O(log jxj) space on its read/write tape for every input x and M has a certificate tape, such that for every x 2 f0; 1g∗, x 2 L () 9u 2 f0; 1gp(jxj) such that M(x; u) = 1 M(x; u) denotes the output of M where x is placed on its input tape and u is placed on its certificate tape. Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes The Certificate Based Definition of NL Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes The Certificate Based Definition of NL Definition: Class NL A language L ⊆ f0; 1g∗ is in NL if there exists a polynomial p : N ! N and a deterministic TM M using at most O(log jxj) space on its read/write tape for every input x and M has a certificate tape, such that for every x 2 f0; 1g∗, x 2 L () 9u 2 f0; 1gp(jxj) such that M(x; u) = 1 M(x; u) denotes the output of M where x is placed on its input tape and u is placed on its certificate tape. Solution: PATH 2 NL Generate the certificate on the certificate tape and verify it. Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Example Is the following in NL? Check whether the following language is in NL? PATH = f< G; s; t > j G is a directed graph in which there is a path from s to tg.