Complexity Theory: Zooming Out Eric Price UT Austin CS 331, Spring - PowerPoint PPT Presentation

Complexity Theory: Zooming Out Eric Price UT Austin CS 331, Spring 2020 Coronavirus Edition Eric Price (UT Austin) Complexity Theory: Zooming Out 1 / 14

Class Outline Complexity classes 1 Computability 2 Eric Price (UT Austin) Complexity Theory: Zooming Out 2 / 14

A few complexity classes P: Polynomial time Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. ◮ Kind of silly: NP ⊆ PP Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. ◮ Kind of silly: NP ⊆ PP (guess x ; if f ( x ) true, return True; if f ( x ) false, flip a coin) Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. ◮ Kind of silly: NP ⊆ PP (guess x ; if f ( x ) true, return True; if f ( x ) false, flip a coin) BPP: Probabilistic polynomial time, failure probability at most 1 / 3. Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. ◮ Kind of silly: NP ⊆ PP (guess x ; if f ( x ) true, return True; if f ( x ) false, flip a coin) BPP: Probabilistic polynomial time, failure probability at most 1 / 3. BQP: Probabilistic quantum polynomial time, failure probability at most 1 / 3. Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. ◮ Kind of silly: NP ⊆ PP (guess x ; if f ( x ) true, return True; if f ( x ) false, flip a coin) BPP: Probabilistic polynomial time, failure probability at most 1 / 3. BQP: Probabilistic quantum polynomial time, failure probability at most 1 / 3. PSPACE: Polynomial space Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. ◮ Kind of silly: NP ⊆ PP (guess x ; if f ( x ) true, return True; if f ( x ) false, flip a coin) BPP: Probabilistic polynomial time, failure probability at most 1 / 3. BQP: Probabilistic quantum polynomial time, failure probability at most 1 / 3. PSPACE: Polynomial space NPSPACE: Nondeterministic, polynomial space Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. ◮ Kind of silly: NP ⊆ PP (guess x ; if f ( x ) true, return True; if f ( x ) false, flip a coin) BPP: Probabilistic polynomial time, failure probability at most 1 / 3. BQP: Probabilistic quantum polynomial time, failure probability at most 1 / 3. PSPACE: Polynomial space NPSPACE: Nondeterministic, polynomial space ◮ NPSPACE = PSPACE: try all proofs. Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. ◮ Kind of silly: NP ⊆ PP (guess x ; if f ( x ) true, return True; if f ( x ) false, flip a coin) BPP: Probabilistic polynomial time, failure probability at most 1 / 3. BQP: Probabilistic quantum polynomial time, failure probability at most 1 / 3. PSPACE: Polynomial space NPSPACE: Nondeterministic, polynomial space ◮ NPSPACE = PSPACE: try all proofs. EXP: Exponential time Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

A few complexity classes P: Polynomial time NP: Nondeterministic polynomial time PP: failure probability < 1 / 2. ◮ Kind of silly: NP ⊆ PP (guess x ; if f ( x ) true, return True; if f ( x ) false, flip a coin) BPP: Probabilistic polynomial time, failure probability at most 1 / 3. BQP: Probabilistic quantum polynomial time, failure probability at most 1 / 3. PSPACE: Polynomial space NPSPACE: Nondeterministic, polynomial space ◮ NPSPACE = PSPACE: try all proofs. EXP: Exponential time NEXP: Nondeterministic exponential time Eric Price (UT Austin) Complexity Theory: Zooming Out 3 / 14

Relations of complexity classes P ⊆ NP ⊆ PSPACE ⊆ EXP ⊆ NEXP ⊆ EXPSPACE ⊆ . . . Eric Price (UT Austin) Complexity Theory: Zooming Out 4 / 14

Relations of complexity classes P ⊆ NP ⊆ PSPACE ⊆ EXP ⊆ NEXP ⊆ EXPSPACE ⊆ . . . Know: P � = EXP , PSPACE � = EXPSPACE . Eric Price (UT Austin) Complexity Theory: Zooming Out 4 / 14

Relations of complexity classes P ⊆ NP ⊆ PSPACE ⊆ EXP ⊆ NEXP ⊆ EXPSPACE ⊆ . . . Know: P � = EXP , PSPACE � = EXPSPACE . That’s about it. Eric Price (UT Austin) Complexity Theory: Zooming Out 4 / 14

Relations of complexity classes P ⊆ NP ⊆ PSPACE ⊆ EXP ⊆ NEXP ⊆ EXPSPACE ⊆ . . . Know: P � = EXP , PSPACE � = EXPSPACE . That’s about it. P ⊆ BPP ⊆ BQP ⊆ PSPACE Eric Price (UT Austin) Complexity Theory: Zooming Out 4 / 14

Relations of complexity classes P ⊆ NP ⊆ PSPACE ⊆ EXP ⊆ NEXP ⊆ EXPSPACE ⊆ . . . Know: P � = EXP , PSPACE � = EXPSPACE . That’s about it. P ⊆ BPP ⊆ BQP ⊆ PSPACE Most people expect: P = BPP , everything else � . Eric Price (UT Austin) Complexity Theory: Zooming Out 4 / 14

Relations of complexity classes P ⊆ NP ⊆ PSPACE ⊆ EXP ⊆ NEXP ⊆ EXPSPACE ⊆ . . . Know: P � = EXP , PSPACE � = EXPSPACE . That’s about it. P ⊆ BPP ⊆ BQP ⊆ PSPACE Most people expect: P = BPP , everything else � . Don’t know NP compared to BPP or BQP (or even if one is inside the other). Eric Price (UT Austin) Complexity Theory: Zooming Out 4 / 14

Prototypical examples P: evaluate a function Eric Price (UT Austin) Complexity Theory: Zooming Out 5 / 14

Prototypical examples P: evaluate a function ◮ Given a circuit f and input x , what is f ( x )? Eric Price (UT Austin) Complexity Theory: Zooming Out 5 / 14

Prototypical examples P: evaluate a function ◮ Given a circuit f and input x , what is f ( x )? NP: solve a puzzle Eric Price (UT Austin) Complexity Theory: Zooming Out 5 / 14

Prototypical examples P: evaluate a function ◮ Given a circuit f and input x , what is f ( x )? NP: solve a puzzle ◮ SAT: given f , determine if ∃ x : f ( x ) = 1? Eric Price (UT Austin) Complexity Theory: Zooming Out 5 / 14

Prototypical examples P: evaluate a function ◮ Given a circuit f and input x , what is f ( x )? NP: solve a puzzle ◮ SAT: given f , determine if ∃ x : f ( x ) = 1? ◮ Think candy crush: is there any sequence of moves to achieve score X ? Eric Price (UT Austin) Complexity Theory: Zooming Out 5 / 14

Prototypical examples P: evaluate a function ◮ Given a circuit f and input x , what is f ( x )? NP: solve a puzzle ◮ SAT: given f , determine if ∃ x : f ( x ) = 1? ◮ Think candy crush: is there any sequence of moves to achieve score X ? ◮ Easy to verify once the solution is found. Eric Price (UT Austin) Complexity Theory: Zooming Out 5 / 14

Prototypical examples P: evaluate a function ◮ Given a circuit f and input x , what is f ( x )? NP: solve a puzzle ◮ SAT: given f , determine if ∃ x : f ( x ) = 1? ◮ Think candy crush: is there any sequence of moves to achieve score X ? ◮ Easy to verify once the solution is found. PSPACE: solve a 2-player game Eric Price (UT Austin) Complexity Theory: Zooming Out 5 / 14

Prototypical examples P: evaluate a function ◮ Given a circuit f and input x , what is f ( x )? NP: solve a puzzle ◮ SAT: given f , determine if ∃ x : f ( x ) = 1? ◮ Think candy crush: is there any sequence of moves to achieve score X ? ◮ Easy to verify once the solution is found. PSPACE: solve a 2-player game ◮ TQBF: ∃ x 1 ∀ x 2 ∃ x 3 · · · ∀ x n : f ( x ) = 1 Eric Price (UT Austin) Complexity Theory: Zooming Out 5 / 14

Prototypical examples P: evaluate a function ◮ Given a circuit f and input x , what is f ( x )? NP: solve a puzzle ◮ SAT: given f , determine if ∃ x : f ( x ) = 1? ◮ Think candy crush: is there any sequence of moves to achieve score X ? ◮ Easy to verify once the solution is found. PSPACE: solve a 2-player game ◮ TQBF: ∃ x 1 ∀ x 2 ∃ x 3 · · · ∀ x n : f ( x ) = 1 ◮ Think chess: do I have a move, so no matter what you do, I can find a move, so no matter, etc., etc., I end up winning? Eric Price (UT Austin) Complexity Theory: Zooming Out 5 / 14