Université Paris 5 – René Descartes LIPADE Algorithmic Complexity Space Complexity Jean-Guy Mailly ([email protected]) 2017 Outline 1 Basics on Space Complexity Main Space Complexity Classes Deterministic and Non-Deterministic Space Space-Time Relations Determining Space Complexity Hardness and Completeness Well-Known Problems J.-G. Mailly | Complexity – Space Space Used by a Turing Machine 2 I If M is a Turing Machine and x an input word, the space used by M on x is the number of different squares visited by M during the computation. I For f : N ! N, we say that a Turing Machine M works with space O(f (n)) if I M stops for every input x I on every input x, M uses on x a space s(n) with s(n) 2 O(f (n)) J.-G. Mailly | Complexity – Space Space Complexity Classes 3 I For f : N ! N, DSPACE(f (n)) is the set of languages which are decided by a deterministic Turing Machine M working with space O(f (n)). I For f : N ! N, NSPACE(f (n)) is the set of languages which are decided by a non-deterministic Turing Machine M working with space O(f (n)). J.-G. Mailly | Complexity – Space Deterministic vs Non-Deterministic Space 4 Intuition: Non-determinism is not “so important” for space complexity::: Theorem [Savitch 1970] For all function f space-constructible, DSPACE(f (n)) ⊆ NSPACE(f (n)) ⊆ DSPACE(f (n)2) J.-G. Mailly | Complexity – Space Complement Classes 5 Proposition For all function f (n) space-constructible, DSPACE(f (n)) = CODSPACE(f (n)) Theorem [Immerman 1988, Szelepcsényi 1988] For all function f (n) ≥ log(n) space-constructible, NSPACE(f (n)) = CONSPACE(f (n)) J.-G. Mailly | Complexity – Space Outline 6 Basics on Space Complexity Main Space Complexity Classes Deterministic and Non-Deterministic Space Space-Time Relations Determining Space Complexity Hardness and Completeness Well-Known Problems J.-G. Mailly | Complexity – Space L ⊆ PSPACE Deterministic Space Complexity 7 I L = DSPACE(log(n)) is the class of decision problems that can be solved by a DTM using logarithmic space S k I PSPACE = k>0 DSPACE(n ) is the class of decision problems that can be solved by a DTM using polynomial space J.-G. Mailly | Complexity – Space Deterministic Space Complexity 7 I L = DSPACE(log(n)) is the class of decision problems that can be solved by a DTM using logarithmic space S k I PSPACE = k>0 DSPACE(n ) is the class of decision problems that can be solved by a DTM using polynomial space L ⊆ PSPACE J.-G. Mailly | Complexity – Space NL ⊆ NPSPACE Non-Deterministic Space Complexity 8 I NL = NSPACE(log(n)) is the class of decision problems that can be solved by a NDTM using logarithmic space S k I NPSPACE = k>0 NSPACE(n ) is the class of decision problems that can be solved by a NDTM using polynomial space J.-G. Mailly | Complexity – Space Non-Deterministic Space Complexity 8 I NL = NSPACE(log(n)) is the class of decision problems that can be solved by a NDTM using logarithmic space S k I NPSPACE = k>0 NSPACE(n ) is the class of decision problems that can be solved by a NDTM using polynomial space NL ⊆ NPSPACE J.-G. Mailly | Complexity – Space Proof: I For any k > 0, NSPACE(nk ) ⊆ DSPACE((nk )2) = DSPACE((n2k ) ⊆ PSPACE, so NPSPACE ⊆ PSPACE k k I For any k > 0, DSPACE(n ) ⊆ NSPACE(n ) ⊆ NPSPACE, so PSPACE ⊆ NPSPACE PSPACE vs NPSPACE 9 Proposition PSPACE = NPSPACE J.-G. Mailly | Complexity – Space PSPACE vs NPSPACE 9 Proposition PSPACE = NPSPACE Proof: I For any k > 0, NSPACE(nk ) ⊆ DSPACE((nk )2) = DSPACE((n2k ) ⊆ PSPACE, so NPSPACE ⊆ PSPACE k k I For any k > 0, DSPACE(n ) ⊆ NSPACE(n ) ⊆ NPSPACE, so PSPACE ⊆ NPSPACE J.-G. Mailly | Complexity – Space Corollary NP ⊆ PSPACE Relations Between Time and Space (1/3) 10 Proposition For any mapping f : N ! N, NTIME(f (n)) ⊆ DSPACE(f (n)) J.-G. Mailly | Complexity – Space Relations Between Time and Space (1/3) 10 Proposition For any mapping f : N ! N, NTIME(f (n)) ⊆ DSPACE(f (n)) Corollary NP ⊆ PSPACE J.-G. Mailly | Complexity – Space Relations Between Time and Space (2/3) 11 Polynomial Hierarchy vs Polynomial Space PH ⊆ PSPACE Intuition For each class of the polynomial hierharchy, there is a complete problem 8i QBF or 9i QBF. All these problems are special instances of TQBF, so they are in PSPACE. J.-G. Mailly | Complexity – Space Relations Between Time and Space (3/3) 12 P NP P Σ2 L ⊆ NL ⊆ P ⊆ coNP ⊆ ∆2 ⊆ P ⊆ · · · ⊆ PH ⊆ PSPACE ⊆ EXP ⊆ NEXP Π2 We also know that some inclusions are strict: I NL ⊂ PSPACE I P ⊂ EXP J.-G. Mailly | Complexity – Space Outline 13 Basics on Space Complexity Main Space Complexity Classes Deterministic and Non-Deterministic Space Space-Time Relations Determining Space Complexity Hardness and Completeness Well-Known Problems J.-G. Mailly | Complexity – Space 0 P I A problem P is C-hard if for every problem P’ in C, P ≤f P, i.e. there is a polynomial reduction from any problem in C to P I A problem is C-complete if it is in C and it is C-hard I This definition works for PSPACE-hardness and PSPACE-completeness I It does not work for L and NL Completeness for Space Complexity Classes 14 The notions of hardness and completeness exist for space complexity classes. I For most of the complexity classes, the definition is exactly the same as hardness and completeness for time complexity J.-G. Mailly | Complexity – Space I A problem is C-complete if it is in C and it is C-hard I This definition works for PSPACE-hardness and PSPACE-completeness I It does not work for L and NL Completeness for Space Complexity Classes 14 The notions of hardness and completeness exist for space complexity classes. I For most of the complexity classes, the definition is exactly the same as hardness and completeness for time complexity 0 P I A problem P is C-hard if for every problem P’ in C, P ≤f P, i.e. there is a polynomial reduction from any problem in C to P J.-G. Mailly | Complexity – Space I This definition works for PSPACE-hardness and PSPACE-completeness I It does not work for L and NL Completeness for Space Complexity Classes 14 The notions of hardness and completeness exist for space complexity classes. I For most of the complexity classes, the definition is exactly the same as hardness and completeness for time complexity 0 P I A problem P is C-hard if for every problem P’ in C, P ≤f P, i.e. there is a polynomial reduction from any problem in C to P I A problem is C-complete if it is in C and it is C-hard J.-G. Mailly | Complexity – Space I It does not work for L and NL Completeness for Space Complexity Classes 14 The notions of hardness and completeness exist for space complexity classes. I For most of the complexity classes, the definition is exactly the same as hardness and completeness for time complexity 0 P I A problem P is C-hard if for every problem P’ in C, P ≤f P, i.e. there is a polynomial reduction from any problem in C to P I A problem is C-complete if it is in C and it is C-hard I This definition works for PSPACE-hardness and PSPACE-completeness J.-G. Mailly | Complexity – Space Completeness for Space Complexity Classes 14 The notions of hardness and completeness exist for space complexity classes. I For most of the complexity classes, the definition is exactly the same as hardness and completeness for time complexity 0 P I A problem P is C-hard if for every problem P’ in C, P ≤f P, i.e. there is a polynomial reduction from any problem in C to P I A problem is C-complete if it is in C and it is C-hard I This definition works for PSPACE-hardness and PSPACE-completeness I It does not work for L and NL J.-G. Mailly | Complexity – Space L P Observation: If P1 ≤f P2, then P1 ≤f P2 NL-Hardness and NL-Completeness 0 0 L I A problem P is NL-hard iff 8P 2 NL, P ≤f P I A problem P is NL-complete iff P is NL-hard and P 2 L Logarithmic Space and Completeness 15 Logarithmic-Space Functional Reduction A logarithmic-space functional reduction f is a total computable function from a problem P1 to a problem P2 such that, for any instance i of P1, I f (i) can be computed in logarithmic-space in the size of i I i is a positive instance of P1 iff f (i) is a positive instance of P2 L Notation: P1 ≤f P2 J.-G. Mailly | Complexity – Space NL-Hardness and NL-Completeness 0 0 L I A problem P is NL-hard iff 8P 2 NL, P ≤f P I A problem P is NL-complete iff P is NL-hard and P 2 L Logarithmic Space and Completeness 15 Logarithmic-Space Functional Reduction A logarithmic-space functional reduction f is a total computable function from a problem P1 to a problem P2 such that, for any instance i of P1, I f (i) can be computed in logarithmic-space in the size of i I i is a positive instance of P1 iff f (i) is a positive instance of P2 L Notation: P1 ≤f P2 L P Observation: If P1 ≤f P2, then P1 ≤f P2 J.-G.