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) ∈ 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 ∀i QBF or ∃i 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 ∀P ∈ NL, P ≤f P I A problem P is NL-complete iff P is NL-hard and P ∈ 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 ∀P ∈ NL, P ≤f P I A problem P is NL-complete iff P is NL-hard and P ∈ 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. Mailly | Complexity – Space 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

NL-Hardness and NL-Completeness 0 0 L I A problem P is NL-hard iff ∀P ∈ NL, P ≤f P I A problem P is NL-complete iff P is NL-hard and P ∈ L

J.-G. Mailly | Complexity – Space To define a meaningful notion of L-completeness, the hardness must be defined with another kind of functional reductions More details in [Garey and Johnson 1979]

What About L-Completeness? 16

Theorem L Any problem in L is L-complete with respect to ≤f !

J.-G. Mailly | Complexity – Space What About L-Completeness? 16

Theorem L Any problem in L is L-complete with respect to ≤f !

To define a meaningful notion of L-completeness, the hardness must be defined with another kind of functional reductions More details in [Garey and Johnson 1979]

J.-G. Mailly | Complexity – Space Complexity of TQBF [ TQBF = (∃nQBF ∪ ∀nQBF) n≥1 TQBF is PSPACE-complete

Canonical PSPACE-Complete Problem 17

Reminder:

I A canonical QBF is a formula Q1X1, Q2X2,... QnXn, φ:

I Qi ∈ {∀, ∃} and Qi 6= Qi+1 I X1,... Xn form a partition of the Boolean variables in φ I φ is a propositional formula

I ∃nQBF: is the QBF ∃X1, ∀X2,... QnXn, φ true?

I ∀nQBF: is the QBF ∀X1, ∃X2,... QnXn, φ true?

J.-G. Mailly | Complexity – Space Canonical PSPACE-Complete Problem 17

Reminder:

I A canonical QBF is a formula Q1X1, Q2X2,... QnXn, φ:

I Qi ∈ {∀, ∃} and Qi 6= Qi+1 I X1,... Xn form a partition of the Boolean variables in φ I φ is a propositional formula

I ∃nQBF: is the QBF ∃X1, ∀X2,... QnXn, φ true?

I ∀nQBF: is the QBF ∀X1, ∃X2,... QnXn, φ true?

Complexity of TQBF [ TQBF = (∃nQBF ∪ ∀nQBF) n≥1 TQBF is PSPACE-complete

J.-G. Mailly | Complexity – Space Complexity of (m, n, k)-STRAT (m − n − k)-STRAT is the decision problem: “Is there a winning strategy for player X in the (m, n, k) game?” (m − n − k)-STRAT is PSPACE-complete

PSPACE vs Games Tic-Tac-Toe 18

I Classical Tic-Tac-Toe: In a 3 × 3 grid, two players must draw some marks (X or O). The winner is the first who has placed three marks in a horizontal, vertical or diagonal row

I Generalized Tic-tac-toe ((m, n, k) game): In a m × n grid, a player wins when he has placed k marks in a row

I A winning strategy at time t for a player is a sequence of moves (drawing a mark in the grid) such that the player wins.

J.-G. Mailly | Complexity – Space PSPACE vs Games Tic-Tac-Toe 18

I Classical Tic-Tac-Toe: In a 3 × 3 grid, two players must draw some marks (X or O). The winner is the first who has placed three marks in a horizontal, vertical or diagonal row

I Generalized Tic-tac-toe ((m, n, k) game): In a m × n grid, a player wins when he has placed k marks in a row

I A winning strategy at time t for a player is a sequence of moves (drawing a mark in the grid) such that the player wins.

Complexity of (m, n, k)-STRAT (m − n − k)-STRAT is the decision problem: “Is there a winning strategy for player X in the (m, n, k) game?” (m − n − k)-STRAT is PSPACE-complete

J.-G. Mailly | Complexity – Space I This can be represented as a QBF formula

PSPACE vs Games Tic-Tac-Toe: Intuition 19

I Determining if there is a winning strategy for player X is like asking the question: Is there a way to draw an X at step 1, such that for all possible moves of the O player at step 2, there is a way to draw an X at step 3, such that for all possible moves of the O player at step 4, there is ... such that X player wins?

J.-G. Mailly | Complexity – Space PSPACE vs Games Tic-Tac-Toe: Intuition 19

I Determining if there is a winning strategy for player X is like asking the question: Is there a way to draw an X at step 1, such that for all possible moves of the O player at step 2, there is a way to draw an X at step 3, such that for all possible moves of the O player at step 4, there is ... such that X player wins?

I This can be represented as a QBF formula

J.-G. Mailly | Complexity – Space Generalized Super Mario Bros. [Demaine et al. 2016] 20

FIGURE: Super Mario Bros. is PSPACE-complete

J.-G. Mailly | Complexity – Space Lemmings [Viglietta 2015] 21

FIGURE: Lemmings is PSPACE-complete

J.-G. Mailly | Complexity – Space NL-Complete Problem in Graphs 22

Reach

Given a directed graph G = hN, Ei and two nodes n1, n2 ∈ N, is there a path from n1 to n2 in G?

Theorem [Papadimitriou 1994] Reach is NL-complete

J.-G. Mailly | Complexity – Space NL-Complete Problem in Logics 23

Reminder on 2SAT 2SAT is a special case of satisfiability problem where only CNF formulas with binary clauses are considered

Theorem [Papadimitriou 1994] 2SAT is NL-complete

J.-G. Mailly | Complexity – Space References 24

[Savitch 1970] W. J. Savitch, Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences, 177–192, 1970. [Immerman 1988] N. Immerman, Nondeterministic space is closed under complementation. SIAM Journal on Computing 17, 935–938, 1988. [Szelepcsényi 1988] R. Szelepcsényi, The Method of Forced Enumeration for Nondeterministic Automata. Acta Informatica vol. 26, no 3, 279–284, 1988.

J.-G. Mailly | Complexity – Space References 25

[Garey and Johnson 1979] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. Section 7.5: Logarithmic Space, 177–181, 1979. [Papadimitriou 1994] C.H Papadimitriou, Computational Complexity. Addison-Wesley, 1994. [Demaine et al. 2016] E.D. Demaine, G. Viglietta and A. Williams, Super Mario Bros. Is Harder/Easier than We Thought. Proceedings of the 8th International Conference on Fun with Algorithms, 2016. [Viglietta 2015] G. Viglietta, Lemmings Is PSPACE-Complete. Theor. Comput. Sci. 586: 120-134, 2015.

J.-G. Mailly | Complexity – Space