[PPT] - Reachability problem for polynomial iteration is PSPACE-complete PowerPoint Presentation

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Reachability problem for polynomial iteration is PSPACE-complete

Reino Niskanen

Department of Computer Science University of Liverpool, UK

11th International Workshop on Reachability Problems

Niskanen Polynomial iteration is PSPACE-complete RP 2017 1 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Introduction

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

p1(x) = x2 + x + 3 p2(x) = x4 + 2x3 + 3x2 + 2x + 1 p3(x) = −x + 5 Can we iterate x = 6 to reach 0? 6

Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

p1(x) = x2 + x + 3 p2(x) = x4 + 2x3 + 3x2 + 2x + 1 p3(x) = −x + 5 Can we iterate x = 6 to reach 0? 6 p3(x)

Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

p1(x) = x2 + x + 3 p2(x) = x4 + 2x3 + 3x2 + 2x + 1 p3(x) = −x + 5 Can we iterate x = 6 to reach 0? 6 p3(x)

Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

p1(x) = x2 + x + 3 p2(x) = x4 + 2x3 + 3x2 + 2x + 1 p3(x) = −x + 5 Can we iterate x = 6 to reach 0? 6 p3(x) p2(x)

Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

p1(x) = x2 + x + 3 p2(x) = x4 + 2x3 + 3x2 + 2x + 1 p3(x) = −x + 5 Can we iterate x = 6 to reach 0? 6 p3(x) p2(x)

Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

p1(x) = x2 + x + 3 p2(x) = x4 + 2x3 + 3x2 + 2x + 1 p3(x) = −x + 5 Can we iterate x = 6 to reach 0? 6 p3(x) p2(x) p1(x)

Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

p1(x) = x2 + x + 3 p2(x) = x4 + 2x3 + 3x2 + 2x + 1 p3(x) = −x + 5 Can we iterate x = 6 to reach 0? 6 p3(x) p2(x) p1(x)

Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

p1(x) = x2 + x + 3 p2(x) = x4 + 2x3 + 3x2 + 2x + 1 p3(x) = −x + 5 Can we iterate x = 6 to reach 0? 6 p3(x) p2(x) p1(x) p3(x)

Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

p1(x) = x2 + x + 3 p2(x) = x4 + 2x3 + 3x2 + 2x + 1 p3(x) = −x + 5 Can we iterate x = 6 to reach 0? 6 p3(x) p2(x) p1(x) p3(x)

Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

How much space is needed? p2(x) = x4 + 2x3 + 3x2 + 2x + 1

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

How much space is needed? p2(x) = x4 + 2x3 + 3x2 + 2x + 1 A lot.. 6 → 1849 → 11700853263801 The representation grows exponentially.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Definitions

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Linear bounded automata

Linear bounded automata is a Turing machine with a finite tape whose length is bounded by a linear function of the size of the input. A configuration is [q, i, w], where q ∈ Q, i is the position of the head, w ∈ {0, 1}n is the word written on the tape. The reachability problem: [q0, 1, 0n] →∗ [qf, 1, 0n]? q3 · · · n Theorem The reachability problem for LBA is PSPACE-complete.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial register machines

Introduced by Finkel, Göller and Haase in MFCS’13 A PRM consists of a graph (V, E) labelled by polynomials in Z[x]. A configuration is [s, z] ∈ V × Z. [s, z] yields [s′, y] if (s, p(x), s′) ∈ E such that p(z) = y. The reachability problem: [s0, 0] →∗ [sf, 0]? p1(x) p

2

( x ) p3(x) p4(x) p5(x) Theorem (FGH 2013) The reachability problem for PRM is PSPACE-complete.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

Can be seen as stateless PRMs. P = {p1(x), p2(x), . . . , pn(x)} ⊆ Z[x]. The reachability problem: Does there exist a finite sequence pi1(x), pi2(x), . . . , pij(x) that maps x0 to xf, i.e., whether pij(pij−1(· · · pi2(pi1(x0)) · · · ) = xf. p1(x) p2(x) p3(x) p4(x) Theorem The reachability problem for polynomial iteration is

PSPACE-complete.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Upper bound

Lemma The reachability problem for polynomial iteration is PSPACE. Proof. The reachability problem is PSPACE even for machines with states.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Upper bound

Lemma The reachability problem for polynomial iteration is PSPACE. Idea of Proof For almost all polynomials p(x), there exists a bound b, such that for any |y| > b, |p(y)| ≥ 2|y|.

Niskanen Polynomial iteration is PSPACE-complete RP 2017 10 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Upper bound

Lemma The reachability problem for polynomial iteration is PSPACE. Idea of Proof For almost all polynomials p(x), there exists a bound b, such that for any |y| > b, |p(y)| ≥ 2|y|. Only polynomials ±x + a, for some a ∈ Z, do not have this

bound. Their behaviour can be simulated by a 1-VASS, for

which the reachability problem is in NP.

Niskanen Polynomial iteration is PSPACE-complete RP 2017 10 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Upper bound

Lemma The reachability problem for polynomial iteration is PSPACE. Idea of Proof For almost all polynomials p(x), there exists a bound b, such that for any |y| > b, |p(y)| ≥ 2|y|. Only polynomials ±x + a, for some a ∈ Z, do not have this

bound. Their behaviour can be simulated by a 1-VASS, for

which the reachability problem is in NP. Moreover, it can be simulated in polynomial space, to which values inside [−b, b] the polynomials ±x + a return to.

Niskanen Polynomial iteration is PSPACE-complete RP 2017 10 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Lower bound

Lemma The reachability problem for polynomial iteration is

PSPACE-hard.

Idea of Proof Follow the proof for PRM by reducing from the reachability of

LBA. Additionally, encode states and state transitions as

polynomials.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Ingredients of the reduction of LBA to PRM

Let p1, . . . , pn ∈ PRIME. We consider an integer x as a residue class r mod p1 · · · pn. The tape word w ∈ {0, 1}n is encoded as an integer r satisfying r ≡ wi mod pi for each i = 1, . . . , n.

q3 · · · n . . .

mod p1 mod p2

. . .

mod pn r ≡ r ≡

. . .

r ≡ Niskanen Polynomial iteration is PSPACE-complete RP 2017 12 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Ingredients of the reduction of LBA to PRM

Let p1, . . . , pn ∈ PRIME. We consider an integer x as a residue class r mod p1 · · · pn. The tape word w ∈ {0, 1}n is encoded as an integer r satisfying r ≡ wi mod pi for each i = 1, . . . , n.

q3 · · · n . . .

mod p1 mod p2

. . .

mod pn r ≡ r ≡

. . .

r ≡

We only consider integers that are solutions to r ≡ b1 mod p1 . . . where bi ∈ {0, 1, 2}. r ≡ bn mod pn,

Niskanen Polynomial iteration is PSPACE-complete RP 2017 12 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Ingredients of the reduction of LBA to PRM

Polynomials that locally modify residue classes.

FLIPi to switch between r ≡ 0 mod pi and r ′ ≡ 1 mod pi EQZEROi to check that r ≡ 0 mod pi EQONEi to check that r ≡ 1 mod pi.

While the other congruences remain untouched.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

The update polynomials

if r ≡ 0 mod pi :

FLIPi(r) ≡

1

mod pi r mod pj if r ≡ 1 mod pi :

FLIPi(r) ≡

mod pi

r mod pj if r ≡ 2 mod pi :

FLIPi(r) ≡

2

mod pi r mod pj.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

The update polynomials

if r ≡ 0 mod pi :

FLIPi(r) ≡

1

mod pi r mod pj if r ≡ 1 mod pi :

FLIPi(r) ≡

mod pi

r mod pj if r ≡ 2 mod pi :

FLIPi(r) ≡

2

mod pi r mod pj.

is realised by pflip,i(x) = a′

2x2 + a′ 1x + a′

a′

2

≡ 3pi+1

2

mod pi a′

2

≡ 0 mod pj

a′

1 ≡ −5 pi+1 2

mod pi a′

1 ≡ 1

mod pj

a′

0 ≡ 1

mod pi a′

0 ≡ 0

mod pj

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

The update polynomials

if r ≡ 0 mod pi :

EQZEROi(r) ≡

mod pi

r mod pj if r ≡ 1, 2 mod pi :

EQZEROi(r) ≡

2

mod pi r mod pj

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

The update polynomials

if r ≡ 0 mod pi :

EQZEROi(r) ≡

mod pi

r mod pj if r ≡ 1, 2 mod pi :

EQZEROi(r) ≡

2

mod pi r mod pj

is realised by peqzero,i(x) = a′

2x2 + a′ 1x + a′

a′

2

≡ −1 mod pi a′

2

≡ 0 mod pj

a′

1 ≡ 3

mod pi a′

1 ≡ 1

mod pj

a′

0 ≡ 0

mod pi a′

0 ≡ 0

mod pj

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

The update polynomials

if r ≡ 1 mod pi :

EQONEi(r) ≡

1

mod pi r mod pj if r ≡ 0, 2 mod pi :

EQONEi(r) ≡

2

mod pi r mod pj

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

The update polynomials

if r ≡ 1 mod pi :

EQONEi(r) ≡

1

mod pi r mod pj if r ≡ 0, 2 mod pi :

EQONEi(r) ≡

2

mod pi r mod pj

is realised by peqone,i(x) = a′

2x2 + a′ 1x + a′

a′

2

≡ 1 mod pi a′

2

≡ 0 mod pj

a′

1 ≡ −2

mod pi a′

1 ≡ 1

mod pj

a′

0 ≡ 2

mod pi a′

0 ≡ 0

mod pj

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Ingredients of the reduction of LBA to PRM

States of PRM contain information on the state of LBA, position

f the head, and which symbol it is reading.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Ingredients of the reduction of LBA to PRM

States of PRM contain information on the state of LBA, position

f the head, and which symbol it is reading.

Connect the states using correct FLIPi, EQZEROi and EQONEi

moves. The machine guesses and verifies the symbols that will

be read next.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Ingredients of the reduction of LBA to PRM

States of PRM contain information on the state of LBA, position

f the head, and which symbol it is reading.

Connect the states using correct FLIPi, EQZEROi and EQONEi

moves. The machine guesses and verifies the symbols that will

be read next. For a move δ(q, 0) = (q′, 1, R) of the LBA, the states and transitions of the PRM (for each i) are:

q, i, 0 q′, i + 1, 0 q′, i + 1, 1

EQZEROi+1 ◦ FLIPi EQONEi+1 ◦ FLIPi

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

LBA to polynomial iteration

Let p1, . . . , pn+n |Q| ∈ PRIME.

q3 · · · n tape content . . .

mod p1 mod p2

. . .

mod pn r ≡ r ≡

. . .

r ≡

. . .

mod pn+1 mod pn+2 mod pn+3

. . .

mod pn+|Q| r ≡ r ≡ r ≡

. . .

r ≡

. . .

mod pn+|Q|+1 mod pn+|Q|+2 mod pn+|Q|+3

. . .

mod pn+2|Q| r ≡ r ≡ r ≡

. . .

r ≡

· · · . . .

mod pn+(n−1)|Q|+1 mod pn+(n−1)|Q|+2 mod pn+(n−1)|Q|+3

. . .

mod pn+n|Q| r ≡ r ≡ r ≡

. . .

r ≡ ← state q1 ← state q2 ← state q3

. . .

← state q|Q| Niskanen Polynomial iteration is PSPACE-complete RP 2017 18 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

LBA to polynomial iteration

Let p1, . . . , pn+n |Q| ∈ PRIME.

q3 · · · n tape content 1st cell 2nd cell nth cell Position of the head: . . .

mod p1 mod p2

. . .

mod pn r ≡ r ≡

. . .

r ≡

. . .

mod pn+1 mod pn+2 mod pn+3

. . .

mod pn+|Q| r ≡ r ≡ r ≡

. . .

r ≡

. . .

mod pn+|Q|+1 mod pn+|Q|+2 mod pn+|Q|+3

. . .

mod pn+2|Q| r ≡ r ≡ r ≡

. . .

r ≡

· · · . . .

mod pn+(n−1)|Q|+1 mod pn+(n−1)|Q|+2 mod pn+(n−1)|Q|+3

. . .

mod pn+n|Q| r ≡ r ≡ r ≡

. . .

r ≡ ← state q1 ← state q2 ← state q3

. . .

← state q|Q| Niskanen Polynomial iteration is PSPACE-complete RP 2017 18 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Simulating moves

To simulate a move of LBA from [qj, i, w] to [qk, i − 1, w′], where wi = 0 and w′

i = 1, using a rule δ(qj, 0) = (qk, 1, L), we need to

Niskanen Polynomial iteration is PSPACE-complete RP 2017 19 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Simulating moves

To simulate a move of LBA from [qj, i, w] to [qk, i − 1, w′], where wi = 0 and w′

i = 1, using a rule δ(qj, 0) = (qk, 1, L), we need to

verify that we are in the correct state qj in position i;

δ(qj, 0) = (qk, 1, L)

EQONEn+j+(i−1)|Q|

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Simulating moves

To simulate a move of LBA from [qj, i, w] to [qk, i − 1, w′], where wi = 0 and w′

i = 1, using a rule δ(qj, 0) = (qk, 1, L), we need to

verify that we are in the correct state qj in position i; move to state qk in position i − 1 from qj in position i;

δ(qj, 0) = (qk, 1, L)

FLIPn+k+(i−2)|Q| ◦ FLIPn+j+(i−1)|Q| ◦ EQONEn+j+(i−1)|Q|

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Simulating moves

To simulate a move of LBA from [qj, i, w] to [qk, i − 1, w′], where wi = 0 and w′

i = 1, using a rule δ(qj, 0) = (qk, 1, L), we need to

verify that we are in the correct state qj in position i; move to state qk in position i − 1 from qj in position i; verify that the symbol in ith position is 0;

δ(qj, 0) = (qk, 1, L)

EQZEROi FLIPn+k+(i−2)|Q| ◦ FLIPn+j+(i−1)|Q| ◦ EQONEn+j+(i−1)|Q|

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Simulating moves

To simulate a move of LBA from [qj, i, w] to [qk, i − 1, w′], where wi = 0 and w′

i = 1, using a rule δ(qj, 0) = (qk, 1, L), we need to

verify that we are in the correct state qj in position i; move to state qk in position i − 1 from qj in position i; verify that the symbol in ith position is 0; rewrite that 0 as 1.

δ(qj, 0) = (qk, 1, L)

FLIPi ◦ EQZEROi FLIPn+k+(i−2)|Q| ◦ FLIPn+j+(i−1)|Q| ◦ EQONEn+j+(i−1)|Q|

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Simulating moves

Applying move δ(q3, 0) = (q1, 1, L) to [q3, 2, 1001 · · · 1].

q3 · · · n tape content 1st cell 2nd cell nth cell Position of the head: . . .

mod p1 mod p2

. . .

mod pn r ≡ r ≡

. . .

r ≡

. . .

mod pn+1 mod pn+2 mod pn+3

. . .

mod pn+|Q| r ≡ r ≡ r ≡

. . .

r ≡

. . .

mod pn+|Q|+1 mod pn+|Q|+2 mod pn+|Q|+3

. . .

mod pn+2|Q| r ≡ r ≡ r ≡

. . .

r ≡

· · · . . .

mod pn+(n−1)|Q|+1 mod pn+(n−1)|Q|+2 mod pn+(n−1)|Q|+3

. . .

mod pn+n|Q| r ≡ r ≡ r ≡

. . .

r ≡ ← state q1 ← state q2 ← state q3

. . .

← state q|Q| Niskanen Polynomial iteration is PSPACE-complete RP 2017 20 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Simulating moves

Applying move δ(q3, 0) = (q1, 1, L) to [q3, 2, 1001 · · · 1].

q1 · · · n tape content 1st cell 2nd cell nth cell Position of the head: . . .

mod p1 mod p2

. . .

mod pn r ≡ r ≡

. . .

r ≡

. . .

mod pn+1 mod pn+2 mod pn+3

. . .

mod pn+|Q| r ≡ r ≡ r ≡

. . .

r ≡

. . .

mod pn+|Q|+1 mod pn+|Q|+2 mod pn+|Q|+3

. . .

mod pn+2|Q| r ≡ r ≡ r ≡

. . .

r ≡

· · · . . .

mod pn+(n−1)|Q|+1 mod pn+(n−1)|Q|+2 mod pn+(n−1)|Q|+3

. . .

mod pn+n|Q| r ≡ r ≡ r ≡

. . .

r ≡ ← state q1 ← state q2 ← state q3

. . .

← state q|Q| Niskanen Polynomial iteration is PSPACE-complete RP 2017 20 / 28

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Final ingredients

Initial integer x0 satisfies x0 ≡ 1 mod pn+1 x0 ≡ 0 mod pj.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Final ingredients

Initial integer x0 satisfies x0 ≡ 1 mod pn+1 x0 ≡ 0 mod pj. If LBA reaches [qf, 1, 0n], then by simulating correctly r ≡ 1 mod pn+|Q| r ≡ 0 mod pj can be reached. Then,

pflip,n+|Q|(peqone,n+|Q|(x)) to reach r ≡ 0 mod pi for all i. p(x) = x ± p1 · · · pn+n|Q| to reach the integer 0.

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Final ingredients

Initial integer x0 satisfies x0 ≡ 1 mod pn+1 x0 ≡ 0 mod pj. If LBA reaches [qf, 1, 0n], then by simulating correctly r ≡ 1 mod pn+|Q| r ≡ 0 mod pj can be reached. Then,

pflip,n+|Q|(peqone,n+|Q|(x)) to reach r ≡ 0 mod pi for all i. p(x) = x ± p1 · · · pn+n|Q| to reach the integer 0.

If LBA does not reach [qf, 1, 0n], then simulating correctly will not result in 0. Simulating incorrectly results in r ≡ 2 mod pi for some i.

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Higher dimensions

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

PRM in higher dimensions

Theorem (Reichert 2015) The reachability problem is undecidable for two-dimensional PRM, where the updates are affine polynomials. Let {(u1, v1), . . . , (un, vn)} ⊆ {0, 1}∗ × {0, 1}∗ be an instance of the PCP . q ⊥ (2|ui|x + ui, 2|vi|x + vi) (x − 1, x − 1) (x − 1, x − 1)

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration in higher dimensions

q ⊥ (2|ui|x + ui, 2|vi|x + vi) (x − 1, x − 1) (x − 1, x − 1)

Let p1, p2 ∈ PRIME. Consider polynomials (2|ui|x + ui, 2|vi|x + vi, peqone,1(x)) for each pair (ui, vi); (x − 1, x − 1, pflip,2(pflip,1(peqone,1(x)))); (x − 1, x − 1, peqone,2(x)); (x, x, pflip,2(peqone,2(x))) and (x, x, x ± p1p2).

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Polynomial iteration in higher dimensions

q ⊥ (2|ui|x + ui, 2|vi|x + vi) (x − 1, x − 1) (x − 1, x − 1)

Let p1, p2 ∈ PRIME. Consider polynomials (2|ui|x + ui, 2|vi|x + vi, peqone,1(x)) for each pair (ui, vi); (x − 1, x − 1, pflip,2(pflip,1(peqone,1(x)))); (x − 1, x − 1, peqone,2(x)); (x, x, pflip,2(peqone,2(x))) and (x, x, x ± p1p2). Theorem The reachability problem for polynomial iteration is undecidable already for three-dimensional polynomials.

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Conclusion

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Summary

Theorem Given P ⊆ Z[x], the reachability problem for polynomial iteration is PSPACE-complete. Model Dimension 1 2 ≥ 3 PRM

PSPACE-complete

U – stateless PRM ? ? ?

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Summary

Theorem Given P ⊆ Z[x], the reachability problem for polynomial iteration is PSPACE-complete. Model Dimension 1 2 ≥ 3 PRM

PSPACE-complete

U – stateless PRM

PSPACE-complete

? U

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Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Future work

Decidability of two-dimensional polynomial iteration. Decidability of polynomial iteration over rational numbers in interval [0, 1]. Complexity of polynomial iteration over rational numbers. Investigate the effect of polynomials of the form ±x + b on the decidability of the reachability.

Niskanen Polynomial iteration is PSPACE-complete RP 2017 27 / 28

SLIDE 56

Introduction Definitions Polynomial iteration Higher dimensions Conclusion

Thank you for your attention!

Niskanen Polynomial iteration is PSPACE-complete RP 2017 28 / 28