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

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

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Introduction Niskanen Polynomial iteration is PSPACE-complete RP 2017 2 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration p 1 ( x ) = x 2 + x + 3 p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 p 3 ( x ) = − x + 5 Can we iterate x = 6 to reach 0? 0 6 Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration p 1 ( x ) = x 2 + x + 3 p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 p 3 ( x ) = − x + 5 Can we iterate x = 6 to reach 0? 0 6 p 3 ( x ) Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration p 1 ( x ) = x 2 + x + 3 p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 p 3 ( x ) = − x + 5 Can we iterate x = 6 to reach 0? 0 6 p 3 ( x ) Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration p 1 ( x ) = x 2 + x + 3 p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 p 3 ( x ) = − x + 5 Can we iterate x = 6 to reach 0? p 2 ( x ) 0 6 p 3 ( x ) Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration p 1 ( x ) = x 2 + x + 3 p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 p 3 ( x ) = − x + 5 Can we iterate x = 6 to reach 0? p 2 ( x ) 0 6 p 3 ( x ) Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration p 1 ( x ) = x 2 + x + 3 p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 p 3 ( x ) = − x + 5 Can we iterate x = 6 to reach 0? p 2 ( x ) p 1 ( x ) 0 6 p 3 ( x ) Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration p 1 ( x ) = x 2 + x + 3 p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 p 3 ( x ) = − x + 5 Can we iterate x = 6 to reach 0? p 2 ( x ) p 1 ( x ) 0 6 p 3 ( x ) Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration p 1 ( x ) = x 2 + x + 3 p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 p 3 ( x ) = − x + 5 Can we iterate x = 6 to reach 0? p 2 ( x ) p 1 ( x ) 0 6 p 3 ( x ) p 3 ( x ) Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration p 1 ( x ) = x 2 + x + 3 p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 p 3 ( x ) = − x + 5 Can we iterate x = 6 to reach 0? p 2 ( x ) p 1 ( x ) 0 6 p 3 ( x ) p 3 ( x ) Niskanen Polynomial iteration is PSPACE-complete RP 2017 3 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration How much space is needed? p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 Niskanen Polynomial iteration is PSPACE-complete RP 2017 4 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration How much space is needed? p 2 ( x ) = x 4 + 2 x 3 + 3 x 2 + 2 x + 1 A lot.. 6 �→ 1849 �→ 11700853263801 The representation grows exponentially. Niskanen Polynomial iteration is PSPACE-complete RP 2017 4 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Definitions Niskanen Polynomial iteration is PSPACE-complete RP 2017 5 / 28

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 q 3 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 · · · n written on the tape. The reachability problem: [ q 0 , 1 , 0 n ] → ∗ [ q f , 1 , 0 n ] ? Theorem The reachability problem for LBA is PSPACE -complete. Niskanen Polynomial iteration is PSPACE-complete RP 2017 6 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial register machines Introduced by Finkel, Göller and Haase in MFCS’13 p 1 ( x ) A PRM consists of a graph ( V , E ) labelled p 3 ( x ) by polynomials in Z [ x ] . ) x ( 2 A configuration is [ s , z ] ∈ V × Z . p 4 ( x ) p [ s , z ] yields [ s ′ , y ] if ( s , p ( x ) , s ′ ) ∈ E such that p ( z ) = y . p 5 ( x ) The reachability problem: [ s 0 , 0 ] → ∗ [ s f , 0 ] ? Theorem (FGH 2013) The reachability problem for PRM is PSPACE -complete. Niskanen Polynomial iteration is PSPACE-complete RP 2017 7 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration Can be seen as stateless PRM s. P = { p 1 ( x ) , p 2 ( x ) , . . . , p n ( x ) } ⊆ Z [ x ] . p 1 ( x ) The reachability problem: Does there exist a p 4 ( x ) p 2 ( x ) finite sequence p i 1 ( x ) , p i 2 ( x ) , . . . , p i j ( x ) that maps x 0 to x f , i.e., whether p 3 ( x ) p i j ( p i j − 1 ( · · · p i 2 ( p i 1 ( x 0 )) · · · ) = x f . Theorem The reachability problem for polynomial iteration is PSPACE -complete. Niskanen Polynomial iteration is PSPACE-complete RP 2017 8 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Polynomial iteration Niskanen Polynomial iteration is PSPACE-complete RP 2017 9 / 28

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

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

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

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

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. Niskanen Polynomial iteration is PSPACE-complete RP 2017 11 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Ingredients of the reduction of LBA to PRM Let p 1 , . . . , p n ∈ P RIME . We consider an integer x as a residue class r mod p 1 · · · p n . The tape word w ∈ { 0 , 1 } n is encoded as an integer r satisfying r ≡ w i mod p i for each i = 1 , . . . , n . mod p 1 q 3 r ≡ r ≡ mod p 2 . . · · · . . . . . . n . r ≡ mod p n Niskanen Polynomial iteration is PSPACE-complete RP 2017 12 / 28

Introduction Definitions Polynomial iteration Higher dimensions Conclusion Ingredients of the reduction of LBA to PRM Let p 1 , . . . , p n ∈ P RIME . We consider an integer x as a residue class r mod p 1 · · · p n . The tape word w ∈ { 0 , 1 } n is encoded as an integer r satisfying r ≡ w i mod p i for each i = 1 , . . . , n . mod p 1 q 3 r ≡ r ≡ mod p 2 . . · · · . . . . . . n . r ≡ mod p n We only consider integers that are solutions to r ≡ b 1 mod p 1 . . . where b i ∈ { 0 , 1 , 2 } . r ≡ b n mod p n , Niskanen Polynomial iteration is PSPACE-complete RP 2017 12 / 28