Reachability Problems in Nondeterministic Polynomial . Maps on the - PowerPoint PPT Presentation

p 3 x . . . . . . . . . . . . . . Nondeterministic Polynomial Map Defjnition (Nondeterministic polynomial map) An n -dimensional (nondeterministic) polynomial map is a tuple Q is a singleton set and q 0 0 6 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . 4 / 27 . . . . R = ( Q , ∆ ) , where ∆ ⊆ Z [ x ] n is a fjnite set of transitions labelled by polynomials with variable x ∈ Z n . p 2 ( x ) p 1 ( x ) p 2 ( x ) p 1 ( x ) p 3 ( x ) p 2 ( x ) p 3 ( x ) p 4 ( x ) Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . Nondeterministic Polynomial Map Defjnition (Nondeterministic polynomial map) An n -dimensional (nondeterministic) polynomial map is a tuple Q is a singleton set and q 0 0 6 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . 4 / 27 . . . . R = ( Q , ∆ ) , where ∆ ⊆ Z [ x ] n is a fjnite set of transitions labelled by polynomials with variable x ∈ Z n . p 2 ( x ) p 1 ( x ) p 2 ( x ) p 1 ( x ) p 3 ( x ) p 2 ( x ) p 3 ( x ) p 3 ( x ) p 4 ( x ) Reachability in Polynomial Maps on Z

. . . . . . . . . . . . Polynomial Register Machine (PRM) . Defjnition (Polynomial register machine) An n -dimensional polynomial register machine ( n -PRM) is a tuple Q is a fjnite set of states and q 0 q 1 q 2 q 3 3 x Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . 5 / 27 . . . . . . . . . . . . R = ( Q , ∆ ) , where ∆ ⊆ Q × Z [ x ] n × Q is a fjnite set of transitions labelled by polynomials with variable x ∈ Z n . − x + 1 x 2 x 2 + 4 2 x + 4 x + 1 Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . Class of Polynomials Defjnition Additive polynomials: Affjne polynomials: Quadratic polynomials: Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . 6 / 27 . . . . . . . . . . . Add Z = { ± x + b | b ∈ Z } , Afg Z [ x ] = { ax + b | a , b ∈ Z } , Quad Z [ x ] = { ax 2 + bx + c | a , b , c ∈ Z } . ∈ Quad Z [ x ] � �� � a n x n + . . . + a 2 x 2 + a 1 x + a 0 ∈ Z [ x ] ± x + a 0 ∈ Add Z � �� � ∈ Afg Z [ x ] Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . Class of Polynomials Defjnition Additive polynomials: Affjne polynomials: Quadratic polynomials: Defjnition (Polynomials without additive polynomials) Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 27 Add Z = { ± x + b | b ∈ Z } , Afg Z [ x ] = { ax + b | a , b ∈ Z } , Quad Z [ x ] = { ax 2 + bx + c | a , b , c ∈ Z } . Afg Z [ x ] \ Add Z = { ax + b ∈ Afg Z [ x ] | a ̸ = ± 1 } , Z [ x ] \ Add Z = { p ( x ) ∈ Z [ x ] | p ( x ) ̸ = ± x + b , where b ∈ Z } . Reachability in Polynomial Maps on Z

. Bell & Potapov showed that with seven 2-d affjne updates of the form . . . . . . . . . Previous Work single linear Diophantine equation over natural numbers. x . ax by c y dy e (variables are not independent, stateless) the reachability problem is undecidable over 2 . [TCS 2008] Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . 7 / 27 . . . . . . . . . . . The additive form (i.e., x ← x + b ) of a map with polynomial updates can be seen as a vector addition systems on Z n . If n = 1, the reachability problem can be reduced to the solution of a Otherwise, the problem is in the form of the n -dimensional VAS on Z n . Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . Previous Work single linear Diophantine equation over natural numbers. Bell & Potapov showed that with seven 2-d affjne updates of the form (variables are not independent, stateless) Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . 7 / 27 The additive form (i.e., x ← x + b ) of a map with polynomial updates can be seen as a vector addition systems on Z n . If n = 1, the reachability problem can be reduced to the solution of a Otherwise, the problem is in the form of the n -dimensional VAS on Z n . { x ← ax + by + c , y ← dy + e the reachability problem is undecidable over Q 2 . [TCS 2008] Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . Previous Work Finkel et al. [MFCS 2013] considered that the reachability problem PSPACE-complete for 1-d polynomials and undecidable for 2-d polynomials with independent variables. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . 7 / 27 for polynomial register machines (with states) on Z n , Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . Previous Work Finkel et al. [MFCS 2013] considered that the reachability problem PSPACE-complete for 1-d polynomials and undecidable for 2-d polynomials with independent variables. Niskanen [RP 2017] showed that the reachability problem is PSPACE-complete in 1-d polynomial maps of degree four and undecidable in 3-d polynomial maps. (stateless) Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . 7 / 27 for polynomial register machines (with states) on Z n , Reachability in Polynomial Maps on Z

x 3 is undecidable and x 2 . . . . . . . . . . . . . . . . In the three-dimensional variant, we are investigating functions of the form First, we will show that The reachability problem for Afg PSPACE-hard for Quad Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . 8 / 27 . . . . . . . . . . . Reachability in Maps over Afg Z [ x ] 3 and Quad Z [ x ] 2  x 1 ← a 1 x 1 + b 1   x 2 ← a 2 x 2 + b 2 , where a i , b i ∈ Z .   x 3 ← a 3 x 3 + b 3 Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . . In the three-dimensional variant, we are investigating functions of the form First, we will show that Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . 8 / 27 . . . . . . . . . . Reachability in Maps over Afg Z [ x ] 3 and Quad Z [ x ] 2  x 1 ← a 1 x 1 + b 1   x 2 ← a 2 x 2 + b 2 , where a i , b i ∈ Z .   x 3 ← a 3 x 3 + b 3 The reachability problem for Afg Z [ x ] 3 is undecidable and PSPACE-hard for Quad Z [ x ] 2 . Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . . Theorem Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . 9 / 27 Undecidability over Afg Z [ x ] 3 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Reachability in Polynomial Maps on Z

u 2 v 2 u 1 v 1 u 3 v 3 u 1 v 1 . . . . . . . . Theorem Proof sketch. . . u 221 12 221 1 v 22 11 22 211 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 / 27 Undecidability over Afg Z [ x ] 3 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Let P = { ( u 1 , v 1 ) , . . . , ( u n , v n ) } ⊆ Σ ∗ × Σ ∗ be an instance of the PCP. Reachability in Polynomial Maps on Z

u 1 v 1 u 3 v 3 u 1 v 1 u 2 v 2 u 1 v 1 u 3 v 3 u 1 v 1 u 2 v 2 Proof sketch. Theorem aa bba ab bba a . . bb . bb baa . u 221 12 221 1 v 22 11 22 211 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 / 27 Undecidability over Afg Z [ x ] 3 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Let P = { ( u 1 , v 1 ) , . . . , ( u n , v n ) } ⊆ Σ ∗ × Σ ∗ be an instance of the PCP. u = v = Reachability in Polynomial Maps on Z

u 3 v 3 u 2 v 2 u 1 v 1 u 3 v 3 u 1 v 1 u 2 v 2 u 1 v 1 bba bb a bba ab bb Proof sketch. Theorem . . aa . baa . u 221 12 221 1 v 22 11 22 211 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 / 27 . . . . . . Undecidability over Afg Z [ x ] 3 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Let P = { ( u 1 , v 1 ) , . . . , ( u n , v n ) } ⊆ Σ ∗ × Σ ∗ be an instance of the PCP. ( u 1 , v 1 ) u = v = Reachability in Polynomial Maps on Z

u 1 v 1 u 3 v 3 u 1 v 1 u 2 v 2 u 1 v 1 u 3 v 3 bba bb a bba ab bb Proof sketch. Theorem . . . aa . baa . u 221 12 221 1 v 22 11 22 211 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 / 27 . . . . . Undecidability over Afg Z [ x ] 3 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Let P = { ( u 1 , v 1 ) , . . . , ( u n , v n ) } ⊆ Σ ∗ × Σ ∗ be an instance of the PCP. ( u 1 , v 1 ) ( u 2 , v 2 ) u = v = Reachability in Polynomial Maps on Z

u 3 v 3 u 1 v 1 u 2 v 2 u 1 v 1 u 3 v 3 bba aa bb a bba ab Proof sketch. Theorem . . . bb . baa . u 221 12 221 1 v 22 11 22 211 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . 9 / 27 . . . . . . . . . . . Undecidability over Afg Z [ x ] 3 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Let P = { ( u 1 , v 1 ) , . . . , ( u n , v n ) } ⊆ Σ ∗ × Σ ∗ be an instance of the PCP. ( u 1 , v 1 ) ( u 2 , v 2 ) ( u 1 , v 1 ) u = v = Reachability in Polynomial Maps on Z

u 1 v 1 u 2 v 2 u 1 v 1 u 3 v 3 aa bb a bba bba ab Proof sketch. Theorem . . . . bb . baa . u 221 12 221 1 v 22 11 22 211 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . 9 / 27 . . . . . . . . . . Undecidability over Afg Z [ x ] 3 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Let P = { ( u 1 , v 1 ) , . . . , ( u n , v n ) } ⊆ Σ ∗ × Σ ∗ be an instance of the PCP. ( u 1 , v 1 ) ( u 2 , v 2 ) ( u 1 , v 1 ) ( u 3 , v 3 ) u = v = Reachability in Polynomial Maps on Z

. . . . . . . . . . . . Theorem . Proof sketch. 221 12 221 1 22 11 22 211 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . 9 / 27 . . . . . . . . . . . . Undecidability over Afg Z [ x ] 3 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Let P = { ( u 1 , v 1 ) , . . . , ( u n , v n ) } ⊆ Σ ∗ × Σ ∗ be an instance of the PCP. ( u 1 , v 1 ) ( u 2 , v 2 ) ( u 1 , v 1 ) ( u 3 , v 3 ) u = v = Reachability in Polynomial Maps on Z

. Theorem . . . . . . . . . . Proof sketch. . 221 12 221 1 22 11 22 211 We can simulate concatenations with affjne functions as follows: Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . 9 / 27 . . . . . . . . . . . . Undecidability over Afg Z [ x ] 3 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Let P = { ( u 1 , v 1 ) , . . . , ( u n , v n ) } ⊆ Σ ∗ × Σ ∗ be an instance of the PCP. ( u 1 , v 1 ) ( u 2 , v 2 ) ( u 1 , v 1 ) ( u 3 , v 3 ) u = v = 3 | u i | σ ( u j ) + σ ( u i ) = σ ( u j u i ) . Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . . Proof sketch. PCP has a solution. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . 9 / 27 Undecidability over Afg Z [ x ] 3 We show that ( 0 , 0 , 1 ) is reachable from ( 0 , 0 , 0 ) if and only if the Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . Proof sketch. PCP has a solution. Defjne the following sets of affjne functions in dimension three: Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . 9 / 27 Undecidability over Afg Z [ x ] 3 We show that ( 0 , 0 , 1 ) is reachable from ( 0 , 0 , 0 ) if and only if the F 1 = { ( 3 | u i | x 1 + σ ( u i ) , 3 | v i | x 2 + σ ( v i ) , 2 x 3 ) | ( u i , v i ) ∈ P for all 1 ⩽ i ⩽ n } , F 2 = { ( 3 | u i | x 1 + σ ( u i ) , 3 | v i | x 2 + σ ( v i ) , 2 x 3 + 1 ) | ( u i , v i ) ∈ P for all 1 ⩽ i ⩽ n } , F 3 = { ( x 1 − 1 , x 2 − 1 , 2 x 3 − 1 ) } . Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . Proof sketch. PCP has a solution. Defjne the following sets of affjne functions in dimension three: 0 1 2 x Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . 9 / 27 . . . . . . . . . . . Undecidability over Afg Z [ x ] 3 We show that ( 0 , 0 , 1 ) is reachable from ( 0 , 0 , 0 ) if and only if the F 1 = { ( 3 | u i | x 1 + σ ( u i ) , 3 | v i | x 2 + σ ( v i ) , 2 x 3 ) | ( u i , v i ) ∈ P for all 1 ⩽ i ⩽ n } , F 2 = { ( 3 | u i | x 1 + σ ( u i ) , 3 | v i | x 2 + σ ( v i ) , 2 x 3 + 1 ) | ( u i , v i ) ∈ P for all 1 ⩽ i ⩽ n } , F 3 = { ( x 1 − 1 , x 2 − 1 , 2 x 3 − 1 ) } . 2 x − 1 2 x + 1 + 1 2 x − 1 x 2 , x ⊥ 2 2 x , 2 x + 1 , 2 x − 1 Reachability in Polynomial Maps on Z

For each edge v i v j (possibly i polynomial f ij x For example, let us try with f 03 x 3 and f 21 x 2 . . . . . . . Simulating State Structure with Affjne Functions Let’s take an any graph for example as follows: 0 1 . 3 . j ) of G , we add an affjne m x i j to the map. 4 x 4 x 2 1. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 / 27 Reachability in Polynomial Maps on Z

For example, let us try with f 03 x 3 and f 21 x . Simulating State Structure with Affjne Functions . . . . . . . . . . 0 Let’s take an any graph for example as follows: . 1 2 3 4 x 4 x 2 1. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 / 27 For each edge ( v i , v j ) (possibly i = j ) of G , we add an affjne polynomial f ij ( x ) = m ( x − i ) + j to the map. Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . Simulating State Structure with Affjne Functions Let’s take an any graph for example as follows: 0 1 2 3 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 10 / 27 For each edge ( v i , v j ) (possibly i = j ) of G , we add an affjne polynomial f ij ( x ) = m ( x − i ) + j to the map. For example, let us try with f 03 ( x ) = 4 x + 3 and f 21 ( x ) = 4 ( x − 2 ) + 1. Reachability in Polynomial Maps on Z

1 , be a one-dimensional PRM with PSPACE-hard reachability problem. For each transition q i p x q j of It is clear that 0 k is reachable from 0 q k 0 . Q q m q 0 , where Q . Let Proof sketch. Theorem . . . . PSPACE -hard. are quadratic . Note that the update polynomials of . , we add two-dimensional function p x m x j m i to the map. if and only if q 0 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 / 27 . . . . . . . PSPACE-hard over Quad Z [ x ] 2 The reachability problem for nondeterministic maps over Quad Z [ x ] 2 is Reachability in Polynomial Maps on Z

For each transition q i p x q j of It is clear that 0 k is reachable from 0 q k 0 . . . . . . . . Theorem PSPACE -hard. Proof sketch. Note that the update polynomials of . are quadratic . . , we add two-dimensional function p x m x j m i to the map. if and only if q 0 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 / 27 PSPACE-hard over Quad Z [ x ] 2 The reachability problem for nondeterministic maps over Quad Z [ x ] 2 is Let R = ( Q , ∆ ) , where Q = { q 0 , . . . , q m − 1 } , be a one-dimensional PRM with PSPACE-hard reachability problem. Reachability in Polynomial Maps on Z

For each transition q i p x q j of It is clear that 0 k is reachable from 0 q k 0 . . . . . . . . . Theorem PSPACE -hard. Proof sketch. . . , we add two-dimensional function p x m x j m i to the map. if and only if q 0 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . 11 / 27 . . . . . . . . . . . . PSPACE-hard over Quad Z [ x ] 2 The reachability problem for nondeterministic maps over Quad Z [ x ] 2 is Let R = ( Q , ∆ ) , where Q = { q 0 , . . . , q m − 1 } , be a one-dimensional PRM with PSPACE-hard reachability problem. Note that the update polynomials of R are quadratic . Reachability in Polynomial Maps on Z

It is clear that 0 k is reachable from 0 q k 0 . . . . . . . . . . . . . . . Theorem PSPACE -hard. Proof sketch. if and only if q 0 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . 11 / 27 . . . . . . . . . . . . PSPACE-hard over Quad Z [ x ] 2 The reachability problem for nondeterministic maps over Quad Z [ x ] 2 is Let R = ( Q , ∆ ) , where Q = { q 0 , . . . , q m − 1 } , be a one-dimensional PRM with PSPACE-hard reachability problem. Note that the update polynomials of R are quadratic . For each transition ( q i , p ( x ) , q j ) of R , we add two-dimensional function ( p ( x ) , m · x + j − m · i ) to the map. Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . Theorem PSPACE -hard. Proof sketch. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . 11 / 27 . . . . . . . . . . . PSPACE-hard over Quad Z [ x ] 2 The reachability problem for nondeterministic maps over Quad Z [ x ] 2 is Let R = ( Q , ∆ ) , where Q = { q 0 , . . . , q m − 1 } , be a one-dimensional PRM with PSPACE-hard reachability problem. Note that the update polynomials of R are quadratic . For each transition ( q i , p ( x ) , q j ) of R , we add two-dimensional function ( p ( x ) , m · x + j − m · i ) to the map. It is clear that ( 0 , k ) is reachable from ( 0 , ℓ ) if and only if [ q ℓ , 0 ] → ∗ R [ q k , 0 ] . Reachability in Polynomial Maps on Z

x n It is easy to see that the reachability problem for maps over Afg . . . . . . . . . . What happens without additive updates? . . is NP-hard by reduction to the Subset Sum Problem (SSP). The NP-hardness proof relies on the use of polynomials of the form x b that correspond to integers in the SSP. Question Does the NP-hardness still hold over the restricted class of maps over Afg x Add ? Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 / 27 Let’s consider a restricted class of maps over Afg Z [ x ] , in the sense that every affjne function in the map is not of the form ± x + b . Reachability in Polynomial Maps on Z

. . . . . . . . . . . . What happens without additive updates? . is NP-hard by reduction to the Subset Sum Problem (SSP). The NP-hardness proof relies on the use of polynomials of the form x b that correspond to integers in the SSP. Question Does the NP-hardness still hold over the restricted class of maps over Afg x Add ? Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . 12 / 27 . . . . . . . . . . . . Let’s consider a restricted class of maps over Afg Z [ x ] , in the sense that every affjne function in the map is not of the form ± x + b . It is easy to see that the reachability problem for maps over Afg Z [ x ] n Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . What happens without additive updates? is NP-hard by reduction to the Subset Sum Problem (SSP). The NP-hardness proof relies on the use of polynomials of the form Question Does the NP-hardness still hold over the restricted class of maps over Afg x Add ? Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . 12 / 27 . . . . . . . . . . . . Let’s consider a restricted class of maps over Afg Z [ x ] , in the sense that every affjne function in the map is not of the form ± x + b . It is easy to see that the reachability problem for maps over Afg Z [ x ] n x + b that correspond to integers in the SSP. Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . What happens without additive updates? is NP-hard by reduction to the Subset Sum Problem (SSP). The NP-hardness proof relies on the use of polynomials of the form Question Does the NP-hardness still hold over the restricted class of maps over Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 12 / 27 Let’s consider a restricted class of maps over Afg Z [ x ] , in the sense that every affjne function in the map is not of the form ± x + b . It is easy to see that the reachability problem for maps over Afg Z [ x ] n x + b that correspond to integers in the SSP. Afg Z [ x ] \ Add Z ? Reachability in Polynomial Maps on Z

1 s i 1 if and only if there is a subset of S such The map reaches s n k . Proof sketch. and s is s k s 1 Let S s be an instance of the SSP, where S . Lemma We construct the set of affjne functions . . . . the target integer. n x F . n i n x 1 i k with target s n k 1 , where n S S is a prime. that its elements add up to s . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 / 27 NP-hardness over Afg Z [ x ] \ Add Z The reachability problem for maps over Afg Z [ x ] \ Add Z is NP -hard. Reachability in Polynomial Maps on Z

1 s i 1 if and only if there is a subset of S such The map reaches s n k . F . . . . . . . Lemma Proof sketch. the target integer. We construct the set of affjne functions n i n x . n x 1 i k with target s n k 1 , where n S S is a prime. that its elements add up to s . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 / 27 NP-hardness over Afg Z [ x ] \ Add Z The reachability problem for maps over Afg Z [ x ] \ Add Z is NP -hard. Let ( S , s ) be an instance of the SSP, where S = { s 1 , . . . , s k , } and s is Reachability in Polynomial Maps on Z

1 if and only if there is a subset of S such The map reaches s n k . . . . . . . . . . . . . . Lemma Proof sketch. the target integer. We construct the set of affjne functions that its elements add up to s . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . 13 / 27 . . . . . . . . . . . . NP-hardness over Afg Z [ x ] \ Add Z The reachability problem for maps over Afg Z [ x ] \ Add Z is NP -hard. Let ( S , s ) be an instance of the SSP, where S = { s 1 , . . . , s k , } and s is F = { n · x + n i − 1 · s i , n · x | 1 ⩽ i ⩽ k } with target s · n k − 1 , where n > max ( S ) · | S | is a prime. Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . Lemma Proof sketch. the target integer. We construct the set of affjne functions that its elements add up to s . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 13 / 27 NP-hardness over Afg Z [ x ] \ Add Z The reachability problem for maps over Afg Z [ x ] \ Add Z is NP -hard. Let ( S , s ) be an instance of the SSP, where S = { s 1 , . . . , s k , } and s is F = { n · x + n i − 1 · s i , n · x | 1 ⩽ i ⩽ k } with target s · n k − 1 , where n > max ( S ) · | S | is a prime. The map reaches s · n k − 1 if and only if there is a subset of S such Reachability in Polynomial Maps on Z

. 6 . . . . . . . . . 2 4 8 . 10 12 2 4 6 8 10 x y Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 / 27 PSPACE Upper Bound over Z [ x ] n \ Add Z p 1 ( x ) = x 2 + 3 x − 12 − 10 − 8 − 6 − 4 − 2 − 2 − 4 − 6 − 8 − 10 Reachability in Polynomial Maps on Z

. 6 . . . . . . . . . 2 4 8 . 10 12 2 4 6 8 10 x y Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 / 27 PSPACE Upper Bound over Z [ x ] n \ Add Z p 1 ( x ) = x 2 + 3 x p 2 ( x ) = 2 x − 4 − 12 − 10 − 8 − 6 − 4 − 2 − 2 − 4 − 6 − 8 − 10 Reachability in Polynomial Maps on Z

. 6 . . . . . . . . . 2 4 8 . 10 12 2 4 6 8 10 x y Observation Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . 14 / 27 . . . . . . . . . . . . PSPACE Upper Bound over Z [ x ] n \ Add Z p 1 ( x ) = x 2 + 3 x p 2 ( x ) = 2 x − 4 − 12 − 10 − 8 − 6 − 4 − 2 − 2 − 4 − 6 − 8 − 10 There exists a bound b ∈ N such that every polynomial in Z [ x ] \ Add Z is monotonically increasing or decreasing in Z \ [− b , b ] . Reachability in Polynomial Maps on Z

We can compute the bound b which is polynomial in size of the input. current value and the computation path in space polynomial in b . Otherwise, due to monotonicity properties of Proof sketch. . . . . . . . . . Theorem . Let z be the target integer. . If z b , we can decide whether the integer z is reachable in PSPACE by applying the given functions since we can store the x Add functions, we do not need to consider the integers outside the interval z z . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 / 27 PSPACE Upper Bound over Z [ x ] n \ Add Z The reachability problem for maps over Z [ x ] n \ Add Z is decidable in PSPACE for any n ⩾ 1 . Reachability in Polynomial Maps on Z

current value and the computation path in space polynomial in b . Otherwise, due to monotonicity properties of . Proof sketch. . . . . . . . . . Theorem We can compute the bound b which is polynomial in size of the input. Let z be the target integer. . If z b , we can decide whether the integer z is reachable in PSPACE by applying the given functions since we can store the x Add functions, we do not need to consider the integers outside the interval z z . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 / 27 PSPACE Upper Bound over Z [ x ] n \ Add Z The reachability problem for maps over Z [ x ] n \ Add Z is decidable in PSPACE for any n ⩾ 1 . Reachability in Polynomial Maps on Z

current value and the computation path in space polynomial in b . Otherwise, due to monotonicity properties of . Proof sketch. . . . . . . . . . Theorem We can compute the bound b which is polynomial in size of the input. Let z be the target integer. . If z b , we can decide whether the integer z is reachable in PSPACE by applying the given functions since we can store the x Add functions, we do not need to consider the integers outside the interval z z . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 / 27 PSPACE Upper Bound over Z [ x ] n \ Add Z The reachability problem for maps over Z [ x ] n \ Add Z is decidable in PSPACE for any n ⩾ 1 . Reachability in Polynomial Maps on Z

Otherwise, due to monotonicity properties of . Theorem . . . . . . . . . . Let z be the target integer. Proof sketch. . We can compute the bound b which is polynomial in size of the input. PSPACE by applying the given functions since we can store the current value and the computation path in space polynomial in b . x Add functions, we do not need to consider the integers outside the interval z z . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . 15 / 27 . . . . . . . . . . . PSPACE Upper Bound over Z [ x ] n \ Add Z The reachability problem for maps over Z [ x ] n \ Add Z is decidable in PSPACE for any n ⩾ 1 . If | z | ⩽ b , we can decide whether the integer z is reachable in Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . Theorem Proof sketch. Let z be the target integer. We can compute the bound b which is polynomial in size of the input. PSPACE by applying the given functions since we can store the current value and the computation path in space polynomial in b . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . 15 / 27 . . . . . . . . . . . PSPACE Upper Bound over Z [ x ] n \ Add Z The reachability problem for maps over Z [ x ] n \ Add Z is decidable in PSPACE for any n ⩾ 1 . If | z | ⩽ b , we can decide whether the integer z is reachable in Otherwise, due to monotonicity properties of Z [ x ] \ Add Z functions, we do not need to consider the integers outside the interval [− z , z ] . Reachability in Polynomial Maps on Z

For each pair u i v i 3 u i F 1 for all 1 3 u i F 2 for some 1 F 3 for all 1 3 v i 3 n , and i 1 i 2 x 3 x 2 1 . n u i 2 n , i i 2 x 3 x 2 1 n u i x 1 x 1 1 u i k DLT 2018 Ko, Niskanen, and Potapov 1, in the fjrst dimension. k j 1 n for all i j 1 , where 1 u i 1 u i 2 v i First construct a word u n . i 1 2 x 3 i x 2 1 n 1 x 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . sets of affjne functions in dimension three: . n , we defjne the following i P , where 1 Let P be an instance of the PCP with n elements. Proof sketch. Theorem . . . . . . . . . . . . . 16 / 27 Undecidability over Afg Q [ x ] 3 \ Add Q The reachability problem for nondeterministic maps over Afg Q [ x ] 3 \ Add Q is undecidable with at least 11 affjne functions over Q . Reachability in Polynomial Maps on Z

For each pair u i v i 3 u i F 1 for all 1 3 u i F 2 for some 1 F 3 for all 1 3 v i 3 n , and i 1 i 2 x 3 x 2 1 . n u i 2 n , i i 2 x 3 x 2 1 n u i x 1 x 1 1 u i k DLT 2018 Ko, Niskanen, and Potapov 1, in the fjrst dimension. k j 1 n for all i j 1 , where 1 u i 1 u i 2 v i First construct a word u n . i 1 2 x 3 i x 2 1 n 1 x 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . sets of affjne functions in dimension three: . n , we defjne the following i P , where 1 Let P be an instance of the PCP with n elements. Proof sketch. Theorem . . . . . . . . . . . . . 16 / 27 Undecidability over Afg Q [ x ] 3 \ Add Q The reachability problem for nondeterministic maps over Afg Q [ x ] 3 \ Add Q is undecidable with at least 11 affjne functions over Q . Reachability in Polynomial Maps on Z

. 2 . . . . . . . Theorem Proof sketch. Let P be an instance of the PCP with n elements. sets of affjne functions in dimension three: 1 3 . 1 First construct a word u u i 1 u i 2 u i k 1 , where 1 i j n for all 1 j k 1, in the fjrst dimension. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 / 27 . . Undecidability over Afg Q [ x ] 3 \ Add Q The reachability problem for nondeterministic maps over Afg Q [ x ] 3 \ Add Q is undecidable with at least 11 affjne functions over Q . For each pair ( u i , v i ) ∈ P , where 1 ⩽ i ⩽ n , we defjne the following ( 3 | u i | · x 1 + σ ( u i ) , ( n + 1 ) · x 2 + i , 2 · x 3 ) ∈ F 1 for all 1 ⩽ i ⩽ n , ( 3 | u i | · x 1 + σ ( u i ) , ( n + 1 ) · x 2 + i , 2 · x 3 + 1 ) ∈ F 2 for some 1 ⩽ i ⩽ n , and ( 1 ) 3 | v i | · ( x 1 − σ ( v i )) , n + 1 · ( x 2 − i ) , 2 · x 3 − 1 ∈ F 3 for all 1 ⩽ i ⩽ n . Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . Theorem Proof sketch. Let P be an instance of the PCP with n elements. sets of affjne functions in dimension three: 1 2 3 1 Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . 16 / 27 . . . Undecidability over Afg Q [ x ] 3 \ Add Q The reachability problem for nondeterministic maps over Afg Q [ x ] 3 \ Add Q is undecidable with at least 11 affjne functions over Q . For each pair ( u i , v i ) ∈ P , where 1 ⩽ i ⩽ n , we defjne the following ( 3 | u i | · x 1 + σ ( u i ) , ( n + 1 ) · x 2 + i , 2 · x 3 ) ∈ F 1 for all 1 ⩽ i ⩽ n , ( 3 | u i | · x 1 + σ ( u i ) , ( n + 1 ) · x 2 + i , 2 · x 3 + 1 ) ∈ F 2 for some 1 ⩽ i ⩽ n , and ( 1 ) 3 | v i | · ( x 1 − σ ( v i )) , n + 1 · ( x 2 − i ) , 2 · x 3 − 1 ∈ F 3 for all 1 ⩽ i ⩽ n . First construct a word u ′ = u i 1 u i 2 · · · u i k − 1 , where 1 ⩽ i j ⩽ n for all 1 ⩽ j ⩽ k − 1, in the fjrst dimension. Reachability in Polynomial Maps on Z

. . . . . . . . . . . . Linear Bounded Automaton . Lemma A linear bounded automaton (LBA) is a Turing by a linear function of the size of the input. on the tape. q 3 n Known fact The reachability problem for LBAs is PSPACE-complete. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . 17 / 27 . . . . . . . . . . . . The reachability problem for maps over Afg Z [ x ] n \ Add Z is PSPACE -hard. machine with a fjnite tape whose length is bounded · · · A confjguration is [ q , i , w ] , where q ∈ Q , i is the position of the head, w ∈ { 0 , 1 } n is the word written The reachability problem: [ q 0 , 1 , 0 n ] → ∗ [ q f , 1 , 0 n ] ? Reachability in Polynomial Maps on Z

. . . . . . . . . . . . Linear Bounded Automaton . Lemma A linear bounded automaton (LBA) is a Turing by a linear function of the size of the input. on the tape. q 3 n Known fact The reachability problem for LBAs is PSPACE-complete. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . 17 / 27 . . . . . . . . . . . . The reachability problem for maps over Afg Z [ x ] n \ Add Z is PSPACE -hard. machine with a fjnite tape whose length is bounded · · · A confjguration is [ q , i , w ] , where q ∈ Q , i is the position of the head, w ∈ { 0 , 1 } n is the word written The reachability problem: [ q 0 , 1 , 0 n ] → ∗ [ q f , 1 , 0 n ] ? Reachability in Polynomial Maps on Z

x k Store the tape content of the LBA . . . . . . . . . . . Proof sketch. Lemma . Reduce the reachability problem of an LBA to the reachability problem for maps over Afg 1 Add . in the fjrst k dimensions and the current state in the last dimension of the affjne map. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . 18 / 27 . . . . . . . . . . . . PSPACE-hardness over Afg Z [ x ] n \ Add Z The reachability problem for maps over Afg Z [ x ] n \ Add Z is PSPACE -hard. Reachability in Polynomial Maps on Z

x k Store the tape content of the LBA . . . . . . . . . . . Proof sketch. Lemma . Reduce the reachability problem of an LBA to the reachability problem for maps over Afg 1 Add . in the fjrst k dimensions and the current state in the last dimension of the affjne map. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . 18 / 27 . . . . . . . . . . . . PSPACE-hardness over Afg Z [ x ] n \ Add Z The reachability problem for maps over Afg Z [ x ] n \ Add Z is PSPACE -hard. Reachability in Polynomial Maps on Z

Store the tape content of the LBA . . . . . . . . . . . . . . . Lemma Proof sketch. in the fjrst k dimensions and the current state in the last dimension of the affjne map. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . 18 / 27 . . . . PSPACE-hardness over Afg Z [ x ] n \ Add Z The reachability problem for maps over Afg Z [ x ] n \ Add Z is PSPACE -hard. Reduce the reachability problem of an LBA A to the reachability problem for maps over Afg Z [ x ] k + 1 \ Add Z . Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . Lemma Proof sketch. current state in the last dimension of the affjne map. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . 18 / 27 PSPACE-hardness over Afg Z [ x ] n \ Add Z The reachability problem for maps over Afg Z [ x ] n \ Add Z is PSPACE -hard. Reduce the reachability problem of an LBA A to the reachability problem for maps over Afg Z [ x ] k + 1 \ Add Z . Store the tape content of the LBA A in the fjrst k dimensions and the Reachability in Polynomial Maps on Z

0 1 k . w 1 w 2 q j i The affjne function corresponding to q j 1 0 q j 2 1 L is q j 1 i q j 2 i Denote w w 1 w 2 Proof sketch. (continue) . w k . The corresponding register value in the affjne map is as follows: . w k k z . Example . x x 2 x 1 x x a x b where a x b corresponds to the edge 1 in G , and 2 x 1 is in the i th dimension. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 / 27 PSPACE-hardness over Afg Z [ x ] n \ Add Z (continue) Let [ q j , i , w ] be the current confjguration of A . Reachability in Polynomial Maps on Z

w 1 w 2 q j i The affjne function corresponding to q j 1 0 q j 2 1 L is q j 1 i q j 2 i z k w k The corresponding register value in the affjne map is as follows: . Proof sketch. (continue) . . . . . Example x . x 2 x 1 x x a x b where a x b corresponds to the edge 1 in G , and 2 x 1 is in the i th dimension. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 / 27 PSPACE-hardness over Afg Z [ x ] n \ Add Z (continue) Let [ q j , i , w ] be the current confjguration of A . Denote w = w 1 w 2 · · · w k ∈ { 0 , 1 } k . Reachability in Polynomial Maps on Z

The affjne function corresponding to q j 1 0 q j 2 1 L is q j 1 i q j 2 i . . . . . . . . Proof sketch. (continue) The corresponding register value in the affjne map is as follows: k Example x 2 x x . 1 x x a x b where a x b corresponds to the edge 1 in G , and 2 x 1 is in the i th dimension. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . 19 / 27 . . . . . . . . . . . PSPACE-hardness over Afg Z [ x ] n \ Add Z (continue) Let [ q j , i , w ] be the current confjguration of A . Denote w = w 1 w 2 · · · w k ∈ { 0 , 1 } k . ( w 1 , w 2 , . . . , w k , z = ( q j , i )) . � �� � Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . Proof sketch. (continue) The corresponding register value in the affjne map is as follows: k Example Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . 19 / 27 . . . . . . . . . . . PSPACE-hardness over Afg Z [ x ] n \ Add Z (continue) Let [ q j , i , w ] be the current confjguration of A . Denote w = w 1 w 2 · · · w k ∈ { 0 , 1 } k . ( w 1 , w 2 , . . . , w k , z = ( q j , i )) . � �� � The affjne function corresponding to ( q j 1 , 0 , q j 2 , 1 , L ) is ( x , . . . , x , 2 x + 1 , x , . . . , x , a · x + b ) , where a · x + b corresponds to the edge (( q j 1 , i ) , ( q j 2 , i − 1 )) in G A , and 2 x + 1 is in the i th dimension. Reachability in Polynomial Maps on Z

x n . If the dimension n is not fjxed, then the reachability problem for maps over . . . . . . . . . Main Results Theorem Corollary . If the dimension n is not fjxed, then the reachability problem for n-ARMs and n-PRMs, where the update polynomials are not of the form x b, is PSPACE -complete. Corollary If the dimension n is not fjxed, then the reachability problem for maps over Add is PSPACE -complete. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 / 27 Afg Z [ x ] n \ Add Z is PSPACE -complete. Reachability in Polynomial Maps on Z

x n . Main Results . . . . . . . . . . Theorem . If the dimension n is not fjxed, then the reachability problem for maps over Corollary If the dimension n is not fjxed, then the reachability problem for n-ARMs PSPACE -complete. Corollary If the dimension n is not fjxed, then the reachability problem for maps over Add is PSPACE -complete. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 / 27 Afg Z [ x ] n \ Add Z is PSPACE -complete. and n-PRMs, where the update polynomials are not of the form ± x + b, is Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . Main Results Theorem If the dimension n is not fjxed, then the reachability problem for maps over Corollary If the dimension n is not fjxed, then the reachability problem for n-ARMs PSPACE -complete. Corollary If the dimension n is not fjxed, then the reachability problem for maps over Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . 20 / 27 Afg Z [ x ] n \ Add Z is PSPACE -complete. and n-PRMs, where the update polynomials are not of the form ± x + b, is Z [ x ] n \ Add Z is PSPACE -complete. Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . . Maps as Language Acceptors Let’s extend our models to operate on words. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . 21 / 27 . . . . . p 1 ( x ) p ( x ) 3 p 2 ( x ) p 4 ( x ) p 5 ( x ) Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . . Maps as Language Acceptors Let’s extend our models to operate on words. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . 21 / 27 . . . . . p 1 ( x ) / a p ( x ) / a 3 p 2 ( x ) / b p 4 ( x ) / b p 5 ( x ) / a Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . . Maps as Language Acceptors Let’s extend our models to operate on words. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . 21 / 27 p 1 ( x ) / a p 3 ( x ) / a p 2 ( x ) / b p 4 ( x ) / b Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . Maps as Language Acceptors Let’s extend our models to operate on words. The word w is accepted if there is a computation path from the initial value to the target value reading w in the map. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . 21 / 27 p 1 ( x ) / a p 3 ( x ) / a p 2 ( x ) / b p 4 ( x ) / b Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . Maps as Language Acceptors Let’s extend our models to operate on words. The word w is accepted if there is a computation path from the initial value to the target value reading w in the map. In this context, the reachability problems of the previous sections can be seen as language emptiness problem. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . 21 / 27 . . . . p 1 ( x ) / a p 3 ( x ) / a p 2 ( x ) / b p 4 ( x ) / b Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . Maps as Language Acceptors Let’s extend our models to operate on words. The word w is accepted if there is a computation path from the initial value to the target value reading w in the map. In this context, the reachability problems of the previous sections can be seen as language emptiness problem. The language accepted by the map is empty if and only if the fjnal confjguration is not reachable from the initial confjguration. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . 21 / 27 p 1 ( x ) / a p 3 ( x ) / a p 2 ( x ) / b p 4 ( x ) / b Reachability in Polynomial Maps on Z

the universality problem is undecidable [Halava & Harju, 1998]. into maps in such way that the second dimension is used to simulate the state transitions of the automaton. For a transition q i a q j z , we construct an affjne function . . . . . . Theorem Proof sketch. Let be an integer weighted automaton over alphabet for which The idea is to encode . . a x 1 z m x 2 j m i to simulate the transition on the map. Then, a word w is accepted by the map if and only if the register values 0 m 1 are reachable from 0 0 while reading word w . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . 22 / 27 . . . . . . Universality is Undecidable over Afg Z [ x ] 2 The universality problem for maps over Afg Z [ x ] 2 is undecidable. Reachability in Polynomial Maps on Z

into maps in such way that the second dimension is used to simulate the state transitions of the automaton. For a transition q i a q j z , we construct an affjne function . . . . . . . . Theorem Proof sketch. The idea is to encode . . a x 1 z m x 2 j m i to simulate the transition on the map. Then, a word w is accepted by the map if and only if the register values 0 m 1 are reachable from 0 0 while reading word w . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 / 27 Universality is Undecidable over Afg Z [ x ] 2 The universality problem for maps over Afg Z [ x ] 2 is undecidable. Let A γ be an integer weighted automaton over alphabet Σ for which the universality problem is undecidable [Halava & Harju, 1998]. Reachability in Polynomial Maps on Z

For a transition q i a q j z , we construct an affjne function dimension is used to simulate the state transitions of the automaton. . . . . . . . . . Theorem Proof sketch. . . a x 1 z m x 2 j m i to simulate the transition on the map. Then, a word w is accepted by the map if and only if the register values 0 m 1 are reachable from 0 0 while reading word w . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . 22 / 27 . . . . . . . . . . . Universality is Undecidable over Afg Z [ x ] 2 The universality problem for maps over Afg Z [ x ] 2 is undecidable. Let A γ be an integer weighted automaton over alphabet Σ for which the universality problem is undecidable [Halava & Harju, 1998]. The idea is to encode A γ into maps in such way that the second Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . Theorem Proof sketch. dimension is used to simulate the state transitions of the automaton. Then, a word w is accepted by the map if and only if the register values 0 m 1 are reachable from 0 0 while reading word w . Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . 22 / 27 . . . Universality is Undecidable over Afg Z [ x ] 2 The universality problem for maps over Afg Z [ x ] 2 is undecidable. Let A γ be an integer weighted automaton over alphabet Σ for which the universality problem is undecidable [Halava & Harju, 1998]. The idea is to encode A γ into maps in such way that the second For a transition ( q i , a , q j , z ) , we construct an affjne function ( a , ( x 1 + z , m · x 2 + j − m · i )) to simulate the transition on the map. Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . Theorem Proof sketch. dimension is used to simulate the state transitions of the automaton. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . 22 / 27 . . . . Universality is Undecidable over Afg Z [ x ] 2 The universality problem for maps over Afg Z [ x ] 2 is undecidable. Let A γ be an integer weighted automaton over alphabet Σ for which the universality problem is undecidable [Halava & Harju, 1998]. The idea is to encode A γ into maps in such way that the second For a transition ( q i , a , q j , z ) , we construct an affjne function ( a , ( x 1 + z , m · x 2 + j − m · i )) to simulate the transition on the map. Then, a word w ∈ Σ ∗ is accepted by the map if and only if the register values ( 0 , m − 1 ) are reachable from ( 0 , 0 ) while reading word w . Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . . Defjnition (Reachability set of a map) The reachability set of F is defjned iteratively: Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . 23 / 27 . . . . Intersection Emptiness is Undecidable over Afg Z [ x ] 2 Let F ⊆ Z [ x ] n be a map over Z [ x ] n and let x 0 ∈ Z n be the initial value. Reach 0 ( F ) = { x 0 } , Reach i ( F ) = { f ( x ) | x ∈ Reach i − 1 ( F ) , f ∈ F } , ∞ ∪ Reach ( F ) = Reach i ( F ) . i = 0 Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . Defjnition (Reachability set of a map) The reachability set of F is defjned iteratively: Lemma Let F and G be two-dimensional affjne maps. It is undecidable whether the intersection of the respective reachability sets is empty or not. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . 23 / 27 . . . . Intersection Emptiness is Undecidable over Afg Z [ x ] 2 Let F ⊆ Z [ x ] n be a map over Z [ x ] n and let x 0 ∈ Z n be the initial value. Reach 0 ( F ) = { x 0 } , Reach i ( F ) = { f ( x ) | x ∈ Reach i − 1 ( F ) , f ∈ F } , ∞ ∪ Reach ( F ) = Reach i ( F ) . i = 0 Reachability in Polynomial Maps on Z

. . . . . . . . . . . . . . . . Defjnition (Reachability set of a map) The reachability set of F is defjned iteratively: Theorem respective languages is empty. Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . 23 / 27 . . . . Intersection Emptiness is Undecidable over Afg Z [ x ] 2 Let F ⊆ Z [ x ] n be a map over Z [ x ] n and let x 0 ∈ Z n be the initial value. Reach 0 ( F ) = { x 0 } , Reach i ( F ) = { f ( x ) | x ∈ Reach i − 1 ( F ) , f ∈ F } , ∞ ∪ Reach ( F ) = Reach i ( F ) . i = 0 Let F , G ⊆ Σ × Afg Z [ x ] 2 and x 0 F , x 0 G and x f F , x f G be the respective initial and target values. It is undecidable whether the intersection of the Reachability in Polynomial Maps on Z

. NP-c. [2] 1 . . . Complexity Landscape Complexity of reachability problems in nondeterministic polynomial maps according to the degrees. dim. degree 1 2 3 4 the leading coeffjcient 1 NP-h. [2]/PSPACE [1] 2 . 1 [2] Haase and Halfon. “Integer Vector Addition Systems with States”. RP 2014 . DLT 2018 Ko, Niskanen, and Potapov 2017 . 3 [3] Niskanen. “Reachability problem for polynomial iteration is PSPACE-complete”. RP MFCS 2013 . 2 [1] Finkel, Göller, and Haase. “Reachability in Register Machines with Polynomial Updates”. undecid. [3] PSPACE-c. [3] 3 undecid. 3 PSPACE-h. [3]/? PSPACE-h./? NP-h. [2]/? 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 / 27 . . . . . . PPPPPPPP a 1 = ± 1 a 1 ∈ Z P Reachability in Polynomial Maps on Z

. polynomial PSPACE-h. [3]/? NP-h. [2]/? 2 PSPACE-c. [3] NP-h. [2]/PSPACE [1] 1 affjne undecid. [3] type dim. Complexity of reachability problems in affjne and polynomial maps Complexity Landscape from Difgerent View . . 3 . . Our goal was also to DLT 2018 Ko, Niskanen, and Potapov and complexity of the reachability problems! b on the decidability x Investigate the efgect of polynomials of the form PSPACE-c. . undecid. PSPACE-c. n NP-h./PSPACE NP-h./PSPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 / 27 with respect to inclusion of polynomials of the form ± x + b . PPPPPPPP a 1 ̸ = ± 1 a 1 ∈ Z a 1 ̸ = ± 1 a 1 ∈ Z P Reachability in Polynomial Maps on Z

. PSPACE-h. [3]/? . . . Complexity Landscape from Difgerent View Complexity of reachability problems in affjne and polynomial maps dim. type affjne polynomial 1 NP-h. [2]/PSPACE [1] PSPACE-c. [3] 2 NP-h. [2]/? 3 . undecid. [3] . . . NP-h./PSPACE NP-h./PSPACE n PSPACE-c. undecid. PSPACE-c. Our goal was also to and complexity of the reachability problems! Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 / 27 . . . . with respect to inclusion of polynomials of the form ± x + b . PPPPPPPP a 1 ̸ = ± 1 a 1 ∈ Z a 1 ̸ = ± 1 a 1 ∈ Z P Investigate the efgect of polynomials of the form ± x + b on the decidability Reachability in Polynomial Maps on Z

x n x 2 is undecidable. It is undecidable whether or not the intersection of the languages If the dimension n is not fjxed, then the reachability problem for maps . . . . . . . . Concluding Remarks Summary . over Afg . Add is PSPACE-complete. The universality problem for maps over Afg accepted by two-dimensional affjne maps is empty. Open problems Complexity of the reachability problem for affjne maps? Decidability of the reachability problem for 2-D affjne maps? Decidability of the reachability problem for 2-D polynomial maps? Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 / 27 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . Reachability in Polynomial Maps on Z

x 2 is undecidable. It is undecidable whether or not the intersection of the languages . . . . . . . . . . . Summary Concluding Remarks . If the dimension n is not fjxed, then the reachability problem for maps The universality problem for maps over Afg accepted by two-dimensional affjne maps is empty. Open problems Complexity of the reachability problem for affjne maps? Decidability of the reachability problem for 2-D affjne maps? Decidability of the reachability problem for 2-D polynomial maps? Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . 26 / 27 . . . . . . . . . . . . . The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . over Afg Z [ x ] n \ Add Z is PSPACE-complete. Reachability in Polynomial Maps on Z

It is undecidable whether or not the intersection of the languages . . . . . . . . . . . Concluding Remarks . . Summary If the dimension n is not fjxed, then the reachability problem for maps accepted by two-dimensional affjne maps is empty. Open problems Complexity of the reachability problem for affjne maps? Decidability of the reachability problem for 2-D affjne maps? Decidability of the reachability problem for 2-D polynomial maps? Ko, Niskanen, and Potapov DLT 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 / 27 The reachability problem for maps over Afg Z [ x ] 3 is undecidable with at least 7 affjne functions over Z . over Afg Z [ x ] n \ Add Z is PSPACE-complete. The universality problem for maps over Afg Z [ x ] 2 is undecidable. Reachability in Polynomial Maps on Z