Computational Complexity of Semigroup Properties Trevor Jack Joint work with Peter Mayr Trevor Jack Semigroup Complexity August 7, 2018 1 / 16
Regularity Preliminaries Notation and Regularity Problem Transformation Semigroups [ n ] := { 1 , ..., n } T n is the semigroup of all unary functions on [ n ] S ≤ T n Trevor Jack Semigroup Complexity August 7, 2018 2 / 16
Regularity Preliminaries Notation and Regularity Problem Transformation Semigroups [ n ] := { 1 , ..., n } T n is the semigroup of all unary functions on [ n ] S ≤ T n General Inquiry: Given generators a 1 , . . . , a k ∈ T n , what is the complexity of verifying certain properties about S = � a 1 , . . . , a n � within: P ⊆ NP ⊆ PSPACE ⊆ EXPTIME? Trevor Jack Semigroup Complexity August 7, 2018 2 / 16
Regularity Preliminaries Notation and Regularity Problem Transformation Semigroups [ n ] := { 1 , ..., n } T n is the semigroup of all unary functions on [ n ] S ≤ T n General Inquiry: Given generators a 1 , . . . , a k ∈ T n , what is the complexity of verifying certain properties about S = � a 1 , . . . , a n � within: P ⊆ NP ⊆ PSPACE ⊆ EXPTIME? Definition b ∈ T n is regular in S if for some s ∈ S , bsb = b . Trevor Jack Semigroup Complexity August 7, 2018 2 / 16
Regularity Preliminaries Notation and Regularity Problem Transformation Semigroups [ n ] := { 1 , ..., n } T n is the semigroup of all unary functions on [ n ] S ≤ T n General Inquiry: Given generators a 1 , . . . , a k ∈ T n , what is the complexity of verifying certain properties about S = � a 1 , . . . , a n � within: P ⊆ NP ⊆ PSPACE ⊆ EXPTIME? Definition b ∈ T n is regular in S if for some s ∈ S , bsb = b . RegularElement Input: a 1 , ..., a k , b ∈ T n Output: Is b regular in � a 1 , ..., a k � ? Trevor Jack Semigroup Complexity August 7, 2018 2 / 16
Regularity Theorems and Proofs RegularElement Theorem and Proof Theorem RegularElement is PSPACE-Complete. Trevor Jack Semigroup Complexity August 7, 2018 3 / 16
Regularity Theorems and Proofs RegularElement Theorem and Proof Theorem RegularElement is PSPACE-Complete. Definition A deterministic finite automata (DFA) has: 1 a set of states Z with a start state and an accept state; and 2 a set of transformations Σ, which map states to states. Trevor Jack Semigroup Complexity August 7, 2018 3 / 16
Regularity Theorems and Proofs RegularElement Theorem and Proof Theorem RegularElement is PSPACE-Complete. Definition A deterministic finite automata (DFA) has: 1 a set of states Z with a start state and an accept state; and 2 a set of transformations Σ, which map states to states. The proof uses the following PSPACE-complete problem (Kozen, 1970): Finite Automata Intersection (FAI) Input: DFA’s A 1 , ..., A ℓ with shared transitions Σ Output: Whether there is w ∈ Σ ∗ accepted by each A i . Trevor Jack Semigroup Complexity August 7, 2018 3 / 16
Regularity Theorems and Proofs Proof Sketch Proof. Given DFAs A 1 , ..., A ℓ with sets of states Z 1 , ..., Z ℓ and shared transitions Σ, define the following transformation semigroup: Transformed Set: Z = � ℓ i =1 Z i along with new state 0. Generators: Σ defined naturally on Z and fixing 0. Add generator h that sends accept states to start states and sends every other state to 0. Then h is regular in this semigroup iff there is a w ∈ Σ ∗ accepted by each A 1 , ..., A ℓ . Hence, RegularElement is PSPACE-hard. RegularElement is in NPSPACE because we can nondeterministically guess the generators that produce an s satisfying bsb = b . So, by Savitch’s Theorem, RegularElement is in PSPACE, and thus PSPACE-complete. Trevor Jack Semigroup Complexity August 7, 2018 4 / 16
Regularity Theorems and Proofs Regular Semigroup Open Problem How hard is it to check that every element in S is regular? Trevor Jack Semigroup Complexity August 7, 2018 5 / 16
Regularity Theorems and Proofs Regular Semigroup Open Problem How hard is it to check that every element in S is regular? A semigroup is completely regular if each element generates a subgroup. Theorem Determining if � a 1 , . . . , a k � ≤ T n is completely regular is in P. Proof requires use of ”transformation graphs” Trevor Jack Semigroup Complexity August 7, 2018 5 / 16
Transformation Graphs Introduction Model Checking Fix u , v semigroup words over variables z 1 , . . . , z m Trevor Jack Semigroup Complexity August 7, 2018 6 / 16
Transformation Graphs Introduction Model Checking Fix u , v semigroup words over variables z 1 , . . . , z m Model( u ≈ v ) Input: a 1 , ..., a k ∈ T n Output: Whether � a 1 , ..., a k � models u ( z 1 , ..., z m ) ≈ v ( z 1 , ..., z m ). Trevor Jack Semigroup Complexity August 7, 2018 6 / 16
Transformation Graphs Introduction Model Checking Fix u , v semigroup words over variables z 1 , . . . , z m Model( u ≈ v ) Input: a 1 , ..., a k ∈ T n Output: Whether � a 1 , ..., a k � models u ( z 1 , ..., z m ) ≈ v ( z 1 , ..., z m ). Example: Band Identity z 1 z 1 ≈ z 1 Trevor Jack Semigroup Complexity August 7, 2018 6 / 16
Transformation Graphs Introduction Model Checking Fix u , v semigroup words over variables z 1 , . . . , z m Model( u ≈ v ) Input: a 1 , ..., a k ∈ T n Output: Whether � a 1 , ..., a k � models u ( z 1 , ..., z m ) ≈ v ( z 1 , ..., z m ). Example: Band Identity z 1 z 1 ≈ z 1 Theorem Model( u ≈ v ) is in P. Trevor Jack Semigroup Complexity August 7, 2018 6 / 16
Transformation Graphs Notation Notation Let W be the set of all prefixes of u and v including the empty word 1. For x ∈ [ n ] , s 1 , . . . , s m ∈ S define evaluations, e ( x , s 1 , . . . , s m ): W → [ n ] , w �→ xw ( s 1 , . . . , s m ) . E ( W , S ) := { e ( x , s 1 , . . . , s m ) : x ∈ [ n ] , s 1 , . . . , s m ∈ S } ⊆ [ n ] W . Then S models u ≈ v iff f ( u ) = f ( v ) for all f ∈ E ( W , S ). Trevor Jack Semigroup Complexity August 7, 2018 7 / 16
Transformation Graphs Notation Notation Let W be the set of all prefixes of u and v including the empty word 1. For x ∈ [ n ] , s 1 , . . . , s m ∈ S define evaluations, e ( x , s 1 , . . . , s m ): W → [ n ] , w �→ xw ( s 1 , . . . , s m ) . E ( W , S ) := { e ( x , s 1 , . . . , s m ) : x ∈ [ n ] , s 1 , . . . , s m ∈ S } ⊆ [ n ] W . Then S models u ≈ v iff f ( u ) = f ( v ) for all f ∈ E ( W , S ). Example: Band Identity E ( W , S ) := { ( x , xs , xs 2 ) : x ∈ [ n ] , s ∈ S } Trevor Jack Semigroup Complexity August 7, 2018 7 / 16
Transformation Graphs Proof Lemmas Lemma Let S = � a 1 , ..., a k � ⊆ T n , d ∈ N , and f ∈ [ n ] d . Then fS can be enumerated in O ( n d k ) time. Trevor Jack Semigroup Complexity August 7, 2018 8 / 16
Transformation Graphs Proof Lemmas Lemma Let S = � a 1 , ..., a k � ⊆ T n , d ∈ N , and f ∈ [ n ] d . Then fS can be enumerated in O ( n d k ) time. Definition The degree-d transformation graph of S = � a 1 , ..., a k � is G d = ( V , E ) having vertices V = [ n ] d and edges E = { ( x , y ) ∈ V 2 : ∃ i ∈ [ k ]( xa i = y } , where S acts on [ n ] d component-wise. Enumerate fS using depth-first search algorithm. There are a maximum of n d k edges and the algorithm traverses each once, hence O ( n d k ) time. Trevor Jack Semigroup Complexity August 7, 2018 8 / 16
Transformation Graphs Proof Lemmas Lemma Let S = � a 1 , ..., a k � ⊆ T n , d ∈ N , and f ∈ [ n ] d . Then fS can be enumerated in O ( n d k ) time. Definition The degree-d transformation graph of S = � a 1 , ..., a k � is G d = ( V , E ) having vertices V = [ n ] d and edges E = { ( x , y ) ∈ V 2 : ∃ i ∈ [ k ]( xa i = y } , where S acts on [ n ] d component-wise. Enumerate fS using depth-first search algorithm. There are a maximum of n d k edges and the algorithm traverses each once, hence O ( n d k ) time. Lemma Let f ∈ [ n ] W . Then f ∈ E ( W , S ) iff ∀ i ∈ [ m ] ∃ g ∈ fS ∀ wz i ∈ W : f ( wz i ) = g ( w ) . Trevor Jack Semigroup Complexity August 7, 2018 8 / 16
Transformation Graphs Extensions Extension of Model( u ≈ v ) strategy We now return to the problem of determining if S := � a 1 , . . . , a k � ≤ T n is completely regular. Lemma a ∈ T n generates a subgroup iff a | Im ( a ) is a permutation. [Proof of lemma omitted] Trevor Jack Semigroup Complexity August 7, 2018 9 / 16
Transformation Graphs Extensions Extension of Model( u ≈ v ) strategy We now return to the problem of determining if S := � a 1 , . . . , a k � ≤ T n is completely regular. Lemma a ∈ T n generates a subgroup iff a | Im ( a ) is a permutation. [Proof of lemma omitted] Let W = { 1 , z , z 2 } and define: e (( x , y ) , s ) : W → [ n ] 2 , w �→ ( x , y ) w ( s ) E ( W , S ) := { e (( x , y ) , s ) : ( x , y ) ∈ [ n ] 2 , s ∈ S } ⊆ [ n ] 6 Then every element of S permutes its images iff for every f ∈ E ( W , S ), f ( z ) �∈ { ( x , x ) : x ∈ [ n ] } ⇒ f ( z 2 ) �∈ { ( x , x ) : x ∈ [ n ] } . Trevor Jack Semigroup Complexity August 7, 2018 9 / 16
Transformation Graphs Extensions Quasi-Identities Open Problem: Quasi-Identities Complexity of whether S models u 1 ( z 1 , ..., z m ) ≈ v 1 ( z 1 , ..., z m ) ⇒ u 2 ( z 1 , ..., z m ) ≈ v 2 ( z 1 , ..., z m )? Trevor Jack Semigroup Complexity August 7, 2018 10 / 16
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