Towards a Propositional Logical Structure of Ambiguous Words in - PowerPoint PPT Presentation

Introduction Automata NWA and Logics Paths as Functions Conclusions Towards a Propositional Logical Structure of Ambiguous Words in Weighted Automata Diego Valota Department of Computer Science University of Milan v alota@di.unimi.it Workshop on Ambiguity 2018

Introduction Automata NWA and Logics Paths as Functions Conclusions Overview Non-deterministic Finite (Weighted) Automata; Degrees of Ambiguity in Finite Automata; (Ravikumar and Ibarra, 1989), (Leung, 1998), (Leung, 2005); Weighted Automata vs Logics; (Droste and Gastin, 2007), (Gerla, 2003-2004), (Schwartz, 2006); � Lukasiewicz logic; Interpreting Paths as Functions.

Introduction Automata NWA and Logics Paths as Functions Conclusions Non-deterministic Finite Automata A Non-deterministic Finite Automaton (NFA) is a tuple A = � Σ , S , I , F , δ � , where: Σ is a finite alphabet; S is a finite set of states ; I ⊆ S is a set of initial states ; F ⊆ S is a set of final states ; δ : S × Σ → 2 S is a transition function .

Introduction Automata NWA and Logics Paths as Functions Conclusions Non-deterministic Finite Automata A Non-deterministic Finite Automaton (NFA) is a tuple A = � Σ , S , I , F , δ � , where: Σ is a finite alphabet; S is a finite set of states ; I ⊆ S is a set of initial states ; F ⊆ S is a set of final states ; δ : S × Σ → 2 S is a transition function . a path is a finite sequence of states s 1 , . . . , s m ∈ S ; a word is a finite sequence w = a 1 · a 2 , · · · a n of length n , of letters a i from Σ; a run of a NFA over a word w is a path s 1 , . . . , s n , such that s 1 ∈ I and s i +1 ∈ δ ( s i , a i ) for any 1 ≤ i ≤ n ; a word w is accepted if the a run s 1 , . . . , s n is such that s n ∈ F ; The language recognized by A , denoted by L ( A ), is the set of words accepted by A .

Introduction Automata NWA and Logics Paths as Functions Conclusions Non-deterministic Finite Automata Every NFA A can be tranformed in a deterministic FA accepting the same language: a 0 a 1 a 1 a 0 a 0 s 0 s 1 s 2 start a 0

Introduction Automata NWA and Logics Paths as Functions Conclusions Non-deterministic Finite Automata Every NFA A can be tranformed in a deterministic FA accepting the same language: a 0 a 0 a 1 a 1 a 0 a 0 s 0 s 1 s 2 start s 0 , s 1 , s 2 a 0 a 0 a 1 a 1 a 0 a 0 s 0 , s 1 s 0 , s 2 s 1 , s 2 a 0 a 1 a 1 a 1 a 1 a 0 s 0 s 1 s 2 start a 0

Introduction Automata NWA and Logics Paths as Functions Conclusions Non-deterministic Finite Automata Every NFA A can be tranformed in a deterministic FA accepting the same language: a 0 a 0 a 1 a 1 a 0 a 0 s 0 s 1 s 2 start s 0 , s 1 , s 2 a 0 a 0 a 1 a 1 In the worst case, from a NFA with n states, we build a DFA with 2 n − 1 states (Leung. a 0 a 0 1998). s 0 , s 1 s 0 , s 2 s 1 , s 2 a 0 a 1 a 1 a 1 a 1 a 0 s 0 s 1 s 2 start a 0

Introduction Automata NWA and Logics Paths as Functions Conclusions Non-deterministic Finite Automata Every NFA A can be tranformed in a deterministic FA accepting the same language: a 0 a 0 a 1 a 1 a 0 a 0 s 0 s 1 s 2 start s 0 , s 1 , s 2 a 0 a 0 a 1 a 1 In the worst case, from a NFA with n states, we build a DFA with 2 n − 1 states (Leung. a 0 a 0 1998). s 0 , s 1 s 0 , s 2 s 1 , s 2 The complexity of such a 0 a 1 a 1 a 1 a 1 operation is tied to the degree of ambiguity of A . a 0 s 0 s 1 s 2 start a 0

Introduction Automata NWA and Logics Paths as Functions Conclusions Degrees of Ambiguity The degree of ambiguity d w of a word w is the number of different accepting paths for w in A . d A ( n ) is the maximum of the degrees of ambiguity of words of length n or less, d A ( n ) = max { d w | w is accepted by A , | w | ≤ n } .

Introduction Automata NWA and Logics Paths as Functions Conclusions Degrees of Ambiguity The degree of ambiguity d w of a word w is the number of different accepting paths for w in A . d A ( n ) is the maximum of the degrees of ambiguity of words of length n or less, d A ( n ) = max { d w | w is accepted by A , | w | ≤ n } . a 0 a 1 a 1 a 0 a 0 s 0 s 1 s 2 start a 0 | w | w paths 3 a 0 a 0 a 0 d w = 2 ��� , →→→ 4 a 0 a 0 a 0 a 0 d w = 3 ���� , →→→ � , � →→→ 5 a 0 a 0 a 0 a 0 a 0 d w = 4 ����� , →→→ �� , �� →→→ , � →→→ � 6 a 0 a 0 a 0 a 0 a 0 a 0 d w = 6 →→→→→→ , ����� , →→→ ��� , ��� →→→ , �� →→→ � , � →→→ ��

Introduction Automata NWA and Logics Paths as Functions Conclusions Degrees of Ambiguity The degree of ambiguity d w of a word w is the number of different accepting paths for w in A . d A ( n ) is the maximum of the degrees of ambiguity of words of length n or less, d A ( n ) = max { d w | w is accepted by A , | w | ≤ n } . a 0 a 1 a 1 a 0 a 0 s 0 s 1 s 2 start a 0 | w | w paths 3 a 0 a 0 a 0 d w = 2 ��� , →→→ 4 a 0 a 0 a 0 a 0 d w = 3 ���� , →→→ � , � →→→ 5 a 0 a 0 a 0 a 0 a 0 d w = 4 ����� , →→→ �� , �� →→→ , � →→→ � 6 a 0 a 0 a 0 a 0 a 0 a 0 d w = 6 →→→→→→ , ����� , →→→ ��� , ��� →→→ , �� →→→ � , � →→→ �� d A ( | w | ) > = 2 ⌊| w | / 3 ⌋ (Leung, 1998)

Introduction Automata NWA and Logics Paths as Functions Conclusions Separating NFA NFA can be classified according to their degree of ambiguity: d ( A ) = sup { d A ( n ) | n ∈ N }

Introduction Automata NWA and Logics Paths as Functions Conclusions Separating NFA NFA can be classified according to their degree of ambiguity: d ( A ) = sup { d A ( n ) | n ∈ N } UFA A is unambiguous if d ( A ) ≤ 1; FNA A is finitely ambiguous if d ( A ) ≤ c , where c is a constant function; PNA A is polinomially ambiguous if d ( A ) ≤ p , where p is a polynomial function; ENA A is exponentially ambiguous if d ( A ) ≤ e , where e is an exponential function;

Introduction Automata NWA and Logics Paths as Functions Conclusions Separating NFA NFA can be classified according to their degree of ambiguity: d ( A ) = sup { d A ( n ) | n ∈ N } UFA A is unambiguous if d ( A ) ≤ 1; FNA A is finitely ambiguous if d ( A ) ≤ c , where c is a constant function; PNA A is polinomially ambiguous if d ( A ) ≤ p , where p is a polynomial function; ENA A is exponentially ambiguous if d ( A ) ≤ e , where e is an exponential function; DFA < p UFA < p FNA ≤ p PNA < p NFA where C ≤ p C ′ when there exists a polynomial p such that for any finite automaton in C with n states, it is possibile to find an equivalent automaton in C ′ with p ( n ) states. References: (Ravikumar and Ibarra, 1989), (Leung, 2005);

Introduction Automata NWA and Logics Paths as Functions Conclusions Separating NFA Ambiguity Matters! Ambiguity is relatead to the succinctness in the number of states. Restricting Ambiguity − → Increases the number of states. Ambiguity influences the tractability of algorithmic issues. For instance, for every n ∈ N it can be determined efficiently if two NFA of ambiguity at most n are equivalent.

Introduction Automata NWA and Logics Paths as Functions Conclusions Weighted Automata A Non-deterministic Weighted Automaton (NWA) is a NFA A = � Σ , S , I , F , δ, e � , s.t.: Σ is a finite alphabet, and S is a finite set of states ; I ⊆ S is a set of initial states , and F ⊆ S is a set of final states ; δ : S × Σ → 2 S is a transition function ; e : S × Σ × S → R is a weight function, assigning to each triple ( s , a , s ′ ) ∈ S × Σ × S a value r ∈ R . A semiring is a structure R = � R , + , · , 0 , 1 � such that: � R , + , 0 � is a commutative monoid, and � R , · , 1 � is a monoid; x · ( y + z ) = x · y + x · z and ( y + z ) · x = y · x + z · x for x , y , z ∈ R ; 0 · x = x · 0 = 0 for each x ∈ R .

Introduction Automata NWA and Logics Paths as Functions Conclusions Weighted Automata A Non-deterministic Weighted Automaton (NWA) is a NFA A = � Σ , S , I , F , δ, e � , s.t.: Σ is a finite alphabet, and S is a finite set of states ; I ⊆ S is a set of initial states , and F ⊆ S is a set of final states ; δ : S × Σ → 2 S is a transition function ; e : S × Σ × S → R is a weight function, assigning to each triple ( s , a , s ′ ) ∈ S × Σ × S a value r ∈ R . A semiring is a structure R = � R , + , · , 0 , 1 � such that: � R , + , 0 � is a commutative monoid, and � R , · , 1 � is a monoid; x · ( y + z ) = x · y + x · z and ( y + z ) · x = y · x + z · x for x , y , z ∈ R ; 0 · x = x · 0 = 0 for each x ∈ R . Let p = s 1 , . . . , s n be an accepting path for a word w = a 1 . . . a n . The behavior of A is a The weight of p is The weight of a word w is map �A� : Σ ∗ → R that sends every w ∈ Σ ∗ to � � � p � = e ( s i , a i , s i +1 ) . � w � = � p � . � w � . 1 ≤ i ≤ n − 1 p ∈ P w