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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 valota@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 → 2S 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 → 2S is a transition function. a path is a finite sequence of states s1, . . . , sm ∈ S; a word is a finite sequence w = a1 · a2, · · · an of length n, of letters ai from Σ; a run of a NFA over a word w is a path s1, . . . , sn, such that s1 ∈ I and si+1 ∈ δ(si, ai) for any 1 ≤ i ≤ n; a word w is accepted if the a run s1, . . . , sn is such that sn ∈ 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: s0 start s1 s2 a0 a0 a0 a1 a0 a1

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: s0 start s1 s2 a0 a0 a0 a1 a0 a1 s0 start s1 s2 s0, s1 s0, s2 s1, s2 s0, s1, s2 a0 a0 a1 a0 a1 a1 a0 a0 a1 a0 a1 a0 a1

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: s0 start s1 s2 a0 a0 a0 a1 a0 a1 In the worst case, from a NFA with n states, we build a DFA with 2n − 1 states (Leung. 1998). s0 start s1 s2 s0, s1 s0, s2 s1, s2 s0, s1, s2 a0 a0 a1 a0 a1 a1 a0 a0 a1 a0 a1 a0 a1

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: s0 start s1 s2 a0 a0 a0 a1 a0 a1 In the worst case, from a NFA with n states, we build a DFA with 2n − 1 states (Leung. 1998). The complexity of such

• peration is tied to the

degree of ambiguity of A. s0 start s1 s2 s0, s1 s0, s2 s1, s2 s0, s1, s2 a0 a0 a1 a0 a1 a1 a0 a0 a1 a0 a1 a0 a1

Introduction Automata NWA and Logics Paths as Functions Conclusions Degrees of Ambiguity

The degree of ambiguity dw of a word w is the number of different accepting paths for w in A. dA(n) is the maximum of the degrees of ambiguity of words of length n or less, dA(n) = max{dw | w is accepted by A, |w| ≤ n}.

Introduction Automata NWA and Logics Paths as Functions Conclusions Degrees of Ambiguity

The degree of ambiguity dw of a word w is the number of different accepting paths for w in A. dA(n) is the maximum of the degrees of ambiguity of words of length n or less, dA(n) = max{dw | w is accepted by A, |w| ≤ n}. s0 start s1 s2 a0 a0 a0 a1 a0 a1 |w| w paths 3 a0a0a0 dw = 2 , →→→ 4 a0a0a0a0 dw = 3 , →→→, →→→ 5 a0a0a0a0a0 dw = 4 , →→→, →→→, →→→ 6 a0a0a0a0a0a0 dw = 6 →→→→→→, , →→→, →→→, →→→, →→→

Introduction Automata NWA and Logics Paths as Functions Conclusions Degrees of Ambiguity

The degree of ambiguity dw of a word w is the number of different accepting paths for w in A. dA(n) is the maximum of the degrees of ambiguity of words of length n or less, dA(n) = max{dw | w is accepted by A, |w| ≤ n}. s0 start s1 s2 a0 a0 a0 a1 a0 a1 |w| w paths 3 a0a0a0 dw = 2 , →→→ 4 a0a0a0a0 dw = 3 , →→→, →→→ 5 a0a0a0a0a0 dw = 4 , →→→, →→→, →→→ 6 a0a0a0a0a0a0 dw = 6 →→→→→→, , →→→, →→→, →→→, →→→ dA(|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{dA(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{dA(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{dA(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 → 2S 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 → 2S 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 = s1, . . . , sn be an accepting path for a word w = a1 . . . an. The weight of p is p =

• 1≤i≤n−1

e(si, ai, si+1). The weight of a word w is w =

• p∈Pw

p . The behavior of A is a map A : Σ∗ → R that sends every w ∈ Σ∗ to w.

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 → 2S 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 = s1, . . . , sn be an accepting path for a word w = a1 . . . an. The weight of p is p =

• 1≤i≤n−1

e(si, ai, si+1). The weight of a word w is w =

• p∈Pw

p . The behavior of A is a map A : Σ∗ → R that sends every w ∈ Σ∗ to w. An R-valued language is a map L : Σ∗ → R. An R-valued language is recognizable if there exists a NWA A s.t. A = L.

Introduction Automata NWA and Logics Paths as Functions Conclusions Weighted Automata

s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 B = {0, 1}, ∨, ∧, 0, 1 s0 start s1 s2 a0|1 a0|5 a0|1 a1|2 a0|1 a1|2 N = {0, 1, 2, . . . }, +, ·, 0, 1

Introduction Automata NWA and Logics Paths as Functions Conclusions Weighted Automata

s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 B = {0, 1}, ∨, ∧, 0, 1 s0 start s1 s2 a0|1 a0|5 a0|1 a1|2 a0|1 a1|2 N = {0, 1, 2, . . . }, +, ·, 0, 1 For every x, y ∈ [0, 1] define: x ⊕ y = min{1, x + y}, x ⊙ y = max{0, x + y − 1}, both ⊕ and ⊙ are commutative.

Introduction Automata NWA and Logics Paths as Functions Conclusions Weighted Automata

s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 B = {0, 1}, ∨, ∧, 0, 1 s0 start s1 s2 a0|1 a0|5 a0|1 a1|2 a0|1 a1|2 N = {0, 1, 2, . . . }, +, ·, 0, 1 For every x, y ∈ [0, 1] define: x ⊕ y = min{1, x + y}, x ⊙ y = max{0, x + y − 1}, both ⊕ and ⊙ are commutative. Then, the structures [0, 1]∧ = [0, 1], ∧, ⊕, 0, 1, [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1, are two commutative semirings.

Introduction Automata NWA and Logics Paths as Functions Conclusions Weighted Automata

s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 B = {0, 1}, ∨, ∧, 0, 1 s0 start s1 s2 a0|1 a0|5 a0|1 a1|2 a0|1 a1|2 N = {0, 1, 2, . . . }, +, ·, 0, 1 For every x, y ∈ [0, 1] define: x ⊕ y = min{1, x + y}, x ⊙ y = max{0, x + y − 1}, both ⊕ and ⊙ are commutative. Then, the structures [0, 1]∧ = [0, 1], ∧, ⊕, 0, 1, [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1, are two commutative semirings. Definitions of degree of ambiguity of a word dw and of automata dA(n) are the same as for NFA.

Introduction Automata NWA and Logics Paths as Functions Conclusions MSO logics

To obtain a logic whose semantics defines the behavior of NWA, (Droste and Gastin, 2007) extend Monadic Second-Order logic adding weights for every logical formula taking values in a semiring R.

Introduction Automata NWA and Logics Paths as Functions Conclusions MSO logics

To obtain a logic whose semantics defines the behavior of NWA, (Droste and Gastin, 2007) extend Monadic Second-Order logic adding weights for every logical formula taking values in a semiring R. This approach has been extended by (Schwartz, 2006), where weights are defined over an MV-algebras instead of a semiring. The variety of MV-algebras gives the algebraic semantics of the propositional many-valued Lukasiewicz logic L.

Introduction Automata NWA and Logics Paths as Functions Conclusions MSO logics

To obtain a logic whose semantics defines the behavior of NWA, (Droste and Gastin, 2007) extend Monadic Second-Order logic adding weights for every logical formula taking values in a semiring R. This approach has been extended by (Schwartz, 2006), where weights are defined over an MV-algebras instead of a semiring. The variety of MV-algebras gives the algebraic semantics of the propositional many-valued Lukasiewicz logic L.

• L can be semantically defined in the following way:

Let Form be the set of formulas over propositional variables x1, x2, , . . . in the language ⊙, ⊕, ¬, ⊥, ⊤. An evaluation is a function µ : Form → [0, 1] ⊆ R with values in the real unit interval such that, for any two α, β ∈ Form, µ(⊥) = 0, µ(⊤) = 1, µ(¬α) = 1 − µ(α), µ(α ⊕ β) = min{1, µ(α) + µ(β)}, µ(α ⊙ β) = max{0, µ(α) + µ(β) − 1}.

Introduction Automata NWA and Logics Paths as Functions Conclusions MSO logics

To obtain a logic whose semantics defines the behavior of NWA, (Droste and Gastin, 2007) extend Monadic Second-Order logic adding weights for every logical formula taking values in a semiring R. This approach has been extended by (Schwartz, 2006), where weights are defined over an MV-algebras instead of a semiring. The variety of MV-algebras gives the algebraic semantics of the propositional many-valued Lukasiewicz logic L.

• L can be semantically defined in the following way:

Let Form be the set of formulas over propositional variables x1, x2, , . . . in the language ⊙, ⊕, ¬, ⊥, ⊤. An evaluation is a function µ : Form → [0, 1] ⊆ R with values in the real unit interval such that, for any two α, β ∈ Form, µ(⊥) = 0, µ(⊤) = 1, µ(¬α) = 1 − µ(α), µ(α ⊕ β) = min{1, µ(α) + µ(β)}, µ(α ⊙ β) = max{0, µ(α) + µ(β) − 1}. Schwartz’s results are based on the relation between MV-algebras and semirings, and the definition of MV-automata defined in the works of B. Gerla (2003-2004).

Introduction Automata NWA and Logics Paths as Functions Conclusions MV-Algebras

A MV-algebra is a structure A = A, ⊕, ⊙, ¬, ⊥, ⊤ such that: x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z x ⊕ y = y ⊕ x, x ⊕ ⊥ = x, x ⊕ ⊤ = ⊤, ¬⊥ = ⊤, ¬¬x = x, ¬(¬x ⊕ ¬y) = x ⊙ y, ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x. In a MV-algebra A we have: x ≤ y iff ¬x ⊕ y = ⊤, x ∨ y = x ⊕ (¬x ⊙ y), x ∧ y = x ⊙ (¬x ⊕ y).

Introduction Automata NWA and Logics Paths as Functions Conclusions MV-Algebras

A MV-algebra is a structure A = A, ⊕, ⊙, ¬, ⊥, ⊤ such that: x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z x ⊕ y = y ⊕ x, x ⊕ ⊥ = x, x ⊕ ⊤ = ⊤, ¬⊥ = ⊤, ¬¬x = x, ¬(¬x ⊕ ¬y) = x ⊙ y, ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x. In a MV-algebra A we have: x ≤ y iff ¬x ⊕ y = ⊤, x ∨ y = x ⊕ (¬x ⊙ y), x ∧ y = x ⊙ (¬x ⊕ y). Theorem (Gerla, 2003) For every MV-algebra A both structures: A∨ = A, ∨, ⊙, ⊥, ⊤ and A∧ = A, ∧, ⊕, ⊥, ⊤, are commutative semirings, and ¬ is an isomorphism between A∨ and A∧.

Introduction Automata NWA and Logics Paths as Functions Conclusions MV-Algebras

A MV-algebra is a structure A = A, ⊕, ⊙, ¬, ⊥, ⊤ such that: x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z x ⊕ y = y ⊕ x, x ⊕ ⊥ = x, x ⊕ ⊤ = ⊤, ¬⊥ = ⊤, ¬¬x = x, ¬(¬x ⊕ ¬y) = x ⊙ y, ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x. In a MV-algebra A we have: x ≤ y iff ¬x ⊕ y = ⊤, x ∨ y = x ⊕ (¬x ⊙ y), x ∧ y = x ⊙ (¬x ⊕ y). Theorem (Gerla, 2003) For every MV-algebra A both structures: A∨ = A, ∨, ⊙, ⊥, ⊤ and A∧ = A, ∧, ⊕, ⊥, ⊤, are commutative semirings, and ¬ is an isomorphism between A∨ and A∧. 1

5 6 4 6 3 6 2 6 1 6

¬ 1 6 ∨ 4 6

• = ¬ 1

6 ∧ ¬ 4 6 ¬ 3 6 ⊙ 4 6

• = ¬ 3

6 ⊕ ¬ 4 6

Introduction Automata NWA and Logics Paths as Functions Conclusions MV-Automata

An MV-Automaton is an NWA A = Σ, S, I, F, δ, e where e takes values over an MV-algebra A = A, ⊕, ⊙, ¬, ⊥, ⊤.

Introduction Automata NWA and Logics Paths as Functions Conclusions MV-Automata

An MV-Automaton is an NWA A = Σ, S, I, F, δ, e where e takes values over an MV-algebra A = A, ⊕, ⊙, ¬, ⊥, ⊤. Let p = s1, . . . , sn be an accepting path for a word w = a1 . . . an. The weight of p is p =

• 1≤i≤n−1

e(si, ai, si+1). The weight of a word w is w =

• p∈Pw

p . The behavior of A is a map A : Σ∗ → A that sends every w ∈ Σ∗ to w. An A-valued language is a map L : Σ∗ → A. An A-valued language is recognizable if there exists a MVA A s.t. A = L.

Introduction Automata NWA and Logics Paths as Functions Conclusions MV-Automata

An MV-Automaton is an NWA A = Σ, S, I, F, δ, e where e takes values over an MV-algebra A = A, ⊕, ⊙, ¬, ⊥, ⊤. Let p = s1, . . . , sn be an accepting path for a word w = a1 . . . an. The weight of p is p =

• 1≤i≤n−1

e(si, ai, si+1). The weight of a word w is w =

• p∈Pw

p . The behavior of A is a map A : Σ∗ → A that sends every w ∈ Σ∗ to w. An A-valued language is a map L : Σ∗ → A. An A-valued language is recognizable if there exists a MVA A s.t. A = L. The class of A-valued languages AΣ∗, can be equipped by pointwise MV-algebras

• perations (Gerla, 2004).

Theorem (Gerla, 2004) A ∨ B = A ∨ B A ⊙ B = A ⊙ B

Introduction Automata NWA and Logics Paths as Functions Conclusions Example with Boolean Semiring and Classical Logic

Take the following NWA over B = {0, 1}, ∨, ∧, 0, 1. s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. Accepting paths p1 =, p2 =→→→, p3 =→→→, p4 =→→→,

Introduction Automata NWA and Logics Paths as Functions Conclusions Example with Boolean Semiring and Classical Logic

Take the following NWA over B = {0, 1}, ∨, ∧, 0, 1. s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0. Accepting paths Weights p1 =, 00000, p1 = 0 p2 =→→→, 11100, p2 = 0 p3 =→→→, 00111, p3 = 0 p4 =→→→, 01110, p4 = 0

Introduction Automata NWA and Logics Paths as Functions Conclusions Example with Boolean Semiring and Classical Logic

Take the following NWA over B = {0, 1}, ∨, ∧, 0, 1. s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0. Accepting paths Weights p1 =, 00000, p1 = 0 p2 =→→→, 11100, p2 = 0 p3 =→→→, 00111, p3 = 0 p4 =→→→, 01110, p4 = 0 · : {0, 1}|w| → {0, 1}.

Introduction Automata NWA and Logics Paths as Functions Conclusions Example with Boolean Semiring and Classical Logic

Take the following NWA over B = {0, 1}, ∨, ∧, 0, 1. s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0. Accepting paths Weights p1 =, 00000, p1 = 0 p2 =→→→, 11100, p2 = 0 p3 =→→→, 00111, p3 = 0 p4 =→→→, 01110, p4 = 0 · : {0, 1}|w| → {0, 1}. Functions f : {0, 1}n → {0, 1} are the elements of the free Boolean algebra

• ver n generators Freen(B).

Free algebras are isomorphic to Lindenbaum algebras. The Lindenbaum algebra of Boolean logic over the language {x1, . . . , xn} is, by construction, Bn = FORMn/ ≡. Where ≡ denotes logical equivalence.

Introduction Automata NWA and Logics Paths as Functions Conclusions Example with Boolean Semiring and Classical Logic

Take the following NWA over B = {0, 1}, ∨, ∧, 0, 1. s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0. Accepting paths Weights p1 =, 00000, p1 = 0 p2 =→→→, 11100, p2 = 0 p3 =→→→, 00111, p3 = 0 p4 =→→→, 01110, p4 = 0 · : {0, 1}|w| → {0, 1}. Functions f : {0, 1}n → {0, 1} are the elements of the free Boolean algebra

• ver n generators Freen(B).

Free algebras are isomorphic to Lindenbaum algebras. The Lindenbaum algebra of Boolean logic over the language {x1, . . . , xn} is, by construction, Bn = FORMn/ ≡. Where ≡ denotes logical equivalence. Hence, w → [ϕ]≡ ∈ Free|w|(B).

Introduction Automata NWA and Logics Paths as Functions Conclusions Example with Boolean Semiring and Classical Logic

Take the following NWA over B = {0, 1}, ∨, ∧, 0, 1. s0 start s1 s2 a0|1 a0|0 a0|1 a1|0 a0|1 a1|0 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0. Accepting paths Weights p1 =, 00000, p1 = 0 p2 =→→→, 11100, p2 = 0 p3 =→→→, 00111, p3 = 0 p4 =→→→, 01110, p4 = 0 · : {0, 1}|w| → {0, 1}. Functions f : {0, 1}n → {0, 1} are the elements of the free Boolean algebra

• ver n generators Freen(B).

Free algebras are isomorphic to Lindenbaum algebras. The Lindenbaum algebra of Boolean logic over the language {x1, . . . , xn} is, by construction, Bn = FORMn/ ≡. Where ≡ denotes logical equivalence. Hence, w → [ϕ]≡ ∈ Free|w|(B). Cardinality issues? The number of Boolean assignments of a formula in n variable is 2n. Let Pw be the set of accepting paths of w = a1, . . . , an. When |Pw| > 2|w|, then there exist (a1, . . . , an), (a1, . . . , an)′ ∈ Pw s.t. (a1 , . . . , an) = (a1 , . . . , an)′.

Introduction Automata NWA and Logics Paths as Functions Conclusions Semiring over the Real Unitary Interval

Take the following NWA over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1. s0 start s1 s2 a0|0.7 a0|0.2 a0|0.7 a1|0.3 a0|0.8 a1|0.3 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0.2. Accepting paths Weights p1 =, 0.2, 0.2, 0.2, 0.2, 0.2, p1 = 0 p2 =→→→, 0.7, 0.7, 0.8, 0.2, 0.2, p2 = 0 p3 =→→→, 0.2, 0.2, 0.7, 0.7, 0.8, p3 = 0.2 p4 =→→→, 0.2, 0.7, 0.7, 0.8, 0.2, p4 = 0

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Introduction Automata NWA and Logics Paths as Functions Conclusions Semiring over the Real Unitary Interval

Take the following NWA over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1. s0 start s1 s2 a0|0.7 a0|0.2 a0|0.7 a1|0.3 a0|0.8 a1|0.3 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0.2. Accepting paths Weights p1 =, 0.2, 0.2, 0.2, 0.2, 0.2, p1 = 0 p2 =→→→, 0.7, 0.7, 0.8, 0.2, 0.2, p2 = 0 p3 =→→→, 0.2, 0.2, 0.7, 0.7, 0.8, p3 = 0.2 p4 =→→→, 0.2, 0.7, 0.7, 0.8, 0.2, p4 = 0 · : [0, 1]|w| → [0, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Introduction Automata NWA and Logics Paths as Functions Conclusions Semiring over the Real Unitary Interval

Take the following NWA over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1. s0 start s1 s2 a0|0.7 a0|0.2 a0|0.7 a1|0.3 a0|0.8 a1|0.3 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0.2. Accepting paths Weights p1 =, 0.2, 0.2, 0.2, 0.2, 0.2, p1 = 0 p2 =→→→, 0.7, 0.7, 0.8, 0.2, 0.2, p2 = 0 p3 =→→→, 0.2, 0.2, 0.7, 0.7, 0.8, p3 = 0.2 p4 =→→→, 0.2, 0.7, 0.7, 0.8, 0.2, p4 = 0 · : [0, 1]|w| → [0, 1] f : ([0, 1]∨)n → [0, 1]∨ What kind of functions are?

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Introduction Automata NWA and Logics Paths as Functions Conclusions Semiring over the Real Unitary Interval

Take the following NWA over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1. s0 start s1 s2 a0|0.7 a0|0.2 a0|0.7 a1|0.3 a0|0.8 a1|0.3 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0.2. Accepting paths Weights p1 =, 0.2, 0.2, 0.2, 0.2, 0.2, p1 = 0 p2 =→→→, 0.7, 0.7, 0.8, 0.2, 0.2, p2 = 0 p3 =→→→, 0.2, 0.2, 0.7, 0.7, 0.8, p3 = 0.2 p4 =→→→, 0.2, 0.7, 0.7, 0.8, 0.2, p4 = 0 · : [0, 1]|w| → [0, 1] f : ([0, 1]∨)n → [0, 1]∨ What kind of functions are? r1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Introduction Automata NWA and Logics Paths as Functions Conclusions Semiring over the Real Unitary Interval

Take the following NWA over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1. s0 start s1 s2 a0|0.7 a0|0.2 a0|0.7 a1|0.3 a0|0.8 a1|0.3 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0.2. Accepting paths Weights p1 =, 0.2, 0.2, 0.2, 0.2, 0.2, p1 = 0 p2 =→→→, 0.7, 0.7, 0.8, 0.2, 0.2, p2 = 0 p3 =→→→, 0.2, 0.2, 0.7, 0.7, 0.8, p3 = 0.2 p4 =→→→, 0.2, 0.7, 0.7, 0.8, 0.2, p4 = 0 · : [0, 1]|w| → [0, 1] f : ([0, 1]∨)n → [0, 1]∨ What kind of functions are? r1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

r1 ⊙ r1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Introduction Automata NWA and Logics Paths as Functions Conclusions Semiring over the Real Unitary Interval

Take the following NWA over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1. s0 start s1 s2 a0|0.7 a0|0.2 a0|0.7 a1|0.3 a0|0.8 a1|0.3 The word w = a0a0a0a0a0 has |w| = 5 and dw = 4. w = 0.2. Accepting paths Weights p1 =, 0.2, 0.2, 0.2, 0.2, 0.2, p1 = 0 p2 =→→→, 0.7, 0.7, 0.8, 0.2, 0.2, p2 = 0 p3 =→→→, 0.2, 0.2, 0.7, 0.7, 0.8, p3 = 0.2 p4 =→→→, 0.2, 0.7, 0.7, 0.8, 0.2, p4 = 0 · : [0, 1]|w| → [0, 1] f : ([0, 1]∨)n → [0, 1]∨ What kind of functions are? r1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

r1 ⊙ r1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

r1 ⊙ r1 ⊙ r1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

What are the characteristics of functions f : ([0, 1]∨)n → [0, 1]∨ over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1?

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

What are the characteristics of functions f : ([0, 1]∨)n → [0, 1]∨ over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1? f is continuous;

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

What are the characteristics of functions f : ([0, 1]∨)n → [0, 1]∨ over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1? f is continuous; there are affine linear polynomials f1, . . . fk with integer coefficients, fi(x1, . . . , xn) = bi + ai1x1 + . . . + ainxn, such that for each point p = (p1 . . . , pn) ∈ [0, 1] there is a j ∈ {1, . . . , k} with f (p) = fj(p).

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

What are the characteristics of functions f : ([0, 1]∨)n → [0, 1]∨ over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1? f is continuous; there are affine linear polynomials f1, . . . fk with integer coefficients, fi(x1, . . . , xn) = bi + ai1x1 + . . . + ainxn, such that for each point p = (p1 . . . , pn) ∈ [0, 1] there is a j ∈ {1, . . . , k} with f (p) = fj(p). f : ([0, 1]∨)n → [0, 1]∨ are piecewise linear functions, that is McNaughton functions (over [0, 1]∨).

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

What are the characteristics of functions f : ([0, 1]∨)n → [0, 1]∨ over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1? f is continuous; there are affine linear polynomials f1, . . . fk with integer coefficients, fi(x1, . . . , xn) = bi + ai1x1 + . . . + ainxn, such that for each point p = (p1 . . . , pn) ∈ [0, 1] there is a j ∈ {1, . . . , k} with f (p) = fj(p). f : ([0, 1]∨)n → [0, 1]∨ are piecewise linear functions, that is McNaughton functions (over [0, 1]∨). The class of McNaughton functions (over [0, 1]) is in bijection with the free n-generated MV algebra, that is the Lindenbaum algebra of Lukasiewicz logic.

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

What are the characteristics of functions f : ([0, 1]∨)n → [0, 1]∨ over [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1? f is continuous; there are affine linear polynomials f1, . . . fk with integer coefficients, fi(x1, . . . , xn) = bi + ai1x1 + . . . + ainxn, such that for each point p = (p1 . . . , pn) ∈ [0, 1] there is a j ∈ {1, . . . , k} with f (p) = fj(p). f : ([0, 1]∨)n → [0, 1]∨ are piecewise linear functions, that is McNaughton functions (over [0, 1]∨). The class of McNaughton functions (over [0, 1]) is in bijection with the free n-generated MV algebra, that is the Lindenbaum algebra of Lukasiewicz logic. From every word w in [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1, we obtain a function fw : ([0, 1]∨)n → [0, 1]∨ such that there exist a corresponding propositional formula ϕfw in Lukasiewicz logic.

Notice that this is valid also in the case of [0, 1]∧ or over a MVA.

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

The full construnction from McNaughton functions to formulas in L is lenghty. Here, we briefly sketch a useful decomposition property of McNaughton functions, that shed a light on how to obtain logical formulas.

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

The full construnction from McNaughton functions to formulas in L is lenghty. Here, we briefly sketch a useful decomposition property of McNaughton functions, that shed a light on how to obtain logical formulas. F1 = {0, 1}, F2 = {0, 1

2 , 1}, F3 = {0, 1 3 , 1 2 , 2 3 , 1},

... for any a1

d1 , a2 d2 in Fi, we add their Farey’s mediant a1+a2 d1+d2 to Fi+1.

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

The full construnction from McNaughton functions to formulas in L is lenghty. Here, we briefly sketch a useful decomposition property of McNaughton functions, that shed a light on how to obtain logical formulas. F1 = {0, 1}, F2 = {0, 1

2 , 1}, F3 = {0, 1 3 , 1 2 , 2 3 , 1},

... for any a1

d1 , a2 d2 in Fi, we add their Farey’s mediant a1+a2 d1+d2 to Fi+1.

The Farey series help us to obtain a unimodular partition U of [0, 1]n such that for a vertex {v} ∈ U, a Schauder hat is the unique continuous function hv s.t.: hv = 1/den (v), and it is linear on every simplex of U. hx = 0 for every simplex S in U such that v ∈ S.

ai di ai di ai di ai di

1 1 1

1 d

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

f (x) =

• 1 − x

x ∈ [0, 2

3 )

2x − 1 x ∈ [ 2

3 , 1]

1 1 1 2 2 3 3

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

f (x) =

• 1 − x

x ∈ [0, 2

3 )

2x − 1 x ∈ [ 2

3 , 1]

1 1 1 2 2 3 3

Introduction Automata NWA and Logics Paths as Functions Conclusions McNaughton functions

f (x) =

• 1 − x

x ∈ [0, 2

3 )

2x − 1 x ∈ [ 2

3 , 1]

1 1 1 2 2 3 3

We can express every McNaughton function as a sum of Schauder hats hv: f =

• kvhv

where f (v) = kv/(denv).

Introduction Automata NWA and Logics Paths as Functions Conclusions Formulas

To each Schauder hat hv it is possibile to associate a L formula ϕhv that takes the role

• f a minterm in

L. That is, every ϕ in L can be constructed as a ⊕-disjunction of minterms ⊕v∈Umv · ϕhv.

Introduction Automata NWA and Logics Paths as Functions Conclusions Formulas

To each Schauder hat hv it is possibile to associate a L formula ϕhv that takes the role

• f a minterm in

L. That is, every ϕ in L can be constructed as a ⊕-disjunction of minterms ⊕v∈Umv · ϕhv. This suggests a slightly different approach:

Introduction Automata NWA and Logics Paths as Functions Conclusions Formulas

To each Schauder hat hv it is possibile to associate a L formula ϕhv that takes the role

• f a minterm in

L. That is, every ϕ in L can be constructed as a ⊕-disjunction of minterms ⊕v∈Umv · ϕhv. This suggests a slightly different approach: Let p be an accepting path for a word in [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1, instead of taking a function fp : ([0, 1]∨)n → [0, 1]∨ as before, we just take a point v ∈ ([0, 1]∨)n such that v = (a1 , . . . , an).

Introduction Automata NWA and Logics Paths as Functions Conclusions Formulas

To each Schauder hat hv it is possibile to associate a L formula ϕhv that takes the role

• f a minterm in

L. That is, every ϕ in L can be constructed as a ⊕-disjunction of minterms ⊕v∈Umv · ϕhv. This suggests a slightly different approach: Let p be an accepting path for a word in [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1, instead of taking a function fp : ([0, 1]∨)n → [0, 1]∨ as before, we just take a point v ∈ ([0, 1]∨)n such that v = (a1 , . . . , an). By taking the corresponding Schauder hat hv we maps p directly to a minterm ϕhvin

• L.

Introduction Automata NWA and Logics Paths as Functions Conclusions Formulas

To each Schauder hat hv it is possibile to associate a L formula ϕhv that takes the role

• f a minterm in

L. That is, every ϕ in L can be constructed as a ⊕-disjunction of minterms ⊕v∈Umv · ϕhv. This suggests a slightly different approach: Let p be an accepting path for a word in [0, 1]∨ = [0, 1], ∨, ⊙, 0, 1, instead of taking a function fp : ([0, 1]∨)n → [0, 1]∨ as before, we just take a point v ∈ ([0, 1]∨)n such that v = (a1 , . . . , an). By taking the corresponding Schauder hat hv we maps p directly to a minterm ϕhvin

• L.

Notice that, this is quite different than in the classical case {0, 1}. Indeed, the number of paths/assignments over {0, 1}n is finite 2n, while in the semirings over [0, 1] we have infinite assignments, allowing us to associate different L formulas to different accepting paths.

Introduction Automata NWA and Logics Paths as Functions Conclusions Further Directions

A commutative integral bounded residuated lattice is an algebra A = (A, ∧, ∨, ⊙, →, ⊥, ⊤) of type (2, 2, 2, 2, 0, 0) such that (A, ∧, ∨, ⊥, ⊤) is a bounded lattice, (A, ⊙, ⊤) is a commutative monoid, and the residuation equivalence, x ⊙ y ≤ z if and only if x ≤ y → z, holds. An MTL algebra A = (A, ∧, ∨, ⊙, →, ⊥, ⊤) is a commutative integral bounded residuated lattice satisfying the prelinearity equation, (x → y) ∨ (y → x) = ⊤ .

Introduction Automata NWA and Logics Paths as Functions Conclusions Further Directions

A commutative integral bounded residuated lattice is an algebra A = (A, ∧, ∨, ⊙, →, ⊥, ⊤) of type (2, 2, 2, 2, 0, 0) such that (A, ∧, ∨, ⊥, ⊤) is a bounded lattice, (A, ⊙, ⊤) is a commutative monoid, and the residuation equivalence, x ⊙ y ≤ z if and only if x ≤ y → z, holds. An MTL algebra A = (A, ∧, ∨, ⊙, →, ⊥, ⊤) is a commutative integral bounded residuated lattice satisfying the prelinearity equation, (x → y) ∨ (y → x) = ⊤ . Every MTL algebra A admits a semiring reduct S∨ = A, ∨, ⊙, ⊥, ⊤. To generalize/apply Gerla’s approach to other MTL algebras, they need also S∧ = A, ∧, ⊕, ⊥, ⊤.

Introduction Automata NWA and Logics Paths as Functions Conclusions Further Directions

A commutative integral bounded residuated lattice is an algebra A = (A, ∧, ∨, ⊙, →, ⊥, ⊤) of type (2, 2, 2, 2, 0, 0) such that (A, ∧, ∨, ⊥, ⊤) is a bounded lattice, (A, ⊙, ⊤) is a commutative monoid, and the residuation equivalence, x ⊙ y ≤ z if and only if x ≤ y → z, holds. An MTL algebra A = (A, ∧, ∨, ⊙, →, ⊥, ⊤) is a commutative integral bounded residuated lattice satisfying the prelinearity equation, (x → y) ∨ (y → x) = ⊤ . Every MTL algebra A admits a semiring reduct S∨ = A, ∨, ⊙, ⊥, ⊤. To generalize/apply Gerla’s approach to other MTL algebras, they need also S∧ = A, ∧, ⊕, ⊥, ⊤. But ⊕ is not always nicely definable (Aguzzoli, Bianchi, Flaminio. 2015). One interesting case, where ⊕ is definable is the variety of algebras corresponding to NM logic, that is Constructive Logic with Strong Negation plus prelinearity.

Introduction Automata NWA and Logics Paths as Functions Conclusions Further Directions

Another interesting (dual) approach to deal with DFA in propositional logic is the one

• f (Aguzzoli, Diaconescu, Flaminio. 2014).

s0 start s1 s2 a0 a0 a0 a1 a0 a1

Introduction Automata NWA and Logics Paths as Functions Conclusions Further Directions

Another interesting (dual) approach to deal with DFA in propositional logic is the one

• f (Aguzzoli, Diaconescu, Flaminio. 2014).

s0 start s1 s2 a0 a0 a0 a1 a0 a1 {s0, s1, s2} {s0, s1} {s0, s2} {s1, s2} {s0} {s1} {s2} ∅

Introduction Automata NWA and Logics Paths as Functions Conclusions Further Directions

Another interesting (dual) approach to deal with DFA in propositional logic is the one

• f (Aguzzoli, Diaconescu, Flaminio. 2014).

s0 start s1 s2 a0 a0 a0 a1 a0 a1 {s0, s1, s2} {s0, s1} {s0, s2} {s1, s2} {s0} {s1} {s2} ∅ Can we generalize this approach to NFA, NWA [0, 1]∧ = [0, 1], ∧, ⊕, 0, 1, or to MVA?

Introduction Automata NWA and Logics Paths as Functions Conclusions Further Directions

Another interesting (dual) approach to deal with DFA in propositional logic is the one

• f (Aguzzoli, Diaconescu, Flaminio. 2014).

s0 start s1 s2 a0 a0 a0 a1 a0 a1 {s0, s1, s2} {s0, s1} {s0, s2} {s1, s2} {s0} {s1} {s2} ∅ Can we generalize this approach to NFA, NWA [0, 1]∧ = [0, 1], ∧, ⊕, 0, 1, or to MVA? (Finite) Duality Theory!

(for MV3 in the picture on the right)

• 1 •2 •1
• 1•2
• 1 •1 •1
• 2•1
• 1•1
• 1
• 2
• 1•1
• 1
• 1
• 1

∅

Introduction Automata NWA and Logics Paths as Functions Conclusions Questions

Interpreting paths as assignments, we have introduced an additional level of ambiguity (different paths may produce the same evaluation); What does it mean? (Disambiguation?) We linked words over NWA to propositional (many-valued) logics formulas; Does words inherits the ”logical structure“ of the Lindenbaum algebra? Is ambiguity on words reflected in the formulas? Can logics tell us something about ambiguous NFA separation?

Introduction Automata NWA and Logics Paths as Functions Conclusions References

• Lukasiewicz logic, MV-Algebras, Semirings and MV-Automata:
• A. Di Nola and B. Gerla. Algebras of

Lukasiewiczs logic and their semiring reducts. In Contemp. Math, volume 377, pages 131–144, 01 2005.

• B. Gerla. Many-valued logic and Semirings. Neural Network World, 13:467–480, 01 2003.
• B. Gerla. Automata over MV-Algebra. In 34th IEEE International Symposium on Multiple-Valued Logic

(ISMVL 2004), 19-22 May 2004, Toronto, Canada, pages 49–54, 2004.

Ambiguity and Separation in NFA:

• H. Leung. Descriptional complexity of NFA of different ambiguity. Intern. J. Foundations Comput. Sci.,

16(5): 975984, 2005.

• B. Ravikumar and O.H. Ibarra. Relating the type of ambiguity of finite automata to the succinctness of

their representation. SIAM Journal of Computing, 18(6): 12631282, 1989.

Weighted Automata:

• M. Droste, W. Kuich, and H. Vogler. Handbook of Weighted Automata. Springer Publishing Company,

Incorporated, 1st edition, 2009.

• S. Eilenberg. Automata, Languages, and Machines. Academic Press, 1974.

MSO Logics and Automata:

• M. Droste and P. Gastin. Weighted Automata and Weighted Logics. Theor. Comput. Sci.,

380(1-2):69–86, 2007.

• S. Schwarz.

Lukasiewicz logic and weighted logics over MV-semirings. Journal of Automata, Languages and Combinatorics, 12(4):485–499, 2007.

Dual approach to Finite Automata:

• S. Aguzzoli, D. Diaconescu, T. Flaminio. A logical descriptor for regular languages via Stone duality.

Proceedings of 11th ICTAC 2014, Bucharest, Romania. LNCS 8687, pp. 25-42, 2014.