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Introduction Automata Intersection Problem Conclusion The Complexity of Intersecting Finite Automata Having Few Final States Michael Blondin 1 2 Andreas Krebs 3 Pierre McKenzie 1 1 DIRO, Universit de Montral 2 LSV, ENS Cachan 3 Universitt

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Introduction Automata Intersection Problem Conclusion

The Complexity of Intersecting Finite Automata Having Few Final States

Michael Blondin1 2 Andreas Krebs3 Pierre McKenzie1

1DIRO, Université de Montréal 2LSV, ENS Cachan 3Universität Tübingen

October 31, 2013

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Definition An automaton is a 5-tuple: Ω (finite set of states) Σ (finite alphabet) δ : Ω × Σ → Ω (transition function) α ∈ Ω (initial state) F ⊆ Ω (final states)

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Definition Transition monoid M(A) of A: {Tσ : σ ∈ Σ} where Tσ(γ) = δ(γ, σ). Example

α β γ ω

1 1 1 0, 1 T011 =

 α β γ ω

β β γ ω

 

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Definition AutoIntb(X) (Automata nonemptiness intersection problem) Input: Automata A1, . . . , Ak on alphabet Σ with M(Ai) ∈ X and at most b final states. Question:

Language(Ai) = ∅?

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Definition AutoIntb(∪mX) (Generalized automata intersection problem) Input: Automata A1,1, . . . , Ak,m

alphabet Σ with M(Ai,j) ∈ X and at most b final states. Question:

Language(Ai,j) = ∅?

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Kozen 77 AutoInt and AutoInt1 are PSPACE−complete. Galil 76 AutoInt is NP−complete when Σ = {a}.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

AutoInt interesting because generalizes: Definition Memb(X) (Membership problem) Input: g, g1, . . . , gk : [m] → [m] such that g1, . . . , gk ∈ X. Question: g ∈ g1, . . . , gk?

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

AutoInt interesting because generalizes: Definition Memb(X) (Membership problem) Input: g, g1, . . . , gk : [m] → [m] such that g1, . . . , gk ∈ X. Question: g ∈ g1, . . . , gk? Connections with graph isomorphism led to deep results on group

problems. It is known that Memb(Groups) ∈ NC.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Definition ACk: languages accepted by Boolean circuits of poly size and depth O(logk n). NCk: similar with gates of indegree 2. NC = AC =

k≥0

NCk

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Definition L: languages accepted by log-space Turing machines. NL: languages accepted by log-space non deterministic Turing machines. ModpL: languages S s.t. w ∈ S iff # accept paths ≡ 0 (mod p) for some NL machine.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Inclusion chain of complexity classes AC0 NC1 L NL ModpL NC2 NC P NP PSPACE

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Inclusion chain of complexity classes AC0 NC1 L NL ModpL NC2 NC P NP PSPACE Contains: binary addition/substraction, star-free languages. Does not contain: parity/majority. Equals: FO(BIT), FO(+, ×) where variables = positions in words.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Inclusion chain of complexity classes AC0 NC1 L NL ModpL NC2 NC P NP PSPACE Contains: binary multiplication/division, regular languages, parity/majority.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Inclusion chain of complexity classes AC0 NC1 L NL ModpL NC2 NC P NP PSPACE Complete problems: undirected connectivity, 2⊕SAT. Contains: problems defined in MSO on graphs of bounded tree-width.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Inclusion chain of complexity classes AC0 NC1 L NL ModpL NC2 NC P NP PSPACE Complete problems: directed connectivity, 2SAT, testing an automaton for emptiness. Equals: coNL.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Inclusion chain of complexity classes AC0 NC1 L NL ModpL NC2 NC P NP PSPACE Complete problems: linear algebra mod p. Equals: coModpL.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Inclusion chain of complexity classes AC0 NC1 L NL ModpL NC2 NC P NP PSPACE Contains: determinant, automata minimization.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Inclusion chain of complexity classes AC0 NC1 L NL ModpL NC2 NC P NP PSPACE Contains: membership in permutation groups.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Inclusion chain of complexity classes AC0 NC1 L NL ModpL NC2 NC P NP PSPACE Complete problems: circuit value problem, linear programming.

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Introduction Automata Intersection Problem Conclusion Definitions Motivation and Prior Work Complexity Classes Our Results

Main result: completeness results for AutoIntb(X) Maximum number of final states 1 2 1 with ∪2 3+ Σ = {a} L L NL NP Z2 × · · · × Z2 ⊕L ⊕L NP NP Zp × · · · × Zp ModpL NP NP NP Abelian groups ∈ NC3 NP NP NP Groups ∈ NC NP NP NP J1 ∈ AC0 NP NP NP * Our classification. * Beaudry 88.

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Complexity of AutoInt2(X) Maximum number of final states 1 2 1 with ∪2 3+ Σ = {a} L L NL NP Z2 × · · · × Z2 ⊕L ⊕L NP NP Zp × · · · × Zp ModpL NP NP NP Abelian groups ∈ NC3 NP NP NP Groups ∈ NC NP NP NP J1 ∈ AC0 NP NP NP

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Theorem AutoInt2(X) is hard for NP for any X beyond Z2 × · · · × Z2. Proof sketch X ⊆ Z2 × · · · × Z2 implies aperiodic monoid or cyclic group Zq, q > 2, in X. Reduction from CIRCUIT–SAT to AutoInt2(X) in both cases.

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Theorem AutoInt2(X) is hard for NP for any X beyond Z2 × · · · × Z2. Proof sketch X ⊆ Z2 × · · · × Z2 implies aperiodic monoid or cyclic group Zq, q > 2, in X. Reduction from CIRCUIT–SAT to AutoInt2(X) in both cases.

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) Given a circuit, we let Σ be the set of gates.

∧ ¬ ∨

Σ = {◦0, ◦1, ◦2, ∧0, ¬0, ∨0, ◦3}

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) Given a circuit, we let Σ be the set of gates.

∧ ¬ ∨

Σ = {◦0, ◦1, ◦2, ∧0, ¬0, ∨0, ◦3}

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) Given a circuit, we let Σ be the set of gates.

∧ ¬ ¬ ¬ ∧ ¬

Σ = {◦0, ◦1, ◦2, ∧0, ¬0, ¬1, ¬2, ∧1, ◦3}

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) For each gate σ, we build automata A such that M(A) = Zq. Strategy: Occurrences of σ mod q encode assignment to σ (0 or 1), Automata verify soundness locally, Intersection represents satisfying assignments.

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) For each σ ∈ Σ, we accept words w such that |w|σ ≡ 0, 1 (mod q). σ σ σ σ

Σ \ {σ} Σ \ {σ} Σ \ {σ} Σ \ {σ} Σ \ {σ}

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) For output gate σ, we accept words w such that |w|σ ≡ 1 (mod q). σ σ σ σ

Σ \ {σ} Σ \ {σ} Σ \ {σ} Σ \ {σ} Σ \ {σ}

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) For each ¬-gate σ with input σ′, we accept words w such that |w|σ + |w|σ′ ≡ 1 (mod q). σ, σ′ σ, σ′ σ, σ′ σ, σ′

Σ \ {σ, σ′} Σ \ {σ, σ′} Σ \ {σ, σ′} Σ \ {σ, σ′} Σ \ {σ, σ′}

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) For each ∧-gate σ with inputs σ′, σ′′, we accept words w such that |w|σ′ + |w|σ′′ − 2 |w|σ ≡ 0, 1 (mod q). σ′σ′′σ σ′ ∧ σ′′ σ′ + σ′′ − 2σ 000 1 001

010 1 1 011

100 1 1 101

110 2 111 1

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) For each ∧-gate σ with inputs σ′, σ′′, we accept words w such that |w|σ′ + |w|σ′′ − 2 |w|σ ≡ 0, 1 (mod q). σ′σ′′σ σ′ ∧ σ′′ = σ σ′ + σ′′ − 2σ ≡ 0, 1 000 ✓ ✓ 001 ✗ ✗ 010 ✓ ✓ 011 ✗ ✗ 100 ✓ ✓ 101 ✗ ✗ 110 ✗ ✗ 111 ✓ ✓

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) Problem when q = 3 since −2 ≡ 1 (mod 3). σ′σ′′σ σ′ ∧ σ′′ = σ σ′ + σ′′ − 2σ ≡ 0, 1 000 ✓ ✓ 001

010 ✓ ✓ 011 ✗ ✗ 100 ✓ ✓ 101 ✗ ✗ 110 ✗ ✗ 111 ✓ ✓

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) Problem when q = 3 since −2 ≡ 1 (mod 3). σ′σ′′σ σ′ ∧ σ′′ = σ σ′ + σ′′ − 2σ ≡ 0, 1 000 ✓ ✓ 001 ✗ ✓ 010 ✓ ✓ 011 ✗ ✗ 100 ✓ ✓ 101 ✗ ✗ 110 ✗ ✗ 111 ✓ ✓

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) When q = 3, we also build |w|σ′ + |w|σ′′ − |w|σ ≡ 0, 1 (mod 3). σ′σ′′σ σ′ ∧ σ′′ σ′ + σ′′ − 2σ σ′ + σ′′ − σ 000 1 001 1 2 010 1 1 011 2 100 1 1 1 101 2 110 2 2 111 1 1

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) When q = 3, we also build |w|σ′ + |w|σ′′ − |w|σ ≡ 0, 1 (mod 3). σ′σ′′σ σ′ ∧ σ′′ = σ σ′ + σ′′ − 2σ ≡ 0, 1 σ′ + σ′′ − σ ≡ 0, 1 000 ✓ ✓ ✓ 001 ✗ ✓ ✗ 010 ✓ ✓ ✓ 011 ✗ ✗ ✓ 100 ✓ ✓ ✓ 101 ✗ ✗ ✓ 110 ✗ ✗ ✗ 111 ✓ ✓ ✓

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Proof sketch: CIRCUIT–SAT reduces to AutoInt2(Zq) ⇒) A satisfying assignment yields a word σb1

1 · · · σbs s

accepted by the automata. ⇐) A word w accepted by the intersection yields a sastisfying assignment σi ← |w|σi mod q.

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Complexity of AutoInt1(Abelian groups) Maximum number of final states 1 2 1 with ∪2 3+ Σ = {a} L L NL NP Z2 × · · · × Z2 ⊕L ⊕L NP NP Zp × · · · × Zp ModpL NP NP NP Abelian groups ∈ NC3 NP NP NP Groups ∈ NC NP NP NP J1 ∈ AC0 NP NP NP

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Definition Let A = (Ω, {σ1, . . . , σs}, δ, α, F) be an abelian group automaton. We define ΦA as:

v ∈ Zs

q : δ(α, σv1 1 · · · σvs s ) = α

where q = lcm(ord(Tσ1), . . . , ord(Tσs)).

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Definition Let UA be a matrix such that its rows are a generating set for ΦA. Let U⊥

A be a matrix such that U⊥ A UT A ≡ 0 (mod q).

Lemma Let x, y ∈ Ns and w = σx1

1 · · · σxs s and w′ = σy1 1 · · · σys s , then

U⊥

A x ≡ U⊥ A y (mod q) ⇔ Tw(α) = Tw′(α).

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Theorem AutoInt1(Abelian groups) ∈ NC3. Proof sketch. Let A1, . . . , Ak be the given automata. We Compute UAi and U⊥

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Theorem AutoInt1(Abelian groups) ∈ NC3. Proof sketch. Let A1, . . . , Ak be the given automata. We Compute UAi and U⊥

Compute wi ∈ Σ∗ such that Twi(αi) = βi

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Theorem AutoInt1(Abelian groups) ∈ NC3. Proof sketch. Let A1, . . . , Ak be the given automata. We Compute UAi and U⊥

Compute wi ∈ Σ∗ such that Twi(αi) = βi Verify ∃x, ∀i ∈ [k], such that U⊥

Aix ≡ U⊥ Ai

  

|wi|σ1 . . . |wi|σs

   (mod qi).

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Complexity of AutoInt2(Z2 × · · · × Z2) Maximum number of final states 1 2 1 with ∪2 3+ Σ = {a} L L NL NP Z2 × · · · × Z2 ⊕L ⊕L NP NP Zp × · · · × Zp ModpL NP NP NP Abelian groups ∈ NC3 NP NP NP Groups ∈ NC NP NP NP J1 ∈ AC0 NP NP NP

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Hint for AutoInt2(Z2 × · · · × Z2) ∈ ⊕L We can remove ∨ of such a system: Bx ≡ b (mod 2) ∨ Bx ≡ b′ (mod 2) by introducing two variables

     

· · · 1 1 B1,1 · · · B1,s b1 b′

. . . ... . . . . . . . . . Bm,1 · · · Bm,s bm b′

             

x1 . . . xm y y ′

       

≡

     

1 . . .

     

(mod 2)

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Introduction Automata Intersection Problem Conclusion AutoInt2(X) is NP−complete AutoInt1(Abelian groups) ∈ NC3 AutoInt2(Z2 × · · · × Z2) is ⊕L−complete

Gap from AutoInt2(Z2) to AutoInt2(Zq) Maximum number of final states 1 2 1 with ∪2 3+ Σ = {a} L L NL NP Z2 × · · · × Z2 ⊕L ⊕L NP NP Zp × · · · × Zp ModpL NP NP NP Abelian groups ∈ NC3 NP NP NP Groups ∈ NC NP NP NP J1 ∈ AC0 NP NP NP

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Introduction Automata Intersection Problem Conclusion

Relationships between algebraic problems and AutoIntb(X) Extensive classification of AutoIntb Close relationship between complexity of Memb and AutoInt1 Surprising gap from AutoInt2(Z2) to AutoInt2(Z3) What is the complexity of AutoInt1(X) for other X such that Memb(X) is in between P and NP?

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Introduction Automata Intersection Problem Conclusion

Thank you! Merci! Danke!

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