Structurally Unambiguous Finite Automata Hing Leung Department of - - PowerPoint PPT Presentation

Structurally Unambiguous Finite Automata Hing Leung Department of Computer Science New Mexico State University Las Cruces, NM 88003, U.S.A. hleung@cs.nmsu.edu

NFA of different ambiguity

◮ Definitions of ambiguity. Given an NFA M.

◮ amb(w) is the number of accepting paths for w. ◮ M is k-ambiguous if ∀w, amb(w) ≤ k. ◮ M is unambiguous (UFA) if M is 1-ambiguous. ◮ M is finitely ambiguous (FNA) if M is k-ambiguous for some k. ◮ M is poly. amb. (PNA) if ∃ poly. p ∀w, amb(w) ≤ p(|w|). ◮ M is exp. amb. (ENA) if ∃ exp. func. f ∀w, amb(w) ≤ f (|w|). ◮ M is strictly exp. amb. if M is exp. amb. but not a PNA. ◮ M is strictly poly. amb. if M is exp. amb. but not a FNA.

◮ MDFA (DFA with multiple initial states) is a FNA

SUFA (structurally unambiguous finite automata)

◮ Def: ∀q ∀x ∃ ≤ 1 path from q0 (start state) to q processing x. ◮ defined in terms of the transition logic of the finite automaton ◮ definition independent of the choice of final states. ◮ UFA and reversal of MDFA are SUFA.

Generalized Structurally Unambiguous Finite Automata (GSUFA)

◮ Definition

◮ Allow more than one starting states. ◮ ∀q, q′, ∀w, there is ≤ 1 path from q to q′ processing w.

◮ SUFA and MDFA are special cases of GSUFA. ◮ Simulate a GSUFA by a SUFA

◮ Let (Q, Σ, δ, S, F) be a GSUFA where S = {s1, . . . , sk}. ◮ Define Mi = (Q, Σ, δ, si, F), 1 ≤ i ≤ k. ◮ Let M to be a nondeterministic union of Mi, 1 ≤ i ≤ k. ◮ Introduce a new start state with ǫ transitions going to Mi’s. ◮ Substitute ǫ moves with direct non-ǫ moves. ◮ Obtain a O(n2)-state SUFA equivalent to the given GSUFA.

◮ Polynomial time for checking if a given NFA is a GSUFA.

◮ Let the given NFA M be (Q = {q1, . . . , qn}, Σ, δ, S, F) ◮ M is a GSUFA iff ∀1 ≤ i, j ≤ n, Mi,j = (Q, Σ, δ, qi, qj) is a UFA ◮ (Stearns and Hunt) Poly time decidable if a given NFA is a UFA.

◮ Polynomial time for checking if a given NFA is a SUFA.

Equivalence and containment problems for SUFA

◮ Stearns and Hunt

◮ Problems for UFA are decidable in poly time. ◮ Problems for k-ambiguous NFA are decidable in poly time.

◮ Equivalence problem for SUFA is PSPACE-complete

◮ DFA intersection-emptyness problem is PSPACE-complete ◮ DFA union-universe problem is PSPACE-complete. ◮ Given an instance of DFA union-universe problem (D1, . . . , Dk) ◮ Poly time reduce to an equivalent MDFA, then to a SUFA. ◮ Ask if SUFA constructed is equivalent to a SUFA denoting Σ∗ ◮ Thus, SUFA equivalence problem is PSPACE-hard. ◮ NFA equivalence is in PSPACE. So is SUFA equivalence.

◮ Containment problem for SUFA is PSPACE-complete

◮ PSPACE-hard: reduce from equivalence problem. ◮ PSPACE: NPSPACE to test for non-containment.

◮ Problems for SUFA with k final states: poly time decidable

Language of some-register-on L

◮ Definition

◮ There are n registers. ◮ Each register holds a value of either 0 or 1 (‘off’ or ‘on’). ◮ Initially, register 1 is on. All other registers are off. ◮ Symbol Ci,j (Meaning: Copy i to j) ◮ An input string consists of a sequence of copy instructions. ◮ An input is in L if some register is on

after the sequence of copy instructions have been performed.

◮ C1,4C4,2C3,1C4,3C1,2 ∈ L

C1,4C4,2C3,2C3,1C2,4 ∈ L

◮ n-state SUFA A for L

◮ Let the state set Q = {1, 2, . . . , n}. ◮ State i is on iff register i is on. ◮ Start state is 1. All states are final. ◮ Transitions: (i, Ci,j, j) and (k, Ci,j, k) if k = j ◮ Nondeterministic move: (i, Ci,j, j) and (i, Ci,j, i) ◮ Reversal is a MDFA.

Transitions: (j, Ci,j, i) and (k, Ci,j, k) if k = j

Smallest UFA for L has 2n − 1 states, where n ≥ 3.

◮ (Schmidt) Given a regular language R and strings xi, yi for

i = 1, 2, . . . , n, let M be a matrix s.t. M[xi, yj] = 1 if xiyj ∈ R and 0 otherwise. Then any UFA for R has ≥ rank(M) states.

◮ Define xQ′ s.t. A reaches Q′ = ∅ on processing xQ′.

◮ xQ′ = C1,q1 . . . C1,qk where 1 ∈ Q′ = {q1, . . . , qk}. ◮ xQ′ = C1,q1Cq,1Cq1,q2 Cq1,q3 . . . Cq1,qk where q ∈ Q − {1, q1}.

◮ Define yQ′′ s.t. AR reaches Q′′ = ∅ on processing yR Q′′.

◮ y R

Q′′ = ǫ if Q′′ = Q.

◮ y R

Q′′ = Cq′

2,q′ 1Cq′ 3,q′ 2 . . . Cq′ h,q′ h−1Cq1,q′ h

where Q − Q′′ = {q′

1, q′ 2, . . . , q′ h}. and q1 ∈ Q′′.

◮ Thus, xQ′yQ′′ ∈ L iff Q′ ∩ Q′′ = ∅. ◮ rank(M) = 2n − 1. Smallest UFA for L has ≥ 2n − 1 states. ◮ Incompletely specificied DFA for L has ≤ 2n − 1 states. ◮ Thus, smallest UFA for L has 2n − 1 states.

The smallest MDFA for L has 2n − 1 states.

◮ Properties of L

◮ All prefixes of L are in L. ◮ S = {w | w −1L = Σ∗} is nonempty. ◮ T = {w | w −1L = ∅ } is nonempty. ◮ Smallest incompletely specified DFA D for L has 2n − 1 states. ◮ One accepting sink state in D. (reached by processing w ∈ S) ◮ ∀u ∈ Σ∗ − S − T, ∃v s.t. (uv)−1L = L ◮ Other 2n − 2 states in D are strongly conncected.

◮ Smallest MDFA for L has ≥ 2n − 2 states.

◮ Suppose the contrary. ∃ k-entry MDFA with < 2n − 2 states. ◮ Treat MDFA as a nondeterministic union of k DFA D1, . . . , Dk. ◮ We can assume every state in Di is a final state. ◮ Aim: design a string t ∈ Σ∗ − S − T such that MDFA aborts. ◮ Design w1 s.t. D1 aborts in processing w1. ◮ Design w2 (may be ǫ) s.t. D2 aborts in processing w1w2. ◮ Define t = w1w2 . . . wk.

◮ Need 2n − 2 states to handle w ∈ Σ∗ − S − T correctly. ◮ Needs one more accepting sink state for processing w ∈ S.

UFA, MDFA and reversal of MDFA are incomparable.

◮ Consider SUFA A, which is the reversal of a MDFA.

◮ reversal of MDFA exponentially more succinct than UFA ◮ reversal of MDFA exponentially more succinct than MDFA

◮ Let L′ = LR. There exists an n-state MDFA for L′

◮ Recall: L requires 2n − 1 states for MDFA to accept ◮ L′ requires 2n − 1 states for the reversal of a MDFA to accept ◮ MDFA exponentially more succinct than reversal of MDFA

◮ Modifying Veloso and Gill’s design, Leung showed

◮ MDFA exponentially more succinct than UFA ◮ n states 0, 1, . . . , n − 1. Smallest UFA has 2n − 1 states. ◮ All states are starting states. States 0, 2, 4, 6, ... are final. ◮ δ(0, 0) = 1, δ(i, 0) = i for 0 < i < n, δ(j, 1) = (j + 1) mod n

◮ Modify L. Modify SUFA A to obtain a UFA (reversal of DFA).

◮ Append a query Qi (Assert:

Register i is on)

◮ Add a new and only final state f . New transitions: (i, Qi, f ) ◮ Equivalent MDFA has ≥ 2n − 1 states. Otherwise,. . . ◮ UFA exponentially more succinct than MDFA ◮ DFA/UFA exponentially more succinct than reversal of MDFA

SUFA simultaneously exponentially more succinct than UFA, MDFA and reversal of MDFA.

◮ Define L′ = LR · b · L where b is a new symbol. ◮ Construct B of 2n states for L′ by connecting AR with A. ◮ replicate n copies of B, each with a different starting state. ◮ nondeterministic union is a SUFA of O(n2) states. ◮ Smallest MDFA for L′ has at least 2n − 1 states.

◮ Suppose the contrary. ∃ a MDFA N for L′ of < 2n − 1 states. ◮ Remove all transitions encountered before N processes b. ◮ Define starting states to be {q | (q′, a, q) is a transition of N}. ◮ Constructed a MDFA of < 2n − 1 states for L. Contradiction.

◮ Smallest reversal of MDFA for L′ has at least 2n − 1 states.

◮ Follows from previous result as (L′)R = L′.

◮ Smallest UFA for L′ has at least 2n − 1 states.

◮ Recall xQ′ and yQ′′ as defined before. ◮ Apply Schmidt’s technique, take x′

Q′ = bxQ′ and y ′ Q′′ = yQ′′.

Why SUFA?

◮ Efficiently implemented on sync. model of parallel computers

◮ one process thread is assigned to each state. ◮ when state is off, corresponding thread waiting to be woken up. ◮ no two messages sending to a thread at the same time. ◮ no buffer needed to hold incoming messages in sync. comp.

◮ Equivalent structural definitions for ENA, PNA and FNA

◮ M is str. ENA iff ∃q, w s.t. ∃ ≥ 2 paths from q to q proc. w ◮ M is str. PNA iff M not str. ENA and p

w

→ p, p

w

→ q, q

w

→ q

◮ M is FNA iff M is not str. ENA and not str. PNA.

◮ UFA is a semantic definition. SUFA is a structural definition. ◮ SUFA, FNA, PNA, ENA: a hierachy based on the struct. def. ◮ UFA, MDFA, Rev MDFA are incomparable w.r.t. desc. comp. ◮ UFA, MDFA, Rev MDFA contained in (separated from) SUFA

Any natural-looking SUFA that is not UFA, MDFA or Rev MDFA?