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Minimal NFA Problems: Understanding Complexity in Automata Theory

Explore the complexity of Non-deterministic Finite Automata (NFA) problems, focusing on the Minimal NFA problem and its NP-completeness results. Dive into decision problems for NFAs that pose computational challenges and discover the intricacies of Unambiguous Finite Automata (UFA). Delve into algorithms, hardness results, and the properties of UFAs, along with their connections to Deterministic Finite Automata (DFA). Unravel the relationships between UFAs and NFAs, and explore the NP-completeness of the DFA UFA problem. Understand the intricacies of state transition sequences, accepting states, and the equivalence of UFAs. Learn about algorithms for determining equivalence between UFAs, ambiguity in NFAs, and the complexity of DFA UFA in the realm of automata theory.

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Minimal NFA Problems: Understanding Complexity in Automata Theory

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  1. Minimal NFA Problems are hardTao Jiang & B. Ravikumar N Andres Parra

  2. Quick Outline • Presentation of the problem • Introduction • UFA: Unambiguous Finite Automata • NP-completeness results

  3. A B Problem: Given a FA of type A, find a minimum equivalent FA of type B DFA DFA NFA DFA NFA NFA DFA NFA ?

  4. Introduction • Several decision problems for NFA are computationally hard. emptiness and finiteness are exceptions. • Very few hardness results involving DFA have been shown

  5. Minimal NFA problem: • Input: FA M of type-A, and integer k • Is there a k-state machine of type-B that accepts L(M)? O(nlogn) time -Hopcroft’s Algorithm- Exponential lower bound in time and space. PSPACE-hard. NP-hard for unary alphabet DFA DFA NFA DFA

  6. Unambiguous FA (UFA) • A UFA is an NFA in which every accepted string has a unique accepting computation a, b a a, b 3-state UFA for *a with  = { a, b} • UFA minimization in NP-complete

  7. Example of NFA and UFA for same L a a a a a a a a a a a 5-state NFA for { ai | i mod 6  0 } 6-state UFA for { ai | i mod 6  0 }

  8. Properties of UFA • TRAN-SEQM(q,k), qQ, k N: Number of state transition sequences of lenght k which take state q into an accepting state. • ACC-SEQM(k), k N: Number of accepting states transition sequences of length k. • ACCM(k), k N: Number of strings of length k accepted by M

  9. Properties of UFA • TRAN-SEQM(q,0) = 1 if q F 0 otherwise • TRAN-SEQM(q,k+1) =

  10. DFA UFA is in NP • There is a deterministic polynomial-time algorithm for deciding whether the two given UFA, M1 and M2 are equivalent • Using proper containment based on difference equations of the ACC functions. • There is a polynomial-time algorithm that, given an NFA M as input, decides whether M is unambiguous • Build an NFA M’ that accepts L’, where L’ is the set of strings that can be derived by at least two different accepting paths. Can be done in polynomial time • M is unambiguous iff L(M’) is empty.

  11. DFA UFA is NP-complete • The vertex Cover problem is reduced to the normal set basis problem. • The Normal set basis problem is reduced to the DFA UFA problem. • Let C and B be collections of sets. B is said to be a normal basis of C if for each c C there is a pairwise disjoint subcollection of B whose union is exactly c.

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