Automata, Languages e Computation: Definizioni e Proprietà - Prof. Satta, Schemi e mappe concettuali di Tecniche E Linguaggi Di Programmazione

Scarica Automata, Languages e Computation: Definizioni e Proprietà - Prof. Satta e più Schemi e mappe concettuali in PDF di Tecniche E Linguaggi Di Programmazione solo su Docsity! Automata, Languages and Computation 1 🤖 Automata, Languages and Computation Definitions: DFA Deterministic Finite Automata): : finite set of states : input symbols alphabet : transition function : initial state : set of final states NFA Nondeterministic Finite Automata): : transition function , where is the set of all subsets of , , , as above NFA Nondeterministic Finite Automata with -transitions): : transition function , where is the set of all subsets of , , , as above CFG Context-Free Grammar): A = (Q, Σ, δ, q ,F)0 Q Σ δ Q× Σ → Q q ∈0 Q F ⊆ Q A = (Q, Σ, δ, q ,F)0 δ Q× Σ → 2Q 2Q Q Q Σ q0 F ε ε A = (Q, Σ, δ, q ,F)0 δ Q× (Σ ∪ {ε}) → 2Q 2Q Q Q Σ q0 F Automata, Languages and Computation 2 : variables (nonterminals) : terminal symbols (language alphabet) : productions having form where is a variable and is a string in : initial symbol PDA Push-Down Automaton): : states : input alphabet : stack alphabet : transition function : initial state : initial stack symbol : set of final states MT Turing Machine): : states : input alphabet : tape symbols ( ) : transition function : initial state : blank symbol ( ) : set of final states G = (V ,T ,P ,S) V T P A → α A α (V ∪ T )∗ S ∈ V P = (Q, Σ, Γ, δ, q ,Z ,F)0 0 Q Σ Γ δ Q× Σ ∪ {ε} × Γ → 2Q×Γ∗ q ∈0 Q Z ∈0 Γ F ⊆ Q M = (Q, Σ, Γ, δ, q ,B,F)0 Q Σ Γ Σ ⊆ Γ δ Q× Γ → Q× Γ × { , } Left L Right R q0 B ∈ Γ B ∈ Σ F ⊆ Q Automata, Languages and Computation 5 Union Concatenation Kleene star ( ) and positive closure ( Homomorphism Reverse Sources of reduction in TMs: (diagonalization) (universal) (halting problem) (not empty) (empty) PCP is undecidable In CFG environment the following problems are undecidable: , with a regex , with a regex , for a fixed alphabet Possible arrangements for and : Both and Both and and (and vice versa) Rice theorem Every property of languages that is not trivial is undecidable. Trivial A property satisfied by all or none of the languages. ∗ +) L d ∈ RE L ∈u RE ∖ REC L ∈h RE ∖ REC L ∈ne RE ∖ REC L e ∈ RE L(G ) ∩1 L(G ) =2 ∅ ? L(G ) =1 L(G ) ?2 L(G ) ⊆1 L(G ) ?2 L(R) ⊆ L(G ) ?1 R L(G ) =1 L(R) ? R L(G ) =1 T  ?∗ T L L̄ L L̄ ∈ REC L L̄ ∈ RE L ∈ RE ∖ REC L̄ ∈ RE P RE RE Automata, Languages and Computation 6 Decidable Property if is recursive. NP algorithms: Hamiltonian circuit Traveling salesman problem P algorithms: Spanning tree Minimum weight spanning tree NP-complete reductions graph: (cyclic since all NP-complete → all equally difficult) PL L