Deeper Connections Between LTL and Alternating Automata

Abstract

Abstract. It is known that Linear Temporal Logic (LTL) has the same expressive power as alternating 1-weak automata (A1W automata, also called alternating linear automata or very weak alternating automata). A translation of LTL formulae into a language equivalent A1W automata has been introduced in [1]. The inverse translation has been developed independently in [2] and [3]. In the first part of the paper we show that the latter translation wastes temporal operators and we propose some improvements of this translation. The second part of the paper draws a direct connection between fragments of the Until-Release hierarchy [4] and alternation depth of nonaccepting and accepting states in A1W automata. We also indicate some corollaries and applications of these results.

An Efficient Normalisation Procedure for Linear Temporal Logic and Very Weak Alternating Automata

“…The argumentation is only slightly more complicated than that of the ψ 1 Uψ 2 case. By induction hypothesis (16) and (17) hold. With the help of Lemma 20 we derive:…”

“…safety ∪ co-safety = AWW G [1] Figure 4. Expressive power of AWWs after Gurumurthy et al [6], and of A1Ws after Pelánek and Strejcek [16].…”

“…Let AWW G [k] denote the class of AWW with at most (k − 1) alternations defined in [6]. Similarly, let A1W PS [k, A] and A1W PS [k, R] denote the classes of A1W with at most (k − 1) alternations and accepting or non-accepting initial state, respectively, defined in [16]. Finally, define A1W Figure 4 shows the results of [6] and [16].…”

“…Similarly, let A1W PS [k, A] and A1W PS [k, R] denote the classes of A1W with at most (k − 1) alternations and accepting or non-accepting initial state, respectively, defined in [16]. Finally, define A1W Figure 4 shows the results of [6] and [16]. We abuse language, and, for example, write Π 2 = A1W PS [2, A] to denote that the class of languages satisfying formulas in Π 2 and the class of languages recognized by automata in A1W PS [2, A] coincide.…”

LTL Semantic Tableaux and Alternating $$\omega $$ ω -automata via Linear Factors

“…The same definition is given by Pelánek and Strejček [8] who note that the resulting automaton is restricted to be a 1-weak alternating automaton. For this class of automata there is a translation back to LTL. We observe that the definition of the transition function in Definition 16 corresponds to the direct definition of partial derivatives in Definition 13.…”