Translation from Classical Two-Way Automata to Pebble Two-Way Automata

Abstract

We study the relation between the standard two-way automata and more powerful devices, namely, two-way finite automata with an additional "pebble" movable along the input tape. Similarly as in the case of the classical two-way machines, it is not known whether there exists a polynomial trade-off, in the number of states, between the nondeterministic and deterministic pebble two-way automata. However, we show that these two machine models are not independent: if there exists a polynomial trade-off for the classical two-way automata, then there must also exist a polynomial trade-off for the pebble two-way automata. Thus, we have an upward collapse (or a downward separation) from the classical two-way automata to more powerful pebble automata, still staying within the class of regular languages. The same upward collapse holds for complementation of nondeterministic two-way machines. These results are obtained by showing that each pebble machine can be, by using suitable inputs, simulated by a classical two-way automaton with a linear number of states (and vice versa), despite the existing exponential blow-up between the classical and pebble two-way machines

On Generating Hierarchical Workflow Nets and their Extensions and Verifying Hierarchicality

“…Still, gaining a full understanding of two-way automata, with or without pebbles, remains a challenging topic (see, e.g., [16]). The classical theory of (one-way) finite-state automata has benefited from a rich algebraic language theory that led, and still leads, to many decision algorithms [41].…”

Two-way pebble transducers for partial functions and their composition

“…One direction of the characterization was shown for k = 1 in the proof of Lemma 1 of [15]. For arbitrary k the characterization was presented for transducers with empty output alphabet, i.e., for automata, in [20] (and rediscovered by this author).…”

“…The constructions involved in this theorem, for the case that the output alphabet is empty, are used in [20] to reduce the descriptional complexity of two-way k-pebble automata to that of two-way (0-pebble) automata: if nondeterministic two-way automata can be simulated by deterministic two-way automata with a polynomial number of states (which is not known, cf. [29]), then the same is true for k-pebble automata.…”

“…Let us now define these specific k-counting transductions. There is one for each input alphabet Σ, called peb k,Σ ; it is the mapping P k in [20].…”

“…Our proofs are based on a reduction of pebble transducers to two-way finite state transducers (without pebbles), which is a straightforward generalization of the reduction of pebble automata to two-way finite state automata in [20]. This allows us to use the well-known facts that a partial function computed by a nondeterministic two-way finite state transducer can also be computed by a deterministic one (see, e.g., Theorem 22 of [12]), and that the deterministic two-way finite state transductions are closed under composition ( [6]).…”