Input-driven Pushdown Automata Research Articles

On the power of pushing or stationary moves for input-driven pushdown automata

Input-driven pushdown automata (IDPDAs) are pushdown automata where the next action on the pushdown store (push, pop, nothing) is solely governed by the input symbol. Nowadays such devices are usually defined such that every push operation pushes exactly one additional symbol on the pushdown store and, in addition, stationary moves are not allowed so that the devices work in real time. Here, we relax this strong definition and consider IDPDAs that may push more than one symbol in one step (push-IDPDA) or may perform stationary moves (stat-IDPDA). We study the computational power of the extended variants both in the deterministic and nondeterministic case, we investigate several decidability questions for the new automata classes, and we obtain interesting representations by inverse homomorphisms. Namely, every (1) deterministic, (2) real-time deterministic, and (3) nondeterministic context-free language can be characterized as the inverse homomorphic image of a language accepted by a (1) stat-IDPDA, (2) push-IDPDA, and (3) nondeterministic push-IDPDA.

Digging input-driven pushdown automata

Input-driven pushdown automata (IDPDA) are pushdown automata where the next action on the pushdown store (push, pop, nothing) is solely governed by the input symbol. Nowadays such devices are usually defined such that popping from the empty pushdown does not block the computation but continues it with empty pushdown. Here, we consider IDPDAs that have a more balanced behavior concerning pushing and popping. Digging input-driven pushdown automata (DIDPDA) are basically IDPDAs that, when forced to pop from the empty pushdown, dig a hole of the shape of the popped symbol in the bottom of the pushdown. Popping further symbols from a pushdown having a hole at the bottom deepens the current hole furthermore. The hole can only be filled up by pushing symbols previously popped. We study the impact of the new behavior of DIDPDAs on their power and compare their capacities with the capacities of ordinary IDPDAs and tinput-driven pushdown automata which are basically IDPDAs whose input may be preprocessed by length-preserving finite state transducers. It turns out that the capabilities are incomparable. We address the determinization of DIDPDAs and their descriptional complexity, closure properties, and decidability questions.

State Complexity of the Quotient Operation on Input-Driven Pushdown Automata

The quotient of a formal language [Formula: see text] by another language [Formula: see text] is the set of all strings obtained by taking a string from [Formula: see text] that ends with a suffix of a string from [Formula: see text], and removing that suffix. The quotient of a regular language by any language is always regular, whereas the context-free languages and many of their subfamilies, such as the linear and the deterministic languages, are not closed under the quotient operation. This paper establishes the closure of the family of languages recognized by input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the quotient operation. A construction of automata representing the result of the operation is given, and its state complexity with respect to nondeterministic IDPDA is shown to be exactly [Formula: see text], where [Formula: see text] and [Formula: see text] are the numbers of states in the automata recognizing [Formula: see text] and [Formula: see text], respectively.

Further closure properties of input-driven pushdown automata

The paper investigates the closure of the language family defined by input-driven pushdown automata (IDPDA) under the following operations: insertion ins(L,K)={xyz|xz∈L,y∈K}, deletion del(L,K)={xz|xyz∈L,y∈K}, square root L={w|ww∈L}, the first half 12L={u|∃v:|u|=|v|,uv∈L} and cyclic shift ▪. For K and L recognized by nondeterministic IDPDA, with m and with n states, respectively, insertion requires exactly mn+2m states, as long as K is well-nested; deletion requires exactly 2n states, for well-nested K; square root requires n3−O(n2) states, for well-nested L; the well-nested subset of the first half is representable with 2O(n2) states; the well-nested subset of the cyclic shift requires exactly 2n2 states. Without the well-nestedness constraints, non-closure is established in each case.

Edit distance neighbourhoods of input-driven pushdown automata

Edit distance ℓ-neighbourhood of a formal language is the set of all strings that can be transformed to one of the strings in this language by at most ℓ insertions and deletions. Both the regular and the context-free languages are known to be closed under this operation, whereas the family recognized by deterministic pushdown automata is not. This paper establishes the closure of the family recognized by input-driven pushdown automata (IDPDA), also known as visibly pushdown automata, under the edit distance neighbourhood operation. For an n-state nondeterministic IDPDA with n stack symbols, an automaton for its edit distance ℓ-neighbourhood using O(nℓ+1) states is constructed, and an asymptotically matching lower bound is established. For an n-state deterministic IDPDA, its 1-neighbourhood in the worst case requires a deterministic IDPDA with at least 2Ω(n2) states.

Input-Driven Double-Head Pushdown Automata

We introduce and study input-driven deterministic and nondeterministic double-head pushdown automata. A double-head pushdown automaton is a slight generalization of an ordinary pushdown automaton working with two input heads that move in opposite directions on the common input tape. In every step one head is moved and the automaton decides on acceptance if the heads meet. Demanding the automaton to work input-driven it is required that every input symbol uniquely defines the action on the pushdown store (push, pop, state change). Normally this is modeled by a partition of the input alphabet and is called a signature. Since our automaton model works with two heads either both heads respect the same signature or each head owes its own signature. This results in two variants of input-driven double-head pushdown automata. The induced language families on input-driven double-head pushdown automata are studied from the perspectives of their language describing capability, their closure properties, and decision problems.

When input-driven pushdown automata meet reversiblity

We investigate subfamilies of context-free languages that share two important properties. The languages are accepted by input-driven pushdown automata as well as by a reversible pushdown automata. So, the languages are input driven and reversible at the same time. This intersection can be defined on the underlying language families or on the underlying machine classes. It turns out that the latter class is properly included in the former. The relationships between the language families obtained in this way and to reversible context-free languages as well as to input-driven languages are studied. In general, a hierarchical inclusion structure within the real-time deterministic context-free languages is obtained. Finally, the closure properties of these families under the standard operations are investigated and it turns out that all language families introduced are anti-AFLs.

Input-driven languages are linear conjunctive

Linear conjunctive grammars define the same family of languages as one-way real-time cellular automata (A. Okhotin, “On the equivalence of linear conjunctive grammars to trellis automata”, RAIRO ITA, 2004), and this family is known to be incomparable with the context-free languages (V. Terrier, “On real-time one-way cellular array”, Theoret. Comput. Sci., 1995). This paper demonstrates the containment of the languages accepted by input-driven pushdown automata (a.k.a. visibly pushdown automata) in the family of linear conjunctive languages, which is established by a direct simulation of an input-driven automaton by a one-way real-time cellular automaton. On the other hand, it is shown that the language families defined by the unambiguous grammars, the LR(k) grammars and the LL(k) grammars are incomparable with the linear conjunctive languages.

Complexity of input-driven pushdown automata

Complexity of input-driven pushdown automata

On the Expressive Power of 2-Stack Visibly Pushdown Automata

Visibly pushdown automata are input-driven pushdown automata that recognize some non-regular context-free languages while preserving the nice closure and decidability properties of finite automata. Visibly pushdown automata with multiple stacks have been considered recently by La Torre, Madhusudan, and Parlato, who exploit the concept of visibility further to obtain a rich automata class that can even express properties beyond the class of context-free languages. At the same time, their automata are closed under boolean operations, have a decidable emptiness and inclusion problem, and enjoy a logical characterization in terms of a monadic second-order logic over words with an additional nesting structure. These results require a restricted version of visibly pushdown automata with multiple stacks whose behavior can be split up into a fixed number of phases. In this paper, we consider 2-stack visibly pushdown automata (i.e., visibly pushdown automata with two stacks) in their unrestricted form. We show that they are expressively equivalent to the existential fragment of monadic second-order logic. Furthermore, it turns out that monadic second-order quantifier alternation forms an infinite hierarchy wrt words with multiple nestings. Combining these results, we conclude that 2-stack visibly pushdown automata are not closed under complementation. Finally, we discuss the expressive power of B\"{u}chi 2-stack visibly pushdown automata running on infinite (nested) words. Extending the logic by an infinity quantifier, we can likewise establish equivalence to existential monadic second-order logic.