Pushdown Stacks Research Articles

The emptiness problem for valence automata over graph monoids

This work studies which storage mechanisms in automata permit decidability of the emptiness problem. The question is formalized using valence automata over graph monoids, an abstract model of automata in which the storage mechanism is specified by a finite graph. In this framework, many important storage mechanisms can be realized. Examples include pushdown stacks, partially blind counters (which behave like Petri net places), blind counters (which may attain negative values), and combinations thereof.We study for which graphs the emptiness problem for valence automata is decidable. A particular model in our framework is that of Petri nets with a pushdown stack. For these, decidability is a long-standing open question and we do not answer it here.However, if one excludes subgraphs corresponding to this model, a characterization can be achieved. Moreover, we provide a description of those storage mechanisms for which decidability remains open. This leads to a model that naturally generalizes both (i) pushdown Petri nets and (ii) another model with high expressiveness: priority multicounter machines introduced by Reinhardt.The cases that are proven decidable constitute a natural and apparently new extension of Petri nets with decidable reachability. We finally present a further decidable generalization that also subsumes a decidable Petri net extension by Atig and Ganty.

"Process and perish" or multiple buffers with push-down stacks?

This commentary raises two issues: (1) Language processing is hastened not only by internal pressures but also externally by turn-taking in language use; (2) the theory requires nested levels of processing, but linguistic levels do not fully nest; further, it would seem to require multiple memory buffers, otherwise there's no obvious treatment for discontinuous structures, or for verbatim recall.

On decision problems for parameterized machines

In this paper, we investigate various decision problems concerning parameterized versions of some classes of machines. Let C ( s , m , t ) be the class of nondeterministic multitape Turing machine (TM) acceptors with a two-way read-only input, at most s states, at most m read–write worktapes, and at most t symbols in the worktape alphabet, where s , m , t are fixed positive integers. There is no restriction on the cardinality of the input alphabet. We are able to show the emptiness, disjointness, and universe (also called universality) problems to be decidable for C ( s , m , t ) . For the class consisting of machines in C ( s , m , t ) that always halt or whose minimal-time accepting computations can be bounded by some recursive function f ( n ) (where n is the input length), the containment and equivalence problems are decidable. These results hold for other machines, e.g., when the worktapes are pushdown stacks (where on every step, each pushdown can only pop the top of the stack or replace the top of the stack by at most two symbols) or when stacks are counters (where on every step, a counter can be incremented by 1, decremented by 1, or remain unchanged, and can be tested for zero). Our results are the best possible in the sense that not parameterizing one of s , m , t (or, in the case of counter machines, allowing the counters to increment by arbitrary integers that may change from machines to machines) makes the universe problem undecidable. We also give a simple characterization of the languages defined by C ( s , m , t ) . Finally, we investigate the applicability of our techniques to machines with multiple input heads or multiple input tapes.

On the power of deep pushdown stacks

Inspired by recent work of Meduna on deep pushdown automata, we consider the computational power of a class of basic program schemes, NPSDS s , based around assignments, while-loops and non-deterministic guessing but with access to a deep pushdown stack which, apart from having the usual push and pop instructions, also has deep-push instructions which allow elements to be pushed to stack locations deep within the stack. We syntactically define sub-classes of NPSDS s by restricting the occurrences of pops, pushes and deep-pushes and capture the complexity classes NP and PSPACE. Furthermore, we show that all problems accepted by program schemes of NPSDS s are in EXPTIME.

On composition and lookahead delegation of [formula omitted]-services modeled by automata

Let M be a class of (possibly nondeterministic) language acceptors with a one-way input tape. A system ( A ; A 1 , … , A r ) of automata in M is composable if for every string w = a 1 ⋯ a n of symbols accepted by A , there is an assignment of each symbol a j in w to one of the A i 's such that for each 1 ⩽ i ⩽ r , the subsequence of w assigned to A i is accepted by A i . For a nonnegative integer k , a k - lookahead delegator for ( A ; A 1 , … , A r ) is a deterministic machine D in M which, knowing (a) the current states of A , A 1 , … , A r and the accessible “local” information of each machine (e.g., the top of the stack if each machine is a pushdown automaton, whether a counter is zero or nonzero if each machine is a multicounter automaton, etc.), and (b) the k lookahead symbols to the right of the current input symbol being processed, can uniquely determine the A i to assign the current symbol. Moreover, every string w accepted by A is also accepted by D ; i.e., the subsequence of string w delegated by D to each A i is accepted by A i . Thus, k -lookahead delegation is a stronger requirement than composability, since the delegator D must be deterministic. A system that is composable may not have a k -delegator for any k . We study the decidability of composability and existence of k -delegators for various classes of machines M . Our results generalize earlier ones (and resolve some open questions) concerning composability of deterministic finite automata as e-services to finite automata that are augmented with unbounded storage (e.g., counters and pushdown stacks) and finite automata with discrete clocks (i.e., discrete timed automata). The results have applications to automated composition of e-services.

Logout and Login

After seven years and 32 columns, the Electronic Seismologist (ES) may have blown a fuse, short-circuited, or just powered down. There ain't no more juice in the circuits. The bit bucket is empty. The do-loop is done. Syntax error and the core-dump file is zero-length. It's time to pass the keyboard to someone with an object-oriented approach; with massive pushdown stacks, megabytes of memory, and an eight-pipe, parallel processor. Now in shutdown mode, the ES produces his last column. However, before your eyes glaze over and you go back to doing something useful, check out the report by the new ES that follows my ramblings. This time you get two ES's for the price of one. Such a deal! As I retire from the publish-or-get-nagged business of providing columns for SRL every other month or so, I feel the need to look back to see how it all started. Don't worry, I won't get maudlin. A quick glance through some of my early columns reveals just how transitory much of this geeky computer stuff really is. My first column in early 1995 was published in the early days of the “World Wide Web” and …

Spinal-Formed Context-Free Tree Grammars

In this paper we introduce a restricted model of context-free tree grammars called spine grammars, and study their formal properties including considerably simple normal forms. Recent research on natural languages has suggested that formalisms for natural languages need to generate a slightly larger class of languages than context-free grammars, and for that reason tree adjoining grammars have been widely studied relating them to natural languages. It is shown that the class of string languages generated by spine grammars coincides with that of tree adjoining grammars. We also introduce acceptors called linear pushdown tree automata, and show that linear pushdown tree automata accept exactly the class of tree languages generated by spine grammars. Linear pushdown tree automata are obtained from pushdown tree automata with a restriction on duplicability for the pushdown stacks.

On the power of programming features

We consider the power of several programming features such as counters, pushdown stacks, queues, arrays, recursion and equality. In this study program schemas are used as the model for computation. The relations between the powers of these features is completely described by a comparison diagram.