Showing papers on "Pushdown automaton published in 2021"

Fast zone-based algorithms for reachability in pushdown timed automata

Abstract: Given the versatility of timed automata a huge body of work has evolved that considers extensions of timed automata. One extension that has received a lot of interest is timed automata with a, possibly unbounded, stack, also called pushdown timed automata (PDTA). While different algorithms have been given for reachability in different variants of this model, most of these results are purely theoretical and do not give rise to efficient implementations. One main reason for this is that none of these algorithms (and the implementations that exist) use the so-called zone-based abstraction, but rely either on the region-abstraction or other approaches, which are significantly harder to implement.

Reactive Synthesis from Visibly Register Pushdown Automata

Abstract: The realizability problem for a given specification \(\mathcal{S}\) is to decide whether there exists an implementation satisfying \(\mathcal{S}\). Although the problem is important in the field of reactive synthesis of recursive programs, the problem has not been studied yet when specification and implementation are given by pushdown computational models. This paper investigates the realizability problem for the cases that a specification and an implementation are given by a pushdown automaton (PDA) and a pushdown transducer (PDT), and a register pushdown automata (RPDA) and a register pushdown transducer (RPDT).

Input-Driven Pushdown Automata on Well-Nested Infinite Strings

Abstract: Automata operating on strings of nested brackets, known as input-driven pushdown automata, and also as visibly pushdown automata, have been studied since the 1980s. They were extended to the case of infinite strings by Alur and Madhusudan (“Visibly pushdown languages”, STOC 2004). This paper investigates the properties of these automata under the assumption that a given infinite string is always well-nested. This restriction enables a complete characterization of the corresponding \(\omega \)-languages in terms of classical \(\omega \)-regular languages and input-driven pushdown automata on finite strings. This characterization leads to a determinization construction for these automata, as well as to the first results on their topological classification.

On linear languages recognized by deterministic biautomata

Abstract: We introduce the notion of a deterministic biautomaton, a device that reads inputs from both ends, and admits at most one computation on every input word. We show that deterministic biautomata recognize a class of languages which is properly included in the class of linear languages, and which is incomparable to the class of languages recognized by deterministic one-turn pushdown automata. We propose three restrictions of the basic model of deterministic biautomata to characterize the classes of languages generated by DL, linear LL(1), and NH-DL grammars. This results in a chain of four subclasses of linear languages in which all inclusions are strict. For these subclasses and basic operations, we show whether or not a particular class is closed under a considered operation. We prove that it is undecidable whether a linear language belongs to any of these four classes and answer some other decidability questions.

Application of Fuzzy Pushdown Automaton on Prediction of Quality Control for Spinning Yarn: Application of FPDA on quality prediction

Abstract: In order to perform better recognition, tracking and control for fuzzy and uncertain thing, this paper will design a suitable fuzzy pushdown automaton (FPDA) control method to solve the problem. Firstly, the control design structure of FPDA and the decision reasoning rules in control are given. Secondly, the application of FPDA in prediction of quality control for spinning yarn is discussed in the practical problem. Finally, the comparison of FPDA and other control methods on the target control is given. The simulation results show that the control speed and the average precision of designed FPDA are faster by12ms and higher by 4.98% than that of traditional method, which its control precision is 96.87%.

A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct

Abstract: We study the expressiveness and succinctness of good-for-games pushdown automata (GFG-PDA) over finite words, that is, pushdown automata whose nondeterminism can be resolved based on the run constructed so far, but independently of the remainder of the input word. We prove that GFG-PDA recognise more languages than deterministic PDA (DPDA) but not all context-free languages (CFL). This class is orthogonal to unambiguous CFL. We further show that GFG-PDA can be exponentially more succinct than DPDA, while PDA can be double-exponentially more succinct than GFG-PDA. We also study GFGness in visibly pushdown automata (VPA), which enjoy better closure properties than PDA, and for which we show GFGness to be EXPTIME-complete. GFG-VPA can be exponentially more succinct than deterministic VPA, while VPA can be exponentially more succinct than GFG-VPA. Both of these lower bounds are tight. Finally, we study the complexity of resolving nondeterminism in GFG-PDA. Every GFG-PDA has a positional resolver, a function that resolves nondeterminism and that is only dependant on the current configuration. Pushdown transducers are sufficient to implement the resolvers of GFG-VPA, but not those of GFG-PDA. GFG-PDA with finite-state resolvers are determinisable.

Token Games and History-Deterministic Quantitative-Automata.

Abstract: A nondeterministic (quantitative) automaton is history deterministic if its nondeterminism can be resolved by only considering the prefix of the word read so far. Due to their good compositional properties, history deterministic automata are useful in solving games and synthesis problems. Deciding whether or not a given nondeterministic automaton is history deterministic (the HDness problem) is generally a difficult task, which might involve an exponential procedure, or even be undecidable, for example for pushdown automata. Token games provide a PTime solution to the HDness problem of B\"uchi and coB\"uchi automata, and it is conjectured that 2-token games characterize HDness for all !-regular automata. We extend token games to the quantitative setting and analyze their potential to help deciding HDness for quantitative automata. In particular, we show that 1-token games characterize HDness for all quantitative (and Boolean) automata on finite words, as well as discounted-sum (DSum) automata on infinite words, and that 2-token games characterize HDness of LimInf and LimSup automata. Using these characterizations, we provide solutions to the HDness problem of Inf and Sup automata on finite words in PTime, for DSum automata on finite and infinite words in NP$\cap$co-NP, for LimSup automata in quasipolynomial time, and for LimInf automata in exponential time, where the latter two are only polynomial for automata with a fixed number of weights.

Higher-order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

Abstract: This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set of elements for which a formula holds), and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems, we provide an effective solution. This is obtained, thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

Generalized fuzzy automata with semantic computing

Abstract: Researches about semantic computing are usually evaluated by experiments, lacking support from the theory of computation. But traditional finite automata are defined on a fixed finite set of states. They will be invalid when a different application needs to change the states. However, semantic computing always asks for more robust automata that can modify itself according to the changes of states. Therefore, in this paper, we redefine traditional automata in more robust forms by semantic computing. The intuitive idea is when the state which an automaton is supposed to enter has changed, we drive automata into a state that is similar to the original one. We can find another similar state in the original state set to replace the changed state by semantic similarity. Based on the theories of semantic similarity, there always exists a similar state. Therefore, once a state has changed, we can always find a similar state to replace it, which means that we can empower automata to adapt to the changes of states. Our new automata bridge the gap between semantic computing and the theory of computation. Furthermore, we also redefine fuzzy finite automata and pushdown automata in more robust forms. Finally, we provide an application about the weather forecast, which indicates how to overcome the limitations of traditional automata.

The No Endmarker Theorem for One-Way Probabilistic Pushdown Automata.

Abstract: In various models of one-way pushdown automata, the explicit use of two designated endmarkers on a read-once input tape has proven to be extremely useful for making a conscious, final decision on the acceptance/rejection of each input word right after reading the right endmarker. With no endmarkers, by contrast, a machine must constantly stay in either accepting or rejecting states at any moment since it never notices the end of the input instance. This situation, however, helps us analyze the behavior of the machine whose tape head makes the consecutive moves on all prefixes of a given extremely long input word. Since those two machine formulations have their own advantages, it is natural to ask whether the endmarkers are truly necessary to correctly recognize languages. In the deterministic and nondeterministic models, it is well-known that the endmarkers are removable without changing the acceptance criteria of each input instance. This paper proves that, for a more general model of one-way probabilistic pushdown automata, the endmarkers are also removable. This is proven by employing probabilistic transformations from an "endmarker" machine to an equivalent "no-endmarker" machine at the cost of double exponential state complexity without compromising its error probability. By setting this error probability appropriately, our proof also provides an alternative proof to both the deterministic and the nondeterministic models as well.

Model Checking Temporal Properties of Recursive Probabilistic Programs

Abstract: Probabilistic pushdown automata (pPDA) are a standard operational model for programming languages involving discrete random choices, procedures, and returns. Temporal properties are useful for gaining insight into the chronological order of events during program execution. Existing approaches in the literature have focused mostly on $\omega$-regular and LTL properties. In this paper, we study the model checking problem of pPDA against $\omega$-visibly pushdown languages that can be described by specification logics such as CaRet and are strictly more expressive than $\omega$-regular properties. With these logical formulae, it is possible to specify properties that explicitly take the structured computations arising from procedural programs into account. For example, CaRet is able to match procedure calls with their corresponding future returns, and thus allows to express fundamental program properties like total and partial correctness.

Detecting Useless Transitions in Pushdown Automata

Abstract: Pushdown automata may contain transitions that are never used in any accepting run of the automaton. We present an algorithm for detecting such useless transitions. A finite automaton that captures the possible stack content during runs of the pushdown automaton, is first constructed in a forward procedure to determine which transitions are reachable, and then employed in a backward procedure to determine which of these transitions can lead to a final state. An implementation of the algorithm is shown to exhibit a favorable performance.

Digging input-driven pushdown automata

Abstract: 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.

The descriptional power of queue automata of constant length

Abstract: We consider the notion of a constant length queue automaton—i.e., a traditional queue automaton with a built-in constant limit on the length of its queue—as a formalism for representing regular languages. We show that the descriptional power of constant length queue automata greatly outperforms that of traditional finite state automata, of constant height pushdown automata, and of straight line programs for regular expressions, by providing optimal exponential and double-exponential size gaps. Moreover, we prove that constant height pushdown automata can be simulated by constant length queue automata paying only by a linear size increase, and that removing nondeterminism in constant length queue automata requires an optimal exponential size blow-up, against the optimal double-exponential cost for determinizing constant height pushdown automata. Finally, we investigate the size cost of implementing Boolean language operations on deterministic and nondeterministic constant length queue automata.

Space Complexity of Stack Automata Models

Abstract: This paper examines several measures of space complexity of variants of stack automata: non-erasing stack automata and checking stack automata. These measures capture the minimum stack size require...

Subcubic Certificates for CFL Reachability

Abstract: Many problems in interprocedural program analysis can be modeled as the context-free language (CFL) reachability problem on graphs and can be solved in cubic time. Despite years of efforts, there are no known truly sub-cubic algorithms for this problem. We study the related certification task: given an instance of CFL reachability, are there small and efficiently checkable certificates for the existence and for the non-existence of a path? We show that, in both scenarios, there exist succinct certificates ($O(n^2)$ in the size of the problem) and these certificates can be checked in subcubic (matrix multiplication) time. The certificates are based on grammar-based compression of paths (for positive instances) and on invariants represented as matrix constraints (for negative instances). Thus, CFL reachability lies in nondeterministic and co-nondeterministic subcubic time. A natural question is whether faster algorithms for CFL reachability will lead to faster algorithms for combinatorial problems such as Boolean satisfiability (SAT). As a consequence of our certification results, we show that there cannot be a fine-grained reduction from SAT to CFL reachability for a conditional lower bound stronger than $n^\omega$, unless the nondeterministic strong exponential time hypothesis (NSETH) fails. Our results extend to related subcubic equivalent problems: pushdown reachability and two-way nondeterministic pushdown automata (2NPDA) language recognition. For example, we describe succinct certificates for pushdown non-reachability (inductive invariants) and observe that they can be checked in matrix multiplication time. We also extract a new hardest 2NPDA language, capturing the "hard core" of all these problems.

Parameterizations of Logarithmic-Space Reductions, Stack-State Complexity of Nonuniform Families of Pushdown Automata, and a Road to the LOGCFL$\subseteq$LOGDCFL/poly Question

Abstract: The complexity class LOGCFL (resp., LOGDCFL) consists of all languages that are many-one reducible to context-free (resp., deterministic context-free) languages using logarithmic space. These complexity classes have been studied over five decades in connection to parallel computation since they are located between Nick's classes $\mathrm{NC}^1$ and $\mathrm{NC}^2$. In contrast, the state complexity of nonuniform finite-automaton families was first discussed in the 1970s and it has been extensively explored lately for various finite-automata families. We extend this old subject to the stack-state complexity (i.e., the total number of inner states plus simultaneously pushable stack symbol series) of nonuniform families of various pushdown automata. We introduce reasonable "parameterizations" of LOGCFL and LOGDCFL and apply them as a technical tool to establish a close connection between the LOGCFL$\subseteq$LOGDCFL/poly question and the polynomial stack-state complexity of nonuniform families of two-way pushdown automata. We also discuss the precise computational complexity of polynomial-size one-way pushdown automata.

On the complexity of the generalized Q2R automaton.

Abstract: We study the dynamic and complexity of the generalized Q2R automaton. We show the existence of non-polynomial cycles as well as its capability to simulate with the synchronous update the classical version of the automaton updated under a block sequential update scheme. Furthermore, we show that the decision problem consisting in determine if a given node in the network changes its state is \textbf{P}-Hard.

Context-Bounded Verification of Thread Pools.

Abstract: Thread pooling is a common programming idiom in which a fixed set of worker threads are maintained to execute tasks concurrently. The workers repeatedly pick tasks and execute them to completion. Each task is sequential, with possibly recursive code, and tasks communicate over shared memory. Executing a task can lead to more new tasks being spawned. We consider the safety verification problem for thread-pooled programs. We parameterize the problem with two parameters: the size of the thread pool as well as the number of context switches for each task. The size of the thread pool determines the number of workers running concurrently. The number of context switches determines how many times a worker can be swapped out while executing a single task - like many verification problems for multithreaded recursive programs, the context bounding is important for decidability. We show that the safety verification problem for thread-pooled, context-bounded, Boolean programs is EXPSPACE-complete, even if the size of the thread pool and the context bound are given in binary. Our main result, the EXPSPACE upper bound, is derived using a sequence of new succinct encoding techniques of independent language-theoretic interest. In particular, we show a polynomial-time construction of downward closures of languages accepted by succinct pushdown automata as doubly succinct nondeterministic finite automata. While there are explicit doubly exponential lower bounds on the size of nondeterministic finite automata accepting the downward closure, our result shows these automata can be compressed. We show that thread pooling significantly reduces computational power: in contrast, if only the context bound is provided in binary, but there is no thread pooling, the safety verification problem becomes 3EXPSPACE-complete.

Pushdown automata and context-free grammars in bisimulation semantics

Abstract: The Turing machine models an old-fashioned computer, that does not interact with the user or with other computers, and only does batch processing. Therefore, we came up with a Reactive Turing Machine that does not have these shortcomings. In the Reactive Turing Machine, transitions have labels to give a notion of interactivity. In the resulting process graph, we use bisimilarity instead of language equivalence. Subsequently, we considered other classical theorems and notions from automata theory and formal languages theory. In this paper, we consider the classical theorem of the correspondence between pushdown automata and context-free grammars. By changing the process operator of sequential composition to a sequencing operator with intermediate acceptance, we get a better correspondence in our setting. We find that the missing ingredient to recover the full correspondence is the addition of a notion of state awareness.

Adaptive Synchronisation of Pushdown Automata

Abstract: We introduce the notion of adaptive synchronisation for pushdown automata, in which there is an external observer who has no knowledge about the current state of the pushdown automaton, but can observe the contents of the stack. The observer would then like to decide if it is possible to bring the automaton from any state into some predetermined state by giving inputs to it in an \emph{adaptive} manner, i.e., the next input letter to be given can depend on how the contents of the stack changed after the current input letter. We show that for non-deterministic pushdown automata, this problem is 2-EXPTIME-complete and for deterministic pushdown automata, we show EXPTIME-completeness. To prove the lower bounds, we first introduce (different variants of) subset-synchronisation and show that these problems are polynomial-time equivalent with the adaptive synchronisation problem. We then prove hardness results for the subset-synchronisation problems. For proving the upper bounds, we consider the problem of deciding if a given alternating pushdown system has an accepting run with at most $k$ leaves and we provide an $n^{O(k^2)}$ time algorithm for this problem.

Parameterized verification of coverability in infinite state broadcast networks

Abstract: Parameterized verification of coverability in broadcast networks with finite state processes has been studied for different types of models and topologies. In this paper, we attempt to develop a theory of broadcast networks in which the processes can be well-structured transition systems. The resulting formalism is called well-structured broadcast networks. For various types of communication topologies, we prove the decidability of coverability in the static case, i.e., when the network topology is not allowed to change. We do this by showing that for these types of static communication topologies, the broadcast network itself is a well-structured transition system, hence proving the decidability of coverability in the broadcast network. We also give an algorithm to decide coverability of well-structured broadcast networks when reconfiguration of links between nodes is allowed. Finally, with minor modifications of this algorithm we prove decidability of coverability when the underlying process is a pushdown automaton.

Collapsible Pushdown Parity Games

Abstract: This article studies a large class of two-player perfect-information turn-based parity games on infinite graphs, namely, those generated by collapsible pushdown automata. The main motivation for studying these games comes from the connections from collapsible pushdown automata and higher-order recursion schemes, both models being equi-expressive for generating infinite trees. Our main result is to establish the decidability of such games and to provide an effective representation of the winning region as well as of a winning strategy. Thus, the results obtained here provide all necessary tools for an in-depth study of logical properties of trees generated by collapsible pushdown automata/recursion schemes.

On the determinization of event-clock input-driven pushdown automata.

Abstract: Input-driven pushdown automata (also known as visibly pushdown automata and as nested word automata) are a subclass of deterministic pushdown automata and a superclass of the parenthesis languages. Nguyen and Ogawa ("Event-clock visibly pushdown automata", SOFSEM 2009) defined a timed extension of these automata under the event-clock model, and showed that this model can be determinized using the method of region construction. This paper defines a further extension of this model with the event clock on the call-return operations, and proposes a new, direct determinization procedure for these automata: an $n$-state nondeterministic automaton with $k$ different clock constraints is transformed to a deterministic automaton with $2^{n^2}$ states, $2^{n^2+k}$ stack symbols and the same clock constraints as in the original automaton. The construction is shown to be asymptotically optimal with respect to both the number of states and the number of stack symbols.

Learning Hierarchical Structures with Differentiable Nondeterministic Stacks.

Abstract: Learning hierarchical structures in sequential data -- from simple algorithmic patterns to natural language -- in a reliable, generalizable way remains a challenging problem for neural language models. Past work has shown that recurrent neural networks (RNNs) struggle to generalize on held-out algorithmic or syntactic patterns without supervision or some inductive bias. To remedy this, many papers have explored augmenting RNNs with various differentiable stacks, by analogy with finite automata and pushdown automata. In this paper, we present a stack RNN model based on the recently proposed Nondeterministic Stack RNN (NS-RNN) that achieves lower cross-entropy than all previous stack RNNs on five context-free language modeling tasks (within 0.05 nats of the information-theoretic lower bound), including a task in which the NS-RNN previously failed to outperform a deterministic stack RNN baseline. Our model assigns arbitrary positive weights instead of probabilities to stack actions, and we provide an analysis of why this improves training. We also propose a restricted version of the NS-RNN that makes it practical to use for language modeling on natural language and present results on the Penn Treebank corpus.

Reversible Top-Down Syntax Analysis

Abstract: Top-down syntax analysis can be based on \(\mathrm {LL}(k)\) grammars. The canonical acceptors for \(\mathrm {LL}(k)\) languages are deterministic stateless pushdown automata with input lookahead of size k. We investigate the computational capacity of reversible computations of such automata. A pushdown automaton with lookahead k is said to be reversible if its predecessor configurations can uniquely be computed by a pushdown automaton with backward input lookahead (lookback) of size k. It is shown that we cannot trade a lookahead for states or vice versa. The impact of having states or a lookahead depends on the language. While reversible pushdown automata with states accept all regular languages, we are going to prove that there are regular languages that cannot be accepted reversibly without states, even in case of an arbitrarily large lookahead. This completes the comparison of reversible with ordinary pushdown automata in our setting. Finally, it turns out that there are problems which can be solved by reversible deterministic stateless pushdown automata with lookahead of size \(k+1\), but not by any reversible deterministic stateless pushdown automaton with lookahead of size k. So, an infinite and tight hierarchy of language families dependent on the size of the lookahead is shown.

Testing Pushdown Systems.

Abstract: Testing on reactive systems is a well-known laborious activity on software development due to their asynchronous interaction with the environment. In this setting model based testing has been employed when checking conformance and generating test suites of such systems using labeled transition system as a formalism as well as the classical ioco conformance relation. In this work we turn to a more complex scenario where the target systems have an auxiliary memory, a stack. We then studied a more powerful model, the Visibly Pushdown Labeled Transition System (VPTS), its variant Input/Output VPTS (IOVPTS), its associated model Visibly Pushdown Automaton (VPA), and aspects of conformance testing and test suite generation. This scenario is much more challenge since the base model has a pushdown stack to capture more complex behaviors which commonly found on reactive systems. We then defined a more general conformance relation for pushdown reactive systems such that it prevents any observable implementation behavior that was not already present in the given specification. Further we gave an efficient algorithm to check conformance in this testing scenario and also showed that it runs in worst case asymptotic polynomial time in the size of both the given specification and the implementation that are put under test.