Content uploaded by Dana S. Scott
Author content
All content in this area was uploaded by Dana S. Scott on Nov 12, 2016
Content may be subject to copyright.
A preview of the PDF is not available
Finite Automata and Their Decision Problems
Abstract
Finite automata are considered in this paper as instruments for classifying finite tapes. Each one-tape automaton defines a set of tapes, a two-tape automaton defines a set of pairs of tapes, et cetera. The structure of the defined sets is studied. Various generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined. Some decision problems concerning automata are shown to be solvable by effective algorithms; others turn out to be unsolvable by algorithms.
... The subset or Rabin-Scott construction which was full described in [1] represents conservative system of choosing between determinism and non-determinism in both aspects, however, lacks the efficiency of complexity in case of deterministic machine operating on the finite set of states, thus, it's obvious that it will lead the number of states as well as number of operations to grow exponentially in time of O(2 n ). ...
... Whereas in (3) deg + (t) denotes the number of incoming edges for the given state t which is artificial by the definition as it wasn't implemented or introduced in prior works [1,2,3], visited(t) is the function denoting the number of visited edges during the closure computation processwe conclude that at each step this function is evaluated to its default value of zero. ...
We present the continuation of studying Extended Regular Expression (ERE) on the view of modified subset construction within the overridden operators like intersection, subtraction, and rewritten complement. As before we have stated that in this case the complexity has a decreasing nature and tendency. We will give the strict definition of the operational part of this modified subset construction which is due to Rabin and Scott. The complexity of algorithm remains a magnitude less than NP-hard problems for which we have given the strict proof of equivalence in the prior work, so this work continues the studying of the comparable proof for a variety of problems to be computationally complex, however, explainable in terms of unified approach like operational calculus. In this calculus the general points of research are given to the representation of modified subset construction with at least two operands which are to be computed by subset construction and in terms of complexity of the effective algorithm they are computed using modified subset construction.
... At first, we will give the definition of the problem for arguments like regular expressions [1] and finite automata which are constructed from these expressions using various types of algorithms like Thompson Construction [2], Berry-Sethi Method [3] and Rabin-Scott subset construction [4]. ...
... Rabin and Scott present the subset or powerset construction to convert existing NFA to DFA [4]. This problem is NP-complete as it lies in EXPTIME and EXPSPACE class on the example given by Schneider Klaus [8]. ...
In this article the final proof of the equivalence of polynomial (P) and non-polynomial classes (NP) are provided within the given Klaus Schneider’s regular expression forms which tend to be
exponential in size and the problem in overall is seen to be in EXPTIME and EXPSPACE and, thus, NP-complete. We provide the full history of this proof with the experimentally obtained results. Our application of the obtained singular algorithm in terms of computational complexity responds to the famous theorem like “P versus NP” which wasn’t solved before and now is resolved in this work. We will also discuss the future of the question from past as the novel proof is given.
... Deterministic finite automata (DFAs) [24,31,10] are computation models that accept or reject some given strings. Before we formally define them, we go through one illustrating example. ...
Despite the success of Large Language Models (LLMs) on various tasks following human instructions, controlling model generation at inference time poses a persistent challenge. In this paper, we introduce Ctrl-G, an adaptable framework that facilitates tractable and flexible control of LLM generation to reliably follow logical constraints. Ctrl-G combines any production-ready LLM with a Hidden Markov Model, enabling LLM outputs to adhere to logical constraints represented as deterministic finite automata. We show that Ctrl-G, when applied to a TULU2-7B model, outperforms GPT3.5 and GPT4 on the task of interactive text editing: specifically, for the task of generating text insertions/continuations following logical constraints, Ctrl-G achieves over 30% higher satisfaction rate in human evaluation compared to GPT4. When applied to medium-size language models (e.g., GPT2-large), Ctrl-G also beats its counterparts for constrained generation by large margins on standard benchmarks. Additionally, as a proof-of-concept study, we experiment Ctrl-G on the Grade School Math benchmark to assist LLM reasoning, foreshadowing the application of Ctrl-G, as well as other constrained generation approaches, beyond traditional language generation tasks.
... Regular expression: A regular expression or regular pattern defines a search pattern in a text in the form of a sequence of characters [82]. In other words, a regular expression defines a set of strings that match it [83]. ...
Machine learning, deep learning, and NLP methods on knowledge graphs are present in different fields and have important roles in various domains from self-driving cars to friend recommendations on social media platforms. However, to apply these methods to knowledge graphs, the data usually needs to be in an acceptable size and format. In fact, knowledge graphs normally have high dimensions and therefore we need to transform them to a low-dimensional vector space. An embedding is a low-dimensional space into which you can translate high dimensional vectors in a way that intrinsic features of the input data are preserved. In this review, we first explain knowledge graphs and their embedding and then review some of the random walk-based embedding methods that have been developed recently.
... How this relates to problems of consciousness is addressed elsewhere[16,23], and is beyond the scope of this paper.12 In any case, the difference between deterministic and non-deterministic seems meaningless when you consider that DFAs and NFAs are equivalent[25]. ...
Simplicity is held by many to be the key to general intelligence. Simpler models tend to “generalise”, identifying the cause or generator of data with greater sample efficiency. The implications of the correlation between simplicity and generalisation extend far beyond computer science, addressing questions of physics and even biology. Yet simplicity is a property of form, while generalisation is of function. In interactive settings, any correlation between the two depends on interpretation. In theory there could be no correlation and yet in practice, there is. Previous theoretical work showed generalisation to be a consequence of “weak” constraints implied by function, not form. Experiments demonstrated choosing weak constraints over simple forms yielded a 110-500% improvement in generalisation rate. Here we show that all constraints can take equally simple forms, regardless of weakness. However if forms are spatially extended, then function is represented using a finite subset of forms. If function is represented using a finite subset of forms, then we can force a correlation between simplicity and generalisation by making weak constraints take simple forms. If function is determined by a goal directed process that favours versatility (e.g. natural selection), then efficiency demands weak constraints take simple forms. Complexity has no causal influence on generalisation, but appears to due to confounding.
In Press: Accepted for publication in the Proceedings of The 17th Conference on Artificial General Intelligence, 2024
... Before continuing to develop this definition, let us briefly comment on our treatment of initial and accepting states, as well as the treatment of determinism and ϵ-transitions. Rabin and Scott's original definition of NDFA [RS59] (also adopted by Eilenberg [Eil74]), takes a set of initial states together with a set of accepting states, while some more recent authors [HMU07,Sip13] take a single initial state together with a set of accepting states. By Rabin and Scott's determinization theorem, a language L ⊆ Σ * may be recognized by a NDFA with a set of initial and accepting states just in case it may be recognized by a DFA with a single initial state and a set of accepting states, so the choice of having a single initial state or a set of initial states does not matter in terms of recognition power. ...
We develop fibrational perspectives on context-free grammars and on finite state automata over categories and operads. A generalized CFG is a functor from a free colored operad (= multicategory) generated by a pointed finite species into an arbitrary base operad: this encompasses classical CFGs by taking the base to be a certain operad constructed from a free monoid, as an instance of a more general construction of an operad of spliced arrows $\mathcal{W}\,\mathcal{C}$ for any category $\mathcal{C}$. A generalized NDFA is a functor satisfying the unique lifting of factorizations and finite fiber properties, from an arbitrary bipointed category or pointed operad: this encompasses classical word automata and tree automata without $\epsilon$-transitions, but also automata over non-free categories and operads. We show that generalized context-free and regular languages satisfy suitable generalizations of many of the usual closure properties, and in particular we give a simple conceptual proof that context-free languages are closed under intersection with regular languages. Finally, we observe that the splicing functor $\mathcal{W} : Cat \to Oper$ admits a left adjoint $\mathcal{C} : Oper \to Cat$, which we call the contour category construction since the arrows of $\mathcal{C}\,\mathcal{O}$ have a geometric interpretation as oriented contours of operations of $\mathcal{O}$. A direct consequence of the contour / splicing adjunction is that every pointed finite species induces a universal CFG generating a language of tree contour words. This leads us to a generalization of the Chomsky-Sch\"utzenberger Representation Theorem, establishing that a subset of a homset $L \subseteq \mathcal{C}(A,B)$ is a CFL of arrows iff it is a functorial image of the intersection of a $\mathcal{C}$-chromatic tree contour language with a regular language.
We provide new examples of groups without rational cross‐sections (also called regular normal forms), using connections with bounded generation and rational orders on groups. Our examples contain a finitely presented HNN‐extension of the first Grigorchuk group. This last group is the first example of finitely presented group with solvable word problem and without rational cross‐sections. It is also not autostackable, and has no left‐regular complete rewriting system.
Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means
of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more
complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions,
one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible neurophysiological
assumptions are equivalent, in the sense that for every net behaving under one assumption, there exists another net which
behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus
are discussed.
Rabin has proved<sup>1,2</sup> that two-way finite automata, which are allowed to move in both directions along their input tape, are equivalent to one-way automata as far as the classification of input tapes is concerned. Rabin's proof is rather complicated and consists in giving a method for the successive elimination of loops in the motion of the machine. The purposeo f this note is to give a short, direct proof of the result.
http://deepblue.lib.umich.edu/bitstream/2027.42/3967/5/bab6288.0001.001.pdf http://deepblue.lib.umich.edu/bitstream/2027.42/3967/4/bab6288.0001.001.txt