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Visibly Counter Languages and the Structure of \mathrm {NC}^{1}
Abstract
We extend the familiar program of understanding circuit complexity in terms of regular languages to visibly counter languages. Like the regular languages, the visibly counter languages are - complete. We investigate what the visibly counter languages in certain constant depth circuit complexity classes are. We have initiated this in a previous work for . We present characterizations and decidability results for various logics and circuit classes. In particular, our approach yields a way to understand , where the regular approach fails.
... Having characterized the regular languages modeled by SSMs, we now consider languages requiring unbounded counting [20], specifically, languages recognized by keeping track of one or more counters, where each character causes a specific increment or decrement to each counter [38,27,70,40]. A prime example is the Dyck-1 language of well-formed strings over "(" and ")"; here a counter is incremented (decremented) whenever an opening (closing) bracket is encountered; a string is well-formed if and only if the counter is 0 at the end of the string. ...
Recently, recurrent models based on linear state space models (SSMs) have shown promising performance in language modeling (LM), competititve with transformers. However, there is little understanding of the in-principle abilities of such models, which could provide useful guidance to the search for better LM architectures. We present a comprehensive theoretical study of the capacity of such SSMs as it compares to that of transformers and traditional RNNs. We find that SSMs and transformers have overlapping but distinct strengths. In star-free state tracking, SSMs implement straightforward and exact solutions to problems that transformers struggle to represent exactly. They can also model bounded hierarchical structure with optimal memory even without simulating a stack. On the other hand, we identify a design choice in current SSMs that limits their expressive power. We discuss implications for SSM and LM research, and verify results empirically on a recent SSM, Mamba.
In this paper, the regular languages of wire linear AC0are characterized as the languages expressible in the two-variable fragment of first-order logic with regular predicates, FO2[reg]. Additionally, they are characterized as the languages recognized by the algebraic class QLDA. The class is shown to be decidable and examples of languages in and outside of it are presented.
The model of deterministic input-driven multi-counter automata is introduced and studied. On such devices, the input letters uniquely determine the operations on the underlying data structure that is consisting of multiple counters. We study the computational power of the resulting language families and compare them with known language families inside the Chomsky hierarchy. In addition, it is possible to prove a proper counter hierarchy depending on the alphabet size. This means that any input alphabet induces an upper bound which depends on the alphabet size only, such that k+1 counters are more powerful than k counters as long as k is less than this bound. The hierarchy interestingly collapses at the level of the bound. Furthermore, we investigate the closure properties of the language families. For input-driven multi-counter automata with 0 or 1 counter, we discuss the computational complexity of their decidable problems. For k≥2 counters, the decidability problems of emptiness, finiteness, universality, inclusion, equivalence, regularity, and context-freeness are shown to be non-semidecidable. Finally, we study descriptional complexity aspects of input-driven multi-counter automata. It is shown that a nondeterministic device can be determinized and that 2n−1 is a necessary and sufficient blow-up in the number of states for the determinization. For the operational state complexity of deterministic input-driven multi-counter automata under Boolean operations, tight bounds on the number of states are established. Finally, it turns out that the size trade-offs between deterministic input-driven multi-counter automata with k+1 and k counters are non-recursive, that is, they are not bounded by any recursive function.
The model of deterministic input-driven multi-counter automata is introduced and studied. On such devices, the input letters uniquely determine the operations on the underlying data structure that is consisting of multiple counters. We study the computational power of the resulting language families and compare them with known language families inside the Chomsky hierarchy. In addition, it is possible to prove a proper counter hierarchy depending on the alphabet size. This means that any input alphabet induces an upper bound which depends on the alphabet size only, such that k+1 counters are more powerful than k counters as long as k is less than this bound. The hierarchy interestingly collapses at the level of the bound. Furthermore, we investigate the closure properties of the language families. Finally, the undecidability of the emptiness problem is derived for input-driven two-counter automata.
Over the last decade a considerable effort was invested into research on implementing string dictionaries. String dictionary is a data structure that bijectively maps a set of strings to a set of integers, and that is used in various index-based applications. A recent paper [18] can be regarded as a reference work on the subject of string dictionary implementations. Although very comprehensive, [18] does not cover the implementation of a string dictionary with the enumerated deterministic finite automaton, a data structure naturally suited for this purpose. We compare the results for the state-of-the-art compressed enumerated automaton with those presented in [18] on the same collection of data sets, and on the collection of natural language word lists. We show that our string dictionary implementation is a competitive variant for different types of data, especially when dealing with large sets of strings, and when strings have more similarity between them. In particular, our method presents as a prominent solution for storing DNA motifs and words of inflected natural languages. We provide the code used for the experiments.
Visibly pushdown automata are special pushdown automata whose stack behavior is driven by the input symbols according to a par- tition of the alphabet. We show that it is decidable for a given visibly pushdown automaton whether it is equivalent to a visibly counter au- tomaton, i.e. an automaton that uses its stack only as counter. In par- ticular, this allows to decide whether a given visibly pushdown language is a regular restriction of the set of well-matched words, meaning that the language can be accepted by a flnite automaton if only well-matched words are considered as input.
Imposing an extensional uniformity condition on a non-uniform circuit complexity class C means simply intersecting C with a uniform class L. By contrast, the usual intensional uniformity conditions require that a resource-bounded machine be able to exhibit the circuits in the circuit family defining C. We say that (C,L) has the "Uniformity Duality Property" if the extensionally uniform class C \cap L can be captured intensionally by means of adding so-called "L-numerical predicates" to the first-order descriptive complexity apparatus describing the connection language of the circuit family defining C. This paper exhibits positive instances and negative instances of the Uniformity Duality Property.
We examine visibly counter languages, which are languages recognized by visibly counter automata (a.k.a. input driven counter automata). We are able to effectively characterize the visibly counter languages in AC0 and show that they are contained in FO[+].
Part 1 Mathematical preliminaries: words and languages automata and regular languages semigroups and homomorphisms. Part 2 Formal languages and formal logic: examples definitions. Part 3 Finite automata: monadic second-order sentences and regular languages regular numerical predicates infinite words and decidable theories. Part 4 Model-theoretic games: the Ehrenfeucht-Fraisse game application to FO [decreasing] application to FO [+1]. Part 5 Finite semigroups: the syntactic monoid calculation of the syntactic monoid application to FO [decreasing] semidirect products categories and path conditions pseudovarieties. Part 6 First-order logic: characterization of FO [decreasing] a hierarchy in FO [decreasing] another characterization of FO [+1] sentences with regular numerical predicates. Part 7 Modular quantifiers: definition and examples languages in (FO + MOD(P))[decreasing] languages in (FO + MOD)[+1] languages in (FO + MOD)[Reg] summary. Part 8 Circuit complexity: examples of circuits circuits and circuit complexity classes lower bounds. Part 9 Regular languages and circuit complexity: regular languages in NC1 formulas with arbitrary numerical predicates regular languages and non-regular numerical predicates special cases of the central conjecture. Appendices: proof of the Krohn-Rhodes theorem proofs of the category theorems.
Recently, S.A. Cook showed that DCFL's can be recognized in O((log n)2) space and polynomial time simultaneously. We study the problem of pebbling mountain ranges (= the height of the pushdown-store as a function of time) and describe a family of pebbling strategies. One such pebbling strategy achieves a simultaneous O((log n)2/log log n) space and polynomial time bound for pebbling mountain ranges. We apply our results to DCFL recognition and show that the languages of input-driven DPDA's can be recognized in space O((log n)2/log log n). For general DCFL's we obtain a parameterized family of recognition algorithms realizing various simultaneous space and time bounds. In particular, DCFL's can be recognized in space O((log n)2) and time O(n2.87) or space O(n log n) and time O(n1.5 log log n) or space O(n/log n) and time O(n(log n)3). More generally, our methods exhibit a general space-time tradeoff for manipulating pushdownstores (e.g. run time stack in block structured programming languages).
A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.
We show that any language recognized by an NC1 circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC1 circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC1. Further, following Ruzzo (J. Comput. System Sci.22 (1981), 365–383) and Cook (Inform. and Control64 (1985) 2–22), if the branching programs are restricted to be ATIME(logn)-uniform, they recognize the same languages as do ATIME(log n)-uniform NC1 circuits, that is, those languages in ATIME(log n). We also extend the method of proof to investigate the complexity of the word problem for a fixed permutation group and show that polynomial size circuits of width 4 also recognize exactly nonuniform NC1.
We propose the class of visibly pushdown languages as embeddings of context-free languages that is rich enough to model program analysis questions and yet is tractable and robust like the class of regular languages. In our definition, the input symbol determines when the pushdown automaton can push or pop, and thus the stack depth at every position. We show that the resulting class Vpl of languages is closed under union, intersection, complementation, renaming, concatenation, and Kleene-*, and problems such as inclusion that are undecidable for context-free languages are Exptime-complete for visibly pushdown automata. Our framework explains, unifies, and generalizes many of the decision procedures in the program analysis literature, and allows algorithmic verification of recursive programs with respect to many context-free properties including access control properties via stack inspection and correctness of procedures with respect to pre and post conditions. We demonstrate that the class Vpl is robust by giving two alternative characterizations: a logical characterization using the monadic second order (MSO) theory over words augmented with a binary matching predicate, and a correspondence to regular tree languages. We also consider visibly pushdown languages of infinite words and show that the closure properties, MSO-characterization and the characterization in terms of regular trees carry over. The main difference with respect to the case of finite words turns out to be determinizability: nondeterministic Büchi visibly pushdown automata are strictly more expressive than deterministic Muller visibly pushdown automata.
We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fan-in circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm. This statement contains as special cases Yao's PARITY result [ Ya 85 ] and Razborov's new MAJORITY result [Ra 86] (MODm gate is an oracle which outputs zero, if the number of ones is divisible by m).
Recently a new connection was discovered between the parallel complexity class NC ¹ and the theory of finite automata in the work of Barrington on bounded width branching programs. There (nonuniform) NC ¹ was characterized as those languages recognized by a certain nonuniform version of a DFA. Here we extend this characterization to show that the internal structures of NC ¹ and the class of automata are closely related.
In particular, using Thérien's classification of finite monoids, we give new characterizations of the classes AC ⁰ , depth- k AC ⁰ , and ACC , the last being the AC ⁰ closure of the mod q functions for all constant q . We settle some of the open questions in [3], give a new proof that the dot-depth hierarchy of algebraic automata theory is infinite [8], and offer a new framework for understanding the internal structure of NC ¹ .
We give several characterizations, in terms of formal logic, semigroup theory, and operations on languages, of the regular languages in the circuit complexity class AC0, thus answering a question of Chandra, Fortune, and Lipton. As a by-product, we are able to determine effectively whether a given regular language is in AC0 and to solve in part an open problem originally posed by McNaughton. Using recent lower-bound results of Razborov and Smolensky, we obtain similar characterizations of the family of regular languages recognized by constant-depth circuit families that include unbounded fan-in mod p addition gates for a fixed prime p along with unbounded fan-in boolean gates. We also obtain logical characterizations for the class of all languages recognized by nonuniform circuit families in which mod m gates (where m is not necessarily prime) are permitted. Comparison of this characterization with our previous results provides evidence for a conjecture concerning the regular languages in this class. A proof of this conjecture would show that computing the bit sum modulo p, where p is a prime not dividing m, is not AC0-reducible to addition mod m, and thus that MAJORITY is not AC0-reducible to addition mod m. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/30017/1/0000385.pdf
: We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when the depth is restricted to k Gamma 1. Our Main Lemma which is of independent interest states that by using a random restriction we can convert an AND of small ORs to an OR of small ANDs and conversely. Warning: Essentially this paper has been published in Advances for Computing and is hence subject to copyright restrictions. It is for personal use only. 1. Introduction Proving lower bounds for the resources needed to compute certain functions is one of the most interesting branches of theoretical computer science. One of the ultimate goals of this branch is of course to show that P 6= NP . However, it seems that we are yet quite far from achieving this goal and that new techniques have to...