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Visibly counter languages and constant depth circuits
Abstract
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[+].
... 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.
... Results showing that some logic classes correspond to some circuit classes abound in the literature, in particular in the corpus of Lange [4,5,10,13,14,16]. The usual pattern is that the quantifier part (say, FO, FO 2 , or more exotic logics with majority quantifiers) corresponds to the allowed circuit gates, while the numerical predicates (say, +1, reg, arb, or multiplication) correspond to the allowed computing power to wire the circuit, the so-called uniformity of the circuit class. ...
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.
... Note that the context-free languages constructed in Section 8 are non-deterministic. Before looking at the full class of deterministic contextfree languages, one might first look at visibly pushdown languages [2] or the even smaller class of visibly one-counter languages [6, 23] . Visibly pushdown languages are defined by pushdown automata, where the current input symbol determines the executed operation on the pushdown. ...
We study the space complexity of querying languages over data streams in the sliding window model. The algorithm has to answer at any point of time whether the content of the sliding window belongs to a fixed regular language. For regular languages, a trichotomy is shown: For every regular language the optimal space requirement is asymptotically either constant, logarithmic, or linear in the size of the sliding window. Moreover, given a DFA for the language, one can check in nondeterministic logspace (and hence polynomial time), which of the three cases holds. For context-free languages the trichotomy no longer holds: For every there exists a context-free language for which the optimal space required to query in the sliding window model is asymptotically . Finally, it is shown that all results also hold for randomized (Monte-Carlo) streaming algorithms.
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.
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.
We obtain results within the area of dense completeness, which describes a close relation between families of formal languages and complexity classes. Previously we were able show that this relation exists between counter languages and but not between the regular languages and .
We narrow the gap between the regular languages and the counter languages by considering visibly counter languages. It turns out that they are not densely complete for . At the same time we found a restricted counter automaton model which is densely complete for .
Besides counter automata we show more positive examples in terms of L-systems.
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.
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).
Two properties, called semi-unboundedness and polynomial proof-size, are identified as key properties shared by the definitions of LOGCFL on several models of computations. The semi-unboundedness property leads to the definition of semi-unbounded fan-in circuit families. These are circuits obtained from unbounded fan-in circuits by restricting the fan-in of gates of one type. A new characterization of LOGCFL is obtained on such a model in which the fan-in of the AND gates are bounded by a constant. This property also suggests new characterizations of LOGCFL on the following models: alternating Turing machines, nondeterministic auxiliary pushdown automata, and bounded fan-in Boolean circuits.
Consider a family of boolean circuitsC1,C2,…,Cn,…, constructed by some uniform, effective procedure operating on inputn. Such a procedure provides a concise representation of a family of parallel algorithms for computing boolean values. A formula of first-order logic may also be viewed as a concise representation of a family of parallel algorithms for evaluating boolean functions. The parallelism is implicit in the quantification (a formula ∀x(x) is true if and only if each of the formulas(a) is true, and all these formulas can be checked simultaneously), and universes of different sizes give rise to boolean functions with different numbers of inputs (the boolean values of the formula's predicates on various combinations of elements of the universe). This note presents an extended first-order logic designed to be exactly equivalent in expressiveness to polynomialsize, constant-depth, unbounded-fan-in circuits constructed by Turing machines of bounded computational complexity.
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.
An alternative definition is given for a family of subsets of a free monoid that has been considered by Trahtenbrot and by McNaughton.
The computation of finite semigroups using unbounded fan-in circuits are considered. There are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset. In the case that the semigroup in fact does not contain a group, then for any primitive recursive function f circuits of size O(nf−1(n)) and constant depth exist for the semigroup product of n elements. The depth depends upon the choice of the primitive recursive function f. The circuits not only compute the semigroup product, but every prefix of the semigroup product. A consequence is that the same bounds apply for circuits computing the sum of two n-bit numbers.
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
this paper a series of languages adequate for expressing exactly those properties checkable in a series of computational complexity classes. For example, we show that a property of graphs (respectively groups, binary strings, etc.) is in polynomial time if and only if it is expressible in the first order language of graphs (respectively groups, binary strings, etc.) together with a least fixed point operator. As another example, a property is in logspace if and only if it is expressible in first order logic together with a deterministic transitive closure operator. The roots of our approach to complexity theory go back to 1974 when Fagin showed that the NP properties are exactly those expressible in second order existential sentences. It follows that second order logic expresses exactly those properties which are in the polynomial time hierarchy. We show that adding suitable transitive closure operators to second order logic results in languages capturing polynomial space and exponential time, respectively. The existence of such natural languages for each important complexity class sheds a new light on complexity theory. These languages reaffirm the importance of the complexity classes as much more than machine dependent issues. Furthermore a whole new approach is suggested. Upper bounds (algorithms) can be produced by expressing the property of interest in one of our languages. Lower bounds may be demonstrated by showing that such expression is impossible.
: 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...