DeepAI

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

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 NC^1 and 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⊆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.

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## 1 Background and an Overview

Let us quickly review necessary background materials and overview the main contribution of this work.

### 1.1 Nonuniform State Complexity Classes

We start with looking into the parallel complexity classes and , which are the collections of all languages that are logarithmic-space many-one reducible (or -m-reducible) to appropriately-chosen deterministic context-free languages and context-free languages, respectively. The class has been extensively studied since its first appearance in 1971 by Cook [3]. It is well-known that , where is the logarithmic-space complexity class and indicates the th Nick’s class. Founded on the arguments in [6, 8], Sudborough [15] characterized as well as in terms of two different machine models with no use of -m-reductions. One of these models is Cook’s auxiliary pushdown automaton model [3]. We remark that, if a working hypothesis known as the linear space hypothesis222The linear space hypothesis (LSH) states that, for any constant , a special -complete problem, called , cannot be deterministically solved in polynomial time using space [18]. [18, 19] is true, then is different from . By further supplementing Karp-Lipton style advice of polynomial length to underlying -m-reduction functions, we obtain advised--m-reductions. These advised reductions naturally induce and respectively from and . In Section 3, we will discuss two different characterizations of those advised complexity classes and . It is not clear at present that is included in .

Toward the question, in this work, we wish to “parameterize” and by introducing a reasonable “parameterization” of the aforementioned advised--m-reductions to define and . Here, a parameterized decision problem over an alphabet refers to a pair of a language and a size parameter , where is a function assigning a “size” to each input [18, 19]. A typical size parameter is the binary length of each input .

To explain the goal of this work, we first review the old results of [1, 14] regarding the question in terms of state complexity of families of two-way finite automata. The “size” of a finite automaton can be measured by the number of inner states used by the automaton and this gives rise to the notion of state complexity. There have been fundamental studies conducted on the state complexity of various finite automata. In the 1970s, Berman and Lingas [1] and Sakoda and Sipser [14] focused particularly on the families of two-way deterministic and nondeterministic finite automata (or 2dfa’s and 2nfa’s, for short) of polynomial state complexities. After a long recess since their initial works, Kapoutsis [10, 11] revitalized the study of the subject and started a systematic study on the nonuniform setting of polynomial state complexities of 2dfa’s and 2nfa’s. Following these works, Kapoutsis [12] and Kapoutsis and Pighizzini [13] later made significant progress, and Yamakami [20, 21, 22] further expanded the study to a wider subject.

The focal points of [12, 13, 14] were set on the nonuniform state complexity classes and of promise problems solved by nonuniform families of 2dfa’s and 2nfa’s333Throughout this paper, we follow the formalism of [20, 21] and fix an input alphabet over all machines in the same family of machines. This point is different from, e.g., [10, 11, 12]. having polynomial state complexity (i.e., using inner states) in clear analogy with the complexity classes and . Similarly to nonuniform circuit families, we here cope with “nonuniform” families of machines that take input strings of “arbitrary” sizes. Berman and Lingas as well as Sakoda and Sipser discovered that a relationship between and is closely connected to another relationship between the space-bounded complexity classes (deterministic logarithmic-space class) and (nondeterministic logarithmic-space class). Later, Kapoutsis and Pighizzini demonstrated that iff , where is an advice version of and is a subclass of whose input instances given to underlying 2nfa’s are restricted to strings of polynomial lengths. This equivalence makes it possible to translate standard advised complexity classes into nonuniform state complexity classes. This phenomenon has been observed also in other nonuniform state complexity classes [20, 21], including classes induced by probabilistic and quantum finite automata. We remark that an important discovery of [20] is the fact that nonuniform state complexity classes are more closely related to parameterized complexity classes, which naturally include standard (non-advised) complexity classes as special cases.

In sharp contrast to and , one-way deterministic and nondeterministic finite automata (or 1dfa’s and 1nfa’s) equipped with polynomially many inner states depict a completely different landscape. The corresponding nonuniform state complexity classes and are proven to be distinct with no assumption (see, e.g., [11]).

### 1.2 The LOGCFL⊆LOGDCFL/poly Question

We have reviewed a logical equivalence between the question and the question. So far, similar equivalences have been observed only for families of various types of “finite automata”. Along this line of study, this work intends to expand the scope of the study of finite automata to deterministic/nondeterministic pushdown automata. Unlike finite automata, pushdown automata rely on both inner states and stack symbols. Those elements are crucial in describing the “size” of pushdown automaton because, by increasing the size of stack alphabet, we can easily reduce the number of inner states down to even . Therefore, the total number of both inner states and simultaneously pushable series of stack symbols is treated distinctively and is referred to as the stack-state complexity throughout this work (see Section 2.3 for its precise definition). For our convenience, we will introduce in Section 2.6 the notations and in direct analogy to and , respectively, using families of two-way deterministic and nondeterministic pushdown automata having polynomial stack-state complexities. Similarly, we introduce and in Section 5 based on the one-way model of pushdown automata. By analogy to the question, a direct application of “parameterizations” of and establishes the following equivalence relationship: iff . This is a pushdown-automaton analogue of the aforementioned result of Kapoutsis and Pighizzini [13]. This result strongly motivates us to conduct an intensive study on and toward answering a long-standing open question concerning the complexities of and .

As for appropriate “parameterizations” of and , since they are defined by an advice form of -m-reductions to languages in and , we will consider a “parameterization” of those reduction functions. Section 3 will further introduce the parameterization of advised--m-reduction functions, from which we can naturally define and . We will demonstrate in Section 4 a close relation between the collapse of to and the collapse of a restricted form of down to .

In contrast to the two-way machine model, we will look into two nonuniform stack-state complexity classes and based on the one-way model of pushdown automata in Section 5 because the one-way model is much easier to handle than the two-way model. This situation is similar to the known separation of [10]. We will claim the clear difference between and . We will actually show a much stronger statement (i.e., and ) than this one.

## 2 Foundations of The Rest of This Work

We will explain the basic notions and notation that the reader needs to read through the rest of this work.

### 2.1 Sets, Numbers, and Alphabets

Given a set , denotes the power set of , i.e., the set of all subsets of . The notation denotes the set of all natural numbers, including . We further set to be . For two integers and with , expresses the integer interval , opposed to real intervals. In particular, when , is abbreviated as . All logarithms are taken to the base and all polynomials are assumed to have nonnegative integer coefficients.

An alphabet is a nonempty finite set of “symbols” or “letters.” A string over alphabet is a finite sequence of symbols in and its length is the total number of symbols in the string. We use the notation for the length of string . The empty string is a unique string of length and is denoted by . Given an alphabet , the notation (resp., ) denotes the set of all strings over of length exactly (resp., at most ). The notation indicates the union . Given a string and an index , we write for the th symbol of . A language over alphabet is a subset of and its complement is , which is succinctly denoted by .

To express a compound pair of strings, we use the track notation of [16]. Given two alphabets and , the notation denotes a new alphabet consisting of all symbols of the form for and . We write each string over this new alphabet as for and , where . For notational convenience, we expand this notation to two strings and of different lengths, using a special symbol not in , as follows: if , then expresses with , and if , then indicates with . Notice that is formally a string over the compound alphabet , which is defined to be the set .

A function (resp., ) is said to be polynomially bounded if there exists a polynomial for which (resp., ) holds for all (resp., ). In contrast, is length-preserving if holds for all . A function is called polynomially honest if there is a polynomial satisfying for any .

### 2.2 FL and FL/poly

A Turing machine considered in this work is equipped with a read-only input tape, a rewritable work tape, and (possibly) a write-once444A tape is write-once if its tape head never moves to the left and, whenever it writes a nonempty symbol, it must move to the right blank cell.

output tape. For the basics of Turing machines, the reader refers to

For later use, we intend to state a useful characterization of . For the sake of completeness, we include the proof of this characterization.

###### Lemma 2.1

For any function , it follows that iff there exist a polynomially-bounded advice function and a function such that for all .

Proof.    (Only If – part) Since , we take a polynomial , an advice function , and an underlying advised DTM such that (i) holds for all and (ii) takes two inputs and written on two separate tapes and eventually produces on its output tape. We combine those two inputs to form a new string . We want to design a new DTM, say, . To ensure the logarithmic-space bound of in the following simulation, first moves its tape head to the right and marks the rightmost tape cell of the work tape by leaving a special symbol . The machine starts with input and simulates on the input string pair given on the input and the advice tapes. To remember the locations of two tape heads of for and , we also use extra work space to store the corresponding tape cell indices because of . We define to be the outcome of on input . In particular, equals for any .

(If – part) Conversely, assume that there are a polynomially-bounded advice function and a function satisfying for any . Take a logarithmic-space DTM that computes . Consider another advised DTM that behaves as follows: on input strings and on an input and an advice tapes, simulate on using an extra counter to remember the length . This requires only extra space. It is easy to check that computes correctly when is provided as .

In the original definitions of both and discussed in Section 1, -functions play a key role as reduction functions. Formally, given two languages over and over , is logarithmic-space many-once reducible (or -m-reducible, for short) to if there exists a function (called a reduction function) in such that, for any , iff . Given a language family , denotes the collection of all languages that are -m-reducible to some languages in . In a similar way, we can define by replacing “” in the above definition with “.” In the presence of advice, we use the term of advised--m-reduction.

### 2.3 Pushdown Automata

Context-free languages are defined by context-free grammars. Those languages are also characterized by one-way nondeterministic pushdown automata (or 1npda’s). A 1npda is formally defined as a nonuple , where is a finite set of inner states, is an input alphabet, and are the left and the right endmarkers, is a stack alphabet, is a transition function with , , is an initial state in , is the bottom marker in , and are sets of accepting and rejecting states in , respectively, with and . Note that is called the push size of . If further satisfies the following deterministic requirement, then it is called a one-way deterministic pushdown automaton (or a 1dpda): (i) for any and (ii) whenever , it follows that for any symbol . When is deterministic, we simply write instead of . A stack content refers to a series of symbols stored sequentially from the bottom to the top in a stack. We express such a stack content as , where and is a topmost symbol. The stack height is the length of this stack content.

A configuration of is a triplet , where , , and . This depicts a circumstance where is in inner state , a tape head is scanning the leftmost symbol of , and is a stack content. The initial configuration is . A transition indicates that, if ’s current configuration is of the form , changes to and replaces by . If , then ’s tape head must move to the right. For two configurations and , means that is obtained from by a single application of (which corresponds one step of ). If we take a finite number of steps (including zero steps), we write .

The value is referred to as the stack-state complexity of . This notion is compared to the state complexity, which indicates , of a finite automaton. Given any string , we say that accepts (resp., rejects) if begins with the initial configuration, reads , enters an inner state in (resp., ), and halts. Given a language , recognizes if (i) for any , accepts and (ii) for any , rejects . To express this language , we often use the notation

. At this moment, we formally introduce two fundamental families

and as the collections of all languages recognized by 1npda’s and by 1dpda’s, respectively.

As for two-way versions of 1npda’s and 1dpda’s, which are succinctly called 2npda’s and 2dpda’s, we modify their aforementioned definition of as follows. A new transition function maps to , where . Assume that is in inner state , scanning on an input tape and on a topmost stack cell. A transition of the form causes to change to , replace by , and move an input-tape head in direction . Note that, when reads , the tape head must stay still, i.e., must take the value .

### 2.4 Advice Extensions of LOGCFL and LOGDCFL

Let us define and , which are respectively advice versions of and , and state a few important characterizations of them. We first review an advice version of .

Consider a compound alphabet composed of two alphabets and . Given a language over and an advice function , we define as the language over . By extending this notation, for a given function , we write to denote the function defined as for all . With the help of these notations, is precisely composed of all languages for length-preserving advice functions and languages . Similarly, we define using a deterministic version of advised-1npda’s, which are called advised-1dpda’s (with no separate advice tape).

###### Definition 2.2

The advised complexity class (resp., ) is defined to be (resp., ).

The advised families and are quite robust classes in the following sense. We further strengthen this robustness in Lemma 2.4.

###### Lemma 2.3

and .

Proof.    We show only the first statement because the second one is similarly proven. Since , we instantly obtain . Conversely, let denote any language over alphabet in . Take a function , a polynomial , and a language over alphabet such that and for all . By Lemma 2.1, we further take a function , a polynomial , and an advice function for advice alphabet satisfying and for all . Note that maps to , where . Moreover, we take a language , a polynomial , and an advice function for advice alphabet such that and for all .

Here, we abbreviate as the string with a separator and we intend to set to be , which is a string over the compound alphabet . Next, we define , where in and in . Finally, let and define , which is a new reduction function from to . Since and is polynomially bounded, follows immediately. Since , we conclude that iff . Therefore, belongs to .

###### Lemma 2.4

and coincide with and , respectively.

For technical reason, we first prove the following characterization lemma.

###### Lemma 2.5

Let be any language. The following statements are logically equivalent.

1. .

2. There exist a polynomially-bounded advice function and a language such that .

The same statements also hold for .

###### Proof.

(1 2) Assume that . Since is in , there exist a function and a 1npda such that . By Lemma 2.1, there exist a function and a polynomially-bounded function satisfying for all . We define as the set . It then follows that .

(2 1) We assume that for a polynomially-bounded advice function and a language . Take a function and a language satisfying . We define for all . By Lemma 2.1, is in . It then follows that . Thus, is in . ∎

Proof of Lemma 2.4.   Hereafter, we intend to verify that . Notice that the deterministic case is similarly proven. Since , we obtain . For the other inclusion, let denote any language in . Take an advised--m-reduction function , a polynomially-bounded advice function , and an advised-1npda such that, for any string , starts with input string on an input tape and advice string on an advice tape, and holds exactly when accepts .

Since , is polynomially bounded, and thus there exists a polynomial satisfying for all . By Lemma 2.1, there exists a function and a polynomially-bounded advice function satisfying . Take another polynomial for which for all . In a way similar to the proof of Lemma 2.3, we set for each . We define as , where is of the form . Note that . We define another advice function as for any . It thus follows that iff accepts . Hence, we obtain . By Lemma 2.5, this implies that is in .

The characterizations given in Lemmas 2.3 and 2.4 leave unstated the use of another plausible complexity class . This is because it is unclear at present that coincides with . See Section 6 for a more discussion.

Concerning and , it is known that iff (see, e.g., [21]). A similar equivalence also holds for and , as shown in the next lemma. This fact will be used in Section 4.2.

###### Lemma 2.6

if and only if .

Proof.    The implication from left to right is trivial since is properly included in . Conversely, assume that . Let denote any language over alphabet in . By Lemma 2.5, there are an advice alphabet , a polynomially-bounded advice function , and a language over the compound alphabet satisfying . For readability, we write for . Our assumption then yields . We then apply Lemma 2.5 for and obtain an advice alphabet , a polynomially-bounded advice function , and a language over the alphabet satisfying . As a new advice function , we set for any and define by treating as a string over . It then follows that . This obviously implies that .

### 2.5 Two-Way Auxiliary Pushdown Automata

Given an advised-aux-2npda , an advice function , and a language , we say that recognizes with the help of if (i) takes standard input and advice string and (ii) for any , if , then accepts, and otherwise, rejects.

The following is an advice version of Sudborough’s characterization of in terms of aux-2npda’s.

###### Lemma 2.7

Let be any language. The following statements are logically equivalent.

1. .

2. There exist a polynomial-time, logarithmic-space advised-aux-2npda and a polynomially-bounded advice function such that recognizes with the help of .

The same statements hold for as well.

Proof.    (1 2) This follows from [15], in which is characterized by logarithmic-space aux-2npda’s running in polynomial time. Let denote any language in . By Lemma 2.5, there exist a language in and an advice function for which . Since , by [15], there exists an aux-2npda that recognizes in polynomial time using logarithmic work space. We thus conclude that, if , then accepts , and otherwise, rejects . Notice that uses a single input tape, which is made up of two tracks. We split these two tracks of the input tape of into two separate tapes, one of which is an advice tape for an advice string. We denote by the obtained machine. Clearly, is an advised-aux-2npda and recognizes with the help of .

(2 1) Assume that is recognized by a certain advised-aux-2npda with a polynomially-bounded advice function in polynomial time using logarithmic space. Recall that has both an input tape and an advice tape with two separate tape heads along them other than an auxiliary tape as well as a stack. Since all tape heads of freely move in two directions, it is possible to treat an input tape and an advice tape as two tracks of a single input tape equipped with a single tape head. It is important to note that this modification requires additional memory bits to remember the locations of the two tape heads of the input and the advice tapes. We therefore obtain a new aux-2npda that takes an input of the form and simulates on the pair of input strings. We define for any and set . By the characterization of [15], belongs to . Since , Lemma 2.5 concludes that is in .

### 2.6 Families of Promise Problems and Stack-State Complexity Classes

Given an alphabet , a promise decision problem over is a pair of sets satisfying that and , where is viewed as a set of “positive” instances and represents a set of “negative” instances of the promise decision problem. Naturally, we expand a single promise decision problem to a “family” of promise decision problems over a single alphabet. Fix an alphabet and let denote such a family of promise decision problems over . It is important to remark that does not depend on the choice of (see, e.g., [20, 21, 22]). All strings in are distinguished as valid or promised strings. At this moment, we demand neither nor for any distinct pair . We say that has a polynomial ceiling if there exists a polynomial satisfying for all indices .

To solve a family of promise decision problems, we use a “family” of pushdown automata. Such a family is expressed as , where each has the form with , where . This machine family is said to solve if, for any index , (i) for all , accepts and (ii) for all , rejects . For any other string outside of , may possibly neither accept nor reject it. moreover, we do not demand any “uniformity” of , that is, any existence of a fixed algorithmic procedure that generates from the description of each machine .

In Section 1, we have already discussed families of 2nfa’s and 2dfa’s of polynomial state complexities. In stark contrast to the state complexity of 2nfa’s and 2dfa’s, we need to consider the stack-state complexity of 2npda’s and 2dpda’s because we can reduce the number of inner states of pushdown automata at will by increasing their stack alphabet size. In this work, we are interested in families of 2npda’s and 2dpda’s having polynomial stack-state complexities. Analogously to the nonuniform classes and , we introduce two complexity classes, and , where the suffix “PD” stands for “pushdown”.

###### Definition 2.8

The nonuniform stack-state complexity class is composed of all nonuniform families of promise decision problems solvable by appropriate families of 2npda’s whose stack-state complexities are bounded from above by a fixed polynomial. Moreover, consists of all families in that have polynomial ceilings. In a similar manner, we define using 2dpda’s instead of 2npda’s.

If we use 1npda’s and 1dpda’s in place of 2npda’s and 2dpda’s, we analogously obtain and , respectively. These nonuniform complexity classes will be extensively discussed in Section 5.

## 3 Parameterizations of LOGCFL/poly and LOGDCFL/poly

Toward the main goal of this work, we intend to parameterize the complexity classes and as well as their advised counterparts and .

### 3.1 Parameterized Complexity Classes

Similarly to and defined in [21, 22], we wish to seek proper “parameterizations” of and , including and as their special cases. For readability, we will follow the basic terminology used in [18, 19, 20, 21, 22]. A parameterized decision problem is a pair of a language and a size parameter . We are particularly interested in size parameters computable using logarithmic space. A log-space size parameter is a function from to for a given alphabet for which its associated function mapping a string to the string of the form belongs to [18]; that is, there is a DTM (equipped with a read-only input tape, a rewritable work tape, and a write-once output tape) such that, for any string , takes as an input and produces on its output tape in time using work space.

How can we define and in a reasonable and systematic way? Since and are defined in terms of -functions in Definition 2.2, we first need to look for a natural parameterization of -functions. Parameterizations of logarithmic-space computation was also discussed in, e.g., [2, 4]. Given a function for two alphabets and and a size parameter , the pair belongs to if is logarithmic-space computable and there exists an advised DTM , an advice function for an advice alphabet , and a polynomial such that takes an input string on its input tape and an advice string on its advice tape, and produces on its output tape within time using space , provided that satisfies . With the use of , we introduce for each language family as the collection of all parameterized problems such that there exist a function and a language satisfying: (i)