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 logarithmicspace manyone reducible (or mreducible) to appropriatelychosen deterministic contextfree languages and contextfree languages, respectively. The class has been extensively studied since its first appearance in 1971 by Cook [3]. It is wellknown that , where is the logarithmicspace 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 mreductions. One of these models is Cook’s auxiliary pushdown automaton model [3]. We remark that, if a working hypothesis known as the linear space hypothesis^{2}^{2}2The 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 KarpLipton style advice of polynomial length to underlying mreduction functions, we obtain advisedmreductions. 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 advisedmreductions 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 twoway 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 twoway 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’s^{3}^{3}3Throughout 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 spacebounded complexity classes (deterministic logarithmicspace class) and (nondeterministic logarithmicspace 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 (nonadvised) complexity classes as special cases.
In sharp contrast to and , oneway 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 LOGCFLLOGDCFL/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 stackstate 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 twoway deterministic and nondeterministic pushdown automata having polynomial stackstate complexities. Similarly, we introduce and in Section 5 based on the oneway 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 pushdownautomaton 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 longstanding open question concerning the complexities of and .
As for appropriate “parameterizations” of and , since they are defined by an advice form of mreductions to languages in and , we will consider a “parameterization” of those reduction functions. Section 3 will further introduce the parameterization of advisedmreduction 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 twoway machine model, we will look into two nonuniform stackstate complexity classes and based on the oneway model of pushdown automata in Section 5 because the oneway model is much easier to handle than the twoway 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 lengthpreserving 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 readonly input tape, a rewritable work tape, and (possibly) a writeonce^{4}^{4}4A tape is writeonce 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
[9] as well as [20, 21]. Given two alphabets and , a function is in if there is a deterministic Turing machine (or a DTM, for short) with a designated writeonce output tape such that, given any input , halts in polynomial time and produces on the output tape using only work space. It is important to note that this space bound is applied only to the work tape. By further supplementing KarpLipton style “advice” to underlying DTMs, we can formulate an advised version of , denoted , by analogy with . This can be done by providing such a DTM (briefly called an advised DTM) with a readonly advice tape, which carries an advice string of length over an appropriate advice alphabet , where indicates any input length. Those advice strings are provided to the advised DTM by an advice function mapping to . Such an advice function is not necessarily computable in general.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 polynomiallybounded 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 logarithmicspace 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 polynomiallybounded advice function and a function satisfying for any . Take a logarithmicspace 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 logarithmicspace manyonce reducible (or mreducible, 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 mreducible 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 advisedmreduction.
2.3 Pushdown Automata
Contextfree languages are defined by contextfree grammars. Those languages are also characterized by oneway 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 oneway 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 stackstate 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 twoway 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 inputtape 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 .
KarpLipton style advice for pushdown automata was discussed in [17] and the language family was introduced there by splitting an input tape of each underlying 1npda into two separate tracks, one of which holds a standard input string and the other holds an advice string of length equal to . For distinction, when advice is given to an underlying 1npda, we call such a machine an advised1npda to emphasize the use of advice. Since an advised1npda moves its tape head only in one direction until it either reads the right endmarker or enters a halting state before the endmarker, the 1npda reads the advice string only once from left to right. This advice model is essentially different from the one equipped with “separate” advice tapes whose heads can freely move in two directions. We will discuss this twotape model later.
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 lengthpreserving advice functions and languages . Similarly, we define using a deterministic version of advised1npda’s, which are called advised1dpda’s (with no separate advice tape).
Now, we are ready to define and using advisedmreductions.
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 .
Unlike the advised1npda’s with no separate advice tape, let us consider another model of advised1npda that holds standard input strings and advice strings on two separate tapes and move its advicetape head in two directions. Formally, a language over alphabet is in if there exist an advice alphabet , a polynomiallybounded advice function , and an advised1npda (equipped with two separate tapes) such that, for any , iff starts with on an input tape and on an advice tape and eventually accepts by moving an advicetape head freely in two directions. Obviously, follows.
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.

.

There exist a polynomiallybounded 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 polynomiallybounded function satisfying for all . We define as the set . It then follows that .
(2 1) We assume that for a polynomiallybounded 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 . ∎
Let us return to the proof of Lemma 2.4.
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 advisedmreduction function , a polynomiallybounded advice function , and an advised1npda 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 polynomiallybounded 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 polynomiallybounded 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 polynomiallybounded 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 TwoWay Auxiliary Pushdown Automata
With the use of reductions in , we have introduced the advised complexity classes and in Section 2.4. Here, we provide another characterization of them with no use of advised mreduction. This will be quite useful in the proof of our main theorem (Theorem 4.1) in Section 4. Recall that Sudborough [15] characterized (as well as ) in terms of Cook’s auxiliary pushdown automata [3]. A twoway nondeterministic auxiliary pushdown automaton (or an aux2npda, for short) is an extension of a 2npda by attaching an additional twoway rewritable work tape. As demonstrated in [15], a language belongs to iff there exists an aux2npda that recognizes in polynomial time using logarithmic work space. We further expand such an aux2npda by augmenting KarpLipton style advice as follows. For the sake of convenience, we call such a machine an advisedaux2npda. An advisedaux2npda uses an extra readonly tape called an advice tape on which an advice string is written. Note that all tape heads of the advisedaux2npda can move in two directions.
Given an advisedaux2npda , 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 aux2npda’s.
Lemma 2.7
Let be any language. The following statements are logically equivalent.

.

There exist a polynomialtime, logarithmicspace advisedaux2npda and a polynomiallybounded 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 logarithmicspace aux2npda’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 aux2npda 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 advisedaux2npda and recognizes with the help of .
(2 1) Assume that is recognized by a certain advisedaux2npda with a polynomiallybounded 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 aux2npda 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 StackState 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 stackstate 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 stackstate 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 stackstate complexity class is composed of all nonuniform families of promise decision problems solvable by appropriate families of 2npda’s whose stackstate 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 logspace 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 readonly input tape, a rewritable work tape, and a writeonce 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 logarithmicspace 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 logarithmicspace 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)