Simplest Non-Regular Deterministic Context-Free Language

We introduce a new notion of $\mathcal{C}$-simple problems for a class $\mathcal{C}$ of decision problems (i.e. languages). A problem is $\mathcal{C}$-simple if it can be reduced to each problem in $\mathcal{C}$; if this problem is, moreover, in $\mathcal{C}$, it can be viewed as a simplest problem in $\mathcal{C}$. The $\mathcal{C}$-simple problems are thus a conceptual counterpart to the common $\mathcal{C}$-hard problems (like, e.g., NP-hard problems) to which conversely any problem in $\mathcal{C}$ reduces. These definitions (of $\mathcal{C}$-simple and $\mathcal{C}$-hard problems) are parametrized by a chosen reduction that does not have a higher computational complexity than the class $\mathcal{C}$ itself. Therefore, it may be said that if a $\mathcal{C}$-hard problem has a (computationally) “easy” solution, then each problem in $\mathcal{C}$ has an “easy” solution. On the other hand, if we prove that a $\mathcal{C}$-simple problem is not “easy”, in particular that it cannot be solved by machines of a type $\mathcal{M}$ that can implement the respective reduction, then all problems in $\mathcal{C}$ are not “easy”, that is, are not solvable by $\mathcal{M}$; this extends a lower-bound result for one problem to the whole class of problems.

In this paper, we consider $\mathcal{C}$ to be the class of non-regular deterministic context-free languages, which we denote by DCFL^′; we thus have DCFL^′ = DCFL $\smallsetminus$ REG (where REG denotes the class of regular languages). We use a truth-table reduction by Mealy machines (which is motivated below). Hence a DCFL^′-simple problem is a language $L_{0}\subseteq\Sigma^{*}$ (over an alphabet $\Sigma$) that can be reduced to each DCFL^′ language $L\subseteq\Delta^{*}$ by a Mealy machine $\mathcal{A}$ with an oracle $L$, denoted $\mathcal{A}^{L}$. More precisely, the finite-state transducer $\mathcal{A}$ transforms a given input word $w\in\Sigma^{*}$ to a prefix $\mathcal{A}(w)\in\Delta^{*}$ of queries for the oracle $L$. In addition, each state $q$ of $\mathcal{A}^{L}$ is associated with a finite tuple $\sigma_{q}=(s_{q1},\ldots,s_{qr_{q}})$ of $r_{q}$ query suffixes from $\Delta^{*}$, and with a truth table $f_{q}:\{0,1\}^{r_{q}}\rightarrow\{0,1\}$. After $\mathcal{A}^{L}$ reads an input word $w$ (translating it to $\mathcal{A}(w)$), by which it enters a state $q$, for each $i\in\{1,2,\dots,r_{q}\}$ it queries whether or not the string $\mathcal{A}(w)\cdot s_{qi}$ is in $L$ (or, equivalently, whether or not $\mathcal{A}(w)$ belongs to the quotient $L/s_{qi}=\{v\in\Delta^{*}\mid v\cdot s_{qi}\in L\}$), and aggregates the answers by the truth table $f_{q}$ for deciding if $w$ is accepted.

This truth-table reduction by Mealy machines proves to be a preorder, denoted as $\leq_{tt}^{\textsc{A}}$. The main technical result of this paper is that the DCFL^′ language $L_{\#}=\{0^{n}1^{n}\mid n\geq 1\}$ (over the binary alphabet $\{0,1\}$) is DCFL^′-simple, since $L_{\#}\leq_{tt}^{\textsc{A}}L$ for each language $L$ in DCFL^′. The class DCFLS of DCFL^′-simple languages comprises REG and is a strict subclass of DCFL; e.g., the DCFL^′ language $L_{R}=\left\{wcw^{R}\mid w\in\{a,b\}^{*}\right\}$ over the alphabet $\{a,b,c\}$ proves to be not DCFL^′-simple. The closure properties of DCFLS are similar to that of DCFL as the class DCFLS is closed under complement and intersection with regular languages, while being not closed under concatenation, intersection, and union.

The above definition of DCFL^′-simple problems has originally been motivated by the analysis of the computational power of neural network (NN) models which is known to depend on the (descriptive) complexity of their weight parameters [8, 11]. The so-called analog neuron hierarchy [9] of binary-state NNs with increasing number of $\alpha$ extra analog-state neurons, denoted as $\alpha$ANN for $\alpha\geq 0$, has been introduced for studying NNs with realistic weights between integers (finite automata) and rational numbers (Turing machines). We use the notation $\alpha$ANN also for the class of languages accepted by $\alpha$ANNs, which can clearly be distinguished by the context. The separation 1ANN $\subsetneq$ 2ANN has been witnessed by the DCFL^′ language $L_{\#}\in$ 2ANN $\setminus$ 1ANN. The proof of $L_{\#}\notin$ 1ANN is rather technical (based on the Bolzano-Weierstrass theorem) which could hardly be generalized to other DCFL^′ languages, while it was conjectured that $L\notin$ 1ANN for all DCFL^′ languages $L$, that is, $\mbox{DCFL${}^{\prime}$}\subseteq(\mbox{2ANN}\,\setminus\,\mbox{1ANN})$ (implying $\mbox{1ANN}\,\cap\,\mbox{DCFL}=\mbox{0ANN}=\mbox{REG}$). An idea how to prove this conjecture is to show that $L_{\#}\notin$ 1ANN is in some sense the simplest problem in the class DCFL^′, namely, to reduce $L_{\#}$ to any DCFL^′ language $L$ by using a reduction that can be carried out by 1ANNs, which are at least as powerful as finite automata. This would imply that $L$ cannot be accepted by any 1ANN since it is at least as hard as $L_{\#}$ that has been proven not to be recognized by 1ANNs.

The idea why $L_{\#}$ should serve as the simplest language in the class DCFL^′ comes from the fact that any reduced context-free grammar $G$ generating a non-regular language $L\subseteq\Delta^{*}$ is self-embedding [3, Theorem 4.10]. This means that there is a so-called self-embedding nonterminal $A$ admitting the derivation $A\Rightarrow^{*}xAy$ for some non-empty strings $x,y\in\Delta^{+}$. Since $G$ is reduced, there are strings $v,w,z\in\Delta^{*}$ such that $S\Rightarrow^{*}vAz$ and $A\Rightarrow^{*}w$ where $S$ is the start nonterminal in $G$, which implies $S\Rightarrow^{*}vx^{m}wy^{m}z\in L$ for every $m\geq 0$. It is thus straightforward to suggest to reduce an input word $0^{m}1^{n}\in\{0,1\}^{*}$ where $m,n\geq 1$, to the string $vx^{m}wy^{n}z\in\Delta^{*}$ (while the inputs outside $0^{+}1^{+}$ are mapped onto some fixed string outside $L$) since $0^{m}1^{n}\in L_{\#}$ entails $vx^{m}wy^{n}z\in L$.

However, the suggested (one-one) reduction from $L_{\#}$ to $L$ is not consistent because $vx^{m}wy^{n}z\in L$ does not necessarily imply $0^{m}1^{n}\in L_{\#}$. For example, consider the DCFL^′ language $L_{1}=\{0^{m}1^{n}\mid 1\leq m\leq n\}$ over the binary alphabet $\Delta=\{0,1\}$ for which there are no words $v,x,w,y,z\in\Delta^{*}$ such that $vx^{m}wy^{n}z\in L_{1}$ would ensure $m=n$. Nevertheless, we can pick two inputs $0^{m}1^{n-1}$ and $0^{m}1^{n}$ instead of one, that is, $x=0$, $y=1$, and $v=w=z=\varepsilon$ ($\varepsilon$ denoting the empty string), which satisfy $0^{m}1^{n}\in L_{\#}$ iff $m=n$ iff $vx^{m}wy^{n-1}z\notin L_{1}$ and $vx^{m}wy^{n}z\in L_{1}$. It turns out that this can be generalized to any DCFL^′ language. Namely, we prove in this paper that for DCFL^′ language $L\subseteq\Delta^{*}$ over any alphabet $\Delta$, there are non-empty words $v,x,w,y,z\in\Delta^{+}$ and a language $L^{\prime}\in\{L,\overline{L}\}$, where $\overline{L}=\Delta^{*}\smallsetminus L$ is the complement of $L$, such that $0^{m}1^{n}\in L_{\#}$ iff $vx^{m}wy^{n-1}z\notin L^{\prime}$ and $vx^{m}wy^{n}z\in L^{\prime}$.

Therefore, the simple many-one (in fact, one-one) reduction from $L_{\#}$ with one query to the oracle $L$ is replaced by a truth-table reduction, that is, by a special Turing reduction in which all its finitely many (in our case two) oracle queries are presented at the same time and there is a Boolean function (a truth table) which, when given the answers to the queries, produces the final answer of the reduction. This truth-table reduction from $L_{\#}$ to $L$ can be implemented by a deterministic finite-state transducer (a Mealy machine) $\mathcal{A}$ with the oracle $L$: It transforms the input $0^{m}1^{n}$ where $m,n\geq 1$ (the inputs outside $0^{+}1^{+}$ are rejected), to the output $vx^{m}wy^{n-1}\in\Delta^{+}$ and carries out two queries to $L$ that arise by concatenation of this output with two fixed suffixes $z$ and $yz$; hence the queries are $vx^{m}wy^{n-1}z\stackrel{{\scriptstyle?}}{{\in}}L$ and $vx^{m}wy^{n}z\stackrel{{\scriptstyle?}}{{\in}}L$. The truth table is defined so that the input $0^{m}1^{n}$ is accepted by $\mathcal{A}^{L}$ iff the two answers to these queries are distinct and at same time, the first answer is negative in the case $L^{\prime}=L$, and positive in the case $L^{\prime}=\overline{L}$, which is equivalent to $0^{m}1^{n}\in L_{\#}$.

It follows that the DCFL^′ language $L_{\#}$ is DCFL^′-simple under the truth-table reduction by Mealy machines. Since this reduction can be implemented by 1ANNs, we achieve the desired stronger separation $\mbox{DCFL${}^{\prime}$}\subseteq(\mbox{2ANN}\,\setminus\,\mbox{1ANN})$ in the analog neuron hierarchy [10]. This result constitutes a non-trivial application of the proposed concept of DCFL^′-simple problem. Moreover, if we could generalize the result to (nondeterministic) context-free languages (CFL), e.g. by proving that some DCFL^′ language is CFL^′-simple (where CFL^′ $=$ CFL$\smallsetminus$ REG), which would imply that $L_{\#}$ is CFL^′-simple by the transitivity of reduction, then we would achieve even stronger separation $\mbox{CFL${}^{\prime}$}\subseteq(\mbox{2ANN}\,\setminus\,\mbox{1ANN})$. We note the interesting fact that $L_{\#}$ cannot be CSL^′-simple (under our reduction), since 1ANN accepts some context-sensitive languages outside CFL [9].

In general, if we show that some $\mathcal{C}$-simple problem under a given reduction cannot be computed by a computational model $\mathcal{M}$ that implements this reduction, then all problems in the class $\mathcal{C}$ are not solvable by $\mathcal{M}$ either. The notion of $\mathcal{C}$-simple problems can thus be useful for expanding known (e.g. technical) lower-bound results for individual problems to the whole classes of problems at once, as it was the case of the DCFL^′-simple problem $L_{\#}\notin\,\mbox{1ANN}$, expanding to $\mbox{DCFL${}^{\prime}$}\cap\,\mbox{1ANN}\,=\emptyset$. It seems worthwhile to explore if looking for $\mathcal{C}$-simple problems in other complexity classes $\mathcal{C}$ could provide effective tools for strengthening known lower bounds.

We remark that the hardest context-free language by Greibach [2] can be viewed as CFL-hard under a special type of our reduction $\leq_{tt}^{\textsc{A}}$. Related line of study concerns the types of reductions used in finite or pushdown automata with oracle. For example, nondeterministic finite automata with oracle complying with many-one restriction have been applied to establishing oracle hierarchies over the context-free languages [7]. For the same purpose, oracle pushdown automata have been used for many-one, truth-table, and Turing reducibilities, respectively, inducing the underlying definitions also to oracle nondeterministic finite automata [13]. In addition, nondeterministic finite automata whose oracle queries are completed by the prefix of an input word that has been read so far and the remaining suffix, have been employed in defining a polynomial-size oracle hierarchy [1].

In the preliminary study [12], some considerations about the simplest DCFL^′ language have appeared, yet without formal definitions of DCFL^′-simple problems, that included only sketches of incomplete proofs of weaker results based on the representation of DCFL by so-called deterministic monotonic restarting automata [5], which have initiated investigations of non-regularity degrees in DCFL [6].

In this paper we achieve a complete argument for $L_{\#}$ to be a DCFL^′-simple problem, within the framework of deterministic pushdown automata (DPDA) by using some ideas on regularity of pushdown processes from [4]. We now give an informal overview of the proof. Given a DPDA $\mathcal{M}$ recognizing a non-regular language $L\subseteq\Delta^{*}$, it is easy to realize that some computations of $\mathcal{M}$ (from the initial configuration) must be reaching configurations where the stack is arbitrarily large while it can be (almost) erased afterwards. Hence the existence of words $v,x,w,y,z\in\Delta^{+}$ such that $vx^{m}wy^{m}z\in L$ for all $m\geq 0$ is obvious. However, we aim to guarantee that for all $m,n$ the equality $m=n$ holds if, and only if, $vx^{m}wy^{n-1}z\notin L^{\prime}$ and $vx^{m}wy^{n}z\in L^{\prime}$, where $L^{\prime}$ is either the language $L$ or its complement. This is not so straightforward but it is confirmed by our detailed analysis (in section 3). We study the computation of $\mathcal{M}$ on an infinite word $a_{1}a_{2}a_{3}\cdots$ that visits infinitely many pairwise non-equivalent configurations. We use a natural congruence property of language equivalence on the set of configurations, and avoid some tedious technical details by a particular use of Ramsey’s theorem. This allows us to extract the required tuple $v,x,w,y,z\in\Delta^{+}$ from the mentioned infinite computation. We note that determinism of $\mathcal{M}$ is essential in the presented proof; we leave open if it can be relaxed to show that $L_{\#}$ is even CFL^′-simple.

The rest of the paper is organized as follows. In section 2 we recall basic definitions and notation regarding DPDA and Mealy machines, introduce the novel concept of DCFL^′-simple problems under truth-table reduction by Mealy machines and show some simple properties of the class DCFLS of DCFL^′-simple problems. In section 3 we present the proof of the main technical result which shows that $L_{\#}$ is DCFL^′-simple. Finally, we summarize the results and list some open problems in section 4.

In this section we define the truth-table reduction by Mealy machines, introduce the notion of DCFL^′-simple problems, show their basic properties, and formulate the main technical result (theorem 1). But first we recall standard definitions of pushdown automata.

A pushdown automaton (PDA) is a tuple $\mathcal{M}=(Q,\Sigma,\Gamma,R,q_{0},X_{0},F)$ where $Q$ is a finite set of states including the start state $q_{0}\in Q$ and the set $F\subseteq Q$ of accepting states, while the finite sets $\Sigma\not=\emptyset$ and $\Gamma\not=\emptyset$ represent the input and stack alphabets, respectively, with the initial stack symbol $X_{0}\in\Gamma$. In addition, the set $R$ contains finitely many transition rules $pX\xrightarrow{a}q\gamma$ with the meaning that $\mathcal{M}$ in state $p\in Q$, on the input $a\in\Sigma_{\varepsilon}=\Sigma\cup\{\varepsilon\}$ (recall $\varepsilon$ denotes the empty string), and with $X\in\Gamma$ as the topmost stack symbol may read $a$, change the state to $q\in Q$, and pop $X$, replacing it by pushing $\gamma\in\Gamma^{*}$.

By a configuration of $\mathcal{M}$ we mean $p\alpha\in Q\times\Gamma^{*}$, and we define relations $\xrightarrow{a}$ for $a\in\Sigma_{\varepsilon}$ on $Q\times\Gamma^{*}$: each rule $pX\xrightarrow{a}q\gamma$ in $R$ induces $pX\alpha\xrightarrow{a}q\gamma\alpha$ for all $\alpha\in\Gamma^{*}$; these relations are naturally extended to $\xrightarrow{w}$ for $w\in\Sigma^{*}$. For a configuration $p\alpha$ we define $\mathcal{L}(p\alpha)=\{w\in\Sigma^{*}\mid p\alpha\xrightarrow{w}q\beta\mbox{ for some }q\in F\mbox{ and }\beta\in\Gamma^{*}\}$, and $\mathcal{L}(\mathcal{M})=\mathcal{L}(q_{0}X_{0})$ is the language accepted by $\mathcal{M}$. A PDA $\mathcal{M}$ is deterministic (a DPDA) if there is at most one rule $pX\xrightarrow{a}..$ for each tuple $p\in Q$, $X\in\Gamma$, $a\in\Sigma_{\varepsilon}$; moreover, if there is a rule $pX\xrightarrow{\varepsilon}..$, then there is no rule $pX\xrightarrow{a}..$ for $a\in\Sigma$. We also use the standard assumption that all $\varepsilon$-steps are popping, that is, in each rule $pX\xrightarrow{\varepsilon}q\gamma$ in $R$ we have $\gamma=\varepsilon$.

The languages accepted by (deterministic) pushdown automata constitute the class of (deterministic) context-free languages; the classes are denoted by DCFL and CFL, respectively, whereas DCFL^′ $=$ DCFL $\smallsetminus$ REG.

In the following theorem we formulate the main technical result: any language in DCFL^′ includes a certain “projection” of the language $L_{\#}=\{0^{n}1^{n}\mid n\geq 1\}$, which means that $L_{\#}$ is in some sense the simplest language in the class DCFL^′. The theorem, whose proof will be presented in section 3, thus provides an interesting property of DCFL^′.

In order to formalize the DCFL^′-simple problems, we now define a Mealy machine $\mathcal{A}$ with an oracle: it is a tuple $\mathcal{A}=(Q,\Sigma,\Delta,\delta,\lambda,q_{0},\{(\sigma_{q},f_{q})\mid q\in Q\})$ where $Q$ is a finite set of states including the start state $q_{0}\in Q$, and the finite sets $\Sigma\not=\emptyset$ and $\Delta\not=\emptyset$ represent the input and output (oracle) alphabets, respectively. Moreover, $\delta:Q\times\Sigma\rightarrow Q$ is a (partial) state-transition function which extends to input strings as $\delta:Q\times\Sigma^{*}\rightarrow Q$ where $\delta(q,\varepsilon)=q$ for every $q\in Q$, while $\delta(q,wa)=\delta(\delta(q,w),a)$ for all $q\in Q$, $w\in\Sigma^{*}$, $a\in\Sigma$. Similarly, $\lambda:Q\times\Sigma\rightarrow\Delta^{*}$ is an output function which extends to input strings as $\lambda:Q\times\Sigma^{*}\rightarrow\Delta^{*}$ where $\lambda(q,\varepsilon)=\varepsilon$ for all $q\in Q$, and $\lambda(q,wa)=\lambda(q,w)\cdot\lambda(\delta(q,w),a)$ for all $q\in Q$, $w\in\Sigma^{*}$, $a\in\Sigma$. In addition, for each $q\in Q$, the tuple $\sigma_{q}=(s_{q1},\ldots,s_{qr_{q}})$ of strings in $\Delta^{*}$ contains $r_{q}$ query suffixes, while $f_{q}:\{0,1\}^{r_{q}}\rightarrow\{0,1\}$ is a truth table that aggregates the answers to the $r_{q}$ oracle queries.

The above Mealy machine $\mathcal{A}$ starts in the start state $q_{0}$ and operates as a deterministic finite-state transducer that transforms an input word $w\in\Sigma^{*}$ to the output string $\mathcal{A}(w)=\lambda(q_{0},w)\in\Delta^{*}$ written to a so-called oracle tape. The oracle tape is a semi-infinite, write-only tape which is empty at the beginning and its contents are only extended in the course of computation by appending the strings to the right. Namely, given a current state $q\in Q$ and an input symbol $a\in\Sigma$, the machine $\mathcal{A}$ moves to the next state $\delta(q,a)\in Q$ and writes the string $\lambda(q,a)\in\Delta^{*}$ to the oracle tape, if $\delta(q,a)$ is defined; otherwise $\mathcal{A}$ rejects the input. After reading the whole input word $w\in\Sigma^{*}$, the machine $\mathcal{A}$ is in the state $p=\delta(q_{0},w)\in Q$, while the oracle tape contains the output $\mathcal{A}(w)=\lambda(q_{0},w)\in\Delta^{*}$.

Finally, the Mealy machine $\mathcal{A}$, equipped with an oracle $L\subseteq\Delta^{*}$, in this case denoted $\mathcal{A}^{L}$, queries the oracle whether $\mathcal{A}(w)$ belongs to the (right) quotient $L/s_{pi}=\{u\in\Delta^{*}\mid u\cdot s_{pi}\in L\}$, for each suffix $s_{pi}$ in $\sigma_{p}$, and the answers are aggregated by the truth table $f_{p}$. Thus, the oracle Mealy machine $\mathcal{A}^{L}$ accepts the input word $w\in\Sigma^{*}$ iff

where $p=\delta(q_{0},w)$ and $\chi_{L/s_{pi}}:\Delta^{*}\rightarrow\{0,1\}$ is the characteristic function of $L/s_{pi}$, that is, $\chi_{L/s_{pi}}(u)=1$ if $u\cdot s_{pi}\in L$, and $\chi_{L/s_{pi}}(u)=0$ if $u\cdot s_{pi}\notin L$. The language accepted by the machine $\mathcal{A}^{L}$ is defined as $\mathcal{L}(\mathcal{A}^{L})=\{w\in\Sigma^{*}\mid w$ is accepted by $\mathcal{A}^{L}\}$.¹¹1Note that the described protocol works also for non-prefix-free languages since for any input prefix that has been read so far, the output value from the truth table determines whether the oracle Mealy machine is in an “accepting” state, deciding about this prefix analogously as a deterministic finite automaton. The truth-table reduction only requires that the given oracle answers do not influence further computation when subsequent input symbols are read.

We say that $L_{1}\subseteq\Sigma^{*}$ is truth-table reducible to $L_{2}\subseteq\Delta^{*}$ by a Mealy machine, which is denoted as $L_{1}\leq_{tt}^{\textsc{A}}L_{2}$, if $L_{1}=\mathcal{L}(\mathcal{A}^{L_{2}})$ for some Mealy machine $\mathcal{A}$ running with the oracle $L_{2}$. The following lemma shows that we can chain these reductions together since the relation $\leq_{tt}^{\textsc{A}}$ is a preorder.

Now we show that the relation $\leq_{tt}^{\textsc{A}}$ is transitive. Let $L_{1}\leq_{tt}^{\textsc{A}}L_{2}$ and $L_{2}\leq_{tt}^{\textsc{A}}L_{3}$ which means $L_{1}=\mathcal{L}(\mathcal{A}_{1}^{L_{2}})\subseteq\Sigma^{*}$ and $L_{2}=\mathcal{L}(\mathcal{A}_{2}^{L_{3}})\subseteq\Delta^{*}$ for some oracle Mealy machines $\mathcal{A}_{1}^{L_{2}}=(Q_{1},\Sigma,\Delta,\delta_{1},\lambda_{1},q_{0}^{1},\{(\pi_{q},g_{q})\mid q\in Q_{1}\})$ and $\mathcal{A}_{2}^{L_{3}}=(Q_{2},\Delta,\Theta,\delta_{2},\lambda_{2},q_{0}^{2},\{(\varrho_{q},h_{q})\mid q\in Q_{2}\})$, respectively. We will construct the oracle Mealy machine $\mathcal{A}^{L_{3}}=(Q,\Sigma,\Theta,\delta,\lambda,q_{0},\{(\sigma_{q},f_{q})\mid q\in Q\})$ such that $L_{1}=\mathcal{L}(\mathcal{A}^{L_{3}})\subseteq\Sigma^{*}$ which implies the transitivity $L_{1}\leq^{\mathcal{A}}L_{3}$. We define $Q=Q_{1}\times Q_{2}$ with $q_{0}=(q_{0}^{1},q_{0}^{2})$, $\delta((q_{1},q_{2}),a)=(\delta_{1}(q_{1},a),\delta_{2}(q_{2},\lambda_{1}(q_{1},a)))$ and $\lambda((q_{1},q_{2}),a)=\lambda_{2}(q_{2},\lambda_{1}(q_{1},a))$ for every $(q_{1},q_{2})\in Q$ and $a\in\Sigma$, which ensures $\mathcal{A}(w)=\lambda(q_{0},w)=\lambda_{2}(q_{0}^{2},\lambda_{1}(q_{0}^{1},w))=\mathcal{A}_{2}(\mathcal{A}_{1}(w))\in\Theta^{*}$ for every $w\in\Sigma^{*}$. For each state $p=(p_{1},p_{2})\in Q$ in $\mathcal{A}$, we define the tuple of query suffixes from $\Theta^{*}$,

where $\pi_{p_{1}}=(s_{p_{1},1},s_{p_{1},2}\ldots,s_{p_{1},r_{p_{1}}})\in\Delta^{r_{p_{1}}}$ and $\varrho_{p_{2}(i)}=(s_{p_{2}(i),1},s_{p_{2}(i),2}\ldots,s_{p_{2}(i),r_{p_{2}(i)}})\in\Theta^{r_{p_{2}(i)}}$ are the query suffixes associated with $p_{1}\in Q_{1}$ and $p_{2}(i)=\delta_{2}(p_{2},s_{p_{1},i})\in Q_{2}$ for $i\in\{1,\ldots,r_{p_{1}}\}$, respectively, and the truth table $f_{p}=g_{p_{1}}(h_{p_{2}(1)},\ldots,h_{p_{2}(r_{p_{1}})})$ aggregates the answers to the corresponding oracle queries, which ensures $L_{1}=\mathcal{L}(\mathcal{A}^{L_{3}})\subseteq\Sigma^{*}$. ∎

We say that a (decision) problem $L_{0}\subseteq\Sigma^{*}$ is DCFL^′-simple if $L_{0}\leq_{tt}^{\textsc{A}}L$ for every non-regular deterministic context-free language $L\subseteq\Delta^{*}$. It follows from theorem 1 that the DCFL^′ language $L_{\#}$ is an example of a DCFL^′-simple problem. In addition, we denote by DCFLS the class of DCFL^′-simple problems and formulate its basic properties.

In order to show that $\mbox{DCFLS}\,\not=\,\mbox{DCFL}$, we prove that the DCFL $L_{R}=\{wcw^{R}\mid w\in\{a,b\}^{*}\}$ over the alphabet $\{a,b,c\}^{*}$ is not DCFL^′-simple. For the sake of contradiction, suppose that $L_{R}\leq_{tt}^{\textsc{A}}L_{\#}$ by a Mealy machine $\mathcal{A}^{L_{\#}}=(Q,\{a,b,c\}^{*},\{0,1\}^{*},\delta,\lambda,q_{0},\{(\sigma_{q},f_{q})\mid q\in Q\})$ with the oracle $L_{\#}=\{0^{n}1^{n}\mid n\geq 1\}$, which means $L_{R}=\mathcal{L}(\mathcal{A}^{L_{\#}})$. Consider all the $2^{k}$ possible prefixes $w\in\{a,b\}^{k}$ of inputs presented to $\mathcal{A}^{L_{\#}}$ that have the length $|w|=k$. These strings can bring $\mathcal{A}^{L_{\#}}$ into a finite number $|\{\delta(q_{0},w)\mid w\in\{a,b\}^{k}\}|\leq|Q|$ of distinct states while the length $|\lambda(q_{0},w)|$ of outputs written to the oracle tape is bounded by $O(k)$. For $\lambda(q_{0},w)$ outside $0^{*}1^{*}$, the acceptance of words $wu$ where $u\in\{a,b,c\}^{*}$, depends only on the truth values $f_{q}(0,\ldots,0)$ associated with the states $q$ from the finite set $Q$, due to $\lambda(q_{0},wu)\notin L_{\#}/s$ for any $s\in\{0,1\}^{*}$. On the other hand, the number of distinct outputs $\lambda(q_{0},w)$ in $0^{*}1^{*}$ is bounded by $O(k)$. This means that for a sufficiently large $k\geq 1$, there must be two distinct prefixes $w_{1},w_{2}\in\{a,b\}^{k}$ such that $\delta(q_{0},w_{1})=\delta(q_{0},w_{2})$ and $\lambda(q_{0},w_{1})=\lambda(q_{0},w_{2})$ in $0^{*}1^{*}$, which results in the contradiction $w_{1}cw_{2}^{R}\in\mathcal{L}(\mathcal{A}^{L_{\#}})\smallsetminus L_{R}$.

Theorem 1 follows from the (more specific) next lemma that we prove in this section.

By $\mathbb{N}$ we denote the set $\{0,1,2,\dots\}$, and by $[i,j]$ the set $\{i,i{+}1,\dots,j\}$ (for $i,j\in\mathbb{N}$).

We note that $p_{0}X_{0}\xrightarrow{v}pX\delta\xrightarrow{x^{m}}pX\gamma^{m}\delta\xrightarrow{w}q\gamma^{m}\delta\xrightarrow{y^{m}}q\delta$ (for each $m\in\mathbb{N}$); hence $vx^{m}wy^{m}z\in L$ iff $z\in\mathcal{L}(q\delta)$ (since $z$ is nonempty). Theorem 1 indeed follows from the lemma: there is $L^{\prime}\in\{L,\overline{L}\}$ such that either $vx^{m}wy^{n}z\in L^{\prime}$ iff $m=n$ (for all $m,n\in\mathbb{N}$), or $vx^{m}wy^{n}z\in L^{\prime}$ iff $m\leq n$ (for all $m,n\in\mathbb{N}$). (In theorem 1 we also stated that $v$ is nonempty. If $v=\varepsilon$ here, then we simply take $vx$ and $yz$ as the new $v,z$, respectively.)