Simplest Non-Regular Deterministic Context-Free Language
aa r X i v : . [ c s . F L ] F e b Simplest Non-Regular Deterministic Context-Free Language
Petr Janˇcar and Jiˇr´ı ˇS´ıma Dept of Computer Science, Faculty of Science, Palack´y University Olomouc, Czechia [email protected] Institute of Computer Science of the Czech Academy of Sciences, Prague, Czechia [email protected]
Abstract
We introduce a new notion of C -simple problems for a class C of decision problems (i.e.languages), w.r.t. a particular reduction. A problem is C -simple if it can be reduced to eachproblem in C . This can be viewed as a conceptual counterpart to C -hard problems to whichall problems in C reduce. Our concrete example is the class of non-regular deterministiccontext-free languages (DCFL ′ ), with a truth-table reduction by Mealy machines (whichproves to be a preorder). The main technical result is a proof that the DCFL ′ language L = { n n | n ≥ } is DCFL ′ -simple, which can thus be viewed as the simplest problemin the class DCFL ′ .This result has already provided an application, to the computational model of neuralnetworks 1ANN at the first level of analog neuron hierarchy. This model was proven notto recognize L , by using a specialized technical argument that can hardly be generalizedto other languages in DCFL ′ . By the result that L is DCFL ′ -simple, w.r.t. the reductionthat can be implemented by 1ANN, we immediately obtain that 1ANN cannot accept anylanguage in DCFL ′ .It thus seems worthwhile to explore if looking for C -simple problems in other classes C under suitable reductions could provide effective tools for expanding the lower-boundresults known for single problems to the whole classes of problems. Keywords : deterministic context-free language, truth-table reduction, Mealy automa-ton, pushdown automaton
We introduce a new notion of C -simple problems for a class C of decision problems (i.e.languages). A problem is C -simple if it can be reduced to each problem in C ; if this problemis, moreover, in C , it can be viewed as a simplest problem in C . The C -simple problems arethus a conceptual counterpart to the common C -hard problems (like, e.g., NP-hard problems)to which conversely any problem in C reduces. These definitions (of C -simple and C -hardproblems) are parametrized by a chosen reduction that does not have a higher computationalcomplexity than the class C itself. Therefore, it may be said that if a C -hard problem has a(computationally) “easy” solution, then each problem in C has an “easy” solution. On theother hand, if we prove that a C -simple problem is not “easy”, in particular that it cannotbe solved by machines of a type M that can implement the respective reduction, then allproblems in C are not “easy”, that is, are not solvable by M ; this extends a lower-boundresult for one problem to the whole class of problems.1n this paper, we consider C to be the class of non-regular deterministic context-freelanguages, which we denote by DCFL ′ ; we thus have DCFL ′ = DCFL r REG (where REGdenotes 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 ⊆ Σ ∗ (overan alphabet Σ) that can be reduced to each DCFL ′ language L ⊆ ∆ ∗ by a Mealy machine A with an oracle L , denoted A L . More precisely, the finite-state transducer A transforms agiven input word w ∈ Σ ∗ to a prefix A ( w ) ∈ ∆ ∗ of queries for the oracle L . In addition, eachstate q of A L is associated with a finite tuple σ q = ( s q , . . . , s qr q ) of r q query suffixes from ∆ ∗ ,and with a truth table f q : { , } r q → { , } . After A L reads an input word w (translatingit to A ( w )), by which it enters a state q , for each i ∈ { , , . . . , r q } it queries whether or notthe string A ( w ) · s qi is in L (or, equivalently, whether or not A ( w ) belongs to the quotient L/s qi = { v ∈ ∆ ∗ | v · s qi ∈ L } ), and aggregates the answers by the truth table f q for decidingif w is accepted.This truth-table reduction by Mealy machines proves to be a preorder, denoted as ≤ A tt .The main technical result of this paper is that the DCFL ′ language L = { n n | n ≥ } (over the binary alphabet { , } ) is DCFL ′ -simple, since L ≤ A tt L for each language L inDCFL ′ . The class DCFLS of DCFL ′ -simple languages comprises REG and is a strict subclassof DCFL; e.g., the DCFL ′ language L R = (cid:8) wcw R | w ∈ { a, b } ∗ (cid:9) over the alphabet { a, b, c } proves to be not DCFL ′ -simple. The closure properties of DCFLS are similar to that of DCFLas the class DCFLS is closed under complement and intersection with regular languages, whilebeing not closed under concatenation, intersection, and union.The above definition of DCFL ′ -simple problems has originally been motivated by theanalysis of the computational power of neural network (NN) models which is known to dependon the (descriptive) complexity of their weight parameters [8, 11]. The so-called analogneuron hierarchy [9] of binary-state NNs with increasing number of α extra analog-stateneurons, denoted as α ANN for α ≥
0, has been introduced for studying NNs with realisticweights between integers (finite automata) and rational numbers (Turing machines). We usethe notation α ANN also for the class of languages accepted by α ANNs, which can clearlybe distinguished by the context. The separation 1ANN ( ′ language L ∈ \ L / ∈ ′ languages, while it was conjectured that L / ∈ ′ languages L , that is,DCFL ′ ⊆ (2ANN \ ∩ DCFL = 0ANN = REG). An idea how toprove this conjecture is to show that L / ∈ ′ , namely, to reduce L to any DCFL ′ language L by using a reduction thatcan be carried out by 1ANNs, which are at least as powerful as finite automata. This wouldimply that L cannot be accepted by any 1ANN since it is at least as hard as L that hasbeen proven not to be recognized by 1ANNs.The idea why L should serve as the simplest language in the class DCFL ′ comes fromthe fact that any reduced context-free grammar G generating a non-regular language L ⊆ ∆ ∗ is self-embedding [3, Theorem 4.10]. This means that there is a so-called self-embeddingnonterminal A admitting the derivation A ⇒ ∗ xAy for some non-empty strings x, y ∈ ∆ + .Since G is reduced, there are strings v, w, z ∈ ∆ ∗ such that S ⇒ ∗ vAz and A ⇒ ∗ w where S is the start nonterminal in G , which implies S ⇒ ∗ vx m wy m z ∈ L for every m ≥
0. It isthus straightforward to suggest to reduce an input word 0 m n ∈ { , } ∗ where m, n ≥
1, tothe string vx m wy n z ∈ ∆ ∗ (while the inputs outside 0 + + are mapped onto some fixed stringoutside L ) since 0 m n ∈ L entails vx m wy n z ∈ L .2owever, the suggested (one-one) reduction from L to L is not consistent because vx m wy n z ∈ L does not necessarily imply 0 m n ∈ L . For example, consider the DCFL ′ language L = { m n | ≤ m ≤ n } over the binary alphabet ∆ = { , } for which there areno words v, x, w, y, z ∈ ∆ ∗ such that vx m wy n z ∈ L would ensure m = n . Nevertheless, wecan pick two inputs 0 m n − and 0 m n instead of one, that is, x = 0, y = 1, and v = w = z = ε ( ε denoting the empty string), which satisfy 0 m n ∈ L iff m = n iff vx m wy n − z / ∈ L and vx m wy n z ∈ L . It turns out that this can be generalized to any DCFL ′ language. Namely,we prove in this paper that for DCFL ′ language L ⊆ ∆ ∗ over any alphabet ∆, there arenon-empty words v, x, w, y, z ∈ ∆ + and a language L ′ ∈ { L, L } , where L = ∆ ∗ r L is thecomplement of L , such that 0 m n ∈ L iff vx m wy n − z / ∈ L ′ and vx m wy n z ∈ L ′ .Therefore, the simple many-one (in fact, one-one) reduction from L with one query tothe oracle L is replaced by a truth-table reduction, that is, by a special Turing reduction inwhich all its finitely many (in our case two) oracle queries are presented at the same timeand 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 beimplemented by a deterministic finite-state transducer (a Mealy machine) A with the oracle L : It transforms the input 0 m n where m, n ≥ + + are rejected), tothe output vx m wy n − ∈ ∆ + and carries out two queries to L that arise by concatenationof this output with two fixed suffixes z and yz ; hence the queries are vx m wy n − z ? ∈ L and vx m wy n z ? ∈ L . The truth table is defined so that the input 0 m n is accepted by A L iff thetwo answers to these queries are distinct and at same time, the first answer is negative in thecase L ′ = L , and positive in the case L ′ = L , which is equivalent to 0 m n ∈ L .It follows that the DCFL ′ language L is DCFL ′ -simple under the truth-table reduction byMealy machines. Since this reduction can be implemented by 1ANNs, we achieve the desiredstronger separation DCFL ′ ⊆ (2ANN \ ′ -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 r REG), whichwould imply that L is CFL ′ -simple by the transitivity of reduction, then we would achieveeven stronger separation CFL ′ ⊆ (2ANN \ L cannot be CSL ′ -simple (under our reduction), since 1ANN accepts some context-sensitivelanguages outside CFL [9].In general, if we show that some C -simple problem under a given reduction cannot becomputed by a computational model M that implements this reduction, then all problems inthe class C are not solvable by M either. The notion of C -simple problems can thus be usefulfor expanding known (e.g. technical) lower-bound results for individual problems to the wholeclasses of problems at once, as it was the case of the DCFL ′ -simple problem L / ∈ ′ ∩ ∅ . It seems worthwhile to explore if looking for C -simpleproblems in other complexity classes C could provide effective tools for strengthening knownlower 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 ≤ A tt . Related line of study concerns the typesof reductions used in finite or pushdown automata with oracle. For example, nondetermin-istic finite automata with oracle complying with many-one restriction have been applied toestablishing 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 reducibil-3ties, respectively, inducing the underlying definitions also to oracle nondeterministic finiteautomata [13]. In addition, nondeterministic finite automata whose oracle queries are com-pleted by the prefix of an input word that has been read so far and the remaining suffix, havebeen employed in defining a polynomial-size oracle hierarchy [1].In the preliminary study [12], some considerations about the simplest DCFL ′ languagehave appeared, yet without formal definitions of DCFL ′ -simple problems, that included onlysketches of incomplete proofs of weaker results based on the representation of DCFL by so-called deterministic monotonic restarting automata [5], which have initiated investigations ofnon-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 onregularity of pushdown processes from [4]. We now give an informal overview of the proof.Given a DPDA M recognizing a non-regular language L ⊆ ∆ ∗ , it is easy to realize that somecomputations of M (from the initial configuration) must be reaching configurations wherethe stack is arbitrarily large while it can be (almost) erased afterwards. Hence the existenceof words v, x, w, y, z ∈ ∆ + such that vx m wy m z ∈ L for all m ≥ m, n the equality m = n holds if, and only if, vx m wy n − z / ∈ L ′ and vx m wy n z ∈ L ′ , where L ′ is either the language L or its complement. This is not sostraightforward but it is confirmed by our detailed analysis (in section 3). We study thecomputation of M on an infinite word a a a · · · that visits infinitely many pairwise non-equivalent configurations. We use a natural congruence property of language equivalenceon the set of configurations, and avoid some tedious technical details by a particular use ofRamsey’s theorem. This allows us to extract the required tuple v, x, w, y, z ∈ ∆ + from thementioned infinite computation. We note that determinism of M is essential in the presentedproof; 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 andnotation regarding DPDA and Mealy machines, introduce the novel concept of DCFL ′ -simpleproblems under truth-table reduction by Mealy machines and show some simple propertiesof the class DCFLS of DCFL ′ -simple problems. In section 3 we present the proof of the maintechnical result which shows that L is DCFL ′ -simple. Finally, we summarize the results andlist some open problems in section 4. ′ -Simple Problem Under Truth-Table Mealy Reduc-tion In this section we define the truth-table reduction by Mealy machines, introduce the notion ofDCFL ′ -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 M = ( Q, Σ , Γ , R, q , X , F ) where Q is a finiteset of states including the start state q ∈ Q and the set F ⊆ Q of accepting states, while thefinite sets Σ = ∅ and Γ = ∅ represent the input and stack alphabets, respectively, with theinitial stack symbol X ∈ Γ. In addition, the set R contains finitely many transition rules pX a −→ qγ with the meaning that M in state p ∈ Q , on the input a ∈ Σ ε = Σ ∪ { ε } (recall ε denotes the empty string), and with X ∈ Γ as the topmost stack symbol may read a , changethe state to q ∈ Q , and pop X , replacing it by pushing γ ∈ Γ ∗ .By a configuration of M we mean pα ∈ Q × Γ ∗ , and we define relations a −→ for a ∈ Σ ε Q × Γ ∗ : each rule pX a −→ qγ in R induces pXα a −→ qγα for all α ∈ Γ ∗ ; these relations arenaturally extended to w −→ for w ∈ Σ ∗ . For a configuration pα we define L ( pα ) = { w ∈ Σ ∗ | pα w −→ qβ for some q ∈ F and β ∈ Γ ∗ } , and L ( M ) = L ( q X ) is the language accepted by M . A PDA M is deterministic (a DPDA) if there is at most one rule pX a −→ .. for each tuple p ∈ Q , X ∈ Γ, a ∈ Σ ε ; moreover, if there is a rule pX ε −→ .. , then there is no rule pX a −→ .. for a ∈ Σ. We also use the standard assumption that all ε -steps are popping, that is, in eachrule pX ε −→ qγ in R we have γ = ε .The languages accepted by (deterministic) pushdown automata constitute the class of (de-terministic) context-free languages ; the classes are denoted by DCFL and CFL, respectively,whereas DCFL ′ = DCFL r REG.In the following theorem we formulate the main technical result: any language in DCFL ′ includes a certain “projection” of the language L = { n n | n ≥ } , which means that L is in some sense the simplest language in the class DCFL ′ . The theorem, whose proof will bepresented in section 3, thus provides an interesting property of DCFL ′ . Theorem 1.
Let L ⊆ ∆ ∗ be a non-regular deterministic context-free language over an alpha-bet ∆ . There exist non-empty words v, x, w, y, z ∈ ∆ + and a language L ′ ∈ { L, L } (where L = ∆ ∗ r L is the complement of L ) such that for all m ≥ and n > we have (cid:0) vx m wy n − z / ∈ L ′ and vx m wy n z ∈ L ′ (cid:1) iff m = n . (1)In order to formalize the DCFL ′ -simple problems, we now define a Mealy machine A withan oracle : it is a tuple A = ( Q, Σ , ∆ , δ, λ, q , { ( σ q , f q ) | q ∈ Q } ) where Q is a finite set ofstates including the start state q ∈ Q , and the finite sets Σ = ∅ and ∆ = ∅ represent theinput and output (oracle) alphabets, respectively. Moreover, δ : Q × Σ → Q is a (partial)state-transition function which extends to input strings as δ : Q × Σ ∗ → Q where δ ( q, ε ) = q for every q ∈ Q , while δ ( q, wa ) = δ ( δ ( q, w ) , a ) for all q ∈ Q , w ∈ Σ ∗ , a ∈ Σ. Similarly, λ : Q × Σ → ∆ ∗ is an output function which extends to input strings as λ : Q × Σ ∗ → ∆ ∗ where λ ( q, ε ) = ε for all q ∈ Q , and λ ( q, wa ) = λ ( q, w ) · λ ( δ ( q, w ) , a ) for all q ∈ Q , w ∈ Σ ∗ , a ∈ Σ. In addition, for each q ∈ Q , the tuple σ q = ( s q , . . . , s qr q ) of strings in ∆ ∗ contains r q query suffixes, while f q : { , } r q → { , } is a truth table that aggregates the answers tothe r q oracle queries.The above Mealy machine A starts in the start state q and operates as a deterministicfinite-state transducer that transforms an input word w ∈ Σ ∗ to the output string A ( w ) = λ ( q , w ) ∈ ∆ ∗ written to a so-called oracle tape. The oracle tape is a semi-infinite, write-onlytape which is empty at the beginning and its contents are only extended in the course ofcomputation by appending the strings to the right. Namely, given a current state q ∈ Q andan input symbol a ∈ Σ, the machine A moves to the next state δ ( q, a ) ∈ Q and writes thestring λ ( q, a ) ∈ ∆ ∗ to the oracle tape, if δ ( q, a ) is defined; otherwise A rejects the input. Afterreading the whole input word w ∈ Σ ∗ , the machine A is in the state p = δ ( q , w ) ∈ Q , whilethe oracle tape contains the output A ( w ) = λ ( q , w ) ∈ ∆ ∗ .Finally, the Mealy machine A , equipped with an oracle L ⊆ ∆ ∗ , in this case denoted A L ,queries the oracle whether A ( w ) belongs to the (right) quotient L/s pi = { u ∈ ∆ ∗ | u · s pi ∈ L } ,for each suffix s pi in σ p , and the answers are aggregated by the truth table f p . Thus, theoracle Mealy machine A L accepts the input word w ∈ Σ ∗ iff f p (cid:16) χ L/s p ( A ( w )) , χ L/s p ( A ( w )) , . . . , χ L/s prp ( A ( w )) (cid:17) = 15here p = δ ( q , w ) and χ L/s pi : ∆ ∗ → { , } is the characteristic function of L/s pi , that is, χ L/s pi ( u ) = 1 if u · s pi ∈ L , and χ L/s pi ( u ) = 0 if u · s pi / ∈ L . The language accepted by themachine A L is defined as L ( A L ) = { w ∈ Σ ∗ | w is accepted by A L } . We say that L ⊆ Σ ∗ is truth-table reducible to L ⊆ ∆ ∗ by a Mealy machine , which isdenoted as L ≤ A tt L , if L = L ( A L ) for some Mealy machine A running with the oracle L .The following lemma shows that we can chain these reductions together since the relation ≤ A tt is a preorder. Lemma 2.
The relation ≤ A tt is reflexive and transitive. Proof:
The relation ≤ A tt is reflexive since L = L ( A L ) ⊆ Σ ∗ for the oracle Mealy machine A L = ( { q } , Σ , Σ , δ, λ, q, { ( σ q , f q ) } ) where δ ( q, a ) = q and λ ( q, a ) = a for every a ∈ Σ, σ q = ( ε ),and f q is the identity.Now we show that the relation ≤ A tt is transitive. Let L ≤ A tt L and L ≤ A tt L whichmeans L = L ( A L ) ⊆ Σ ∗ and L = L ( A L ) ⊆ ∆ ∗ for some oracle Mealy machines A L =( Q , Σ , ∆ , δ , λ , q , { ( π q , g q ) | q ∈ Q } ) and A L = ( Q , ∆ , Θ , δ , λ , q , { ( ̺ q , h q ) | q ∈ Q } ),respectively. We will construct the oracle Mealy machine A L = ( Q, Σ , Θ , δ, λ, q , { ( σ q , f q ) | q ∈ Q } ) such that L = L ( A L ) ⊆ Σ ∗ which implies the transitivity L ≤ A L .We define Q = Q × Q with q = ( q , q ), δ (( q , q ) , a ) = ( δ ( q , a ) , δ ( q , λ ( q , a )))and λ (( q , q ) , a ) = λ ( q , λ ( q , a )) for every ( q , q ) ∈ Q and a ∈ Σ, which ensures A ( w ) = λ ( q , w ) = λ ( q , λ ( q , w )) = A ( A ( w )) ∈ Θ ∗ for every w ∈ Σ ∗ . For each state p = ( p , p ) ∈ Q in A , we define the tuple of query suffixes from Θ ∗ , σ p = (cid:0) λ ( p , s p ,i ) · s p ( i ) ,j (cid:12)(cid:12) i = 1 , . . . , r p , j = 1 , . . . , r p ( i ) (cid:1) where π p = ( s p , , s p , . . . , s p ,r p ) ∈ ∆ r p and ̺ p ( i ) = ( s p ( i ) , , s p ( i ) , . . . , s p ( i ) ,r p i ) ) ∈ Θ r p i ) are the query suffixes associated with p ∈ Q and p ( i ) = δ ( p , s p ,i ) ∈ Q for i ∈ { , . . . , r p } , respectively, and the truth table f p = g p ( h p (1) , . . . , h p ( r p ) ) aggregates theanswers to the corresponding oracle queries, which ensures L = L ( A L ) ⊆ Σ ∗ .We say that a (decision) problem L ⊆ Σ ∗ is DCFL ′ -simple if L ≤ A tt L for every non-regular deterministic context-free language L ⊆ ∆ ∗ . It follows from theorem 1 that the DCFL ′ language L is an example of a DCFL ′ -simple problem. In addition, we denote by DCFLSthe class of DCFL ′ -simple problems and formulate its basic properties. Corollary 3 (of theorem 1) . The non-regular deterministic context-free language L = { n n | n ≥ } is DCFL ′ -simple. Proof:
Let L ⊆ ∆ ∗ be any DCFL ′ language. According to theorem 1, there are v, x, w, y, z ∈ ∆ + and L ′ ∈ { L, L } such that condition (1) holds for L ′ . We define theMealy machine A L = ( { q , q , q } , { , } , ∆ , δ, λ, q , { ( σ q , f q ) | q ∈ Q } ) with the oracle L , as δ ( q ,
0) = δ ( q ,
0) = q , δ ( q ,
1) = δ ( q ,
1) = q , λ ( q ,
0) = vx , λ ( q ,
0) = x , λ ( q ,
1) = w , λ ( q ,
1) = y , σ q = ( z, yz ), f q = f q = 0, f q (0 ,
0) = f q (1 ,
1) = 0, and f q (1 ,
0) = 1 − f q (0 , f q (0 ,
1) = 1 iff L ′ = L . It is easy to verify that L = L ( A L ), which implies L ≤ A tt L . Note that the described protocol works also for non-prefix-free languages since for any input prefix thathas been read so far, the output value from the truth table determines whether the oracle Mealy machineis in an “accepting” state, deciding about this prefix analogously as a deterministic finite automaton. Thetruth-table reduction only requires that the given oracle answers do not influence further computation whensubsequent input symbols are read. L is DCFL ′ -simple. Proposition 4.
1. REG ( DCFLS.2. DCFLS ( DCFL, and L R = { wcw R | w ∈ { a, b } ∗ } ∈ DCFL r DCFLS.3. The class DCFLS is closed under complement and intersection with regular languages.4. The class DCFLS is not closed under concatenation, intersection and union.
Proof: [Sketch.] For any regular language L , consider a Mealy machine A L with the DCFL ′ -simple oracle L , that simulates a deterministic finite automaton recognizing L , while its constant truthtables produce 1 iff associated with the accept states. Hence, L ≤ A tt L which means L isDCFL ′ -simple according to lemma 2 and corollary 3 which also implies REG = DCFLS. We first observe that DCFLS ⊆ DCFL. Let L ∈ DCFLS be any DCFL ′ -simple languagewhich ensures L ≤ A tt L by an oracle Mealy machine A L . The machine A L can be simulatedby a DPDA M which extends a suitable DPDA M (e.g. with no ε -transitions) accepting L = L ( M ), so that the finite control of M implements the finite-state transducer A whoseoutput is presented online as an input to M . Moreover, for each state q of A , the finitecontrol of M evaluates the truth table f q which aggregates the answers to the queries with r q suffixes associated with q , by inspecting at most constant number of topmost stack symbols.Hence L = L ( M ) ∈ DCFL.In order to show that DCFLS = DCFL, we prove that the DCFL L R = { wcw R | w ∈{ a, b } ∗ } over the alphabet { a, b, c } ∗ is not DCFL ′ -simple. For the sake of contradiction,suppose that L R ≤ A tt L by a Mealy machine A L = ( Q, { a, b, c } ∗ , { , } ∗ , δ, λ, q , { ( σ q , f q ) | q ∈ Q } ) with the oracle L = { n n | n ≥ } , which means L R = L ( A L ). Consider all the 2 k possible prefixes w ∈ { a, b } k of inputs presented to A L that have the length | w | = k . Thesestrings can bring A L into a finite number |{ δ ( q , w ) | w ∈ { a, b } k }| ≤ | Q | of distinct stateswhile the length | λ ( q , w ) | of outputs written to the oracle tape is bounded by O ( k ). For λ ( q , w ) outside 0 ∗ ∗ , the acceptance of words wu where u ∈ { a, b, c } ∗ , depends only on thetruth values f q (0 , . . . ,
0) associated with the states q from the finite set Q , due to λ ( q , wu ) / ∈ L /s for any s ∈ { , } ∗ . On the other hand, the number of distinct outputs λ ( q , w ) in 0 ∗ ∗ is bounded by O ( k ). This means that for a sufficiently large k ≥
1, there must be two distinctprefixes w , w ∈ { a, b } k such that δ ( q , w ) = δ ( q , w ) and λ ( q , w ) = λ ( q , w ) in 0 ∗ ∗ ,which results in the contradiction w cw R ∈ L ( A L ) r L R . The class DCFLS is closed under complement since the truth tables can be negated.Furthermore, any oracle Mealy machine be can modified so that it simulates another givenfinite automaton in parallel and is forced to reject if this automaton rejects, which showsDCFLS to be closed under intersection with regular languages. Observe that ( L ) is not DCFL ′ -simple under truth-table reduction. In addition, L = { m m n | m, n ≥ } and L = { m n n | m, n ≥ } are DCFL ′ -simple while L ∩ L isnot context-free. The proof for union follows from 3 and De Morgan’s law.7 Proof of the Main Result (Theorem 1)
Theorem 1 follows from the (more specific) next lemma that we prove in this section.By N we denote the set { , , , . . . } , and by [ i, j ] the set { i, i +1 , . . . , j } (for i, j ∈ N ). Lemma 5.
Let M = ( Q, Σ , Γ , R, p , X , F ) be a DPDA where L = L ( p X ) is non-regular(hence L belongs to DCFL ′ ). There are v ∈ Σ ∗ , x, w, y, z ∈ Σ + , p, q ∈ Q , X ∈ Γ , γ ∈ Γ + , δ ∈ Γ ∗ such that the following four conditions hold:1. p X v −→ pXδ and pX x −→ pXγ ,which entails the infinite (stack increasing) computation p X v −→ pXδ x −→ pXγδ x −→ pXγγδ x −→ pXγγγδ x −→ · · · ; (2) pX w −→ q ;3. qγ y −→ q ,hence qγ ℓ δ ′ y ℓ −→ qδ ′ for all ℓ ∈ N and δ ′ ∈ Γ ∗ ;4. one of the following cases is valid (depending on whether z ∈ L ( qδ ) or z
6∈ L ( qδ ) ):(a) L ( qγ k δ ) ∋ y ℓ z iff k = ℓ (for all k, ℓ ∈ N ), or L ( qγ k δ ) ∋ y ℓ z iff k ≤ ℓ (for all k, ℓ ∈ N );(b) L ( qγ k δ ) ∋ y ℓ z iff k = ℓ (for all k, ℓ ∈ N ), or L ( qγ k δ ) ∋ y ℓ z iff k > ℓ (for all k, ℓ ∈ N ). We note that p X v −→ pXδ x m −−→ pXγ m δ w −→ qγ m δ y m −−→ qδ (for each m ∈ N ); hence vx m wy m z ∈ L iff z ∈ L ( qδ ) (since z is nonempty). Theorem 1 indeed follows from thelemma: there is L ′ ∈ { L, L } such that either vx m wy n z ∈ L ′ iff m = n (for all m, n ∈ N ), or vx m wy n z ∈ L ′ iff m ≤ n (for all m, n ∈ N ). (In theorem 1 we also stated that v is nonempty.If v = ε here, then we simply take vx and yz as the new v, z , respectively.) Proof of Lemma 5
In the rest of this section we provide a proof of lemma 5, assuminga fixed DPDA M = ( Q, Σ , Γ , R, p , X , F ) where L = L ( p X ) is non-regular. The proofstructure is visible from the auxiliary claims that we state and prove on the way. Convention.
W.l.o.g. we assume that M always reads the whole input w ∈ Σ ∗ from p X . This can be accomplished in the standard way, by adding a special bottom-of-stacksymbol ⊥ and a (non-accepting) fail-state. (Each empty-stack configuration qε becomes q ⊥ ,and each originally stuck computation enters the fail-state where it loops. We also recall thatall ε -steps are popping, and thus infinite ε -sequences are impossible.) Hence for any infiniteword a a a · · · in Σ ω there is the unique infinite computation of M starting in p X ; itstepwise reads the whole infinite word a a a · · · .The left quotient of L by u ∈ Σ ∗ is the set u \ L = { v ∈ Σ ∗ | uv ∈ L } ; concatenation haspriority over \ , hence u u \ L = ( u u ) \ L . (The next claim is valid for any non-regular L .) Claim 6.
We can fix an infinite word a a a · · · in Σ ω ( a i ∈ Σ ) such that a a · · · a i \ L = a a · · · a j \ L for all i = j . roof: Let us consider the labelled transition system T = ( LQ ( L ) , Σ , ( a −→ ) a ∈ Σ ) where LQ ( L ) = { u \ L | u ∈ Σ ∗ } and a −→ = { ( L ′ , a \ L ′ ) | L ′ ∈ LQ ( L ) } . (We recall that L ′ = u \ L entails a \ L ′ = ua \ L .) Since L is non-regular, the set of states reachable from L = ε \ L in T is infinite. The out-degree of states in T is finite (in fact, bounded by | Σ | ), hence anapplication of K¨onig’s lemma yields an infinite acyclic path L a −→ L a −→ L a −→ · · · .We call a configuration pα of M unstable if α = Y β and R contains a rule pY ε −→ q (werecall that ε -steps are only popping); otherwise pα is stable . Since M is a deterministic PDA,for each unstable pα we can soundly define the stable successor of pα as the unique stableconfiguration p ′ α ′ where pα ε −→ p ′ α ′ ( α ′ being a suffix of α ). The path pα ε −→ p ′ α ′ might (not)go via an accepting state (in F ), hence L ( pα ) = L ( p ′ α ′ ) or L ( pα ) = { ε } ∪ L ( p ′ α ′ ). (We notethat the configurations in the computation (2) that start with pX are necessarily stable.) Claim 7.
Each configuration is visited at most twice by the computation of M from p X on a a a · · · that is fixed by claim 6. (3) Proof:
The computation (3) is infinite, stepwise reading the whole word a a a · · · , and itcan be presented as r γ a −→ r γ a −→ r γ a −→ · · · (for r γ = p X )where each r i γ i is stable; each segment r i γ i a i +1 −−−→ r i +1 γ i +1 starts with a (visible) a i +1 -stepthat is followed by a (maybe empty) sequence of (popping) ε -steps via unstable configurations.Since such an ε -sequence might go through an accepting state, we can have r i γ i = r j γ j for i = j though a a · · · a i \ L = a a · · · a j \ L ; in this case L contains precisely one of the words a a · · · a i and a a · · · a j , and the languages a a · · · a i \ L and a a · · · a j \ L differ just on ε . Nevertheless, this reasoning entails that we cannot have r i γ i = r j γ j = r ℓ γ ℓ for pairwisedifferent i, j, ℓ .Since each segment r i γ i a i +1 −−−→ r i +1 γ i +1 visits any unstable configuration at most once and r i +1 γ i +1 is the stable successor for all unstable configurations in the segment, we deduce thatalso each unstable configuration can be visited at most twice in the computation (3). Claim 8.
The computation (3) on a a a · · · can be “stair-factorized”, that is, written p X v −→ p X α v −→ p X α α v −→ p X α α α v −→ · · · (4) so that for each i ∈ N we have v i ∈ Σ + and p i X i v i −→ p i +1 X i +1 α i +1 where α i +1 is a nonemptysuffix of the right-hand side of a rule in R (i.e., a nonempty suffix of γ in a rule pX a −→ qγ ). Proof:
We consider the computation (3), and call a stable configuration pXβ a level ,with position i ∈ N , if p X a ··· a i −−−−→ pXβ and all configurations visited by the computation pXβ a i +1 a i +2 ··· −−−−−−−→ after pXβ have the stack longer than | Xβ | ; we note that each level pXβ hasa unique position pos ( pXβ ). Since each configuration is visited at most twice in (3), theset of levels is infinite, with elements p ′ X ′ , p X β , p X β , . . . where 0 ≤ pos ( p ′ X ′ ) < pos ( p X β ) < pos ( p X β ) < · · · . The computation (3) can thus be presented as p X v ′ −→ p ′ X ′ v ′′ −→ p X β v −→ p X β v −→ p X β v −→ · · · | v ′ | = pos ( p ′ X ′ ), and | v v · · · v j − | = pos ( p j X j β j ) for j ≥
1, putting v = v ′ v ′′ .Each segment pXβ v −→ p ′ X ′ β ′ between two neighbouring levels can be obviously written as pXβ a −→ qγ γ β v ′ −→ p ′ X ′ γ β where pX a −→ qγ γ is a rule in R , both γ and γ are nonempty, v = av ′ , and qγ v ′ −→ p ′ X ′ . Hence the validity of the claim is clear.We define the natural equivalence relation ∼ on the set of configurations of M : we put pα ∼ qβ if L ( pα ) = L ( qβ ).We fix the presentation (4), calling p i X i α i α i − · · · α the level-configurations (for all i ∈ N ).Since we have L ( p i X i α i α i − · · · α ) r { ε } = ( v v · · · v i − \ L ) r { ε } , there cannot be threelevel-configurations in the same ∼ -class (i.e., in the same equivalence class w.r.t. ∼ ). Henceany infinite set of level-configurations represents infinitely many ∼ -classes. Now we showa congruence-property that might enable to shorten a level-configuration while keeping its ∼ -class. We use the notation DS ( pα ) (the “down-states” of pα ), putting DS ( pα ) = { q | pα w −→ q for some w ∈ Σ ∗ } . Claim 9. If qγ ∼ qγ ′ for each q ∈ DS ( pβ ) , then pβγ ∼ pβγ ′ . Proof:
Let us consider w ∈ Σ ∗ . If w ∈ L ( pβ ), then w ∈ L ( pβµ ) for all µ ∈ Γ ∗ . If w
6∈ L ( pβ )and there is no prefix v of w such that pβ v −→ q , then w
6∈ L ( pβµ ) for all µ ∈ Γ ∗ . If w
6∈ L ( pβ )and w = vv ′ where pXβ v −→ q (necessarily for some q ∈ DS ( pXβ )), then w ∈ L ( pβµ ) iff v ′ ∈ L ( qµ ). Hence the claim is clear.The next claim is an immediate corollary. Claim 10.
Any computation p X w −→ pXβ w −→ pXβ β w −→ p ′ X ′ β β β where pX w −→ pXβ ( w ∈ Σ + ), pX w −→ p ′ X ′ β , and qβ β ∼ qβ for each q ∈ DS ( p ′ X ′ β ) can be shortenedto p X w −→ pXβ w −→ p ′ X ′ β β where p ′ X ′ β β ∼ p ′ X ′ β β β . The i -th level-configuration in (4) is reached by the computation p X v v ··· v i − −−−−−−→ p i X i α i α i − · · · α . It can happen that there are j , j , 0 ≤ j < j ≤ i such that p j X j = p j X j and qα j α j − · · · α ∼ qα j α j − · · · α for all q ∈ DS ( p i X i α i α i − · · · α j +1 ).In this case we can shorten the computation as in claim 10, where v j v j +1 · · · v j − cor-responds to the omitted w . The resulting shorter computation might be possible to berepeatedly shortened further (if it can be presented so that the conditions of claim 10 aresatisfied). Now for each i ≥ p i, X i, v i, −−→ p i, X i, α i, v i, −−→ p i, X i, α i, α i, · · · v i,ni − −−−−→ p i,n i X i,n i α i,n i α i,n i − · · · α i, (5)that has arisen by a maximal sequence of the above shortenings of the prefix p X v v ··· v i − −−−−−−→ p i X i α i α i − · · · α of (4).Hence p i, X i, = p X , p i,n i X i,n i = p i X i , α i,n i , α i,n i − , . . . , α i, is a subsequence of α i , α i − , . . . , α , and p i,n i X i,n i α i,n i α i,n i − · · · α i, ∼ p i X i α i α i − · · · α . Claim 11.
For each ℓ ∈ N there is i such that n i > ℓ (where n i is from (5)). roof: As already discussed, the set of level-configurations represents infinitely many ∼ -classes. The last configurations of computations (5) represent the same infinite set of ∼ -classes, and their lengths thus cannot be bounded; since the lengths of all α i,j are bounded(they are shorter than the longest right-hand sides of the rules in R ), the claim is clear.Now we come to a crucial claim in our proof of lemma 5. Besides the notation DS ( pα )we also introduce ES ( pα ) (the by- ε -reached down-states of pα ), by putting ES ( pα ) = { q | pα ε −→ q } .Hence ES ( pα ) ⊆ DS ( pα ), and | ES ( pα ) | ≤ M ).We recall that pα ∼ qβ means L ( pα ) = L ( qβ ). To handle the special case of the emptyword ε , we also define a (much) coarser equivalence ∼ : we put pα ∼ qβ if ε either belongsto both L ( pα ) and L ( qβ ), or belongs to none of them. Claim 12.
There is a constant B ∈ N determined by the DPDA M such that for all i ∈ N where n i > B the final configuration in (5) can be written as p i,n i X i,n i α i,n i α i,n i − · · · α i, = ¯ p ¯ Xβγδ where the following conditions hold:1. γ = α i,j α i,j − · · · α i,j ′ +1 where n i ≥ j > j ′ ≥ n i − B and p i,j X i,j = p i,j ′ X i,j ′ (and β = α i,n i α i,n i − · · · α i,j +1 , δ = α i,j ′ α i,j ′ − · · · α i, );2. the sets DS (¯ p ¯ Xβ ) and DS (¯ p ¯ Xβγ ) are equal, further being denoted by ¯ Q ;3. for each q ∈ ¯ Q , if ES ( qγ ) = { q ′ } , then ES ( q ′ γ ) = { q ′ } (and q ′ ∈ ¯ Q );4. each q ′ ∈ ¯ Q belongs to DS ( qγ ) for some self-containing q ∈ ¯ Q , where q ∈ ¯ Q is self-containing if q ∈ DS ( qγ ) ;5. there is a state q ′ ∈ ¯ Q for which q ′ γδ q ′ δ and q ′ γδ ∼ q ′ δ . Proof:
We fix some i with n i larger than a constant B determined by M as described below(there are such i by claim 11). For convenience we put p i,n i X i,n i = ¯ p ¯ X , n i = n , and α i,j = ¯ α j ,hence the final configuration in (5) is p i,n i X i,n i α i,n i α i,n i − · · · α i, = ¯ p ¯ X ¯ α n ¯ α n − · · · ¯ α . Weview the n +1 prefixes¯ p ¯ X, ¯ p ¯ X ¯ α n , ¯ p ¯ X ¯ α n ¯ α n − , ¯ p ¯ X ¯ α n ¯ α n − ¯ α n − , . . . , ¯ p ¯ X ¯ α n ¯ α n − · · · ¯ α as the vertices of a complete graph with coloured edges.For ¯ p ¯ X ¯ α n ¯ α n − · · · ¯ α = ¯ p ¯ Xµνρ , where µ = ¯ α n ¯ α n − · · · ¯ α j +1 , ν = ¯ α j ¯ α j − · · · ¯ α j ′ +1 , and ρ = ¯ α j ′ ¯ α j ′ − · · · ¯ α , n ≥ j > j ′ ≥
0, the edge between the vertices ¯ p ¯ Xµ and ¯ p ¯ Xµν has thefollowing tuple as its colour : (cid:16) p i,j X i,j , p i,j ′ X i,j ′ , DS (¯ p ¯ Xµ ) , DS (¯ p ¯ Xµν ) , ( DS ( qν ) , ES ( qν )) q ∈ DS (¯ p ¯ Xµ ) , Q , Q (cid:17) where Q = { q ′ ∈ DS (¯ p ¯ Xµ ) | q ′ νρ q ′ ρ } and Q = { q ′ ∈ Q | q ′ νρ ∼ q ′ ρ } (and p i,j X i,j , p i,j ′ X i,j ′ are taken from (5)).Since the set of colours is bounded (by a constant determined by M ), Ramsey’s theoremyields a bound B guaranteeing that there is a monochromatic clique of size 3 among thevertices ¯ p ¯ X , ¯ p ¯ X ¯ α n , ¯ p ¯ X ¯ α n ¯ α n − , . . . , ¯ p ¯ X ¯ α n ¯ α n − · · · ¯ α n − B . (We have soundly chosen i so that n = n i is bigger than B .) We fix such a monochromatic clique MC , denoting its 3 vertices as11 p ¯ Xβ , ¯ p ¯ Xβγ , ¯ p ¯ Xβγ ¯ γ , and its colour as C = ( p ′ X ′ , p ′ X ′ , ¯ Q, ¯ Q, ( D q , E q ) q ∈ ¯ Q , Q ′ , Q ′ ).This is sound, since the fact that both edges { ¯ p ¯ Xβ, ¯ p ¯ Xβγ } and { ¯ p ¯ Xβγ, ¯ p ¯ Xβγ ¯ γ } have the samecolour entails that the first component in this colour is the same as the second component,and the third component is the same as the fourth component.We now show that the conditions 1–5 are satisfied for the presentation of ¯ p ¯ X ¯ α n ¯ α n − · · · ¯ α as ¯ p ¯ Xβγδ , where δ = ¯ γ ¯ α k ¯ α k − · · · ¯ α for the respective k .Conditions 1 and 2 are trivial (due to the colour C ).Condition 3: Let q ∈ ¯ Q and ES ( qγ ) = { q ′ } (hence also q ′ ∈ ¯ Q ). Then E q = ES ( qγ ) = ES ( qγ ¯ γ ) = { q ′ } (since MC is monochromatic). This entails ES ( q ′ ¯ γ ) = { q ′ } , hence E q ′ = { q ′ } ,which in turn entails ES ( q ′ γ ) = { q ′ } .Condition 4: We first note a general fact: DS ( pµν ) = S q ∈ DS ( pµ ) DS ( qν ). Since ¯ Q = DS (¯ p ¯ Xβ ) = DS (¯ p ¯ Xβγ ) = DS (¯ p ¯ Xβγ ¯ γ ), for each q ′ ∈ ¯ Q there is thus q ∈ ¯ Q such that q ′ ∈ D q . We also have the following “transitivity”: if q , q , q ∈ ¯ Q , q ∈ D q , and q ∈ D q ,then q ∈ D q (since MC is monochromatic). For any q ′ ∈ ¯ Q there is clearly a “chain” q ′ = q , q , q , . . . , q ℓ where ℓ > q j ∈ D q j +1 for all j ∈ [1 , ℓ − q j = q ℓ for some j < ℓ .By the above transitivity, q ℓ is self-containing ( q ℓ ∈ D q ℓ and thus q ℓ ∈ DS ( q ℓ γ )) and q ′ ∈ D q ℓ (hence q ′ ∈ DS ( q ℓ γ )).Condition 5: For any three configurations at least two belong to the same ∼ -class.Since the edges among the vertices ¯ p ¯ Xβ , ¯ p ¯ Xβγ , ¯ p ¯ Xβγ ¯ γ have the same Q ′ in their colour C , we get that Q ′ = Q ′ , and thus also q ′ γδ ∼ q ′ δ for all q ′ ∈ ¯ Q such that q ′ γδ q ′ δ .Now if for all q ′ ∈ ¯ Q we had q ′ γδ ∼ q ′ δ (which includes the case ¯ Q = ∅ ), then we wouldget a contradiction with our choice of (5) since it could have been shortened as in claim 10.Now we are already close to lemma 5: Claim 13.
There are v ∈ Σ ∗ , x, w, y, z ∈ Σ + , p, q ∈ Q , X ∈ Γ , γ ∈ Γ + , δ ∈ Γ ∗ such that p X v −→ pXδ , pX x −→ pXγ , pX w −→ q , qγ y −→ q , and • either z ∈ L ( qδ ) and z
6∈ L ( qγ ℓ δ ) for all ℓ > , • or z
6∈ L ( qδ ) and z ∈ L ( qγ ℓ δ ) for all ℓ > . Proof:
We fix one ¯ p ¯ Xβγδ guaranteed by claim 12 (satisfying the respective conditions 1–5).There are v ∈ Σ ∗ , x, w, y, ¯ z ∈ Σ + , p, q ∈ Q , X ∈ Γ, γ ∈ Γ + , δ ∈ Γ ∗ , q ′ ∈ DS ( qγ ) such that p X v −→ pXδ , pX x −→ pXγ , pX w −→ q , qγ y −→ q , and L ( q ′ γδ ) and L ( q ′ δ ) differ on ¯ z (i.e., ¯ z ∈ ( L ( q ′ γδ ) r L ( q ′ δ )) ∪ ( L ( q ′ δ ) r L ( q ′ γδ )).(Indeed: The respective computation (5) can be written p X v −→ pXδ x −→ pXγδ w ′ −→ ¯ p ¯ Xβγδ where x and γ are nonempty. The claimed q ′ and [nonempty] ¯ z are guaranteed by 5 inclaim 12, and q is a respective self-containing state from 4. Since q ∈ DS (¯ p ¯ Xβ ) and q ∈ DS ( qγ ), we get pXγδ w ′ w ′′ −−−→ qγδ y −→ qδ , where w ′′ = ε . We also have y = ε , since otherwise DS ( qγ ) = ES ( qγ ) = { q } , q ′ = q , and we could not have qγδ qδ and qγδ ∼ qδ .)Since q ′ ∈ DS ( qγ ), we can fix z ′ such that qγ z ′ −→ q ′ . Hence the languages L ( qγγδ ) and L ( qγδ ) differ on z = z ′ ¯ z ; more generally, L ( qγ ℓ +1 γδ ) and L ( qγ ℓ γδ ) differ on y ℓ z for all ℓ ≥ ℓ we have z ∈ L ( qγ ℓ δ ).We recall that ¯ Q = DS (¯ p ¯ Xβ ) = DS (¯ p ¯ Xβγ ); hence S ¯ q ∈ ¯ Q DS (¯ qγ ) = ¯ Q . Since q ∈ ¯ Q , weget that DS ( qγ d ) ⊆ ¯ Q for all d ∈ N (by induction). We now distinguish two cases:12. For each prefix z of z and each d ≤ | z | we have: if qγ d z −→ ¯ q , then ES (¯ qγ ) = ∅ .2. There are a prefix z of z , d ≤ | z | , and ¯ q, q ′′ ∈ ¯ Q such that qγ d z −→ ¯ q and ES (¯ qγ ) = { q ′′ } .In the case 1 we clearly have either ∀ ℓ > | z | : z ∈ L ( qγ ℓ δ ) or ∀ ℓ > | z | : z
6∈ L ( qγ ℓ δ ) (here δ plays no role). In the case 2 we recall that ¯ qγ ε −→ q ′′ entails that ¯ qγ k δ ε −→ q ′′ δ for all k ≥ ES ( q ′′ γ ) = { q ′′ } by 3 in claim 12). Hence we have either ∀ ℓ > | z | + 1 : z ∈ L ( qγ ℓ δ ) or ∀ ℓ > | z | + 1 : z
6∈ L ( qγ ℓ δ ).Since L ( qγ δ ) and L ( qγ δ ) differ on z , we deduce that there is ℓ ≥ z ∈ L ( qγ ℓ δ ) and z
6∈ L ( qγ ℓ δ ) for all ℓ > ℓ , or z
6∈ L ( qγ ℓ δ ) and z ∈ L ( qγ ℓ δ ) for all ℓ > ℓ .Hence for ¯ δ = γ ℓ δ we have either z ∈ L ( q ¯ δ ) and z
6∈ L ( qγ ℓ ¯ δ ) for all ℓ >
0, or z
6∈ L ( q ¯ δ ) and z ∈ L ( qγ ℓ ¯ δ ) for all ℓ >
0. Since for ¯ v = vx ℓ we have p X v −→ pX ¯ δ , the claim is proven.Claim 13 is a weaker version of lemma 5; it shows that there is L ′ ∈ { L, L } such that vx m wy m z ∈ L ′ and vx m wy n z L ′ for m > n . To handle the case m < n , we have to find outfor which ℓ we have y ℓ z ∈ L ( qδ ). We thus look at the computation from qδ on the infiniteword y ω (recalling our convention that this computation is infinite, stepwise reading the word yyy · · · ), and use the obvious fact that after a prefix this computation becomes “periodic”(either cycling among finitely many configurations, or increasing the stack forever). Claim 14.
For any configuration qδ and words y, z there are numbers k ≥ and p > (“period”) such that for all ℓ ≥ k the remainder ( ℓ mod p ) determines whether or not L ( qδ ) ∋ y ℓ z . Proof:
We assume y = ε (otherwise the claim is trivial). For the infinite computation from qδ on yyy · · · there are obviously k ≥ k >
0, ¯ q ∈ Q , and ρ, µ, ν ∈ Γ ∗ such that thecomputation can be written qδ y k −−→ ¯ qρν y k −−→ ¯ qρµν y k −−→ ¯ qρµµν y k −−→ ¯ qρµµµν y k −−→ · · · where¯ qρ y k −−→ ¯ qρµ . (We have µ = ε if the computation visits only finitely many configurations, andotherwise we consider the stair-factorization of the computation.)For each j ∈ [0 , k −
1] we put ¯ qρ y j −→ ¯ qρ j , and we have two possible cases:1. There is d ≥ d ≥ d performing z from ¯ qρ j µ d ν does not reach ν atthe bottom.2. There are d ≥
0, a prefix z ′ of z , q ′ ∈ Q , and ¯ d ∈ [1 , | Q | ] such that ¯ qρ j µ d z ′ −→ q ′ and q ′ µ ¯ d ε −→ q ′ .In the case 1 either L ( qδ ) ∋ y d · k + j z for all d ≥ d , or L ( qδ ) y d · k + j z for all d ≥ d .In the case 2, for each d ≥ q ′ µ d ε −→ q d where q d = q d if d ≡ d (mod ¯ d ). Hencefor each d ≥ d , the (non)membership of y d · k + j z in L ( qδ ) is determined by ( d mod ¯ d ).The claim is thus clear.Now we finish the proof of lemma 5. We take the notation from claim 13; for the respective qδ, y, z we add k, p from claim 14. Let k be a multiple of p that is bigger than k . We nowview x k , y k , γ k as new x, y, γ , respectively. Claims 13 and 14 now yield the statement oflemma 5. 13 Conclusion and Open Problems
In this paper, we have introduced a new notion of the C -simple problem that reduces toeach problem in C , being thus a conceptual counterpart to the C -hard problem to which eachproblem in C reduces. We have illustrated this concept on the definition of the DCFL ′ -simpleproblem that reduces to each DCFL ′ language under the truth-table reduction by Mealymachines. We have proven that the DCFL ′ language L = { n n | n ≥ } is DCFL ′ -simple,and thus represents the simplest languages in the class DCFL ′ . This result finds its applicationin expanding the known lower bound for L , namely that L cannot be recognized by theneural network model 1ANN, to all DCFL ′ languages. Moreover, the class DCFLS of DCFL ′ -simple problems containing the regular languages is a strict subclass of DCFL and has similarclosure properties as DCFL.We note that the hardest context-free language L by Greibach [2], where each L inCFL is an inverse homomorphic image of L or L r { ε } , can be viewed as CFL-hard w.r.t.a many-one reduction based on Mealy machines realizing the respective homomorphisms.Our aims in the definition of DCFL ′ -simple problems cannot be achieved by such a many-one reduction, hence we have generalized it to a truth-table reduction. We can alternativelyconsider a general Turing reduction that is implemented by a Mealy machine which queriesthe oracle at special query states, each associated with a corresponding query suffix, while itsnext transition from the query state depends on the given oracle answer. The oracle Mealymachine then accepts an input word if it reaches an accept state after reading the input.The language L proves to be DCFL ′ -simple under this Turing reduction allowing for anunbounded number of online oracle queries; this can be shown by claim 13 (a weaker versionof lemma 5).It is natural to try extending our result to non-regular nondeterministic (or at least un-ambiguous) context-free languages, by possibly showing that L is CFL ′ -simple. Anotherimportant challenge for further research is looking for C -simple problems for other complexityclasses C and suitable reductions. This could provide an effective tool for strengthening lower-bounds results known for single problems to the whole classes of problems, which deserves adeeper study. Acknowledgements
Presented research has been partially supported by the Czech Science Foundation, grantGA19-05704S, and by the institutional support RVO: 67985807 (J. ˇS´ıma). J. ˇS´ıma alsothanks Martin Pl´atek for his intensive collaboration at the first stages of this research.
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