Grammars with Restricted Derivation Trees - Bulletin of the ACM ...

Grammars with Restricted Derivation TreesJiří Koutný ∗Department of Information SystemsFaculty of Information TechnologyBrno Universtity of TechnologyBožetěchova 1/2, 616 66 Brno, Czech Republicikoutny@fit.vutbr.czAbstractThis extended abstract of the doctoral thesis studies theoreticalproperties of grammars with restricted derivationtrees. After presenting the state of the art concerning thisinvestigation area, the research is focused on the threemain kinds of the restrictions placed upon the derivationtrees. First, it introduces completely new investigationarea represented by cut-based restriction and examinesthe power of the grammars restricted in this way. Second,it investigates several new properties of path-basedrestriction placed upon the derivation trees. Specifically,it studies the impact of erasing productions on the powerof grammars with restricted path and introduces two correspondingnormal forms. Then, it describes a new relationbetween grammars with restricted path and somepseudoknots. Next, it presents a counterargument to thepower of grammars with controlled path that has beenconsidered as well-known so far. Finally, it introduces ageneralization of path-based restriction to not just onebut several paths. The model generalized in this way isstudied, namely its pumping, closure, and parsing properties.Categories and Subject DescriptorsF.4.3 [Mathematical Logic and Formal Languages]:Formal Languages—Classes defined by grammars or automata,Operations on languagesKeywordstree controlled grammars, level controlled grammars, pathcontrolled grammars, paths controlled grammars, cut controlledgrammars, ordered cut controlled grammars, regulatedrewriting, restricted derivation trees∗ Recommended by thesis supervisor: Prof. AlexanderMeduna. Defended at Faculty of Information Technology,Brno University of Technology, Brno on September 18,2012c⃝ Copyright 2012. All rights reserved. Permission to make digitalor hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies show this notice onthe first page or initial screen of a display along with the full citation.Copyrights for components of this work owned by others than ACMmust be honored. Abstracting with credit is permitted. To copy otherwise,to republish, to post on servers, to redistribute to lists, or to useany component of this work in other works requires prior specific permissionand/or a fee. Permissions may be requested from STU Press,Vazovova 5, 811 07 Bratislava, Slovakia.Koutný, J.: Grammars with Restricted Derivation Trees. InformationSciences and Technologies Bulletin of the ACM Slovakia, Vol. 4, No. 4(2012) 15-231. IntroductionThe formal language theory is an inherent part of thetheoretical computer science particularly concerned withthe study of the formal models. The formal models aremathematical objects used to describe the formal languages.The fundamental models include grammars andautomata. The former are used to generate words andthe latter accept them.Grammars are the kind of the rewriting models that startfrom a specified symbol (i.e., start symbol). Then, thesymbol is modified according to the given set of rewritingproductions. Each production is composed of twocomponents—the left-hand side and the right-hand sideof a production. The application of a production on aword is done by rewriting a symbol equivalent to the lefthandside of a production by its right-hand side in theword. This process is known as a derivation step. Duringthe computation of a derivation step, just one symbol isrewritten in the word. Given a start symbol of a grammar,a derivation step is computed repeatedly by applying theproductions from the given set. Once the resulting wordis composed of the symbols that cannot be rewritten anymore,the process of applying derivation steps ends andthe resulting word belongs to the language of the grammar.Essentially, the grammars are composed of a finitely manysymbols that are rewritten by finitely many production infinitely many derivation steps. In this way, the grammarsrepresent a finite description of even infinite languages.By the notion of infinite languages are meant those languagesthat contains infinitely many words. Since themost of the languages commonly used in practice are infinite,the grammars represent a powerful tool how to dealwith them. In the formal language theory, there exists ahuge variety of grammars which essentially differ in twodomains. Specifically, in the complexity of the productionsand in the way how to select appropriate productionto be applied in a derivation step.Generally, the complexity of rewriting productions can beseen from two angles—theoretical and practical. Theoreticalviewpoint: As little as possible restrictions placed onthe form of the rewriting productions in a rewriting modelis desirable. More specifically, the more complex rewritingproductions are, the larger class of languages may themodel generate. In other words, to generate complex languages,complex productions are needed. By the notionof a form of a production, namely the number of the symbolson its left-hand side is meant. Practical viewpoint:Grammars are theoretical models that are implemented

Information Sciences and Technologies Bulletin of the ACM Slovakia, Vol. 4, No. 4 (2012) 15-23 17basic properties of tree controlled grammars—namely, themembership problem and the power.Based on the original definition of a tree controlled grammar,Păun studies the modifications where many wellknowntypes of both controlled grammars and control languagesare considered in [33]. More precisely, Păun studiescontrolling the levels of the derivation trees of a regulargrammar by several types of a control language, controllingthe levels of the derivation trees of a context-freegrammar without erasing productions by several types ofa control language, and controlling the levels of the derivationtrees of a context-free grammar by a finite language.It is well-known that tree controlled grammars with acontext-free grammar controlled by a regular languagecharacterize the class of recursively enumerable languages.Thus, the question arises whether or not it is possible toachieve the same power as tree controlled grammars havewhen the levels of the derivation trees are restricted by asubregular control language. This problem is studied byDassow and Truthe in [13], where many types of subregularlanguages are considered. Dassow and Truthe studyprimarily controlling the levels of the derivation trees ofa context-free grammar by two types of a language suchthat one is a subset of the other and controlling the levelsof the derivation trees of a context-free grammar bymany different types of subregular languages. The sameauthors, Dassow and Truthe, also study hierarchies ofsubregularly tree controlled languages in [11] and [12].They present controlling the levels of the derivation treesof a context-free grammar by the union of monoids, byregular languages with restricted descriptional complexity,and by the language accepted by a deterministic finiteautomaton with at most given number of states.Stiebe in [40] states that there is a tree controlled grammarfor every linearly bounded queue automaton. Then,Stiebe proves that controlling the levels of the derivationtrees of a context-free grammar by the language acceptedby a minimal finite automaton with at most five statescharacterize the class of context-sensitive languages. If,additionally, erasing productions in a controlled grammarare allowed, controlling the levels of the derivation treesof a context-free grammar by the language accepted by aminimal finite automaton with at most five states characterizesthe class of recursively enumerable languages.Turaev, Dassow, and Selamat in [41] examine tree controlledgrammars with bounded nonterminal complexityand demonstrate that seven nonterminals in a tree controlledgrammar are enough to generate any recursivelyenumerable language. Then, they establish that threenonterminals in a tree controlled grammar are enoughto generate any regular language and any regular simplematrix language can be generated by a tree controlledgrammar with three nonterminals. Finally, they demonstratethat three nonterminals in a tree controlled grammarare enough to generate any linear language. Thesame authors in [42] state several further nonterminalcomplexity-relatedproperties of tree controlled grammars.2.2 Path Based RestrictionAs an attempt to increase the power of context-free grammarswithout changing the basic formalism and loosingsome basic properties of context-free languages (decidability,efficient parsing, etc.), Marcus, Martín-Vide, Mitrana,and Păun in [26] study a new type of a restrictionin a derivation: a derivation tree in a context-free grammaris accepted if it contains a path described by a controllanguage. More precisely, they consider two context-freegrammars, G and G ′ . A word, w, generated by G belongsto the language defined by G and G ′ if there is a derivationtree, t, for w in G such that there exists a path oft described by the language of G ′ . Based on this restriction,they introduce a path controlled grammar and studyseveral properties of this model. Specifically, they studycontrolling a path of the derivation trees of several typesof grammars by a regular language and controlling a pathof the derivation trees of a regular grammar by a linear orcontext-free language. Then, they establish two kinds ofpumping properties depending on the type of a controlledgrammar, some consequences to the closure properties ofpath controlled grammars, and a basic parsing propertyfor path controlled grammars. They also investigate thepower of path controlled grammars. However, there existsa serious problem with the correctness of the proof theypresent.As a continuation of the investigation of path-based restrictions,Martín-Vide and Mitrana study parsing propertiesof path controlled grammars, closure properties ofpath controlled grammars, and several decision problemsfor path controlled grammars in [27] and [28].2.3 Goals of the ThesisAs it clearly follows from the previous sections, levelbasedrestriction is well-studied and the most of the importantquestions have been answered. On the otherhand, in the case of path-based restriction many basicproperties including the generative power have not beensuccessfully investigated yet. Moreover, several other restrictionsplaced upon the derivation trees have not yetbeen introduced at all. Indeed, the restrictions placedupon the cuts of the derivation tree as well as upon severalpaths of the derivation trees represent completely newinvestigation areas. Thus, the goals of the doctoral thesisconsist in three investigation areas.• First, to introduce new investigation area representedby cut-based restrictions and establish the generativepower of the model restricted in this way.• Second, to establish several new results in the investigationof one-path-restricted grammars introducedin [26].• Third, to generalize one-path-restricted model intoseveral paths and investigate several its properties.3. New Results in Restrictions Placed upon DerivationTreesFirst we reformulate the fundamental definitions so thatall derivation-tree-based restrictions can be studied usingthe same notation. Then, we present new results in thederivation-tree based restrictions investigation area.Since restriction placed upon a level, a path, and a cut is,in essence, a restriction placed upon a derivation tree, weuse a slightly modified but equivalent formulation of thedefinitions stated in [26], [27], and [28]. Consequently,aforementioned modifications allow us to investigate allderivation-tree-based restrictions using the same terminology—i.e.,restriction on the levels (see [9], [13], [33],

Information Sciences and Technologies Bulletin of the ACM Slovakia, Vol. 4, No. 4 (2012) 15-23 19ord-cut-TC(CF, REG) = { ord-cut L(G, R) : (G, R) isa tree controlled grammar in which G is a context-freegrammar and R ∈ REG} and the class of tree controlledlanguages with ε-free controlled grammar under orderedcuts control is defined asord-cut-TC ε(CF ε, REG) = { ord-cut L(G, R) : (G, R) isa tree controlled grammar in which G is an ε-free contextfreegrammar and R ∈ REG}.3.1.2 ResultsConcerning cut-based restriction placed upon the derivationtrees, we have introduced two fundamental types ofsuch kind of a restriction and thus, we have opened anew investigation area in derivation-tree-restricted models.Next, we have proved that both restrictions increasethe power of context-free grammars so they characterizeRE:ord-cut-TC(CF, REG) = cut-TC(CF, REG) = RE.An important open problem consists of the investigationof cut controlled grammars where ε-productions are forbidden.Consequently, the grammars restricted in thisway should be placed into the relation with some otherwell-known language families, such as CS, and the decidingthe question whether or not:ord-cut-TC ε(CF ε, REG) = CS,cut-TC ε (CF ε , REG) = CS.Next problem is represented by the descriptional complexityof ord-cut-TC(CF, REG) and cut-TC(CF, REG).The results presented above are based on the transformationof an unrestricted grammar into Pentonnen normalform. However, using Geffert normal form, the numberof nonterminals in the resulting cut controlled grammarwould be reduced.Another future research idea is represented by the controllingthe cuts of the derivation trees in which severaltypes of subregular control languages are considered. Inthis way, the question whether or not a kind of a subregurallanguage is enough to increase the power of controlledgrammar properly. Consequently, the relation betweenthe power of level-based and cut-based models restrictedin this way would be founded out.3.2 Path Based RestrictionIn this section, we introduce a path-based restriction ontree-controlled grammars that is equivalent to the modelintroduced in [26], [27], and [28]. Next, we formally definethe pseudoknot structure represented as a language. Weintroduce new results related to the normal forms and thepresence of erasing productions in a controlled grammar.Then, this section presents a relationship between biologyand the formal language theory in the form of word representationof pseudoknots generated by path controlledgrammars. Last section of this part being a counterargumentagainst the proof of the power of path controlledgrammars that has been considered as correct so far.3.2.1 DefinitionsDefinition 3.13. Let (G, R) be a tree controlled grammar.The language that (G, R) generates under the pathcontrol by R is denoted by path L(G, R) and defined by thefollowing equivalence:For all x ∈ T ∗ , x ∈ path L(G, R) if and only if thereis a derivation tree, t ∈ G△(x), such that there is a path,p, of t with word(p) ∈ R.Definition 3.14. For X, Y ∈ {LIN, CF}, the class oftree controlled languages under the path control is definedaspath-TC(X, Y) = { path L(G, R) : (G, R) is a tree controlledgrammar in which G ∈ G X and R ∈ Y} and theclass of tree controlled languages with ε-free controlledgrammar under path control is defined aspath-TC ε (CF ε, Y) = { path L(G, R) : (G, R) is a treecontrolled grammar in which G is an ε-free context-freegrammar and R ∈ Y}.Definition 3.15. Let (G, R) be a tree controlled grammarthat generates the language under path control by R,where G = (V, T, P, S). (G, R) is in 1 st normal form ifevery production, r : A → x ∈ P , is of the form A ∈ V −Tand x ∈ T ∪ (V − T ) ∪ (V − T ) 2 .Definition 3.16. Let (G, R) be a tree controlled grammarthat generates the language under path control by R,where G = (V, T, P, S). (G, R) is in 2 nd normal form ifevery production, r : A → x ∈ P , is of the form A ∈ V −Tand x ∈ T ∪ ((V ∪ {E}) − T ) 2 where E ∩ V = ∅ andE → ε ∈ P . The alphabet of G should now include E,with E ∉ V .Definition 3.17. Let Σ be an alphabet. The followinglanguages over Σ are pseudoknots:1. {xyx R y R : x, y ∈ Σ ∗ },{u 1xu 2yu 3x R u 4y R u 5 : x, y, u i ∈ Σ ∗ , 1 ≤ i ≤ 5},2. {xyx R zz R y R : x, y, z ∈ Σ ∗ },{u 1xu 2yu 3x R u 4zu 5z R u 6y R u 7 : x, y, z, u i ∈ Σ ∗ , 1 ≤i ≤ 7},3. {xyx R zy R z R : x, y, z ∈ Σ ∗ },{u 1xu 2yu 3x R u 4zu 5y R u 6z R u 7 : x, y, z, u i ∈ Σ ∗ , 1 ≤i ≤ 7},4. {xyzx R y R z R : x, y, z ∈ Σ ∗ },{u 1xu 2yu 3zu 4x R u 5y R u 6z R u 7 : x, y, z, u i ∈ Σ ∗ , 1 ≤i ≤ 7}.Note that presented pseudoknots form obviously non-context-freelanguages.3.2.2 ResultsAs a continuation of the investigation of path-based restrictionsintroduced in [26] and studied in [27] and [28],we have considered the impact of ε-productions in pathcontrolled grammars to the power and we have statedthat ε-productions can be removed from a path controlledgrammar without affecting its language:path-TC(CF, CF) = path-TC ε (CF ε , CF).

20 Koutný, J.: Grammars with Restricted Derivation TreesWe have established two Chomsky-like normal forms forpath controlled grammars and we have formulated algorithmsthat transform a path controlled grammar into itsnormal form:• Let L ∈ path-TC(CF, CF). Then, there existsa tree controlled grammar, (G, R), in 1 st normalform such that L = path L(G, R).• Let L ∈ path-TC(CF, CF). Then, there existsa tree controlled grammar, (G, R), in 2 nd normalform such that L = path L(G, R).A future investigation idea consists of the modifying ageneral parsing methods that are based on Chomsky normalform such that it will be able to parse path controlledgrammars in a polynomial time.Another practically motivated idea is represented the relationbetween path controlled grammars and the theoryof pseudoknots. We have demonstrated several typicalpseudoknots used in biology represented by the words ofnon-context-free languages. We have demonstrated somepseudoknots belong to path-TC(LIN, LIN):{xyx R y R : x, y ∈ Σ ∗ } ∈ path-TC(LIN, LIN),{xyx R zz R y R : x, y, z ∈ Σ ∗ } ∈ path-TC(LIN, LIN),{xyx R zy R z R : x, y, z ∈ Σ ∗ } ∈ path-TC(LIN, LIN).Apparently, there is a huge variety of another pseudoknotstructures in biology. For example, {xyzx R y R z R :x, y, z ∈ Σ ∗ } and it is an open question whether or notthose pseudoknots can be generated by tree controlledgrammars with linear components that generate the languageunder path control.The last presented result deals with a reflection on thepower of path controlled grammars that has been consideredas well-known (see [26]) for more than last tenyears. However, we have presented an argument againstthe correctness of the proof given in [26] that statespath-TC(CF, CF) ⊆ MAT.We have concluded the counterargument by stating thatthe aforementioned inclusion still may hold; however, itcannot be proved in the way given in [26]. More precisely,we have found the language tha can be generated by agrammar with controlled path. However, this languagecannot be generated by the grammar obtained by theconstruction introduced in [26]. Apparently, the powerof path controlled grammar still represents an open problem.3.3 Several Paths Based RestrictionThis section being a generalization of path-restricted rewritingmodel to a restriction placed upon not just onebut several paths. Then, it presents several properties ofthe model restricted in this way. Specifically, the power ofa kind of all-paths-restricted rewriting model, closure andpumping properties in relation to the number of controlledpaths, and the approximation of the power for n-pathrestricted model. The last section of this part presents anapplication related result concerning the parsing of pathrestricted languages.3.3.1 DefinitionsDefinition 3.18. Let (G, R) be a tree controlled grammar.The language that (G, R) generates under the allpathscontrol by R is denoted by all-paths L(G, R) and definedby the following equivalence:For all x ∈ T ∗ , x ∈ all-paths L(G, R) if and only ifthere is a derivation tree, t ∈ G △(x), such that for allpaths, s, of t, word(s) ∈ R.Definition 3.19. The class of tree controlled languagesunder all paths control is defined asall-path-TC(CF, REG) = { all-paths L(G, R) : (G, R) isa tree controlled grammar in which G is a context-freegrammar and R ∈ REG}.Definition 3.20. Let (G, R) be a tree controlled grammar.The language of tree controlled grammar undernot common n-path control by R, n ≥ 1, is denoted bync-n-pathL(G, R) and defined by the following equivalence:For all x ∈ T ∗ , x ∈ nc-n-path L(G, R) if there existsa derivation tree, t ∈ G △(x), such that there is a set, Q t ,of n paths of t such that for each path, p ∈ Q t , word(p) ∈R.Definition 3.21. For X, Y ∈ {REG, LIN, CF}, theclass of tree controlled languages under not-common n-path control is defined asnc-n-path-TC(X, Y) = { nc-n-path L(G, R) : (G, R) isa tree controlled grammar with G ∈ G X and R ∈ Y}.Definition 3.22. Let p 1 , p 2 be any different two pathsof a derivation tree, t. Then, p 1 and p 2 contain at leastone common node (the root of t, root(t)), and p 1 ends ina different leaf of t than p 2. Let cmn(p 1, p 2) denote themaximal number of consecutive common nodes of p 1 andp 2.Definition 3.23. Let Q t be a nonempty set of somepaths of a derivation tree, t. The paths of Q t are dividedin a common node of t if and only if for some k ≥ 1,cmn(p 1, p 2) = k for every pair (p 1, p 2) ∈ Q 2 t . Let all pathsof Q t be divided in a common node of t. If Q t = {p},then m Qt = | word(p)|, otherwise m Qt ≥ 1 denotes themaximal number of consecutive common nodes of all pathsin Q t .Definition 3.24. Let (G, R) be a tree controlled grammar.The language of tree controlled grammar under n-path control by R, n ≥ 1, is denoted by n-path L(G, R) anddefined by the following equivalence:For all x ∈ T ∗ , x ∈ n-path L(G, R) if there existsa derivation tree, t ∈ G△(x), such that there is a set, Q t,of n paths of t that are divided in a common node of t andfor each path, p ∈ Q t, word(p) ∈ R.Definition 3.25. For X, Y ∈ {REG, LIN, CF}, theclass of tree controlled languages under n-path control isdefined asn-path-TC(X, Y) = { n-path L(G, R) : (G, R) is a treecontrolled grammar with G ∈ G X and R ∈ Y}.

Information Sciences and Technologies Bulletin of the ACM Slovakia, Vol. 4, No. 4 (2012) 15-23 21ConventionsHereafter, tree controlled grammars that generate the languageunder the n-path control are referred to as n-pathtree controlled grammars.Note that if we consider 0 controlled paths (i.e., n =0 and consequently card(Q t ) = 0) in the definition ofn-pathL(G, R), then, clearly, the power of such a modelequals CF.Definition 3.26. Let (G, R) be a tree controlled grammar.Consider n-path L(G, R), for n ≥ 1. If for eachword, z ∈ n-path L(G, R), there exist a derivation tree,t ∈ G△(z), a set of its paths, Q t, m Qt ≥ 1, and a partition,word(p) = uvwxy, for each path, p ∈ Q t, satisfyingthe premise of the pumping lemma for linear languagessuch that it holds• 1 ≤ m Qt ≤ |u|,then n-path L(G, R) is I-n-path L(G, R),• |u| < m Qt ≤ |uv|,then n-path L(G, R) is II-n-path L(G, R),• |uv| < m Qt ≤ |uvw|,then n-path L(G, R) is III-n-path L(G, R),• |uvw| < m Qt ≤ |uvwx|,then n-path L(G, R) is IV -n-path L(G, R),• |uvwx| < m Qt ≤ |uvwxy|,then n-path L(G, R) is V -n-path L(G, R).Definition 3.27. For i ∈ {I, II, III, IV, V}, n ≥ 1,the class of i-n-path tree controlled languages is definedasi-n-path-TC(CF, LIN) = { i-n-path L(G, R) : (G, R) istree controlled grammar in which G is a context-free grammarand R ∈ LIN}.3.3.2 ResultsIt is well-know that path controlled grammars where thecontrolling grammar is regular characterize the same languageclass as its controlled grammar (see [26]) do. Wehave proved that the power of context-free grammars remainsunchanged even if we restrict all paths in theirderivation trees by regular languages:CF = all-path-TC(CF, REG).Since for each context-free grammar, there is a regularlanguage that describes all paths in all its derivation trees;and there is no regular language which increases its powerwhen used to restrict the paths, if we consider tree controlledgrammars (G, R) with R ∈ REG, then, obviously,the power of such a model equals CF for any n ≥ 1.Therefore, we investigate the properties of tree controlledgrammar with non-regular control language. More specifically,we study tree controlled grammars that generatestheir languages under n-path control with linear controllanguages.We have introduced a generalization of path controlledgrammars so that they generate the language under therestriction placed on not just one but several paths. Consequently,we have found some subsets of n-path controlledgrammars so their languages satisfy pumping premisessimilar to well-known premises stated by pumpinglemmata for CF, LIN, and REG—more precisely:• If L ∈ I-n-path-TC(CF, LIN), n ≥ 1, then thereare two constants, k, q ≥ 0, such that each word,z ∈ L, with |z| ≥ k can be written asz = u 1v 1u 2v 2 . . . u 4nv 4nu 4n+1with 0 < |v 1 v 2 . . . v 4n | ≤ q and for all i ≥ 1,u 1 v i 1u 2 v i 2 . . . u 4n v i 4nu 4n+1 ∈ L.• If L ∈ III-n-path-TC(CF, LIN), n ≥ 1, thenthere are two constants,k, q ≥ 0, such that eachword, z ∈ L, with |z| ≥ k can be written asz = u 1 v 1 u 2 v 2 . . . u 2n+2 v 2n+2 u 2n+3with 0 < |v 1v 2 . . . v 2n+2| ≤ q and for all i ≥ 1,u 1v i 1u 2v i 2 . . . u 2n+2v i 2n+2u 2n+3 ∈ L.• If L ∈ V-n-path-TC(CF, LIN), n ≥ 1, then thereare two constants,k, q ≥ 0, such that each word,z ∈ L, with |z| ≥ k can be written asz = u 1 v 1 u 2 v 2 u 3 v 3 u 4 v 4 u 5with 0 < |v 1v 2v 3v 4| ≤ q and for all i ≥ 1,u 1v i 1u 2v i 2u 3v i 3u 4v i 4u 5 ∈ L.A natural question that still remains open is whether ornot there are similar pumping properties also for the languagesof II-n-path-TC(CF, LIN) and IV-n-path-TC(CF, LIN).We have also proved some closure properties, that is• for n ≥ 1, i ∈ {I, II, III, IV, V} and T G , T R ∈{REG, LIN, CF},n-path-TC(T G, T R), i-n-path-TC(T G, T R)are closed under intersection with regular languages,union, and non-erasing homomorphism,• for n ≥ 1, i ∈ {I, III, V}, i-n-path-TC(CF, LIN)are not closed under concatenation, intersection, andcomplement.Since n-path controlled grammars are a natural generalizationof grammars with just one path controlled, wehave studied several properties that are well-known forpath controlled grammars in the case of controlling givenn paths. Most importantly, we have tried to establishthe power for n-path controlled grammar. We have foundthe approximation of the power that can be applied alsoon grammars with just one path controlled. However,this approximation does not say too much since it is wellknownthat PSC is not closed under erasing homomorphism.Thus, we have informally concluded that: ”Weeither have the power to check what we need but not to removeit (using PSC) or vice versa (using MAT).” Moreprecisely, we have stated the following:Let L ∈ n-path-TC(CF, CF), for n ≥ 1. Thenthere exists L ′ ∈ PSC with L = h(L ′ ) for some homomorphismh.

22 Koutný, J.: Grammars with Restricted Derivation TreesFinally, we have studied several parsing properties of pathbasedrestriction which is indisputably one of the most importantlanguage-class-characterizing property from thepractical viewpoint. Formally, we have studied a polynomialtime parsing possibilities and we have stated that:• For a tree controlled grammar, (G, R) with an unambiguouscontext-free grammar, G, and a linearcontrol language, R, the membershipx ∈ nc-n-path L(G, R),n ≥ 1, is decidable in O(|x| k ), for some k ≥ 0.• For a tree controlled grammar, (G, R), where G isa context-free grammar and R ∈ LIN, there is atree controlled grammar, (G ′ , R ′ ), such that G ′ doesnot contain unit productions and nc-n-path L(G, R) =nc-n-pathL(G ′ , R ′ ), n ≥ 1.• For tree controlled grammar (G, R) where G is m-ambiguous LR grammar, m ≥ 1, and an unambiguouslanguage R ∈ LIN, the membership x ∈nc-n-pathL(G, R), n ≥ 1, is decidable in O(|x| k ), forsome k ≥ 0.The significant disadvantage concerning n-path tree controlledgrammars is that the number of n paths satisfyingthe properties of Def. 3.24 is strictly limited by the lengthof the right-hand sides of the productions of underlyingcontext-free grammar. That is, given a general contextfreegrammar, G, and a linear language, R, controllingthe paths, the membership of a certain language mightbe decidable. However, given the same context-free languageas L(G) as a context-free grammar, H, in Chomskynormal form together with R, we might not be able tofind suitable path restriction to obtain the same language.On the other hand, the derivation trees of tree controlledgrammars that generates their languages under n-pathcontrol by a linear language are constructed exactly as incontext-free grammars and, in addition, we have to checksome of their paths. Thus, there is actually great possibilityto use well-known parsing methods for context-freelanguages to construct the derivation trees and to checktheir paths. However, in this viewpoint, n-path tree controlledgrammars seems to be a quite fragile formalismsince it requires a context-free grammar to have a productionwith at least n nonterminals on the right-handside which ensures the division of n paths in a commonnode. Moreover, it means that any attempt to use a parsingmethod that transforms a context-free grammar intoChomsky normal form will basically destroy any path restrictionwith n ≥ 3. Moreover, several nice properties ofcontext-free grammars have been lost—e.g., decompositionbased on pumping lemma for linear languages is potentiallyambiguous and thus, the membership problemfor i-n-path-TC, i ∈ {I, II, III, IV, V}, is potentiallyambiguous also.There are still many questions to be answered, namelythe power of grammars with path or paths controlled nonregularly,further closure properties, decision properties,etc. However, there are several other more general variantsof path-based restriction. Indeed, a modification ofthe formalism such that the paths do not have to be dividedin a common node of a derivation tree, or a variantwhere a path in a tree does not have to start in the rootand end in a leaf of the tree have not been investigatedyet.Acknowledgements. We gladly acknowledge the supportof the CZ.1.05/1.1.00/02.0070, CZ.1.07/2.4.00/12.0017, FIT-S-10-2, FIT-S-11-2, FR2581/2010/G1, GD102/09/H042, MEB041003, and MSM0021630528 grants.References[1] S. Abraham. Some questions of language theory. In Proceedingsof the 1965 conference on Computational linguistics, 1965.[2] A. V. Aho. Indexed grammars - an extension of context-freegrammars. Journal of the ACM, 15:25, 1968.[3] A. V. Aho, R. Sethi, and J. D. Ullman. Compilers principles,techniques, and tools. Addison-Wesley, 1986.[4] A. V. Aho and J. D. Ullman. The theory of parsing, translation,and compiling. Prentice-Hall, Inc., 1972.[5] A. Bondy. Graph Theory. Springer, 2010.[6] R. V. Book. 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