Umeå universitets logga

The string correction problem looks at minimal ways to modify one stringinto another using fixed operations, such as for example inserting a symbol, deleting asymbol and interchanging the positions of two symbols (a “swap”). This has been generalizedto trees in various ways, but unfortunately having operations to insert/deletenodes in the tree and operations that move subtrees, such as a “swap” of adjacent subtrees,makes the correction problem for trees intractable. In this paper we investigatewhat happens when we have a tree edit distance problem with only swaps. We callthis problem tree swap distance, and go on to prove that this correction problem isNP-complete. This suggests that the swap operation is fundamentally problematic inthe tree case, and other subtree movement models should be studied.

This paper considers a characterization of the context-free non-regular languages, conjecturing that there for all such languages exists a fixed string thatcan be pumped to exhibit infinitely many equivalence classes. A proof is given only for a special case, but the general statement is conjectured to hold. The conjecture is then shown to imply that the shuffle of two context-free languagesis not context-free.

Denna avhandling diskuterar utökningar av klassiska formalismer inom formella språk, till exempel reguljära uttryck och kontextfria grammatiker. Utökningarna handlar på ett eller annat sätt omordning, och ett särskilt fokus ligger på att göra utökningarna på ett sätt som dels har intressanta spatiala/ordningsrelaterade effekter och som dels bevarar den effektiva parsningen som är möjlig för de ursprungliga klassiska formalismerna. Detta står i kontrast till att ta det större steget upp i Chomsky-hierarkin till de kontextkänsliga språken, vilket medför ett svårt parsningsproblem.

Ett omedelbart exempel på en sådan utökning är s.k. shuffle-formalismer. Dessa utökar existerande formalismer genom att introducera operatorer som godtyckligt sammanflätar strängar från argumentspråk. Om shuffle-operator introduceras till de reguljära uttrycken ger det inte förmågan att känna igen t.ex. det kontextfria språket aⁿbⁿ, men det fångar istället vissa språk som inte är kontextfria, till exempel språket som består av alla strängar som innehåller lika många a:n, b:n och c:n. Sättet på vilket dessa utökningar påverkar parsningsproblemet är mångfacetterat. Utöver dessa shuffle-operatorer tas också formalismer där delsträngar kan upprepas, formalismer där delsträngar flyttas runt, och formalismer som begränsar hur delsträngar får konkateneras upp. Formalismerna som tas upp här har dock vissa egenskaper gemensamma.

1. De är nära besläktade med de klassiska reguljära och kontextfria formalismerna. De arbetar stegvis, och konstruerar strängar genom successiva applikationer av individuellt enkla regler.
2. Varje steg genererar ett konstant antal symboler och modifierar inte det som redan genererats. Det vill säga, strängar byggs additivt och längden på dem kan inte explodera (i kontrast till t.ex. Lindenmayer-system). Alla språk som tas upp kommer att ha en semi-linjär Parikh-avbildning.
3. De har någon instressant spatial/ordningsrelaterad egenskap. Exempelvis sättet på vilket shuffle-operatorer sammanflätar annars oberoende deriveringar.
4. Alla formalismerna är tänkta att vara begränsade nog att det är resonabelt att ha effektiv parsning som mål.

Denna avhandling kommer att ge intuitiva förklaring av ett antal formalismer som uppfyller ovanstående krav, och kommer att skissa en blandning av resultat relaterade till parsningsproblemet för dem. Detta bör ses som förberedande inför läsning av de mer djupgående och komplexa resultaten och förklaringarna i de artiklar som finns inkluderade som appendix.

The work presented in this thesis discusses various formalisms for representing the addition of order-controlling and order-relaxing mechanisms to existing formal language models. An immediate example is shuffle expressions, which can represent not only all regular languages (a regular expression is a shuffle expression), but also features additional operations that generate arbitrary interleavings of its argument strings. This defines a language class which, on the one hand, does not contain all context-free languages, but, on the other hand contains an infinite number of languages that are not context-free. Shuffle expressions are, however, not themselves the main interest of this thesis. Instead we consider several formalisms that share many of their properties, where some are direct generalisations of shuffle expressions, while others feature very different methods of controlling order. Notably all formalisms that are studied here

• have a semi-linear Parikh image,
• are structured so that each derivation step generates at most a constant number of symbols (as opposed to the parallel derivations in for example Lindenmayer systems),
• feature interesting ordering characteristics, created either by derivation steps that may generate symbols in multiple places at once, or by multiple generating processes that produce output independently in an interleaved fashion, and
• are all limited enough to make the question of efficient parsing an interesting and reasonable goal.

This vague description already hints towards the formalisms considered; the different classes of mildly context-sensitive devices and concurrent finite-state automata.

This thesis will first explain and discuss these formalisms, and will then primarily focus on the associated membership problem (or parsing problem). Several parsing results are discussed here, and the papers in the appendix give a more complete picture of these problems and some related ones.

Formal language models which employ shuffling, or interleaving, of strings are of interest in many areas of computer science. Notable examples include system verification, plan recognition, and natural language processing. Membership problems for the shuffle of languages are especially interesting. It is known that deciding membership for shuffles of regular languages can be done in polynomial time, and that deciding (non-uniform) membership in the shuffle of two deterministic context-free languages is NP-complete. In this paper we narrow the gap by showing that the non-uniform membership problem for the shuffle of two deterministic *linear* context-free languages is NP-complete.

Whereas Perl-compatible regular expression matchers typically exhibit some variation of leftmost-greedy semantics, those conforming to the posix standard are prescribed leftmost-longest semantics. However, the posix standard leaves some room for interpretation, and Fowler and Kuklewicz have done experimental work to confirm differences between various posix matchers. The Boost library has an interesting take on the posix standard, where it maximises the leftmost match not with respect to subexpressions of the regular expression pattern, but rather, with respect to capturing groups. In our work, we provide the first formalisation of Boost semantics, analyze the complexity of regular expression matching when using Boost semantics, and provide efficient algorithms for both online and multipass matching.

A regular unranked tree folding consists of a regular unranked tree language and a folding operation that merges, i.e., folds, selected nodes of a tree to form a graph; the combination is a formal device for representing graph languages. If, in the process of folding, the order among edges is discarded so that the result is an unordered graph, then two applications of a fold operation is enough to make the associated parsing problem NP-complete. However, if the order is kept, then the problem is solvable in non-uniform polynomial time. In this paper we address the remaining case where only one fold operation is applied, but the order among edges is discarded. We show that under these conditions, the problem is solvable in non-uniform polynomial time.

A regular unranked tree folding consists of a regular unranked tree language and a folding operation that merges (i.e., folds) selected nodes of a tree to form a graph; the combination is a formal device for representing graph languages. If, in the process of folding, the order among edges is discarded so that the result is an unordered graph, then two applications of a fold operation are enough to make the associated parsing problem NP-complete. However, if the order is kept, then the problem is solvable in non-uniform polynomial time. In this paper, we address the remaining case, where only one fold operation is applied, but the order among the edges is discarded. We show that, under these conditions, the problem is solvable in non-uniform polynomial time.

Language models that use interleaving, or shuffle, operators have applications in various areas of computer science, including system verification, plan recognition, and natural language processing. We study the complexity of the membership problem for such models, in other words, how difficult it is to determine if a string belongs to a language or not. In particular, we investigate how interleaving can be introduced into models that capture the context-free languages.

We introduce a fold operation that realises a tree-to-graph transduction by merging selected nodes in the input tree to form a possibly cyclic output graph. The work is motivated by the increasing use of graph-based representations in semantic parsing. We show that a suitable class of graphs languages can be generated by applying the fold operation to regular unranked tree languages. We investigate two versions of the fold operation, one that preserves a depth-first ordering between the edges, and one that does not. Finally, we demonstrate that the time complexity for the associated non-uniform membership problem is solvable in polynomial time for the order-preserving version, and NP-complete for the order-cancelling one.