Markus L. Schmid

Pattern Matching with Variables: A Multivariate Complexity Analysis

In the context of this paper, a pattern is a string that con- tains variables and terminals. A pattern α matches a terminal word w if w can be obtained by uniformly substituting the variables of α by termi- nal words. It is a well-known fact that deciding whether a given terminal word matches a given pattern is an NP-complete problem. In this work, we consider numerous parameters of this problem and for all possible combinations of these parameters, we investigate the question whether or not the variant obtained by bounding these parameters by constants can be solved ef fi ciently.

On the Parameterised Complexity of String Morphism Problems

Given a source string u and a target string w, to decide whether w can be obtained by applying a string morphism on u (i. e., uniformly replacing the symbols in u by strings) constitutes an 𝓝𝓟

Contextual array grammars and array P systems

\mathcal {NP}

Pattern Matching with Variables: Fast Algorithms and New Hardness Results.

-complete problem. We present a multivariate analysis of this problem (and its many variants) from the viewpoint of parameterised complexity theory, thereby pinning down the sources of its computational hardness. Our results show that most parameterised variants of the string morphism problem are fixed-parameter intractable and, apart from some very special cases, tractable variants can only be obtained by considering a large part of the input as parameters, namely the length of w and the number of different symbols in u.

Pattern matching with variables

Contextual array grammars, with selectors not having empty cells, are considered. A P system model, called contextual array P system, that makes use of array objects and contextual array rules, is introduced and its generative power for the description of picture arrays is examined. A main result of the paper is that there is a proper infinite hierarchy with respect to the classes of languages described by contextual array P systems. Such a hierarchy holds as well in the case when the selector is also endowed with the #−sensing ability.

Characterising REGEX languages by regular languages equipped with factor-referencing

A pattern (i. e., a string of variables and terminals) maps to a word, if this is obtained by uniformly replacing the variables by terminal words; deciding this is NP-complete. We present efficient algorithms\footnote{The computational model we use is the standard unit-cost RAM with logarithmic word size. Also, all logarithms appearing in our time complexity evaluations are in base 2.} that solve this problem for restricted classes of patterns. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors; this shows that the injective version (i.e., different variables are replaced by different words) of the above matching problem is NP-complete even for very restricted cases.

Scanning Pictures the Boustrophedon Way

A pattern α, i.e., a string that contains variables and terminals, matches a terminal word w if w can be obtained by uniformly substituting the variables of α by terminal words. Deciding whether a given terminal word matches a given pattern is NP-complete and this holds for several natural variants of the problem that result from whether or not variables can be erased, whether or not the patterns are required to be terminal-free or whether or not the mapping of variables to terminal words must be injective. We consider numerous parameters of this problem (i.e., number of variables, length of w, length of the words substituted for variables, number of occurrences per variable, cardinality of the terminal alphabet) and for all possible combinations of the parameters (and variants described above), we answer the question whether or not the problem is still NP-complete if these parameters are bounded by constants.

Jumping Finite Automata: Characterizations and Complexity

A (factor-)reference in a word is a special symbol that refers to another factor in the same word; a reference is dereferenced by substituting it with the referenced factor. We introduce and investigate the class ref-REG of all languages that can be obtained by taking a regular language R and then dereferencing all possible references in the words of R. We show that ref-REG coincides with the class of languages defined by regular expressions as they exist in modern programming languages like Perl, Python, Java, etc. (often called REGEX languages).

A note on the complexity of matching patterns with variables

We are introducing and discussing finite automata working on rectangular-shaped arrays i.e., pictures in a boustrophedon reading mode. We prove close relationships with the well-established class of regular matrix picture languages. We derive several combinatorial, algebraic and decidability results for the corresponding class of picture languages. For instance, we show pumping and interchange lemmas for our picture language class. We also explain similarities and differences to the status of decidability questions for classical finite string automata. For instance, the non-emptiness problem for our picture-processing automaton models turns out to be NP-complete. Finally, we sketch possible applications to character recognition.

Characterization and complexity results on jumping finite automata

We characterize the class of languages described by jumping finite automata (i. e., finite automata, for which the input head after reading (and consuming) a symbol, can jump to an arbitrary position of the remaining input) in terms of special shuffle expressions. We can characterize some interesting subclasses of this language class. The complexity of parsing these languages is also investigated.