Theoretical Aspects Of Computer Science Research Articles

Count-free Weisfeiler–Leman and group isomorphism

We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include: • Direct products of non-Abelian simple groups. • Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an [Formula: see text]-generated solvable group with solvability class [Formula: see text]. This notably includes instances where the complement is an [Formula: see text]-generated nilpotent group. This problem was previously known to be in [Formula: see text] [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in Proc. 28th Symp. Theoretical Aspects of Computer Science, Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to [Formula: see text] [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in 24th Int. Symp. Fundamentals of Computation Theory, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247]. • Graphical groups of class 2 and exponent [Formula: see text] [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, J. Symb. Logic 46(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J.-Y. Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica 12(4) (1992) 389–410], respectively. In particular, our work improves upon previous results of Brachter and Schweitzer [On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786]. Notably, each of these families was previously known to be identified by the counting variant of the more powerful Weisfeiler–Leman Version II algorithm. We finally show that the q-ary count-free pebble game is unable to even distinguish Abelian groups. This extends the result of Grochow and Levet (ibid), who established the result in the case of [Formula: see text]. The general theme is that some counting appears necessary to place [Formula: see text] into [Formula: see text].

Fault Tolerant Approximate BFS Structures with Additive Stretch

This paper addresses the problem of designing a $$\beta $$ -additive fault-tolerant approximate BFS (or FT-ABFS for short) structure, namely, a subgraph H of the network G such that subsequent to the failure of a single edge e, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, whose distances satisfy $$\mathrm{dist}(s,v,H{\setminus } \{e\}) \le \mathrm{dist}(s,v,G{\setminus } \{e\})+\beta $$ for every $$v \in V$$ . It was shown in Parter and Peleg (SODA, 2014), that for every $$\beta \in [1, O(\log n)]$$ there exists an n-vertex graph G with a source s for which any $$\beta $$ -additive FT-ABFS structure rooted at s has $$\Omega (n^{1+\epsilon (\beta )})$$ edges, for some function $$\epsilon (\beta ) \in (0,1)$$ . In particular, 3-additive FT-ABFS structures admit a lower bound of $$\Omega (n^{5/4})$$ edges. In this paper we present the first upper bound, showing that there exists a poly-time algorithm that for every n-vertex unweighted undirected graph G and source s constructs a 4-additive FT-ABFS structure rooted at s with at most $$O(n^{4/3})$$ edges. The main technical contribution of our algorithm is in adapting the path-buying strategy used in Baswana et al. (ACM Trans Algorithms 7:A5, 2010) and Cygan et al. (Proceedings of the 30th symposium on theoretical aspects of computer science, pp 209–220, 2013) to failure-prone settings.

Preface of the Special Issue on Theoretical Aspects of Computer Science (2018)

This special issue contains eight articles which are based on extended abstracts that were presented at the 35th Symposium on Theoretical Aspects of Computer Science (STACS), which was held at the University of Caen from February 28th to March 3rd, 2018. These extended abstracts were chosen among the top papers of those that were selected for presentation at STACS 2018 in a highly competitive peer-review process (after which only 54 papers out of 186 submissions were accepted). Compared with the original extended abstracts, the articles have been extended with a description of the context, full proofs and additional results. They underwent a further rigorous reviewing process, following the TOCS Journal standards, completely independent from the selection process of STACS 2018. The topics of the chosen papers cover a rich set of areas within Theoretical Computer Science, that is, algorithmic game theory, automata theory, circuit complexity, efficient algorithms, logic and model checking, Markov chains, pa-rameterized complexity, and proof complexity. Kernelization (provably effective and efficient preprocessing) is a central theme in parameterized complexity. The article Computing Hitting Set Kernels by AC 0-Circuits by Max Bannach and Till Tantau presents for the first time kernels for the d-Hitting Set problem that can be computed in constant parallel time, thereby refuting a previous conjecture. To this end, the authors introduce a new, generalized notion of hypergraph sunflowers. Altogether, this gives an important contribution to parameterized circuit complexity.

Guest Editorial: Special Issue on Theoretical Aspects of Computer Science

This special issue contains seven articles which are based on extended abstracts that were presented at the 34th Symposium on Theoretical Aspects of Computer Science (STACS), which was held at the Leibnizhaus in Hannover from March 8th to 11th, 2017. These extended abstracts were chosen among the top papers of those that were selected for presentation at STACS 2017 in a highly competitive peer-review process (after which only 54 papers out of 212 submissions were accepted). Compared with the original extended abstracts, the articles have been extended with a description of the context, full proofs and additional results. They underwent a further rigorous reviewing process, following the TOCS Journal standard, completely independent from the selection process of STACS 2017. The topics of the chosen papers cover combinatorics on words, circuit complexity , computational geometry, (parameterized) algorithms for graph problems , Markov chains, and stochastic games. The article On Long Words Avoiding Zimin Patterns by Arnaud Carayol and Stefan Goller extends a classical result on combinatorics on words concerning unavoidable patterns, so called Zimin patterns. The authors study the question of the smallest length (depending on alphabet size) of a word that guarantees that a certain pattern appears-a very active research area in recent years. The main contribution here is a non-elementary lower bound on some of these minimal lengths.

Maximum Matching on Trees in the Online Preemptive and the Incremental Graph Models

We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the online preemptive and the incremental graph models. In the Online Preemptive model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded, and all rejections are permanent. In this model, the complexity of the problems is settled for deterministic algorithms (McGregor, in: Proceedings of the 8th international workshop on approximation, randomization and combinatorial optimization problems, and proceedings of the 9th international conference on randomization and computation: algorithms and techniques, APPROX’05/RANDOM’05, Springer, Berlin, pp. 170–181, 2005; Varadaraja, in: Automata, languages and programming: 38th international colloquium, ICALP 2011, Zurich, Switzerland, proceedings, part I, pp. 379–390, 2011. https://doi.org/10.1007/978-3-642-22006-7_32 ). Epstein et al. (in: 30th international symposium on theoretical aspects of computer science, STACS 2013, Kiel, Germany, pp. 389–399, 2013. https://doi.org/10.4230/LIPIcs.STACS.2013.389 ) gave a 5.356-competitive randomized algorithm for MWM, and also proved a lower bound on the competitive ratio of $$(1+\ln 2) \approx 1.693$$ for MCM. The same lower bound applies for MWM. In the Incremental Graph model, at each step an edge is added to the graph, and the algorithm is supposed to quickly update its current matching. Gupta (in: 34th international conference on foundation of software technology and theoretical computer science, FSTTCS 2014, 15–17 Dec 2014, New Delhi, India, pp. 227–239, 2014. https://doi.org/10.4230/LIPIcs.FSTTCS.2014.227 ) proved that for any $$\epsilon \le 1/2$$ , there exists an algorithm that maintains a $$(1+\epsilon )$$ -approximate MCM for an incremental bipartite graph in an amortized update time of $$O\left( \frac{\log ^2 n}{\epsilon ^4}\right) $$ . No $$(2-\epsilon )$$ -approximation algorithm with a worst case update time of O(1) is known in this model, even for special classes of graphs. In this paper we show that some of the results can be improved for trees, and for some special classes of graphs. In the online preemptive model, we present a 64 / 33-competitive randomized algorithm (which uses only two bits of randomness) for MCM on trees. Inspired by the above mentioned algorithm for MCM, we present the main result of the paper, a randomized algorithm for MCM with a worst case update time of O(1), in the incremental graph model, which is 3 / 2-approximate (in expectation) on trees, and 1.8-approximate (in expectation) on general graphs with maximum degree 3. Note that this algorithm works only against an oblivious adversary. We derandomize this algorithm, and give a $$(3/2+\epsilon )$$ -approximate deterministic algorithm for MCM on trees, with an amortized update time of $$O(1/\epsilon )$$ . We also present a minor result for MWM in the online preemptive model, a 3-competitive randomized algorithm (that uses only O(1) bits of randomness) on growing trees (where the input revealed upto any stage is always a tree, i.e. a new edge never connects two disconnected trees).

How Much Does a Treedepth Modulator Help to Obtain Polynomial Kernels Beyond Sparse Graphs?

In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarský et al. (J Comput Syst Sci 84:219–242, 2017) proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a c-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c, where $$c \ge 1$$ is a fixed integer. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs. In this article we answer this question by finding two very natural such problems: we prove that Vertex Cover admits a polynomial kernel on general graphs for any integer $$c \ge 1$$ , and that Dominating Set does not for any integer $$c \ge 2$$ even on degenerate graphs, unless $$\text {NP} \subseteq \text {coNP}/\text {poly}$$ . For the positive result, we build on the techniques of Jansen and Bodlaender (Proceedings of the 28th symposium on theoretical aspects of computer science (STACS), volume 9 of LIPIcs, pp 177–188, 2011), and for the negative result we use a polynomial parameter transformation for $$c\ge 3$$ and an or-cross-composition for $$c = 2$$ . As existing results imply that Dominating Set admits a polynomial kernel on degenerate graphs for $$c = 1$$ , our result provides a dichotomy about the existence of polynomial kernels for Dominating Set on degenerate graphs with this parameter.

Editorial: Special Issue on “Theoretical Aspects of Computer Science” (STACS 2015)

Editorial: Special Issue on “Theoretical Aspects of Computer Science” (STACS 2015)

Half-integrality, LP-branching, and FPT Algorithms

A recent trend in parameterized algorithms is the application of polytope tools to fixed-parameter tractable (FPT) algorithms [e.g., Cygan et al., FOCS 2011, 52nd Annual Symposium on Foundations of Computer Science, IEEE, 2011, pp. 150--159; Narayanaswamy et al., STACS 2012, Symposium on Theoretical Aspects of Computer Science, 2012, pp. 338--349]. Although this approach has yielded significant speedups for a range of important problems, it requires the underlying polytope to have very restrictive properties, including half-integrality and Nemhauser--Trotter-style persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover [Nemhauser and Trotter, Math. Program., 8 (1975), pp. 232--248] and Node Multiway Cut [Garg et al., J. Alg., 50 (2004), pp. 49--61]. Taking a slightly different approach, we view half-integrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from $\{0,1\}^V$ to $\{0,1/2,1\}^V$ such that the new problem admits a polynomial-time exact solution. Using tools from constraint satisfaction problems [in particular Thapper and Živný, FOCS 2012, 53rd Annual Symposium on Foundations of Computer Science, IEEE, 2012, pp. 669--678] to study the existence of such relaxations, we are able to provide a much broader class of half-integral polytopes with the required properties. Our results unify and significantly extend the previously known cases, and yield a range of new and improved FPT algorithms, including an $O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover and an $O^*(4^k)$-time algorithm for Group Feedback Vertex Set where the group is given by oracle access. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. [Algorithmica, 74 (2016), pp. 630--642]. Additionally, we propose a network-flow-based approach to solve several cases of the relaxation problem. This gives the first linear-time FPT algorithm to edge-deletion Unique Label Cover.

Preface of STACS 2013 Special Issue

This special issue contains six articles based on extended abstracts that were presented at the 30th Symposium on Theoretical Aspects of Computer Science (STACS), which was held at the Christian-Albrechts-Universitat zu Kiel, Germany, from February 27th to March 2nd, 2013. These extended abstracts were among the top papers of those chosen for presentation at STACS 2013 in a highly competitive peer-review process: the members of the program committee with chairs Thomas Wilke and Natacha Portier selected only 54 papers out of 254 submissions. Compared with the original extended abstracts that appeared in the conference proceedings the articles in this issue have been extended by full proofs and additional results. They underwent a further rigorous reviewing process, following the TOCS standard, completely independent of the selection process of STACS 2013. The paper The Arithmetic Complexity of Tensor Contraction considers the fundamental arithmetic complexity class VP. This class was originally defined by Valiant, in the framework of his theory of arithmetic complexity, as the class of sequences of polynomials with polynomially bounded degree that can be computed by polynomially sized circuits. Although of quite some interest, as the analog of deterministic polynomial time in arithmetic complexity, this class is not too well understood so far. This is partly due to the fact that no really natural characterization of it has been known. In the paper polynomials in VP are characterized as functions obtainable as

Producibility in hierarchical self-assembly

Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly ? is producible. The algorithm can be optimized to use $$O(|\alpha | \log ^2 |\alpha |)$$O(|?|log2|?|) time. Cannon et al. (STACS 2013: proceedings of the thirtieth international symposium on theoretical aspects of computer science. pp 172---184, 2013) showed that the problem of deciding if an assembly ? is the unique producible terminal assembly of a tile system $${\mathcal {T}}$$T can be solved in $$O(|\alpha |^2 |{\mathcal {T}}| + |\alpha | |{\mathcal {T}}|^2)$$O(|?|2|T|+|?||T|2) time for the special case of noncooperative temperature 1 systems. It is shown that this can be improved to $$O(|\alpha | |{\mathcal {T}}| \log |{\mathcal {T}}|)$$O(|?||T|log|T|) time. Finally, it is shown that if two assemblies are producible, and if they can be overlapped consistently--i.e., if the positions that they share have the same tile type in each assembly--then their union is also producible .

Optimal Collapsing Protocol for Multiparty Pointer Jumping

In this paper, we study the pointer jumping problem under the one-way number-on-the-forehead (NOF) multiparty communication model. This problem is widely considered to be a candidate for proving strong lower bounds under the NOF model, and has applications to proving lower bounds for many other problems. We investigate the maximum communication complexity of collapsing protocols for pointer jumping, where each player sees all layers behind her and only the composition of layers ahead of her. We present a collapsing protocol in which every player communicates at most $n-\frac{1}{2}\log_{2} n+1$ bits, which tightly matches the lower bound of $n-\frac{1}{2}\log_{2} n-2$ given by Brody and Chakrabarti (in Proc. 25th Annual Symposium on Theoretical Aspects of Computer Science (STACS), pp. 145---156, 2008). Actually, in our protocol only three players need to communicate information: the first player sends log2(n+1) bits, the second to last player sends $n-\frac{1}{2}\log_{2} n+1$ bits, and the last player just outputs the answer. A natural question is whether the log2(n+1) bits communicated by the first player is necessary for achieving a low maximum communication complexity. We make progress towards this question by proving that in any collapsing protocol for the 3-player pointer jumping problem, if the first player only sends one bit, then the second player must communicate at least n?2 bits.

Preface of Special Issue on Theoretical Aspects of Computer Science

This special issue contains nine articles which are based on extended abstracts that were presented at the 28th Symposium on Theoretical Aspects of Computer Science (STACS), which was held at Technische Universitat Dortmund, Germany, March 10– 12, 2011. These extended abstracts were among the top papers of those that were chosen for presentation at STACS 2011 in a highly competitive peer-review process (after which only 54 papers out of 271 submission were accepted). Compared with the original extended abstracts the articles have been strongly revised and extended by full proofs and additional results. They underwent a further rigorous reviewing process, following the TOCS standard, completely independent from the selection process of STACS 2011. Three articles are about formal languages. The first two investigate size issues for automata and (extended) regular expressions, respectively, while the third one investigates the syntactic monoids of regular languages over infinite alphabets. The article Series Parallel Digraphs with Loops by Stefan Gulan studies the question which kinds of finite automata can be translated into regular expressions of linear size. It gives a very comprehensive answer to this question by showing that such linear translations are possible from automata with a certain underlying graph structure, series parallel loop graphs, into regular expressions and vice versa. Furthermore it characterizes this graph class (within the class of hammocks, i.e., all graphs with dis-

Theoretical Aspects of Computer Science

Theoretical Aspects of Computer Science

On a measure of distance for quantum strategies

The present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels. Characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem. A full, formal proof of strong duality is presented for the semidefinite optimization problem in question. This norm and its properties are employed to generalize a state discrimination result of Gutoski and Watrous [In Proceedings of the 22nd Symposium on Theoretical Aspects of Computer Science (STACS’05), Lecture Notes in Computer Science, Vol. 3404 (Springer, 2005), pp. 605–616. The generalized result states that for any two convex sets S0, S1 of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of S0 ∈ S0 from any choice of S1 ∈ S1 with bias proportional to the minimal distance between the sets S0 and S1 as measured by this norm. A similar discrimination result for channels then follows as a special case.

Two-stage Robust Network Design with Exponential Scenarios

We study two-stage robust variants of combinatorial optimization problems on undirected graphs, like Steiner tree, Steiner forest, and uncapacitated facility location. Robust optimization problems, previously studied by Dhamdhere et al. (Proc. of 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pp. 367---378, 2005), Golovin et al. (Proc. of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS), 2006), and Feige et al. (Proc. of the 12th International Integer Programming and Combinatorial Optimization Conference, pp. 439---453, 2007), are two-stage planning problems in which the requirements are revealed after some decisions are taken in Stage 1. One has to then complete the solution, at a higher cost, to meet the given requirements. In the robust k-Steiner tree problem, for example, one buys some edges in Stage 1. Then k terminals are revealed in Stage 2 and one has to buy more edges, at a higher cost, to complete the Stage 1 solution to build a Steiner tree on these terminals. The objective is to minimize the total cost under the worst-case scenario. In this paper, we focus on the case of exponentially many scenarios given implicitly. A scenario consists of any subset of k terminals (for k-Steiner tree), or any subset of k terminal-pairs (for k-Steiner forest), or any subset of k clients (for facility location). Feige et al. (Proc. of the 12th International Integer Programming and Combinatorial Optimization Conference, pp. 439---453, 2007) give an LP-based general framework for approximation algorithms for a class of two stage robust problems. Their framework cannot be used for network design problems like k-Steiner tree (see later elaboration). Their framework can be used for the robust facility location problem, but gives only a logarithmic approximation. We present the first constant-factor approximation algorithms for the robust k-Steiner tree (with exponential number of scenarios) and robust uncapacitated facility location problems. Our algorithms are combinatorial and are based on guessing the optimum cost and clustering to aggregate nearby vertices. For the robust k-Steiner forest problem on trees and with uniform multiplicative increase factor for Stage 2 (also known as inflation), we present a constant approximation. We show APX-hardness of the robust min-cut problem (even with singleton-set scenarios), resolving an open question of (Dhamdhere et al. in Proc. of 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pp. 367---378, 2005) and (Golovin et al. in Proc. of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS), 2006).

Trichotomies in the Complexity of Minimal Inference

We study the complexity of the propositional minimal inference problem. Although the complexity of this problem has been already extensively studied before because of its fundamental importance in nonmonotonic logics and commonsense reasoning, no complete classification of its complexity was found. We classify the complexity of four different and well-studied formalizations of the problem in the version with unbounded queries, proving that the complexity of the minimal inference problem for each of them has a trichotomy (between P, coNP-complete, and Π2P-complete). One of these results finally settles with a positive answer the trichotomy conjecture of Kirousis and Kolaitis (Theory Comput. Syst. 37(6):659–715, 2004). In the process we also strengthen and give a much simplified proof of the main result from Durand and Hermann (Proceedings 20th Symposium on Theoretical Aspects of Computer Science (STACS 2003), pp. 451–462, 2003).

Improved Approximation Guarantees for Weighted Matching in the Semi-streaming Model

We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke [Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science, 2008, pp. 669–680] by devising a deterministic approach whose performance guarantee is . In addition, we study preemptive online algorithms, a class of algorithms related to one-pass semi-streaming algorithms, where we are allowed to maintain only a feasible matching in memory at any point in time. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges that is not necessarily a feasible matching. We conclude by presenting an empirical study, conducted in order to compare the practical performance of our approach to that of previously suggested algorithms.

An Efficient Quantum Algorithm for the Hidden Subgroup Problem in Nil-2 Groups

In this paper we show that the hidden subgroup problem in nil-2 groups, that is in groups of nilpotency class at most 2, can be solved efficiently by a quantum procedure. The algorithm is an extension of our earlier method for extraspecial groups in Ivanyos et al. (Proceedings of the 24th Symposium on Theoretical Aspects of Computer Science (STACS), vol. 4393, pp. 586–597, 2007), but it has several additional features. It contains a powerful classical reduction for the hidden subgroup problem in nilpotent groups of constant nilpotency class to the specific case where the group is a p-group of exponent p and the subgroup is either trivial or cyclic. This reduction might also be useful for dealing with groups of higher nilpotency class. The quantum part of the algorithm uses well chosen group actions based on some automorphisms of nil-2 groups. The right choice of the actions requires the solution of a system of quadratic and linear equations. The existence of a solution is guaranteed by the Chevalley-Warning theorem, and we prove that it can also be found efficiently.

Preface: Special Issue on Theoretical Aspects of Computer Science (STACS)

This special issue contains seven papers from the 26th International Symposium on Theoretical Aspects of Computer Science (STACS), held February 26–28, 2009, at the University of Freiburg, Germany. Based on their initial ratings, these papers were judged by the program committee to be among the best submissions to the conference. Authors were invited to provide complete journal versions of the STACS paper. These submissions underwent the full review process of this journal. We thank the authors for submitting their papers to this issue, and we thank the referees for many thorough reviews of the manuscripts.

Second-Order Abstract Categorial Grammars as Hyperedge Replacement Grammars

Second-order abstract categorial grammars (de Groote in Association for computational linguistics, 39th annual meeting and 10th conference of the European chapter, proceedings of the conference, pp. 148---155, 2001) and hyperedge replacement grammars (Bauderon and Courcelle in Math Syst Theory 20:83---127, 1987; Habel and Kreowski in STACS 87: 4th Annual symposium on theoretical aspects of computer science. Lecture notes in computer science, vol 247, Springer, Berlin, pp 207---219, 1987) are two natural ways of generalizing "context-free" grammar formalisms for string and tree languages. It is known that the string generating power of both formalisms is equivalent to (non-erasing) multiple context-free grammars (Seki et al. in Theor Comput Sci 88:191---229, 1991) or linear context-free rewriting systems (Weir in Characterizing mildly context-sensitive grammar formalisms, University of Pennsylvania, 1988). In this paper, we give a simple, direct proof of the fact that second-order ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for tree-generating hyperedge replacement grammars given by Engelfriet and Maneth (Graph transformation. Lecture notes in computer science, vol 1764. Springer, Berlin, pp 15---29, 2000) can then be used to show that the tree generating power of second-order ACGs is exactly the same as that of hyperedge replacement grammars.