Approximating Language Edit Distance Beyond Fast Matrix Multiplication: Ultralinear Grammars Are Where Parsing Becomes Hard!

In 1975, a breakthrough result of L. Valiant showed that parsing context free grammars can be reduced to Boolean matrix multiplication, resulting in a running time of O(n^omega) for parsing where omega <= 2.373 is the exponent of fast matrix multiplication, and n is the string length. Recently, Abboud, Backurs and V. Williams (FOCS 2015) demonstrated that this is likely optimal; moreover, a combinatorial o(n^3) algorithm is unlikely to exist for the general parsing problem. The language edit distance problem is a significant generalization of the parsing problem, which computes the minimum edit distance of a given string (using insertions, deletions, and substitutions) to any valid string in the language, and has received significant attention both in theory and practice since the seminal work of Aho and Peterson in 1972. Clearly, the lower bound for parsing rules out any algorithm running in o(n^omega) time that can return a nontrivial multiplicative approximation of the language edit distance problem. Furthermore, combinatorial algorithms with cubic running time or algorithms that use fast matrix multiplication are often not desirable in practice. To break this n^omega hardness barrier, in this paper we study additive approximation algorithms for language edit distance. We provide two explicit combinatorial algorithms to obtain a string with minimum edit distance with performance dependencies on either the number of non-linear productions, k^*, or the number of nested non-linear production, k, used in the optimal derivation. Explicitly, we give an additive O(k^*gamma) approximation in time O(|G|(n^2 + (n/gamma)^3)) and an additive O(k gamma) approximation in time O(|G|(n^2 + (n^3/gamma^2))), where |G| is the grammar size and n is the string length. In particular, we obtain tight approximations for an important subclass of context free grammars known as ultralinear grammars, for which k and k^* are naturally bounded. Interestingly, we show that the same conditional lower bound for parsing context free grammars holds for the class of ultralinear grammars as well, clearly marking the boundary where parsing becomes hard

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

On the finiteness of picture languages of synchronous deterministic chain code picture systems

Chain Code Picture Systems are LINDENMAYER systems over a special alphabet. The strings generated are interpreted as pictures. This leads to Chain Code Picture Languages. In this paper, synchronous deterministic Chain Code Picture Systems (sDOL systems) are studied with respect to the finiteness of their picture languages. First, a hierarchy of abstractions is developed, in which the interpretation of a string as a picture passes through a multilevel process. Second, on the basis of this hierarchy, an algorithm is designed which decides the finiteness or infiniteness of any sDOL system in polynomial time

Improved bounds for testing Dyck languages

In this paper we consider the problem of deciding membership in Dyck languages, a fundamental family of context-free languages, comprised of well-balanced strings of parentheses. In this problem we are given a string of length $n$ in the alphabet of parentheses of $m$ types and must decide if it is well-balanced. We consider this problem in the property testing setting, where one would like to make the decision while querying as few characters of the input as possible. Property testing of strings for Dyck language membership for $m=1$, with a number of queries independent of the input size $n$, was provided in [Alon, Krivelevich, Newman and Szegedy, SICOMP 2001]. Property testing of strings for Dyck language membership for $m \ge 2$ was first investigated in [Parnas, Ron and Rubinfeld, RSA 2003]. They showed an upper bound and a lower bound for distinguishing strings belonging to the language from strings that are far (in terms of the Hamming distance) from the language, which are respectively (up to polylogarithmic factors) the $2/3$ power and the $1/11$ power of the input size $n$. Here we improve the power of $n$ in both bounds. For the upper bound, we introduce a recursion technique, that together with a refinement of the methods in the original work provides a test for any power of $n$ larger than $2/5$. For the lower bound, we introduce a new problem called Truestring Equivalence, which is easily reducible to the $2$-type Dyck language property testing problem. For this new problem, we show a lower bound of $n$ to the power of $1/5$

Constraint LTL Satisfiability Checking without Automata

This paper introduces a novel technique to decide the satisfiability of formulae written in the language of Linear Temporal Logic with Both future and past operators and atomic formulae belonging to constraint system D (CLTLB(D) for short). The technique is based on the concept of bounded satisfiability, and hinges on an encoding of CLTLB(D) formulae into QF-EUD, the theory of quantifier-free equality and uninterpreted functions combined with D. Similarly to standard LTL, where bounded model-checking and SAT-solvers can be used as an alternative to automata-theoretic approaches to model-checking, our approach allows users to solve the satisfiability problem for CLTLB(D) formulae through SMT-solving techniques, rather than by checking the emptiness of the language of a suitable automaton A_{\phi}. The technique is effective, and it has been implemented in our Zot formal verification tool.Comment: 39 page