Recognition Of Context-free Languages Research Articles

Algorithm partition and parallel recognition of general context-free languages using fixed-size VLSI architecture

Recognition of general context-free languages has been very important in language processing and syntactic pattern recognition. Many researchers have attempted to speed-up the recognition procedure. Recently, several VLSI implementations of context-free language recognition have been proposed. The main disadvantage of these proposed VLSI systems is that the size of the processor array is critical to the performance. That is, an n × n upper triangular array can only process strings with length not longer than n. In this paper we propose two algorithms which can recognize general context-free languages without restriction on the length of input string. The first one has time complexity O( n 4 k 2 ) using k × k processing elements to recognize n input symbols; if k = n, its time complexity will be O( n 2). The second one, if using k × k processing elements to recognize n input symbols, has time complexity O( n 3 k 2 ) ; if k = n, the time complexity will be O( n). If there are several such modules, we can perform parallel recognition of general context-free languages. Since these algorithms essentially speed-up the dynamic programming procedure by using highly pipelining and parallelism of VLSI architecture, the proposed algorithms may also be useful for other problems solvable by dynamic programming.

Fast recognition of pushdown automaton and context-free languages

We prove: (1) every language accepted by a two-way nondeterministic pushdown automaton can be recognized on a random access machine in O ( n 3 /log n ) time; (2) every language accepted by a loop-free two-way nondeterministic pushdown automaton can be recognized in O ( n 3 /log 2 n ) time; (3) every context-free language can be recognized on-line in O ( n 3 /log 2 n ) time. We improve the results of Aho, Hopcroft, and Ullman ( Inform. Contr. 13 , 1968, 186–206), Rytter ( Inform. Process. Lett. 16 , 1983, 127–129), and Graham, Harrison, and Ruzzo ( ACM Trans. Programm. Lang. Systems 2 , No. 3, 1980, 415–462).

On efficient recognition of transductions and relations

We give some new or improved algorithms for recognizing transductions, relations and languages defined by pushdown machines, counter machines, etc. For example, we show that transductions defined by κ-tape one-way nondeterministic pushdown acceptors with outputs can be recognized in CFL( n κ ) time, where CFL( n) is the multitape TM or unit-cost random access machine (RAM) complexity of context-free language recognition. When the pushdown store makes only one-turn or the pushdown store is replaced by a counter, we have RAM algorithms that run in O(n 2κ/ log 2κ (2κ−1) n) and O(n κ+1/ log (κ+1) κ n) time respectively. Thus, linear context-free languages and one-counter languages are recognizable in O( n 2/log 2 n) time, which is an improvement over previously known algorithms. We also show that relations accepted by κ-tape two-way nondeterministic finite-state acceptors can be recognized in O( n κ ) time on a RAM, which improves another known result. Our methods use translation, dynamic programming with ‘precomputation’ and depth-first search.

Some Area-Time Tradeoffs for VLSI

Area-time bounds on VLSI circuits for context-free language recognition, for the evaluation of propositional calculus formulae and for set equality and disjointness questions, are considered. In all cases, a lower bound $AT^{2\alpha } = \Omega (n^{1 + \alpha } )$ is proved, where A is the chip area, T the execution time, and $0 \leqq \alpha \leqq 1$. Similar results were known for computations with $\Omega (n)$-bit outputs, but the computations considered here have only 1-bit outputs. Upper bounds are also discussed.

VLSI architectures for high speed recognition of context-free languages and finite-state languages

This paper presents two VLSI architectures for the recognition of context-free languages and finite-state languages. The architecture for context-free languages consists of n(n+1)/2 identical cells and it is capable of recognizing an input string of length n in 2n time units. The architecture for finite-state languages consists of n cells and it can recognize a string of length n in constant time. Since both architectures have characteristics such as modular layout, simple constrol and dataflow pattern, high degree of multiprocessing and/or pipelining, etc., they are very suitable for VLSI implementation.

On uniform circuit complexity

We argue that uniform circuit complexity introduced by Borodin is a reasonable model of parallel complexity. Three main results are presented. First, we show that alternating Turing machines are also a surprisingly good model of parallel complexity, by showing that simultaneous size/depth of uniform circuits is the same as space/time of alternating Turing machines, with depth and time within a constant factor and likewise log(size) and space. Second, we apply this to characterize NC, the class of polynomial size and polynomial-in-log depth circuits, in terms of tree-size bounded alternating TMs and other models. In particular, this enables us to show that context-free language recognition is in NC. Third, we investigate various definitions of uniform circuit complexity, showing that it is fairly insensitive to the choice of definition.

Tree-size bounded alternation

The size of an accepting computation tree of an alternating Turing machine (ATM) is introduced as a complexity measure. We present a number of applications of tree-size to the study of more traditional complexity classes. Tree-size on ATMs is shown to closely correspond to time on nondeterministic TMs and on nondeterministic auxiliary pushdown automata. One application of the later is a useful new characterization of the class of languages log-space-reducible to context-free languages. Surprising relationships with parallel-time complexity are also demonstrated. ATM computations using at most space S( n) and tree-size Z( n) (simultaneously) can be simulated in alternating space S( n) and time S( n) · log Z( n) (simultaneously). Several well-known simulations, e.g., Savitch's theorem, are special cases of this result. It also leads to improved parallel complexity bounds for many problems in terms of both time and number of “processors.” As one example we show that context-free language recognition in time O(log 2 n) is possible on several parallel models. Further, this bound is achievable with only a polynomial number of processors, in contrast to all previously known sub-linear time CFL recognizers.

Spatial complexity of recognition of context-free languages

For the recognition of context-free languages (CF-languages), use is made of Turing machines with two tapes: the input tape and the operational tape. Let n be the length of the input tape. As a measure of complexity we choose the length of the operational tape L(n). We denote by M (respectively, N) the class of languages which are recognized on deterministic (nondeterministic) Turing machines with L(n) ~< log2n. It is known that all CF-languages enter in class N [4, 7] and are recognized by deterministic Turing machines with L(n) ~< (log2n) 2 [3], while class M contains certain subclasses of CF-languages (for example, the Dyck language, bounded CFlanguages [7]). It is also noted in [7] that all the examples of CF-languages known to the authors belong to class M. The following two problems are well known:

Speed of Recognition of Context-Free Languages by Array Automata

The recognition speed of context-free languages (CFL’s) using arrays of finite state machines is considered. It is shown that CFL’s can be recognized by 2-dimensional arrays in linear time and by 1-dimensional arrays in time $n^2 $.

Marker automata

This paper deals with finite automata augmented with markers which the automata can move about on their input tapes. The concept of augmenting markers to automata was first introduced by Blum and Hewitt [1] in two-dimensional automata. Kreider, Ritchie, and Springsteel [6, 7, 8, 12] investigated recognition of context-free languages by (one-dimensional) automata with markers. In this paper, we investigate some fundamental properties of marker automata and study their relationships to other types of automata and languages. The main result in this paper is the establishment of an infinite hierarchy of languages recognizable by deterministic and deterministic, halting marker automata. It also turns out that, because of the equivalence of finite marker automata and multi-head automata, the study of three-marker automata becomes very interesting due to results of Hartmanis [4] and Savitch [10].

The Hardest Context-Free Language

There is a context-free language $L_0 $ such that every context-free language is an inverse homomorphic image of $L_0 $ or $L_0 - \{ e\} $. Hence the time complexity of recognition of $L_0 $ is the least upper bound for time complexity of recognition of context-free languages. A similar result holds for quasirealtime Turing machine languages. Several languages are given such that deterministic and nondeterministic polynomial time acceptance are equivalent if and only if any one of them is deterministic polynomial time acceptable.

Generation, recognition and parsing of context-free languages by means of recursive graphs

A device called recursive graph defining context-free languages is described. Previously such a device was used byConway [1] for a compiler design.Conway called it “transition diagram”. Roughly, a recursive graph is a finite set of finite state graphs, which are used in a recursive manner. A recursive graph covers the essential features of both standard devices: It describes the syntactical structure as grammars do, and it represents a method for recognition and parsing as push-down automata do. A notion ofk-determinacy is introduced for recursive graphs and for a restricted kind of them, called simple recursive graphs. Thek-deterministic simple recursive graphs are more general thanLL (k) grammars [5] with respect to parsing-power, but equal with respect to language generation power. The more generalk-deterministic recursive graphs cannot parse the full set ofLR (k)-grammars [4], but they can recognize the full set ofLR (k)-languages. The set of languages recognized by (simple) recursive graphs is the set of context-free languages.

P. M. LewisII, R. E. Stearns, and J. Hartmanis. Memory bounds for recognition of context-free and context-sensitive languages. Sixth Annual Symposium on Switching Circuit Theory and Logical Design, University of Michigan, Ann Arbor, Mich., The Institute of Electrical and Electronics Engineers, Inc., New York 1965, pp. 191–202. See Errata, ibid., p. 190.

P. M. LewisII, R. E. Stearns, and J. Hartmanis. Memory bounds for recognition of context-free and context-sensitive languages. Sixth Annual Symposium on Switching Circuit Theory and Logical Design, University of Michigan, Ann Arbor, Mich., The Institute of Electrical and Electronics Engineers, Inc., New York 1965, pp. 191–202. See Errata, ibid., p. 190. - Volume 37 Issue 3

Sheila A. Greibach. The unsolvability of the recognition of linear context-free languages. Journal of the Association for Computing Machinery, vol. 13 (1966), pp. 582–587.

Sheila A. Greibach. The unsolvability of the recognition of linear context-free languages. Journal of the Association for Computing Machinery, vol. 13 (1966), pp. 582–587.

A machine-oriented recognition algorithm for context-free languages

A variant of the Cocke-Younger algorithm for recognition of context-free languages permits faster operation by a factor proportional to word length on computers with parallel bit-manipulation capabilities.

A note on computing time for the recognition of context-free languages by a single-tape Turing machine

It is shown that (1) every context-free language is T(n) = n 4 -recognizable by a single-tape Turing machine and (2) every linear context-free language is T(n) = n 2 -recognizable by a single-tape Turing machine. It follows from (2) and a result obtained by Hartmanis (1968) that T(n) = n 2 is a least upper time bound in which every linear context-free language can be recognized by a single-tape Turing machine.

Recognition and parsing of context-free languages in time n3

A recognition algorithm is exhibited whereby an arbitrary string over a given vocabulary can be tested for containment in a given context-free language. A special merit of this algorithm is that it is completed in a number of steps proportional to the “cube” of the number of symbols in the tested string. As a byproduct of the grammatical analysis, required by the recognition algorithm, one can obtain, by some additional processing not exceeding the “cube” factor of computational complexity, a parsing matrix—a complete summary of the grammatical structure of the sentence. It is also shown how, by means of a minor modification of the recognition algorithm, one can obtain an integer representing the ambiguity of the sentence, i.e., the number of distinct ways in which that sentence can be generated by the grammar. The recognition algorithm is then simulated on a Turing Machine. It is shown that this simulation likewise requires a number of steps proportional to only the “cube” of the test string length.

The Unsolvability of the Recognition of Linear Context-Free Languages

The problem of whether a given context-free language is linear is shown to be recursively undecidable.