Alexander Okhotin

Boolean grammars

A new generalization of context-free grammars is introduced: Boolean grammars allow the use of all set-theoretic operations as an integral part of the formalism of rules. Rigorous semantics for these grammars is defined by language equations in a way that allows to generalize some techniques from the theory of context-free grammars, including Chomsky normal form, Cocke-Kasami-Younger cubic-time recognition algorithm and some limited extension of the notion of a parse tree, which together allow to conjecture practical applicability of the new concept.

Conjunctive grammars over a unary alphabet: undecidability and unbounded growth

It has recently been proved (Jez, DLT 2007) that conjunctive grammars (that is, context-free grammars augmented by conjunction) generate some nonregular languages over a one-letter alphabet. The present paper improves this result by constructing conjunctive grammars for a larger class of unary languages. The results imply undecidability of a number of decision problems of unary conjunctive grammars, as well as nonexistence of an r.e. bound on the growth rate of generated languages. An essential step of the argument is a simulation of a cellular automaton recognizing positional notation of numbers using language equations.

Conjunctive Grammars and Systems of Language Equations

This paper studies systems of language equations that are resolved with respect to variables and contain the operations of concatenation, union and intersection. Every system of this kind is proved to have a least fixed point, and the equivalence of these systems to conjunctive grammars is established. This allows us to obtain an algebraic characterization of the language family generated by conjunctive grammars.

State complexity of power

The number of states in a deterministic finite automaton (DFA) recognizing the language L^k, where L is regular language recognized by an n-state DFA, and k>=2 is a constant, is shown to be at most n2^(^k^-^1^)^n and at least (n-k)2^(^k^-^1^)^(^n^-^k^) in the worst case, for every n>k and for every alphabet of at least six letters. Thus, the state complexity of L^k is @Q(n2^(^k^-^1^)^n). In the case k=3 the corresponding state complexity function for L^3 is determined as 6n-384^n-(n-1)2^n-n with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of L^k is demonstrated to be nk. This bound is shown to be tight over a two-letter alphabet.

On the equivalence of linear conjunctive grammars and trellis automata

This paper establishes computational equivalence of two seemingly unrelated concepts: linear conjunctive grammars and trellis automata. Trellis automata, also studied under the name of one-way real-time cellular automata, have been known since early 1980s as a purely abstract model of parallel computers, while linear conjunctive grammars, introduced a few years ago, are linear context-free grammars extended with an explicit intersection operation. Their equivalence implies the equivalence of several other formal systems, including a certain restricted class of Turing machines and a certain type of language equations, thus giving further evidence for the importance of the language family they all generate.

Conjunctive Grammars over a Unary Alphabet: Undecidability and Unbounded Growth

It has recently been proved (Jeż, DLT 2007) that conjunctive grammars (that is, context-free grammars augmented by conjunction) generate some non-regular languages over a one-letter alphabet. The present paper improves this result by constructing conjunctive grammars for a larger class of unary languages. The results imply undecidability of a number of decision problems of unary conjunctive grammars, as well as non-existence of a recursive function bounding the growth rate of the generated languages. An essential step of the argument is a simulation of a cellular automaton recognizing positional notation of numbers using language equations.

State complexity of cyclic shift

The cyclic shift of a language L, defined as SHIFT(L) = {vu | uv ∈ L}, is an operation known to preserve both regularity and context-freeness. Its descriptional complexity has been addressed in Maslovs pioneering paper on the state complexity of regular language operations [Soviet Math. Dokl. 11 (1970) 1373-1375], where a high lower bound for partial DFAs using a growing alphabet was given. We improve this result by using a fixed 4-letter alphabet, obtaining a lower bound (n - 1)! 2 (n-1)(n-2) , which shows that the state complexity of cyclic shift is 2 n2+n log n-o(n) for alphabets with at least 4 letters. For 2- and 3-letter alphabets, we prove 2 ⊖(n2) state complexity. We also establish a tight 2n 2 + 1 lower bound for the nondeterministic state complexity of this operation using a binary alphabet.

A recognition and parsing algorithm for arbitrary conjunctive grammars

Conjunctive grammars are basically context-free grammars with an explicit set intersection operation added to the formalism of rules. This paper presents a cubic-time recognition and parsing algorithm for this family of grammars, which is applicable to an arbitrary conjunctive grammar without any initial transformations.The algorithm is in fact an extension of the context-free recognition and parsing algorithm due to Graham, Harrison and Ruzzo, and it retains the cubic time complexity of its prototype. It is shown that for the case of linear conjunctive grammars this algorithm can be modified to work in quadratic time and use linear space.The given algorithm is then applied to solve the membership problem for conjunctive grammars in polynomial time, and subsequently to prove the problems P-completeness, as well as P-completeness of the membership problem for linear conjunctive grammars.

Decision problems for language equations with Boolean operations

The paper studies resolved systems of language equations that allow the use of all Boolean operations in addition to concatenation. Existence and uniqueness of solutions are shown to be their nontrivial properties, these properties are given characterizations by first order formulae, and the position of the corresponding decision problems in the arithmetical hierarchy is determined. The class of languages defined by components of unique solutions of such systems is shown to coincide with the class of recursive languages.

On stateless multihead automata: Hierarchies and the emptiness problem

We look at stateless multihead finite automata in their two-way and one-way, deterministic and nondeterministic variations. The transition of a k-head automaton depends solely on the symbols currently scanned by its k heads, and every such transition moves each head one cell left or right, or instructs it to stay. We show that stateless (k+4)-head two-way automata are more powerful than stateless k-head two-way automata. In the one-way case, we prove a tighter result: stateless (k+1)-head one-way automata are more powerful than stateless k-head one-way automata. Finally, we show that the emptiness problem for stateless 2-head two-way automata is undecidable.