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20
The Equational Theory of Fixed Points with Applications to Generalized Language Theory
, 2001
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Path algorithms on regular graphs
"... Abstract. We consider standard algorithms of finite graph theory, like for instance shortest path algorithms. We present two general methods to polynomially extend these algorithms to infinite graphs generated by deterministic graph grammars. 1 ..."
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Abstract. We consider standard algorithms of finite graph theory, like for instance shortest path algorithms. We present two general methods to polynomially extend these algorithms to infinite graphs generated by deterministic graph grammars. 1
Combining Equational Tree Automata Over AC and ACI Theories ⋆
"... Abstract. In this paper, we study combining equational tree automata in two different senses: (1) whether decidability results about equational tree automata over disjoint theories E1 and E2 imply similar decidability results in the combined theory E1 ∪ E2; (2) checking emptiness of a language obtai ..."
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Abstract. In this paper, we study combining equational tree automata in two different senses: (1) whether decidability results about equational tree automata over disjoint theories E1 and E2 imply similar decidability results in the combined theory E1 ∪ E2; (2) checking emptiness of a language obtained from the Boolean combination of regular equational tree languages. We present a negative result for the first problem. Specifically, we show that the intersectionemptiness problem for tree automata over a theory containing at least one AC symbol, one ACI symbol, and 4 constants is undecidable despite being decidable if either the AC or ACI symbol is removed. Our result shows that decidability of intersectionemptiness is a nonmodular property even for the union of disjoint theories. Our second contribution is to show a decidability result which implies the decidability of two open problems: (1) If idempotence is treated as a rule f(x, x) → x rather than an equation f(x, x) = x, is it decidable whether an AC tree automata accepts an idempotent normal form? (2) If E contains a single ACI symbol and arbitrary free symbols, is emptiness decidable for a Boolean combination of regular Etree languages? 1
Systems].
"... We present a generic aproach to the static analysis of concurrent programs with procedures. We model programs as communicating pushdown systems. It is known that typical dataflow problems for this model are undecidable, because the emptiness problem for the intersection of contextfree languages, wh ..."
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We present a generic aproach to the static analysis of concurrent programs with procedures. We model programs as communicating pushdown systems. It is known that typical dataflow problems for this model are undecidable, because the emptiness problem for the intersection of contextfree languages, which is undecidable, can be reduced to them. In this paper we propose an algebraic framework for defining abstractions (upper approximations) of contextfree languages. We consider two classes of abstractions: finitechain abstractions, which are abstractions whose domains do not contain any infinite chains, and commutative abstractions corresponding to classes of languages that contain a word if and only if they contain all its permutations. We show how to compute such approximations by combining automata theoretic techniques with algorithms for solving systems of polynomial inequations in Kleene algebras.
Parikh’s Theorem: A simple and direct automaton construction
"... Parikh’s theorem states that the Parikh image of a contextfree language is semilinear or, equivalently, that every contextfree language has the same Parikh image as some regular language. We present a very simple construction that, given a contextfree grammar, produces a finite automaton recogniz ..."
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Parikh’s theorem states that the Parikh image of a contextfree language is semilinear or, equivalently, that every contextfree language has the same Parikh image as some regular language. We present a very simple construction that, given a contextfree grammar, produces a finite automaton recognizing such a regular language. The Parikh image of a word w over an alphabet {a1,..., an} is the vector (v1,..., vn) ∈ Nn such that vi is the number of occurrences of ai in w. For example, the Parikh image of a1a1a2a2 over the alphabet {a1, a2, a3} is (2, 2, 0). The Parikh image of a language is the set of Parikh images of its words. Parikh images are named after Rohit Parikh, who in 1966 proved a classical theorem of formal language theory which also carries his name. Parikh’s theorem [1] states that the Parikh image of any contextfree language is semilinear. Since semilinear sets coincide with the Parikh images of regular languages, the theorem is equivalent to the statement that every contextfree language has the same Parikh image as some regular language. For instance, the language {anbn  n ≥ 0} has the same Parikh image as (ab)∗. This statement is also often referred to as Parikh’s theorem, see e.g. [10], and in fact it has been considered a more natural formulation [14]. Parikh’s proof of the theorem, as many other subsequent proofs [8, 14, 13, 9, 10, 2], is constructive: given a contextfree grammar G, the proof produces (at least implicitly) an automaton or regular expression whose language has the same Parikh image as L(G). However, these constructions are relatively complicated, not given explicitly, or yield crude upper bounds: automata of size O(nn) for grammars in Chomsky normal form with n variables (see Section 4 for a detailed discussion). In this note we present an explicit and very simple construction yielding an automaton with O(4n) states, for a lower bound of 2n.