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The complexity class BPP (defined by Gill) contains problems that can be solved in polynomial time with bounded error probability. A new and simple characterization of BPP is given. It is shown that a language L is in BPP iff (x ∈ L → ∃⁺y∀zP(x, y, z)) ∧ (x ∉ L → ∀y∃⁺ z ¬ P(x, y, z)) for a polynomial-time predicate P and for |y|, |z| ⩽ poly (|x|). The formula ∃⁺ yP(y) with the random quantifier ∃⁺ means that the probability Pr({y|P(y)}) ⩾+ ɛ for a fixed ɛ. This characterization allows a simple proof that BPP ⊆ ZPP^NP, which strengthens the result of (Lautemann, Inform. Process. Lett. 17 (1983), 215–217; Sipser, in “Proceedings, 15th Annu. ACM Sympos. Theory of Comput.,” 1983, pp. 330–335) that BPP ⊆ Σ₂^p ∩ Π₂^p. Several other results about probabilistic classes can be proved using similar techniques, e.g., NP^R ⊆ ZPP^NP and Σ₂^p,BPP = Σ₂^p.

Two automata models are introduced that play, with respect to attribute grammars and attribute-evaluation for them, the same role as pushdown automata have with respect to context-free grammars and their parsing. It is shown, in fact, that these automata define the same class of string-to-value translations as attribute grammars. Their class of tree-to-value translations seems instead to be larger than that of attribute grammars and the difference is overcome by means of (a special type of) context-free grammar interpretations. An extended model of attribute grammar is presented that is as powerful as the automata with respect to tree-to-value translations.

For a Jordan curve C in the plane nowhere tangent to the x axis, let x₁, x₂,…, x_n be the abscissas of the intersection points of C with the x axis, listed in the order the points occur on C. We call x₁, x₂,…, x_n a Jordan sequence. In this paper we describe an O(n)-time algorithm for recognizing and sorting Jordan sequences. The problem of sorting such sequences arises in computational geometry and computational geography. Our algorithm is based on a reduction of the recognition and sorting problem to a list-splitting problem. To solve the list-splitting problem we use level-linked search trees.

We prove that every set of partial recursive functions which can be identified by an inductive inference machine is included in some identifiable function set with index set in Σ₃ ∩ Π₃. An identifiable set is presented with index set in Σ₂ ∩ Π₃ but neither in Σ₂ nor in Π₂. Furthermore we show that there is no nonempty identifiable set with index set in Σ₁. In Π₁ it is possible to locate this king of set. In the last part of the paper we show that the problem to identify all partial recursive functions and the halting problem are of the same degree of unsolvability.

For recursive sets A, define a complexity theoretic version of the ordinary recursion theoretic jump by setting A′ equal to the canonical NP^A-complete set. Thus A < ^P_T A′ iff P^A ≠ NP^A. The nth jump, A⁽ⁿ⁾, is defined by iteration. A jumps n times if A < ^P_T A′ < ^P_T… < ^P_T A⁽ⁿ⁾. It is straightforward that the jump operation is monotone. Post's theorem holds for the (relativized) polynomial hierarchy. We establish the following analogues of results in ordinary recursion theory: all relationships between pairs of polynomial Turing degrees and their jumps consistent with monotonicity can be realized by degrees which jump at least twice. For example, there are polynomially incomparable A and B with A′ ≡ ^P_T B′. Moreover, if for each recursive D the set of E such that D join E jumps at least n times is effectively comeager, then these relationships can be realized by degrees jumping at least n times. We also relativize some well-known results by showing that if A′ is polynomially many-one reducible to the join of A and a (co-)sparse set, then P^A = NP^A; and if A′ is polynomially Turing reducible to the join of A and an NP^A-(co-)sparse set, then the relativized polynomial hierarchy collapses to Δ^P,A₂.

We characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe. We also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation. This proves that the fixed point query hierarchy suggested by Chandra and Harel collapses at the first fixed point level. It is also a general result showing that in finite model theory one application of fixed point suffices.

This paper investigates the circular retrieval problem and the k-nearest neighbor problem, for sets of n points in the Euclidean plane. Two similar data structures each solve both problems. A deterministic structure uses space O(n(log n log log n)²), and a probabilistic structure uses space O(n log² n). For both problems, these two structures answer a query that returns k points in O(k + log n) time.

We propose a translation method of finite terms of CCS into formulas of a modal language representing their class of observational congruence. For this purpose, we define a modal language and a function associating with any finite term of CCS a formula of the language, satisfied by the term. Furthermore, this function is such that two terms are congruent if and only if the corresponding formulas are equivalent. The translation method consists in associating with operations on terms (action, +) operations on the corresponding formulas. This work is a first step towards the definition of a modal language with modalities expressing both possibility and inevitability and which is compatible with observational congruence.

A syntax directed proof system which allows to prove liveness properties of while-programs is introduced. The proof system is proved to be arithmetically sound and complete in the sense of Harel (“Lecture Notes in Comput. Sci. Vol. 68,” Springer-Verlag, Berlin/New York, 1979). The results of the paper generalize a corresponding result Pneuli (“Prc. 18th Sympos. FOCS” IEEE, Providence, R. I., 1977) proves for unstructured programs. The proof system decomposes into two parts. The first part allows to prove simple safety properties. These properties are used as axioms in the second proof system which deals with liveness properties. The completeness proof is constructive and provides a heuristic for proving specific liveness properties.

We discuss the mathematical foundations of specifications, theories, and models with higher types. Higher type theories are presented by specifications either using the language of cartesian closure or a typed λ-calculus. We prove equivalence of both the specification methods, the main result being the equivalence of cartesian closure and a typed λ-calculus. Then we investigate “intensional” and extensional” models (the distinction is similar to that between λ-algebras and (λ)-models). We prove completeness of higher type theories with regard to intensional models as well as existence of free intensional models. For extensional models we prove that completeness and existence of an initial models implies that the theory itself already is the initial model. As a consequence intensional models seem to be better suited for the purposes of data type specification.