We prove that, for every theory T which is given by an Lω1,ω sentence, T has less than 2א0 many countable models if and only if we have that, for every X ∈ 2 on a cone of Turing degrees, every X-hyperarithmetic model of T has an X-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaught’s conjecture.

In many practical situations, we would like to compute the set of all possible values that satisfy given constraints. It is known that even for computable (constructive) constraints, computing such set is not always algorithmically possible. One reason for this algorithmic impossibility is that sometimes, the dependence of the desired set on the parameters of the problem is not continuous, whil...

To the Editor:—In a recent commentary, Fleisher stated that even if ST segment trend monitoring has identified accurately a significant ST segment change, ST segment depression may not represent myocardial ischemia in younger individuals or in those with a low probability of coronary artery disease (CAD). In support of this contention, the argument is made that structural CAD usually is absent ...

In Eurocrypt’98 [1], Okamoto et al. exhibited a new trapdoor function based on the use of a special moduli (pq) allowing easy discrete logarithm computations. The authors proved that the scheme’s resistance to chosen-plaintext attacks is equivalent to factoring n. Unfortunately, the proposed scheme suffers from not being a permutation (the expansion rate is = 3), and hence cannot be used for pu...

It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional (‘type-theoretical’) axiom of choice ACint, which is provable in Martin-Löf’s type theory, and a weak extensionality principle (Ext−), which is provable in Martin-Löf’s extensional type theory. In particular, EM ⇔ ACext h...

In the recent years, significant progress has been made in explaining apparent hardness ofimproving over naive solutions for many fundamental polynomially solvable problems. Thiscame in the form of conditional lower bounds – reductions to one of problems assumed to behard. These include 3SUM, All-Pairs Shortest Paths, SAT and Orthogonal Vectors, and others.In the (min,+)-convolu...

This paper describes an algorithm for the construction of a certain one-dimensional cellular automaton equivalent to a given D0L system. A cellular automaton is considered equivalent to an L-system if both generates the same words in the same order. Our cellular automata produce the same words and in the same order as the given D0L system for a finite number of derivations. There is no constrai...

This paper shows the first practical semantically secure public-key encryption scheme such that its one-wayness is equivalent to general factoring in the standard model (in the sense of IND-CPA). Next our proof technique is applied to Rabin-Parlier encryption scheme and a variant of RSA-Paillier encryption scheme to prove their exactly tight one-wayness. We finally present the first KEM which i...

A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H ′ are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H ′. In this paper, we study the problem of determining which graphs are Ramsey-equivalent to the complete graph Kk. A famous theorem of Nešetřil and Rödl implies that any graph H which ...