], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding

We consider the restriction of the guarded fragment to the two-variable case where, in addition, binary relations may be specified as transitive. We show that (i) this very restricted form of the guarded fragment without equality is undecidable and that (ii) when allowing non-unary relations

, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature.

in which linear orders are the only non-unary relations. Recently, a contrasting result has been shown, that the finite satisfiability problem for FO 2 with two linear orders and with no additional non-unary relations is decidable. We observe that our proof can be adapted to some interesting fragments

Σ ∈ N in unary, and the hierarchy's maximum depth D ∈ N, the setup algorithm first generates a bilinear instance 1. Select 7 + 5 D + D 2 random integers modulo p (some of them forcibly non-zero): 2. Publish G and the system parameters Pub ∈ G T × G 2 (1+D) (2+D) given by: 3. Retain the master

, Halpern, and Koller [22], in the general case, asymptotic conditional probabilities do not always exist, and most questions relating to this issue are highly undecidable. These results, however, all depend on the assumption that ` can use a nonunary predicate symbol. Liogon'kii [31] shows that if we

DFCA that can be obtained from a given minimal DFA with n states by merging the similar states in the given DFA. We compute an upper bound for this number and prove that in the worst case (for a non-unary alphabet) it is d

succinct than finite automata. In order to state this result, we investigate the conversion of deterministic pushdown automata into context-free grammars. We prove that in the unary case the number of variables in the resulting grammar is strictly smaller than the number of variables needed in the case

. The case of unary languages is one of them and as a consequence, we show how to decide for two given finite subsets of nonnegative integers, whether they are the n-th root of a common set, for some n ≥ 1. We also show that it is decidable whether a finite collection of finite languages is a Parikh code

-calculi, and non-deterministic SPCF. We prove universality and full abstraction results for these models by reduction to the case of unary PCF, for which we give a simple new argument to show that any order-extensional and sequential model is universal.