Mathematics

The present paper is devoted to study some completeness properties of transitive binary relational set, i.e., a set together with a transitive binary relation (so called t-set).

The minimal Kolmogorov complexity of a total computable function that exceeds everywhere all total computable functions of complexity at most n , is 2 n+O(1) . If we replace "everywhere" by "for all sufficiently large inputs", the answer is n+O(1) .

We consider the Lambek calculus, or non-commutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an ω -rule, and prove that the derivability problem in this calculus is Π 0 1 -hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with unique type assignment, without Lambek's non-emptiness restriction imposed (cf. Safiullin 2007).

Non-monotone inductive definitions were studied in the late 1960's and early 1970's with the aim of understanding connections between the complexity of the formulas defining the induction steps and the size of the ordinals measuring the duration of the inductions. In general, any type 2 functional will generate an inductive process, and in this paper we will view non-monotone induction as a functional of type 3. We investigate the associated computation theory inherited from the Kleene schemes and we investigate the nature of the associated companion of sets with codes computable in non-monotone induction. The interest in this functional is motivated from observing that constructions via non-monotone induction appear as natural in classical analysis in its original form. There are two groups of results: We establish strong closure properties of the least ordinal without a code computable in non-monotone induction, and we provide a characterisation of the class of functionals of type 3 computable from non-monotone induction, a characterisation in terms of sequential operators working in transfinite time. We will also see that the full power of non-monotone induction is required when this principle is used to construct functionals witnessing the compactness of the Cantor space and of closed, bounded intervals.

We introduce the computable FS-jump, an analog of the classical Friedman--Stanley jump in the context of equivalence relations on N . We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

In this paper, we will provide a method to compute the density of tautologies among the set of well-formed formulae consisting of m variables, a negation symbol and an implication symbol, which has a possibility to be applied for other logical systems. This paper contains computational numerical values of the density of tautologies for two, three, and four variable cases. Also, for certain quadratic systems, we will introduce the s -cut concept to make a better approximation when we compute the ratio by brute-force counting, and discover a fundamental relation between generating functions' values on the singularity point and ratios of coefficients, which can be understood as another intepretation of the Szegő lemma for such quadratic systems. With this relation, we will provide an asymptotic lower bound m −1 −(7/4) m −3/2 +O( m −2 ) of the density of tautologies as m goes to the infinity.

We present three classes of abstract prearithmetics, { A M } M?? , { A ?�M,M } M?? , and { B M } M?? . The first one is weakly projective with respect to the conventional nonnegative real Diophantine arithmetic R + =( R + ,+,?, ??R + ) , while the other two are weakly projective with respect to the conventional real Diophantine arithmetic R=(R,+,?, ??R ) . In addition, we have that every A M and every B M are a complete totally ordered semiring, while every A ?�M,M is not. We show that the weak projection of any series in R + converges in A M , for any M?? , and that the weak projection of any non-oscillating series in R converges in A ?�M,M , for any M?? , and in B M , for all M??R + . We also prove that working in A M and in A ?�M,M , for any M?? , allows to overcome a version of the paradox of the heap, while working in B M does not.

We introduce a formal logical language, called conditional probability logic (CPL), which extends first-order logic and which can express probabilities, conditional probabilities and which can compare conditional probabilities. Intuitively speaking, although formal details are different, CPL can express the same kind of statements as some languages which have been considered in the artificial intelligence community. We also consider a way of making precise the notion of lifted Bayesian network, where this notion is a type of (lifted) probabilistic graphical model used in machine learning, data mining and artificial intelligence. A lifted Bayesian network (in the sense defined here) determines, in a natural way, a probability distribution on the set of all structures (in the sense of first-order logic) with a common finite domain D . Our main result is that for every "noncritical" CPL-formula φ( x ¯ ) there is a quantifier-free formula φ ∗ ( x ¯ ) which is "almost surely" equivalent to φ( x ¯ ) as the cardinality of D tends towards infinity. This is relevant for the problem of making probabilistic inferences on large domains D , because (a) the problem of evaluating, by "brute force", the probability of φ( x ¯ ) being true for some sequence d ¯ of elements from D has, in general, (highly) exponential time complexity in the cardinality of D , and (b) the corresponding probability for the quantifier-free φ ∗ ( x ¯ ) depends only on the lifted Bayesian network and not on D . The main result has two corollaries, one of which is a convergence law (and zero-one law) for noncritial CPL-formulas.

We continue the research of the relation ∣ ˜ on the set βN of ultrafilters on N , defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of = ∼ -equivalence classes, where F = ∼ G means that F and G are mutually ∣ ˜ -divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that = ∼ -equivalent ultrafilters do not necessarily have the same residue modulo m∈N . Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we also introduce a strengthening of ∣ ˜ and show that it also behaves well in relation to the congruence relation.

A join-semilattice L is said to be conjunctive if it has a top element 1 and it satisfies the following first-order condition: for any two distinct a,b∈L , there is c∈L such that either a∨c≠1=b∨c or a∨c=1≠b∨c . Equivalently, a join-semilattice is conjunctive if every principal ideal is an intersection of maximal ideals. We present simple examples showing that a conjunctive join-semilattice may fail to have any prime ideals. (Maximal ideals of a join-semilattice need not be prime.) We show that every conjunctive join-semilattice is isomorphic to a join-closed subbase for a compact T 1 -topology on MaxL , the set of maximal ideals of L . The representation is canonical in that when applied to a join-closed subbase for a compact T 1 -space X , the space produced by the representation is homeomorphic with X . We say a join-semilattice morphism ϕ:L→M is conjunctive if ϕ −1 (w) is an intersection of maximal ideals of L whenever w is a maximal ideal of M . We show that every conjunctive morphism between conjunctive join-semilattices is induced by a multi-valued function from MaxM to MaxL . A base for a topological space is said to be annular if it is a lattice, and Wallman if it is annular and for any point u in any basic open U , there a basic open V that misses u and together with U covers X . It is easy to show that every Wallman base is conjunctive. We give an example of a conjunctive annular base that is not Wallman. Finally, we examine the free distributive lattice dL over a conjunctive join semilattice L . In general, it is not conjunctive, but we show that a certain canonical, algebraically-defined quotient of dL is isomorphic to the sub-lattice of the topology of the representation space that is generated by L . We describe numerous applications.