First-Order Concatenation Theory with Bounded Quantifiers

We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.Comment: arXiv admin note: text overlap with arXiv:1804.0636

Hilbert’s Tenth Problem for Term Algebras with a Substitution Operator

We introduce a first-order theory of finite full binary trees and show that the analogue of Hilbert’s Tenth Problem is undecidable by constructing a many-to-one reduction of Post’s Correspondence Problem

Weak Essentially Undecidable Theories of Concatenation

In the language {0,1,∘,⪯}, where 0 and 1 are constant symbols, ∘ is a binary function symbol and ⪯ is a binary relation symbol, we formulate two theories, WD and D, that are mutually interpretable with the theory of arithmetic R and Robinson arithmetic Q, respectively. The intended model of WD and D is the free semigroup generated by {00,11} under string concatenation extended with the prefix relation. The theories WD and D are purely universally axiomatised, in contrast to Q which has the Π2-axiom ∀x[ x=0∨∃y[ x=Sy ] ]

Weak Sequential Theories of Finite Full Binary Trees

We study a first-order theory of finite full binary trees with an axiom schema of open induction. We show that this theory is sequential by constructing a direct interpretation of Adjunctive Set Theory in a very weak finitely axiomatized subtheory. We show that weakening the latter theory by removal of an axiom which states that the subtree relation is transitive gives a theory that directly interprets Vaught’s weak set theory, a non-finitely axiomatizable fragment of Adjunctive Set Theory

Aldol condensation with 2,5-diketopiperazines

2,5-diketopiperazines (DKPs) are naturally occurring cyclopeptides obtained by the condensation of two amino acids. They are found alone or among other natural products produced by fungi, bacteria, the plant kingdom and mammals One way of obtaining substituted DKPS is by aldol condensation of 1,4-Diacetylpiperazine-2,5-dione with aldehydes. In this project, we wanted to study the effect of various bases on this reaction for different aldehydes. The best base would be applied to reaction with ketones, which has not been reported. DBU was found to be to optimal base in reaction with aldehydes, but works poorly for ketones because 1,4-Diacetylpiperazine-2,5-dione is unstable under strongly basic conditions over time

First-Order Concatenation Theory with Bounded Quantifiers

We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results

There are no minimal essentially undecidable theories

We show that there is no theory that is minimal with respect to interpretability among recursively enumerable essentially undecidable theories