Ian M. Hodkinson

Decidable fragments of first-order temporal logics

Abstract In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator (Since or Until) have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment of classical first-order logic. This reduction is then used to single out a number of decidable fragments of first-order temporal logics and of two-sorted first-order logics in which one sort is intended for temporal reasoning. Besides standard first-order time structures, we consider also those that have only finite first-order domains, and extend the results mentioned above to temporal logics of finite domains. We prove decidability in three different ways: using decidability of monadic second-order logic over the intended flows of time, by an explicit analysis of structures with natural numbers time, and by a composition method that builds a model from pieces in finitely many steps.

Complete representations in algebraic logic

A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any xed dimension are elementary.

An Axiomatization of the Temporal Logic with Until and Since over the Real Numbers

A Hilbert style axiomatization of the temporal logic with connectives Until and Since for the real numbers is presented. We prove independence of the axioms, and completeness for this semantics with respect to single formulas.

Step by step-building representations in algebraic logic

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is co-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this approach are looked at, and include the step by step method. ?

On the computational complexity of decidable fragments of first-order linear temporal logics

We study the complexity of some fragments of first-order temporal logic over natural numbers time. The one-variable fragment of linear first-order temporal logic even with sole temporal operator /spl square/ is EXPSPACE-complete (this solves an open problem of J. Halpern and M. Vardi (1989)). So are the one-variable, two-variable and monadic monodic fragments with Until and Since. If we add the operators O/sup n/, with n given in binary, the fragment becomes 2EXPSPACE-complete. The packed monodic fragment has the same complexity as its pure first-order part - 2EXPTIME-complete. Over any class of flows of time containing one with an infinite ascending sequence - e.g., rationals and real numbers time, and arbitrary strict linear orders - we obtain EXPSPACE lower bounds (which solves an open problem of M. Reynolds (1997)). Our results continue to hold if we restrict to models with finite first-order domains.

Monodic Packed Fragment with Equality is Decidable

We prove decidability of satisfiability of sentences of the monodic packed fragment of first-order temporal logic with equality and connectives Until and Since, in models with various flows of time and domains of arbitrary cardinality. We also prove decidability over models with finite domains, over flows of time including the real order.

A toolkit enabling efficient, scalable and reproducible gene tagging in trypanosomatids

One of the first steps in understanding a proteins function is to determine its localization; however, the methods for localizing proteins in some systems have not kept pace with the developments in other fields, creating a bottleneck in the analysis of the large datasets that are generated in the post-genomic era. To address this, we developed tools for tagging proteins in trypanosomatids. We made a plasmid that, when coupled with long primer PCR, can be used to produce transgenes at their endogenous loci encoding proteins tagged at either terminus or within the protein coding sequence. This system can also be used to generate deletion mutants to investigate the function of different protein domains. We show that the length of homology required for successful integration precluded long primer PCR tagging in Leishmania mexicana. Hence, we developed plasmids and a fusion PCR approach to create gene tagging amplicons with sufficiently long homologous regions for targeted integration, suitable for use in trypanosomatids with less efficient homologous recombination than Trypanosoma brucei. Importantly, we have automated the primer design, developed universal PCR conditions and optimized the workflow to make this system reliable, efficient and scalable such that whole genome tagging is now an achievable goal.

Loosely Guarded Fragment of First-Order Logic has the Finite Model Property

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.

Atom structures of cylindric algebras and relation algebras

For any finite n 3 there are two atomic n-dimensional cylindric algebras with the same atom structure, with one representable, the other, not. Hence, the complex algebra of the atom structure of a representable atomic cylindric algebra is not always representable, so that the class RCAn of representable n-dimensional cylindric algebras is not closed under completions. Further, it follows by an argument of Venema that RCAn is not axiomatisable by Sahlqvist equations, and hence nor by equations where negation can only occur in constant terms. Similar results hold for relation algebras.

A construction of cylindric and polyadic algebras from atomic relation algebras

Given a simple atomic relation algebra