84 research outputs found
Characterizing specification languages which admit initial semantics
AbstractThe paper proposes an axiomatic approach to specification languages, and introduces notions of reducibility and equivalence as tools for their study and comparison. Algebraic specification languages are characterized up to equivalence. They are shown to be limited in expressive power by implicational languages
Weak second order characterizations of various program verification systems
AbstractWe show the equivalence of Leivant's characterization of Floyd-Hoare Logic in weak second order logic (Leivant (1985)) with both Csirmaz's (1980) and Sain's (1985) characterizations of Floyd-Hoare logic in Nonstandard Logics of Programs. Our method allows us to spell out the precise role of the comprehension axiom in weak second order logic. We then prove similar results for other program verification systems (suggested by Burstall and Pnueli) and identify exactly the comprehension axioms corresponding to those systems
The parametrized complexity of knot polynomials
AbstractWe study the parametrized complexity of the knot (and link) polynomials known as Jones polynomials, Kauffman polynomials and HOMFLY polynomials. It is known that computing these polynomials is ♯P hard in general. We look for parameters of the combinatorial presentation of knots and links which make the computation of these polynomials to be fixed parameter tractable, i.e., in the complexity class FPT. If the link is explicitly presented as a closed braid, the number of its strands is known to be such a parameter. In a generalization thereof, if the link is explicitly presented as a combination of compositions and rotations of k-tangles the link is called k-algebraic, and its algebraicity k is such a parameter. The previously known proofs that, for this parameter, the link polynomials are in FPT uses the so called skein modules, and is algebraic in its nature. Furthermore, it is not clear how to find such an algebraic presentation from a given link diagram. We look at the treewidth of two combinatorial presentation of links: the crossing diagram and its shading diagram, a signed graph. We show that the treewidth of these two presentations and the algebraicity of links are all linearly related to each other. Furthermore, we characterize the k-algebraic links using the pathwidth of the crossing diagram. Using this, we can apply algorithms for testing fixed treewidth to find k-algebraic presentations in polynomial time. From this we can conclude that also treewidth and pathwidth are parameters of link diagrams for which the knot polynomials are FPT. For the Kauffman and Jones polynomials (but not for the HOMFLY polynomials) we get also a different proof for FPT via the corresponding result for signed Tutte polynomials
On the complexity of Generalized Chromatic Polynomials
J. Makowsky and B. Zilber (2004) showed that many variations of graph colorings,
called CP-colorings in the sequel, give rise to graph polynomials. This is true in
particular for harmonious colorings, convex colorings, mcct-colorings, and rainbow
colorings, and many more. N. Linial (1986) showed that the chromatic polynomial
ďż˝(G;X) is #P-hard to evaluate for all but three values X = 0, 1, 2, where evaluation
is in P.
This dichotomy includes evaluation at real or complex values, and has the further
property that the set of points for which evaluation is in P is finite. We investigate
how the complexity of evaluating univariate graph polynomials that arise from CPcolorings
varies for different evaluation points. We show that for some CP-colorings
(harmonious, convex) the complexity of evaluation follows a similar pattern to the chromatic
polynomial. However, in other cases (proper edge colorings, mcct-colorings,
H-free colorings) we could only obtain a dichotomy for evaluations at non-negative
integer points. We also discuss some CP-colorings where we only have very partial
results
Journeying through Dementia: the story of a 14 year design-led research enquiry
Consider a linear ordering equipped with a finite sequence of monadic
predicates. If the ordering contains an interval of order type \omega or
-\omega, and the monadic second-order theory of the combined structure is
decidable, there exists a non-trivial expansion by a further monadic predicate
that is still decidable.Comment: 18 page
Invariant Definability (Extended Abstract)
) J.A. Makowsky ? Department of Computer Science Technion---Israel Institute of Technology IL-32000 Haifa, Israel [email protected] Abstract. We define formally the notion of invariant definability in a logic L and study it systematically. We relate it to other notions of definability (implicit definability, \Delta-definability and definability with built-in relations) and establish connections between them. In descriptive complexity theory, invariant definability is mostly used with a linear order (or a successor relation) as the auxiliary relation. We formulate a conjecture which spells out the special role linear order plays in capturing complexity classes with logics and prove two special cases. 1 Introduction This paper initiates an analysis of the notion of invariant definability special cases of which have been used in various contexts of finite model theory and descriptive complexity theory. We assume the reader is familiar with the basics of complexity theory as given ..
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