7,221 research outputs found
Logics of Finite Hankel Rank
We discuss the Feferman-Vaught Theorem in the setting of abstract model
theory for finite structures. We look at sum-like and product-like binary
operations on finite structures and their Hankel matrices. We show the
connection between Hankel matrices and the Feferman-Vaught Theorem. The largest
logic known to satisfy a Feferman-Vaught Theorem for product-like operations is
CFOL, first order logic with modular counting quantifiers. For sum-like
operations it is CMSOL, the corresponding monadic second order logic. We
discuss whether there are maximal logics satisfying Feferman-Vaught Theorems
for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th
birthday. The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-23534-9_1
Cellular Automata are Generic
Any algorithm (in the sense of Gurevich's abstract-state-machine
axiomatization of classical algorithms) operating over any arbitrary unordered
domain can be simulated by a dynamic cellular automaton, that is, by a
pattern-directed cellular automaton with unconstrained topology and with the
power to create new cells. The advantage is that the latter is closer to
physical reality. The overhead of our simulation is quadratic.Comment: In Proceedings DCM 2014, arXiv:1504.0192
Evolving MultiAlgebras unify all usual sequential computation models
It is well-known that Abstract State Machines (ASMs) can simulate
"step-by-step" any type of machines (Turing machines, RAMs, etc.). We aim to
overcome two facts: 1) simulation is not identification, 2) the ASMs simulating
machines of some type do not constitute a natural class among all ASMs. We
modify Gurevich's notion of ASM to that of EMA ("Evolving MultiAlgebra") by
replacing the program (which is a syntactic object) by a semantic object: a
functional which has to be very simply definable over the static part of the
ASM. We prove that very natural classes of EMAs correspond via "literal
identifications" to slight extensions of the usual machine models and also to
grammar models. Though we modify these models, we keep their computation
approach: only some contingencies are modified. Thus, EMAs appear as the
mathematical model unifying all kinds of sequential computation paradigms.Comment: 12 pages, Symposium on Theoretical Aspects of Computer Scienc
Spectra of Monadic Second-Order Formulas with One Unary Function
We establish the eventual periodicity of the spectrum of any monadic
second-order formula where:
(i) all relation symbols, except equality, are unary, and
(ii) there is only one function symbol and that symbol is unary
Interactive Small-Step Algorithms I: Axiomatization
In earlier work, the Abstract State Machine Thesis -- that arbitrary
algorithms are behaviorally equivalent to abstract state machines -- was
established for several classes of algorithms, including ordinary, interactive,
small-step algorithms. This was accomplished on the basis of axiomatizations of
these classes of algorithms. Here we extend the axiomatization and, in a
companion paper, the proof, to cover interactive small-step algorithms that are
not necessarily ordinary. This means that the algorithms (1) can complete a
step without necessarily waiting for replies to all queries from that step and
(2) can use not only the environment's replies but also the order in which the
replies were received
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