52 research outputs found
Uniqueness of fixpoints of single-step operators determined by Belnap's four-valued logic
Recently, Hitzler and Seda showed how a domain-theoretic proof can be given of the fact
that, for a locally hierarchical program, the single-step operator TP , de�ned in two-valued
logic, has a unique �xed point. Their approach employed a construction which turned a ScottErshov domain into a generalized ultrametric space. Finally, a �xed-point theorem of PriessCrampe and Ribenboim was applied to TP to establish the result. In this paper, we extend
these methods and results to the corresponding well-known single-step operators �P and P
determined by P and de�ned, respectively, in three-valued and four-valued logic
Classes of Logic Programs which Possess Unique Supported Models
Logic programming is concerned with the use of logic as a programming language. The main manifestation of this computing paradigm is in the various versions of Prolog which are now available, in which computation is viewed as deduction from sets of Horn clauses, although there is also growing interest in the related form known as answer set programming, see [10]. The reference [1] contains a good survey of the growth of logic programming over the last twenty-five years both as a stand-alone programming language and as a software component of large information systems. One advantage a logic program P has over conventional imperative and object oriented programs is that it has a natural machine-independent meaning, namely, its logical meaning. This is often referred to as its declarative semantics, and is usually taken to be some \u27standard\u27 model canonically associated with P. Unfortunately, it is often the case that there are many possible choices for the standard model, some even taken in many-valued logic, which do not in general coincide and all of which have a claim to be \u27the natural choice\u27 depending on one\u27s view of non-monotonic reasoning
Computational Processes and Incompleteness
We introduce a formal definition of Wolfram's notion of computational process
based on cellular automata, a physics-like model of computation. There is a
natural classification of these processes into decidable, intermediate and
complete. It is shown that in the context of standard finite injury priority
arguments one cannot establish the existence of an intermediate computational
process
The Pagoda Sequence: a Ramble through Linear Complexity, Number Walls, D0L Sequences, Finite State Automata, and Aperiodic Tilings
We review the concept of the number wall as an alternative to the traditional
linear complexity profile (LCP), and sketch the relationship to other topics
such as linear feedback shift-register (LFSR) and context-free Lindenmayer
(D0L) sequences. A remarkable ternary analogue of the Thue-Morse sequence is
introduced having deficiency 2 modulo 3, and this property verified via the
re-interpretation of the number wall as an aperiodic plane tiling
A Concrete View of Rule 110 Computation
Rule 110 is a cellular automaton that performs repeated simultaneous updates
of an infinite row of binary values. The values are updated in the following
way: 0s are changed to 1s at all positions where the value to the right is a 1,
while 1s are changed to 0s at all positions where the values to the left and
right are both 1. Though trivial to define, the behavior exhibited by Rule 110
is surprisingly intricate, and in (Cook, 2004) we showed that it is capable of
emulating the activity of a Turing machine by encoding the Turing machine and
its tape into a repeating left pattern, a central pattern, and a repeating
right pattern, which Rule 110 then acts on. In this paper we provide an
explicit compiler for converting a Turing machine into a Rule 110 initial
state, and we present a general approach for proving that such constructions
will work as intended. The simulation was originally assumed to require
exponential time, but surprising results of Neary and Woods (2006) have shown
that in fact, only polynomial time is required. We use the methods of Neary and
Woods to exhibit a direct simulation of a Turing machine by a tag system in
polynomial time
On the boundaries of solvability and unsolvability in tag systems. Theoretical and Experimental Results
Several older and more recent results on the boundaries of solvability and
unsolvability in tag systems are surveyed. Emphasis will be put on the
significance of computer experiments in research on very small tag systems
Logic Programs and Connectionist Networks
Graphs of the single-step operator for first-order logic programs—displayed in the real plane—exhibit self-similar structures known from topological dynamics, i.e., they appear to be fractals, or more precisely, attractors of iterated function systems. We show that this observation can be made mathematically precise. In particular, we give conditions which ensure that those graphs coincide with attractors of suitably chosen iterated function systems, and conditions which allow the approximation of such graphs by iterated function systems or by fractal interpolation. Since iterated function systems can easily be encoded using recurrent radial basis function networks, we eventually obtain connectionist systems which approximate logic programs in the presence of function symbols
Multi-Head Finite Automata: Characterizations, Concepts and Open Problems
Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg,
1966). Since that time, a vast literature on computational and descriptional
complexity issues on multi-head finite automata documenting the importance of
these devices has been developed. Although multi-head finite automata are a
simple concept, their computational behavior can be already very complex and
leads to undecidable or even non-semi-decidable problems on these devices such
as, for example, emptiness, finiteness, universality, equivalence, etc. These
strong negative results trigger the study of subclasses and alternative
characterizations of multi-head finite automata for a better understanding of
the nature of non-recursive trade-offs and, thus, the borderline between
decidable and undecidable problems. In the present paper, we tour a fragment of
this literature
Playing With Population Protocols
Population protocols have been introduced as a model of sensor networks
consisting of very limited mobile agents with no control over their own
movement: A collection of anonymous agents, modeled by finite automata,
interact in pairs according to some rules.
Predicates on the initial configurations that can be computed by such
protocols have been characterized under several hypotheses.
We discuss here whether and when the rules of interactions between agents can
be seen as a game from game theory. We do so by discussing several basic
protocols
A General Notion of Useful Information
In this paper we introduce a general framework for defining the depth of a
sequence with respect to a class of observers. We show that our general
framework captures all depth notions introduced in complexity theory so far. We
review most such notions, show how they are particular cases of our general
depth framework, and review some classical results about the different depth
notions
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