Abstract:
We consider a model of analog computation which can recognize various languages in real time. We encode an input word as a point in R d by composing iterated maps, and then apply inequalities to the resulting point to test for membership in the language. Each class of maps and inequalities, such as quadratic functions with rational coefficients, is capable of recognizing a particular class of languages; for instance, linear and quadratic maps can have both stack-like and queue-like memories. We use methods equivalent to the VapnikChervonenkis dimension to separate some of our classes from each other, e.g. linear maps are less powerful than quadratic or piecewise-linear ones, polynomials are less powerful than elementary (trigonometric and exponential) maps, and deterministic polynomials of each degree are less powerful than their non-deterministic counterparts. Comparing these dynamical classes with various discrete language classes helps illuminate how iterated maps can store and retrieve information in the continuum, the extent to which computation can be hidden in the encoding from symbol sequences into continuous spaces, and the relationship between analog and digital computation in general. We relate this model to other models of analog computation; for instance, it can be seen as a real-time, constant-space, off-line version of Blum, Shub and Smale's real-valued machines. 1
Citations
|
7716
|
Computers and Intractability: A Guide to the Theory of NP-Completeness
– Garey, Johnson
- 1979
|
|
2771
|
Introduction to Automata Theory, Languages and Computation
– Hopcroft, Ullman
- 1979
|
|
1588
|
Computational Complexity
– Papadimitriou
- 1994
|
|
554
|
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields
– Guckenheimer, Holmes
- 1983
|
|
536
|
Learnability and the Vapnik-Chervonenkis Dimension
– Blumer, Ehrenfeucht, et al.
- 1989
|
|
303
|
On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines
– Blum, Shub, et al.
- 1989
|
|
193
|
The induction of dynamical recognizers
– Pollack
- 1991
|
|
185
|
Seminumerical Algorithms
– Knuth
- 1969
|
|
102
|
The dynamics of discrete-time computation, with application to recurrent neural networks and finite state machine extraction
– Casey
- 1996
|
|
91
|
Transductions and Context-Free Languages. Teubner Studienbucher
– Berstel
- 1979
|
|
67
|
Unpredictability and undecidability in dynamical systems
– Moore
- 1990
|
|
52
|
Lower bounds for approximation by nonlinear manifolds
– Warren
- 1968
|
|
50
|
Language as a Dynamical System
– Elman
- 1995
|
|
34
|
Learning to count without a counter: A case study of dynamics and activation landscapes in recurrent networks
– Wiles, Elman
- 1995
|
|
30
|
Descriptive complexity theory over the real numbers
– Gradel, Meer
- 1996
|
|
29
|
Computing over the reals with addition and order
– Koiran
- 1994
|
|
23
|
Quasi-realtime languages
– Book, Greibach
- 1970
|
|
22
|
On the power of real Turing machines over binary inputs
– Cucker, Grigoriev
- 1997
|
|
20
|
Closed-form analytic maps in one and two dimensions can simulate universal turing machines
– Koiran, Moore
- 1999
|
|
19
|
Computability properties of low-dimensional dynamical systems
– Koiran, Cosnard, et al.
- 1994
|
|
19
|
A recurrent network that performs a contextsensitive prediction task
– Steijvers, Grunwald
- 1996
|
|
18
|
On digital nondeterminism
– Cucker, Matamala
- 1996
|
|
16
|
Real number models under various sets of operations
– Meer
- 1993
|
|
15
|
Real-time definable languages
– Rosenberg
- 1967
|
|
12
|
New real-time simulations of multihead tape units
– Leong, Seiferas
- 1981
|
|
11
|
An infinite hierarchy of intersections of context-free languages
– Liu, Weiner
- 1973
|
|
8
|
Differential fields, machines over the real numbers and automata
– Michaux
- 1991
|
|
6
|
QRT FIFO automata, breadth-first grammars and their relations
– Cherubini, Citrini, et al.
- 1991
|
|
6
|
Learning and Extracting Finite-State Automata with Second-Order Recurrent Neural Networks
– Giles, Miller, et al.
- 1992
|
|
5
|
Intersections of some families of languages
– Brandenburg
- 1986
|
|
4
|
Smooth one-dimensional maps of the interval and the real line capable of universal computation." Santa Fe Institute Working Paper 9301
– Moore
- 1993
|
|
4
|
The rational index: a complexity measure for languages
– Boasson, Courcelle, et al.
- 1981
|
|
3
|
A note on undecidable properties of formal languages
– Greibach
- 1968
|
|
2
|
G.Z.Sun, "Using prior knowledge in an NNPDA to learn languages
– Das, Giles
- 1993
|
|
2
|
Bounding the Vapnik-Chevonenkis dimension of concept classes parametrized by real numbers
– Goldberg, Jerrum
- 1995
|
|
2
|
A weak version of the Blum, Shub and Smale model." DIMACS
– Koiran
- 1994
|
|
1
|
Computational complexity of piecewise linear maps of the interval
– Bartlett, Garzon
|
|
1
|
Generalized one-sided shifts and maps of the interval." Nonlinearity 4
– Moore
- 1991
|
|
1
|
Her er en lille gris. Gyldendal
– Nilsson, Hald
- 1994
|
|
1
|
Analog Computation Via Neural Networks. " Theoretical Computer Science 131
– Siegelmann, Sontag
- 1994
|
|
1
|
A class of measures on formal languages." Acta Informatica 9
– Paradaens, Vincke
- 1977
|
