Stream Differential Equations: Specification Formats and Solution Methods

@article{Hansen2017StreamDE,
  title={Stream Differential Equations: Specification Formats and Solution Methods},
  author={Helle Hvid Hansen and Clemens Kupke and Jan J. M. M. Rutten},
  journal={Log. Methods Comput. Sci.},
  year={2017},
  volume={13}
}
Streams, or infinite sequences, are infinite objects of a very simple type, yet they have a rich theory partly due to their ubiquity in mathematics and computer science. Stream differential equations are a coinductive method for specifying streams and stream operations, and their theory has been developed in many papers over the past two decades. In this paper we present a survey of the many results in this area. Our focus is on the classification of different formats of stream differential… Expand

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References

SHOWING 1-10 OF 78 REFERENCES
Stream Differential Equations: concrete formats for coinductive definitions
In this article we give an accessible introduction to stream differential equations, ie., equations that take the shape of differential equations from analysis and that are used to define infiniteExpand
Elements of Stream Calculus (An Extensive Exercise in Coinduction)
  • J. Rutten
  • Computer Science, Mathematics
  • MFPS
  • 2001
TLDR
A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics. Expand
Concrete stream calculus: An extended study
  • R. Hinze
  • Computer Science
  • Journal of Functional Programming
  • 2010
TLDR
This paper redevelops the theory of recurrences, finite calculus and generating functions using streams and stream operators, building on the cornerstone of unique solutions. Expand
A coinductive calculus of streams
  • J. Rutten
  • Mathematics, Computer Science
  • Mathematical Structures in Computer Science
  • 2005
TLDR
A coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers) is developed, which can be used to formulate both coinduction proofs and definitions. Expand
Proving Equality of Streams Automatically
TLDR
This paper presents a tool Streambox, a tool that can prove equality of a wide range of examples fully automatically, and investigates techniques for proving equality of streams suitable for automation. Expand
A tutorial on coinductive stream calculus and signal flow graphs
  • J. Rutten
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 2005
TLDR
This paper presents an application of coinductive stream calculus to signal flow graphs based on Z-transforms (a discrete version of Laplace transforms) and transfer functions. Expand
Well-definedness of Streams by Transformation and Termination
  • H. Zantema
  • Computer Science
  • Log. Methods Comput. Sci.
  • 2010
TLDR
This work proposes a transformation from a stream specification to a term rewriting system (TRS) in such a way that termination of the resulting TRS implies that the stream specification is well-defined, that is, admits a unique solution. Expand
Representations of Stream Processors Using Nested Fixed Points
TLDR
Representations of continuous functions on infinite streams of dis- crete values are defined, both in the case of discrete-valued functions, and in the cases of stream-valuedfunction representatives, which are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. Expand
A Sound and Complete Calculus for Finite Stream Circuits
  • S. Milius
  • Mathematics, Computer Science
  • 2010 25th Annual IEEE Symposium on Logic in Computer Science
  • 2010
TLDR
It is proved that a final locally finite (dimensional) coalgebra is, equivalently, an initial iterative algebra of the category of real vector spaces and makes the connection to existing work on the semantics of recursive specifications. Expand
On the Final Coalgebra of Automatic Sequences
TLDR
This paper shows that the set of automatic sequences carries a final coalgebra structure, consisting of the operations of head, even, and odd, which will allow it to be shown that automatic sequences are to streams what rational languages are to (arbitrary) languages. Expand
...
1
2
3
4
5
...