25 research outputs found
Logic for exact real arithmetic
Continuing earlier work of the first author with U. Berger, K. Miyamoto and
H. Tsuiki, it is shown how a division algorithm for real numbers given as a
stream of signed digits can be extracted from an appropriate formal proof. The
property of being a real number represented as a stream is formulated by means
of coinductively defined predicates, and formal proofs involve coinduction. The
proof assistant Minlog is used to generate the formal proofs and extract their
computational content as terms of the underlying theory, a form of type theory
for finite or infinite data. Some experiments with running the extracted term
are described, after its translation to Haskell
Dyadic Subbases and Representations of Topological Spaces
We explain topological properties of the embedding-based approach to
computability on topological spaces. With this approach, he considered
a special kind of embedding of a topological space into Plotkin\u27s
, which is the set of infinite sequences of .
We show that such an embedding can also be characterized by a dyadic
subbase, which is a countable subbase such that are regular open
and and are exteriors of each other. We survey properties
of dyadic subbases which are related to efficiency properties of the
representation corresponding to the embedding
Domain Representations Induced by Dyadic Subbases
We study domain representations induced by dyadic subbases and show that a
proper dyadic subbase S of a second-countable regular space X induces an
embedding of X in the set of minimal limit elements of a subdomain D of
. In particular, if X is compact, then X is a retract of
the set of limit elements of D
A Stream Calculus of Bottomed Sequences for Real Number Computation
AbstractA calculus XPCF of 1ā„-sequences, which are infinite sequences of {0,1,ā„} with at most one copy of bottom, is proposed and investigated. It has applications in real number computation in that the unit interval I is topologically embedded in the set Ī£ā„,1Ļ of 1ā„-sequences and a real function on I can be written as a program which inputs and outputs 1ā„-sequences. In XPCF, one defines a function on Ī£ā„,1Ļ only by specifying its behaviors for the cases that the first digit is 0 and 1. Then, its value for a sequence starting with a bottom is calculated by taking the meet of the values for the sequences obtained by filling the bottom with 0 and 1. The validity of the reduction rule of this calculus is justified by the adequacy theorem to a domain-theoretic semantics. Some example programs including addition and multiplication are shown. Expressive powers of XPCF and related languages are also investigated
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Mathematical Logic: Proof Theory, Constructive Mathematics
The workshop āMathematical Logic: Proof Theory, Constructive Mathematicsā was centered around proof-theoretic aspects of core mathematics and theoretical computer science as well as homotopy type theory and logical aspects of computational complexity
Extracting nondeterministic concurrent programs
We introduce an extension of intuitionistic fixed point logic by a modal operator facilitating the extraction of nondeterministic concurrent programsfrom proofs. We apply this extension to program extraction in computable analysis, more precisely, to computing with Tsuiki's infinite Gray code for real numbers
A coinductive approach to computing with compact sets
Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first author it was shown how to extract certified algorithms working with the signed digit representations from constructiveproofs. In this paper we lay the foundation for doing a similar thing with nonempty compact sets. It turns out that a representation by streams of finitely many digits is impossible and instead trees are needed
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Mathematical Logic: Proof theory, Constructive Mathematics
The workshop āMathematical Logic: Proof Theory, Constructive Mathematicsā was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
Extracting total Amb programs from proofs
We present a logical system CFP (Concurrent Fixed Point Logic) supporting the
extraction of nondeterministic and concurrent programs that are provably total
and correct. CFP is an intuitionistic first-order logic with inductive and
coinductive definitions extended by two propositional operators: Rrestriction,
a strengthening of implication, and an operator for total concurrency. The
source of the extraction are formal CFP proofs, the target is a lambda calculus
with constructors and recursion extended by a constructor Amb (for McCarthy's
amb) which is interpreted operationally as globally angelic choice and is used
to implement nondeterminism and concurrency. The correctness of extracted
programs is proven via an intermediate domain-theoretic denotational semantics.
We demonstrate the usefulness of our system by extracting a nondeterministic
program that translates infinite Gray code into the signed digit
representation. A noteworthy feature of CFP is the fact that the proof rules
for restriction and concurrency involve variants of the classical law of
excluded middle that would not be interpretable computationally without Amb.Comment: 39 pages + 4 pages appendix. arXiv admin note: text overlap with
arXiv:2104.1466