26,359 research outputs found
Non-primitive Recursive Function Definitions
This paper presents an approach to the problem of introducingnon-primitive recursive function definitions in higher order logic. Arecursive specification is translated into a domain theory version, wherethe recursive calls are treated as potentially non-terminating. Once wehave proved termination, the original specification can be derived easily.A collection of algorithms are presented which hide the domain theoryfrom a user. Hence, the derivation of a domain theory specificationhas been automated completely, and for well-founded recursive functionspecifications the process of deriving the original specification from thedomain theory one has been automated as well, though a user mustsupply a well-founded relation and prove certain termination propertiesof the specification. There are constructions for building well-foundedrelations easily
Sub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov
complexity, and (3) martingales. We show these notions can be equivalently
defined with prefix-free Kolmogorov complexity. We prove that one direction of
van Lambalgen's theorem holds for relative computability, but the other
direction fails. We discuss statistical properties of these notions of
randomness
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
Data Definitions in the ACL2 Sedan
We present a data definition framework that enables the convenient
specification of data types in ACL2s, the ACL2 Sedan. Our primary motivation
for developing the data definition framework was pedagogical. We were teaching
undergraduate students how to reason about programs using ACL2s and wanted to
provide them with an effective method for defining, testing, and reasoning
about data types in the context of an untyped theorem prover. Our framework is
now routinely used not only for pedagogical purposes, but also by advanced
users.
Our framework concisely supports common data definition patterns, e.g. list
types, map types, and record types. It also provides support for polymorphic
functions. A distinguishing feature of our approach is that we maintain both a
predicative and an enumerative characterization of data definitions.
In this paper we present our data definition framework via a sequence of
examples. We give a complete characterization in terms of tau rules of the
inclusion/exclusion relations a data definition induces, under suitable
restrictions. The data definition framework is a key component of
counterexample generation support in ACL2s, but can be independently used in
ACL2, and is available as a community book.Comment: In Proceedings ACL2 2014, arXiv:1406.123
Wittgenstein on Pseudo-Irrationals, Diagonal Numbers and Decidability
In his early philosophy as well as in his middle period, Wittgenstein holds a purely
syntactic view of logic and mathematics. However, his syntactic foundation of logic
and mathematics is opposed to the axiomatic approach of modern mathematical logic.
The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and operations; its aim is to reduce mathematics to operations. This paper illustrates Wittgenstein’s approach by examining his discussion of irrational numbers
- …