A constructive interpretation of Ramsey's theorem via the product of selection functions

a

What\u27s So Special About Kruskal\u27s Theorem and the Ordinal \u3cem\u3eT\u3c/em\u3e\u3csub\u3eo\u3c/sub\u3e? A Survey of Some Results in Proof Theory

This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Krusal\u27s tree theorem, and in particular the connection with the ordinal Ƭo. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen Hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard\u27s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the tree theorem , as well as a finite miniaturization of Kruskal\u27s theorem due to Harvey Friedman. These versions of Kruskal\u27s theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called natural independence phenomena , which are considered by more logicians as more natural than the mathematical incompleteness results first discovered by Gödel. Kruskal\u27s tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bandix completion procedures. There is also a close connection between a certain infinite countable ordinal called Ƭoand Kruskal\u27s theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence

Analysis of methods for extraction of programs from non-constructive proofs

The present thesis compares two computational interpretations of non-constructive proofs: refined A-translation and Gödel's functional "Dialectica" interpretation. The behaviour of the extraction methods is evaluated in the light of several case studies, where the resulting programs are analysed and compared. It is argued that the two interpretations correspond to specific backtracking implementations and that programs obtained via the refined A-translation tend to be simpler, faster and more readable than programs obtained via Gödel's interpretation. Three layers of optimisation are suggested in order to produce faster and more readable programs. First, it is shown that syntactic repetition of subterms can be reduced by using let-constructions instead of meta substitutions abd thus obtaining a near linear size bound of extracted terms. The second improvement allows declaring syntactically computational parts of the proof as irrelevant and that this can be used to remove redundant parameters, possibly improving the efficiency of the program. Finally, a special case of induction is identified, for which a more efficient recursive extracted term can be defined. It is shown the outcome of case distinctions can be memoised, which can result in exponential improvement of the average time complexity of the extracted program

Metalevel and reflexive extension in mechanical theorem proving

In spite of many years of research into mechanical assistance for mathematics it is still much more difficult to construct a proof on a machine than on paper. Of course this is partly because, unlike a proof on paper, a machine checked proof must be formal in the strictest sense of that word, but it is also because usually the ways of going about building proofs on a machine are limited compared to what a mathematician is used to. This thesis looks at some possible extensions to the range of tools available on a machine that might lend a user more flexibility in proving theorems, complementing whatever is already available.In particular, it examines what is possible in a framework theorem prover. Such a system, if it is configured to prove theorems in a particular logic T, must have a formal description of the proof theory of T written in the framework theory F of the system. So it should be possible to use whatever facilities are available in F not only to prove theorems of T, but also theorems about T that can then be used in their turn to aid the user in building theorems of T.The thesis is divided into three parts. The first describes the theory FS₀, which has been suggested by Feferman as a candidate for a framework theory suitable for doing meta-theory. The second describes some experiments with FS₀, proving meta-theorems. The third describes an experiment in extending the theory PRA, declared in FS₀, with a reflection facility.More precisely, in the second section three theories are formalised: propositional logic, sorted predicate logic, and the lambda calculus (with a deBruijn style binding). For the first two the deduction theorem and the prenex normal form theorem are respectively proven. For the third, a relational definition of beta-reduction is replaced with an explicit function.In the third section, a method is proposed for avoiding the work involved in building a full Godel style proof predicate for a theory. It is suggested that the language be extended with quotation and substitution facilities directly, instead of providing them as definitional extensions. With this, it is possible to exploit an observation of Solovay's that the Lob derivability conditions are sufficient to capture the schematic behaviour of a proof predicate. Combining this with a reflection schema is enough to produce a non-conservative extension of PRA, and this is demonstrated by some experiments

The Logic of Principia Mathematica

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1995.Includes bibliographical references (p. 181-184).by Darryl Jung.Ph.D

The External Tape Hypothesis: a Turing machine based approach to cognitive computation

The symbol processing or "classical cognitivist" approach to mental computation suggests that the cognitive architecture operates rather like a digital computer. The components of the architecture are input, output and central systems. The input and output systems communicate with both the internal and external environments of the cognizer and transmit codes to and from the rule governed, central processing system which operates on structured representational expressions in the internal environment. The connectionist approach, by contrast, suggests that the cognitive architecture should be thought of as a network of interconnected neuron-like processing elements (nodes) which operates rather like a brain. Connectionism distinguishes input, output and central or "hidden" layers of nodes. Connectionists claim that internal processing consists not of the rule governed manipulation of structured symbolic expressions, but of the excitation and inhibition of activity and the alteration of connection strengths via message passing within and between layers of nodes in the network. A central claim of the thesis is that neither symbol processing nor connectionism provides an adequate characterization of the role of the external environment in cognitive computation. An alternative approach, called the External Tape Hypothesis (ETH), is developed which claims, on the basis of Turing's analysis of routine computation, that the Turing machine model can be used as the basis for a theory which includes the environment as an essential part of the cognitive architecture. The environment is thought of as the tape, and the brain as the control of a Turing machine. Finite state automata, Turing machines, and universal Turing machines are described, including details of Turing's original universal machine construction. A short account of relevant aspects of the history of digital computation is followed by a critique of the symbol processing approach as it is construed by influential proponents such as Allen Newell and Zenon Pylyshyn among others. The External Tape Hypothesis is then developed as an alternative theoretical basis. In the final chapter, the ETH is combined with the notion of a self-describing Turing machine to provide the basis for an account of thinking and the development of internal representations

From axiomatization to generalizatrion of set theory

The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within the mathematical approach, in the light of the significance of Cohen's Independence results