Results 1  10
of
15
Constructivism and Proof Theory
, 2003
"... Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. F ..."
Abstract

Cited by 162 (4 self)
 Add to MetaCart
Introduction to the constructive point of view in the foundations of mathematics, in
particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics
due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation
and formalization of logic is described. For constructive (intuitionistic)
arithmetic, Kleene’s realizability interpretation is given; this provides an example
of the possibility of a constructive mathematical practice which diverges from classical
mathematics. The crucial notion in intuitionistic analysis, choice sequence, is
briefly described and some principles which are valid for choice sequences are discussed.
The second half of the article deals with some aspects of proof theory, i.e.,
the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions
are outlined: his introduction of the socalled Gentzen systems which use
sequents instead of formulas and his result on firstorder arithmetic showing that
(suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in
firstorder arithmetic.
Using Reflection to Explain and Enhance Type Theory
 Proof and Computation, volume 139 of NATO Advanced Study Institute, International Summer School held in Marktoberdorf, Germany, July 20August 1, NATO Series F
, 1994
"... The five lectures at Marktoberdorf on which these notes are based were about the architecture of problem solving environments which use theorem provers. Experience with these systems over the past two decades has shown that the prover must be extensible, yet it must be kept safe. We examine a way to ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
The five lectures at Marktoberdorf on which these notes are based were about the architecture of problem solving environments which use theorem provers. Experience with these systems over the past two decades has shown that the prover must be extensible, yet it must be kept safe. We examine a way to safely add new decision procedures to the Nuprl prover. It relies on a reflection mechanism and is applicable to any tacticoriented prover with sufficient reflection. The lectures explain reflection in the setting of constructive type theory, the core logic of Nuprl.
Cycling in proofs and feasibility
 Transactions of the American Mathematical Society
, 1998
"... Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual b ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.
Turning Cycles into Spirals
, 1999
"... Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14 ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Introduction The structure of LK proofs presents intriguing combinatorial aspects which turn out to be very difficult to study [6,8]. It is wellknown that as soon as one wants to intervene over the structure of a proof to simplify it, the complexity of the proof might increase enormously [16,12,14]. There is a link between the presence of cut formulas with nested quantifiers and the nonelementary expansion needed to prove a theorem without the help of such formulas. If one considers the graph defined by tracing the flow of occurrences of formulas (in the sense of [2]) for proofs allowing a nonelementary compression, one Preprint submitted to Elsevier Preprint 7 November 1997 finds that such graphs contain cycles [5] or almost cyclic structures[6]. These cycles codify in a small space (i.e. a proof with a small number of lines) all the information which is present in the proof once cuts on formulas wit
The Cost of a Cycle is a Square
, 1999
"... The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be nonelementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be nonelementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with O(k n+1 ) lines. In particular, there is a quadratic time algorithm which eliminates a single cycle from a proof. These results are motivated by the search for general methods on proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search.
Looking From the Inside and From the Outside
, 1998
"... Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for actual points to move around inside them, perhaps to parameterize them in some way. The concept of formal proofs has the interesting feature that it provides opportunities for both perspectives. The inner perspective has been largely overlooked, but in fact lengths of proofs lead to new ways to measure the information content of mathematical objects. The disparity between minimal lengths of proofs with and without "lemmas" provides an indication of internal symmetry of mathematical objects and their descriptions.
Modularity, Reuse and Hierarchy: Measuring Complexity by Measuring Structure and Organization
"... To develop better complexity measures, a reasonable approach is to base them on those principles of design that designers use. Modularity, reuse and hierarchy (MR&H) have been identified by engineers as useful principles for designing complex systems, and these characteristics can be seen in Nat ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
To develop better complexity measures, a reasonable approach is to base them on those principles of design that designers use. Modularity, reuse and hierarchy (MR&H) have been identified by engineers as useful principles for designing complex systems, and these characteristics can be seen in Nature. Here we develop metrics for each of MR&H, and then use them to develop several metrics of structure and organization. Further, we propose to evaluate complexity measures both empirically and on a set of abstract objectconstruction examples. After applying these tests to a handful of previously defined complexity measures, as well as ones we define here, we find that only two of our measures pass both sets of tests. Key words: design, evolutionary algorithm, evolutionary design, complexity, structure, organization 1
ON FORMALLY MEASURING AND ELIMINATING EXTRANEOUS NOTIONS IN PROOFS
, 2008
"... ABSTRACT. Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentz ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
ABSTRACT. Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen’s cutelimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures. Many mathematicians and philosophers of mathematics think that it’s somehow valuable for a proof to be “pure”, that is, for it not to use notions extraneous to what is being proved. Not worrying about why that is for now, we would like to make better sense of the suggestion. What does it mean for a notion used in a proof to be extraneous to the theorem being proved? One way of making this sharper would be to develop a syntactic way of evaluating extraneousness. I want to consider such a proposal, using Gerhard Gentzen’s cutelimination theorem. I will argue that there are serious obstacles to making this proposal work. 1. FOUR CLAIMS CONCERNING EXTRANEOUSNESS Bertrand’s postulate states that for every natural number n ≥ 1, there is a prime number between n and 2n. It is so named because it was verified by computation for
Streams and Strings in Formal Proofs
"... Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. ..."
Abstract
 Add to MetaCart
Streams are acyclic directed subgraphs of the logical flow graph of a proof and represent bundles of paths with the same origin and the same end. Streams can be described with a natural algebraic formalism which allows to explain in algebraic terms the evolution of proofs during cutelimination. In our approach, "logic" is often forgotten and combinatorial properties of graphs are taken into account to explain logical phenomena.