Results 11  20
of
33
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
Intuitionistic Choice and Restricted Classical Logic
, 2000
"... Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in all finite types together with various forms of the axiom of choice and a numerical omniscience schema (NOS) which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theore ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in all finite types together with various forms of the axiom of choice and a numerical omniscience schema (NOS) which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theoretic strength of such systems can be determined by functional interpretation based on a nonconstructive operator and his wellknown results on the strength of this operator from the 70's. In this note we consider a weaker form LNOS (lesser numerical omniscience schema) of NOS which su#ces to derive the strong form of binary Konig's lemma studied by Coquand/Palmgren and gives rise to a new and mathematically strong semiclassical system which, nevertheless, can proof theoretically be reduced to primitive recursive arithmetic PRA. The proof of this fact relies on functional interpretation and a majorization technique developed in a previous paper. # Basic Research in Computer Science, Centre...
A Note on Spector's QuantifierFree Rule of Extensionality
 Arch. Math. Logic
, 1999
"... In this note we show that the socalled weakly extensional arithmetic in all nite types, which is based on a quanti erfree rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem for add ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
In this note we show that the socalled weakly extensional arithmetic in all nite types, which is based on a quanti erfree rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem for additional axioms. This holds already for 1  axioms. Previously, only the failure of the stronger deduction theorem for deductions from (possibly open) assumptions (with parameters kept xed) was known.
On weak Markov's principle
 MLQ MATH. LOG. Q
, 2002
"... We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We show that the socalled weak Markov's principle (WMP) which states that every pseudopositive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved within the framework of Bishopstyle mathematics (which has been open for about 20 years). The underivability even holds if the ine#ective schema of full comprehension (in all types) for negated formulas (in particular for #free formulas) is added which allows to derive the law of excluded middle for such formulas.
On the Computational Content of the BolzanoWeierstraß Principle
, 2009
"... We will apply the methods developed in the field of ‘proof mining’ to the BolzanoWeierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation (combined with nega ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
We will apply the methods developed in the field of ‘proof mining’ to the BolzanoWeierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation (combined with negative translation) as well as the monotone functional interpretation of BW for the product space ∏i∈N[−k i, k i] (with the standard product metric). This results in optimal program and bound extraction theorems for proofs based on fixed instances of BW, i.e. for BW applied to fixed sequences in ∏i∈N[−k i, k i].
Hybrid Functional Interpretations
"... Abstract. We show how different functional interpretations can be combined via a multimodal linear logic. A concrete hybrid of Kreisel’s modified realizability and Gödel’s Dialectica is presented, and several small applications are given. We also discuss how the hybrid interpretation relates to var ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. We show how different functional interpretations can be combined via a multimodal linear logic. A concrete hybrid of Kreisel’s modified realizability and Gödel’s Dialectica is presented, and several small applications are given. We also discuss how the hybrid interpretation relates to variants of Dialectica and modified realizability with noncomputational quantifiers. 1
Unifying functional interpretations
 Notre Dame J. Formal Logic
"... Abstract. The purpose of this article is to present a parametrised functional interpretation. Depending on the choice of the parameter relations one obtains wellknown functional interpretations, such as Gödel’s Dialectica interpretation, DillerNahm’s variant of the Dialectica interpretation, Kohle ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. The purpose of this article is to present a parametrised functional interpretation. Depending on the choice of the parameter relations one obtains wellknown functional interpretations, such as Gödel’s Dialectica interpretation, DillerNahm’s variant of the Dialectica interpretation, Kohlenbach’s monotone interpretations, Kreisel’s modified realizability and Stein’s family of functional interpretations. A functional interpretation consists of a formula translation and a proof translation. We show that all these interpretation only differ on two choices: firstly, on “how much ” of the counterexamples for A became witnesses for ¬A when defining the formula translation, and, secondly, “how much ” of the witnesses of A one is interested in when defining the proof translation.
Light Functional Interpretation
 Lecture Notes in Computer Science, 3634:477 – 492, July 2005. Computer Science Logic: 19th International Workshop, CSL
, 2005
"... an optimization of Gödel’s technique towards the extraction of (more) efficient programs from (classical) proofs ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
an optimization of Gödel’s technique towards the extraction of (more) efficient programs from (classical) proofs