On the non-confluence of cut-elimination

2011 ◽  
Vol 76 (1) ◽  
pp. 313-340 ◽  
Author(s):  
Matthias Baaz ◽  
Stefan Hetzl
Keyword(s):  
Classical Logic ◽  
Normal Forms ◽  
Strong Sense ◽  
Cut Elimination ◽  
First Order ◽  
Elementary Number

AbstractWe study cut-elimination in first-order classical logic. We construct a sequence of polynomial-length proofs having a non-elementary number of different cut-free normal forms. These normal forms are different in a strong sense: they not only represent different Herbrand-disjunctions but also differ in their prepositional structure.This result illustrates that the constructive content of a proof in classical logic is not uniquely determined but rather depends on the chosen method for extracting it.

scholarly journals On the No-Counterexample Interpretation

1997 ◽  
Vol 4 (42) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Normal Forms ◽  
Formal System ◽  
Modus Ponens ◽  
Order Type ◽  
Cut Elimination ◽  
Gamma Delta ◽  
Functional Interpretation ◽  
First Order ◽  
Epsilon Substitution ◽  
Definition Of

In [15],[16] Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano<br />arithmetic. In particular he proved, using a complicated epsilon-substitution method (due to<br />W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic (PA) one can find ordinal recursive functionals Phi_A of order type < epsilon_0 which realize the<br />Herbrand normal form A^H of A.<br /> Subsequently more perspicuous proofs of this fact via functional interpretation (combined<br />with normalization) and cut-elimination were found. These proofs however do not carry out the n.c.i. as a local proof interpretation and don't respect the modus<br />ponens on the level of the n.c.i. of formulas A and A -> B. Closely related to this phenomenon is the fact that both proofs do not establish the condition (delta) and - at<br />least not constructively - (gamma) which are part of the definition of an `interpretation of a<br />formal system' as formulated in [15].<br />In this paper we determine the complexity of the n.c.i. of the modus ponens rule for<br />(i) PA-provable sentences,<br />(ii) for arbitrary sentences A;B in L(PA) uniformly in functionals satisfying the n.c.i. of (prenex normal forms of) A and A -> B; and<br />(iii) for arbitrary A;B in L(PA) pointwise in given alpha(< epsilon_0)-recursive functionals satisfying the n.c.i. of A and A -> B. <br /> This yields in particular perspicuous proofs of new uniform versions of the conditions ( gamma), (delta). <br /> Finally we discuss a variant of the concept of an interpretation presented in [17] and<br />show that it is incomparable with the concept studied in [15],[16]. In particular we show<br />that the n.c.i. of PA_n by alpha(<omega_n(omega))-recursive functionals (n >= 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (delta).

A SUFFICIENT CONDITION FOR THE UNIQUENESS OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF GENERALIZED HOPF SINGULARITIES

2004 ◽  
Vol 14 (09) ◽  
pp. 3337-3345 ◽  
Author(s):  
JIANPING PENG ◽  
DUO WANG
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Sufficient Condition ◽  
First Order ◽  
Recursive Formulas ◽  
Unique Normal Forms

A sufficient condition for the uniqueness of the Nth order normal form is provided. A new grading function is proposed and used to prove the uniqueness of the first-order normal forms of generalized Hopf singularities. Recursive formulas for computation of coefficients of unique normal forms of generalized Hopf singularities are also presented.

The consistency of arithmetic

2021 ◽  
pp. 268-311
Author(s):  
Paolo Mancosu ◽  
Sergio Galvan ◽  
Richard Zach
Keyword(s):  
Simple Proof ◽  
Sequent Calculus ◽  
Cut Elimination ◽  
Successor Function ◽  
Cut Rule ◽  
Original Proof ◽  
First Order ◽  
Successive Elimination ◽  
Pure Logic ◽  
Classical Arithmetic

This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of interest are not purely logical ones, but rather formalized versions of mathematical theories, and in particular the first-order version of classical arithmetic built on top of the sequent calculus. Classical arithmetic goes beyond pure logic in that it contains a number of specific axioms for, among other symbols, 0 and the successor function. In particular, it contains the rule of induction, which is the essential rule characterizing the natural numbers. Proving a cut-elimination theorem for this system is hopeless, but something analogous to the cut-elimination theorem can be obtained. Indeed, one can show that every proof of a sequent containing only atomic formulas can be transformed into a proof that only applies the cut rule to atomic formulas. Such proofs, which do not make use of the induction rule and which only concern sequents consisting of atomic formulas, are called simple. It is shown that simple proofs cannot be proofs of the empty sequent, i.e., of a contradiction. The process of transforming the original proof into a simple proof is quite involved and requires the successive elimination, among other things, of “complex” cuts and applications of the rules of induction. The chapter describes in some detail how this transformation works, working through a number of illustrative examples. However, the transformation on its own does not guarantee that the process will eventually terminate in a simple proof.

scholarly journals A Certain Kind of Formal Theories

1965 ◽  
Vol 25 ◽  
pp. 59-86 ◽  
Author(s):  
Katuzi Ono
Keyword(s):  
Classical Logic ◽  
Axiom System ◽  
Logical System ◽  
First Order ◽  
Formal Theories

A common feature of formal theories is that each theory has its own system of axioms described in terms of some symbols for its primitive notions together with logical symbols. Each of these theories is developed by deduction from its axiom system in a certain logical system which is usually the classical logic of the first order.

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