MoL199902: Grabmayer, Clemens (1999) CutElimination in the Implicative Fragment $>G3mi$ of an Intuitionistic $G3$GentzenSystem and its Computational Meaning. [Report]
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Abstract
(Abstract)
Clemens Grabmayer
[Master's Thesis at the Institute for Logic, Language and Computation (ILLC),
Universiteit van Amsterdam, October 1999]
NOTATION: Superscripts and subscripts are indicated in this .txt file
in a LATEXstyle manner with the help of symbols ^ and _ , such that
e.g. G^+_v stands for the symbol G with superscript + and subscript v;
G^{++} means the symbol G with the superscriptstring ++ .
> means the implicationsign in logical formulas, ==> the implication
sign used in mathematical proofs and between the antecedent and the
succedent of sequents in Gentzen systems; __ stands for the sign for
"bottom", the logical symbol for falsehood. Greek symbols are here
referred to simply as "Phi" or "Gamma" (these letters in notcapitalized
form would be referred to as "phi", "gamma").
This thesis consists of two chapters, the first of which is centered
around a very recent article [Vest99] by R. Vestergaard, whereas the second
is concerned with the topic of strong cutelimination in the Gentzensystems
>G3mi and >__G3i.
R. Vestergaard in [Vest99] presents a "computational anomaly" in a typed
Gentzensystem (here called:) G^+_v, which is based on the formulation of the
intuitionistic and minimal G3systems G3[mi] without explicit weakening and
contractionrules in the book [TS96] by A.S. Troelstra and H. Schwichtenberg.
G^+_v is moreover tailormade to the purposes of
(1) allowing to view a typed derivationterm t^C in the succedent of the
conclusion Gamma ==> t:C of a G^+_v derivation D very directly as
the "computational meaning" of D (which corresponds to the "image" D^*
of D as a naturaldeduction derivation with conclusion C from marked
open assumptionclasses occuring among the annotated formulas in Gamma),
(2) facilitating to carry through cutelimination as a stepwise process of
local transformations by the use of explicit rules of G^+_v for contraction
and inversion during the procedure (the effects of weakening during
cutelimination are taken to be trivial and ignored in [Vest99]),
(3) making it furthermore possible, that cutelimination for G^+_v can still
be done in quite the same way as for (the special case of the implicative
fragment >G3mi of) the systems G3[mi], which is described implicitly in
[TS96].
Vestergaard then essentially presents a sequence {D_n}_{n\in N} of distinct
derivations D_n in G^+_v + Cut, which by the cutelimination procedure related
to that for G3[mi] in [TS96] all reduce to the same derivation D' , but
where the "computational meanings" of D_n are mutually different. This is
what in [Vest99] is called a "computational anomaly".
Here a variantsystem G^+ of G^+_v is considered, in which the effects of
applications of a (mulitiple) weakeningrule are also accounted for in the
derivationterms forming the succedents of sequents. The relation between
G_0^+ derivations D
(The system G_0^+ is defined from G^+ by restricting the rules of the system
to its logical rules L> and R>, thereby dropping the structural
rules weakening and contraction as well as the additional rule inversion
of G^+.)
and their "computational meanings" Phi(D) is made more visual by the
definition and use of a map Phi (similar to one that was essentially first
described in [Pra65] by D. Prawitz) from derivations in (G_0^+ + Cut) to
derivations in >N[mi]^*, a typed version of a naturaldeduction calculus
>N[mi] for the implicative fragment of intuitionistic or minimal logic.
Vestergaard's example of an "anomaly" leads basically to the same sequence of
cutelimination reductions also in the setting of the system G^+, but it does
not look quite convincingly there any more. In fact, if one step during
cutelimination for upwardspermutation of contraction is altered slightly to
a more careful form, this "anomaly" disappears completely. Nevertheless,
a modified, but different example of an "anomaly" is given here for the
situation arising from the reformulation of this step. (A certain possibility
of a still more refined version of this step for upwardspermutation of
contraction is not covered by the new example.)
Also a closer analysis of the obviously problematic cutelimination step
in G^+, (one case occuring for) upwardspermutation of contraction, is given
in the case of the underlying untyped system >G3mi. This analysis leads to
the interpretation, that while cutelimination in a >G3mi derivation D
(done in the way as given implicitly in [TS96] for G3[mi]) does not
correspond to normalization on the related naturaldeduction derivation
Phi(D), this procedure might nevertheless have a more special property
of allowing to extract from a given derivation in G^+  in a certain
specifiable sense  "just what is needed for its particular conclusion"
([Fef75]) in this system. It might therefore be a special feature of this
system, that cutelimination allowes to do more simplifications than are
possible by normalization on the naturaldeduction images.
Furthermore a remark of Vestergaard is discussed, that his example of an
"anomaly" would be avoided, if a variant Gentzensystem of G^+ with the
antecedents of sequents consisting of s e t s instead of multisets of
formulas were considered. This seems not directly possible. Instead a related
typed system >G2'mi^{e*} is presented here, which looks to be a good
candidate of a typed >G3mi like system, in which "anomalies" like
Vestergaard's are avoided.
In the second chapter of the thesis the topic of strong cutelimination
is investigated for the systems >G3mi and >__G3i. That is the question,
whether cutelimination steps from a fixed list L can be applied to a given
derivation D containing Cut in one of these systems with respect to arbitrary
cuts occuring in D (cuts that need not be topmost in D), such that despite
this freedome of choice for applicable cutelimination steps from L a cutfree
derivation D' is always reached after the execution of finitely many such
steps, when starting from D.
Strong cutelimination theorems for LJlike systems sometimes follow from
results about strongnormalization (like that in [Pra71] by D. Prawitz) with
the help of results by J. Zucker in [Zu74] and G. Pottinger in [Pott77]
about the precise nature of the connection between cutelimination in such
sequentcalculi and normalization for the respective naturaldeduction systems.
But Vestergaard's result may be looked upon as a strong indication for the
possibility, that the very smooth and direct connection described in
[Zu74] and [Pott77] between normalization on the side of the naturaldeduction
calculi and cutelimination for Gentzensystems does not exist with respect
to the G3[mi] Gentzensystems for perhaps substantial reasons. It is
therefore at least not in an obvious manner possible to arrive at a strong
cutelimination result for >G3mi and >__G3i in the way mentioned before.
Instead, ideas from a proof by A.G. Dragalin in [Drag79] for a Strong Form
of a CutElimination Theorem for LJ and LK (with respect to a list of
exclusively such cutelimination steps that have already been presented by
G. Gentzen in [Gen35]) are used here to prove a strong cutelimination
theorem for >G3mi and >__G3i .
Additionally a generalization of this result with respect to a larger list
of cutelimination steps, where now limited permutations of the structural
rules weakening and contraction and also of inversion and Cut over each other
are permitted, is discussed and a proof for an in this respect more general
version of the before shown strong cutelimination theorem for >G3mi and
>__G3i is sketched.
REFERENCES
[Drag79] Dragalin, A.G.: Mathematical Intuitionism (Introduction to
Proof Theory), Moscow 1979; Translations of Mathematical Monographs,
Volume 67, American Mathematical Society, Providence, Rhode Island,
1988.
[Fef75] Feferman, S.: Review of [Pra71] in: Journal of Symbolic Logic 2,
Vol. 40, 1975, p. 232234.
[Gen35] Gentzen, G.: "Untersuchungen ueber das logische Schliessen I, II",
Mathematische Zeitschrift 39, 1935, S. 176210, 405431.
[Pott77] Pottinger, G.: "Normalization as a homomorphic image of
Cutelimination", Annals of Mathematical Logic 12, 1977, p. 323357.
[Pra65] Prawitz, D.: Natural Deduction (A ProofTheoretical Study),
Almqvist & Wiksell, Stockholm, Goeteborg, Uppsala, 1965.
[Pra71] Prawitz, D.: "Ideas and Results in ProofTheory", p. 235307 in:
Fenstad, J.E.[ed]: Proceedings of the 2nd Scandinavian Logic
Symposium, NorthHolland Publishing Company, Amsterdam, London, 1971.
[TS96] Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory,
Cambridge Tracts in Theoretical Computer Science, Cambridge University
Press, 1996.
[Vest99] Vestergaard, R.: "Revisiting Kreisel: A Computational Anomaly in
the TroelstraSchwichtenberg G3iSystem", March 1999, available from
R. Vestergaard's Homepage: http://www.cee.hw.ac.uk/~jrvest/ .
[Zu74] Zucker, J.: "The Correspondence between CutElimination and
Normalization", Annals of Mathematical Logic 7, 1974, p. 1112, 156.
Item Type:  Report 

Report Nr:  MoL199902 
Series Name:  Master of Logic Thesis (MoL) Series 
Year:  1999 
Date Deposited:  12 Oct 2016 14:38 
Last Modified:  12 Oct 2016 14:38 
URI:  https://eprints.illc.uva.nl/id/eprint/705 
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