Cut-Elimination in the Implicative Fragment $->G3mi$ of an Intuitionistic $G3$-Gentzen-System and its Computational Meaning Clemens Grabmayer 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 LATEX-style 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 superscript-string ++ . --> means the implication-sign 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 not-capitalized 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 cut-elimination in the Gentzen-systems -->G3mi and -->_|_G3i. R. Vestergaard in [Vest99] presents a "computational anomaly" in a typed Gentzen-system (here called:) G^+_v, which is based on the formulation of the intuitionistic and minimal G3-systems G3[mi] without explicit weakening- and contraction-rules 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 derivation-term 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 natural-deduction derivation with conclusion C from marked open assumption-classes occuring among the annotated formulas in Gamma), (2) facilitating to carry through cut-elimination 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 cut-elimination are taken to be trivial and ignored in [Vest99]), (3) making it furthermore possible, that cut-elimination 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 cut-elimination 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 variant-system G^+ of G^+_v is considered, in which the effects of applications of a (mulitiple-) weakening-rule are also accounted for in the derivation-terms 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 natural-deduction 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 cut-elimination 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 cut-elimination for upwards-permutation 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 upwards-permutation of contraction is not covered by the new example.) Also a closer analysis of the obviously problematic cut-elimination step in G^+, (one case occuring for) upwards-permutation of contraction, is given in the case of the underlying untyped system -->G3mi. This analysis leads to the interpretation, that while cut-elimination 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 natural-deduction 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 cut-elimination allowes to do more simplifications than are possible by normalization on the natural-deduction images. Furthermore a remark of Vestergaard is discussed, that his example of an "anomaly" would be avoided, if a variant Gentzen-system 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 cut-elimination is investigated for the systems -->G3mi and -->_|_G3i. That is the question, whether cut-elimination 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 cut-elimination steps from L a cut-free derivation D' is always reached after the execution of finitely many such steps, when starting from D. Strong cut-elimination theorems for LJ-like systems sometimes follow from results about strong-normalization (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 cut-elimination in such sequent-calculi and normalization for the respective natural-deduction 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 natural-deduction calculi and cut-elimination for Gentzen-systems does not exist with respect to the G3[mi] -Gentzen-systems for perhaps substantial reasons. It is therefore at least not in an obvious manner possible to arrive at a strong cut-elimination 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 Cut-Elimination Theorem for LJ and LK (with respect to a list of exclusively such cut-elimination steps that have already been presented by G. Gentzen in [Gen35]) are used here to prove a strong cut-elimination theorem for -->G3mi and -->_|_G3i . Additionally a generalization of this result with respect to a larger list of cut-elimination 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 cut-elimination 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. 232--234. [Gen35] Gentzen, G.: "Untersuchungen ueber das logische Schliessen I, II", Mathematische Zeitschrift 39, 1935, S. 176-210, 405-431. [Pott77] Pottinger, G.: "Normalization as a homomorphic image of Cut-elimination", Annals of Mathematical Logic 12, 1977, p. 323--357. [Pra65] Prawitz, D.: Natural Deduction (A Proof-Theoretical Study), Almqvist & Wiksell, Stockholm, Goeteborg, Uppsala, 1965. [Pra71] Prawitz, D.: "Ideas and Results in Proof-Theory", p. 235-307 in: Fenstad, J.E.[ed]: Proceedings of the 2nd Scandinavian Logic Symposium, North-Holland 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 Troelstra-Schwichtenberg G3i-System", March 1999, available from R. Vestergaard's Homepage: http://www.cee.hw.ac.uk/~jrvest/ . [Zu74] Zucker, J.: "The Correspondence between Cut-Elimination and Normalization", Annals of Mathematical Logic 7, 1974, p. 1-112, 156.