8.2.1
More Complex Calculi
The most obvious area of future study is tweaking, or perhaps reformulating, the criteria laid down in section5.5so that more exotic calculi can be analysed using the methods presented here. A different structure, such as sets or lists, could be considered for the contexts of rules. Only context-sharing rules are permitted at present. Some calculi, however, contain both context-sharing and context-splitting rules, notably the linear logic of [Girard,1987]. Investigation of the invertibility of such rules would be a worthwhile avenue down which one could venture. The danger, of course, is that the criteria for both classifying the calculus itself, and for providing sufficient conditions for invertibility within that calculus, become too convoluted. One must avoid the situation where it is more straightforward to check
invertibility directly rather than checking a series of conditions which prove invertibility indirectly.
The first choice for extension would be first-order calculi. In order to define such calculi in a similar fashion to section5.5, we would require substitution to be defined for metasequents and other meta notions. It is possible to avoid this definitional approach if we instead think of a rule as the infinite set of its inferences. Then, substitution needs only to be defined for the formulae and multisets. However, this would then no longer be an extension of section
5.5. Of course, one could rework those sections to follow this alternative approach. Another problem associated with first-order calculi is with formalisation. The formulae, and hence also the sequents and rules, in chapter7are indexed by type variables. Currently this is not possible inNominal Isabelle(see appendixB).
Some calculi, such as hGKi, the Gentzen-Kleene calculus for intuitionistic logic with head formulae, have additional structure on the left of the sequent arrow in the form of a stoup or head. A reserved place, which can be filled by a formula, highlights the principal formula of a left inference. This extra structure simplifies analysis: if we come across a derivation which ends with:
A0⊃A1; Γ⇒B
then this can only have had L⊃or a right rule as the last inference used. There is the additional structural rule calledDereliction, which moves a formula from the context to the stoup position. To prove invertibility, one may have to use instances ofDereliction, which will make strong invertibility unlikely.
Interestingly, the left premiss of the ruleL⊃is not invertible forhGKi, butfor a different reason to why it is not invertible in G3i. InG3i, the base case of the proof by induction failed: there is no reason Γ⊕A0⊃A1⇒Bshould be derivable for general Γ andB. However,
with stoups the base case is treated byex falso: if the derivation ends withA0⊃A1; Γ⇒B
then itcannot also end withP; Γ⇒P, whereP is some propositional variable. The proof now fails because of the interplay betweenR⊃andL⊃. When the last inference used isR⊃, the derivation ends with:
A0⊃A1; Γ, B⇒C
A0⊃A1; Γ⇒B⊃C
We can apply the induction hypothesis to the premiss, but this gives a copy of B in the context which is unwanted and impossible to remove:
A0⊃A1; Γ, B⇒C
−; Γ, B, A0⊃A1⇒A0 ih
More study is needed to identify which combinations of rules, or rather which properties of rules, will create the above situation so that it can be restricted.
8.2.2
Other Formalisms
The next area of interest is that of other formalisms. Sequent calculus has its roots with Gentzen in the 1930s. Since then, many formalisms have sprung up to answer questions which the sequent calculus was unable to answer. Such things include proof nets [Girard,
1987] and deep inference [Br¨unnler,2006]. Both were developed to remove the redundancy present in sequent calculi, however invertibility-type questions can still be asked about such systems.
8.2.3
Related Problems
Another option one could investigate is to widen the scope from invertibility to permutability. Permutability is a more general problem, and is concerned with when two inferences can be switched without altering the provability of a sequent. For example, inG3cp, it is possible to switch the order of theL∧andR∨inferences in the following derivations:
Γ, A, B⇒A, B,∆ Γ, A∧B⇒A, B,∆ L∧ Γ, A∧B⇒A∨B,∆ R∨ Γ, A, B⇒A, B,∆ Γ, A, B⇒A∨B,∆ R∨ Γ, A∧B⇒A∨B,∆ L∧
In this case, we say L∧ permutes over R∨. This relates to invertibility in the following way: if a rule can be permuted over any other rule, then the initial rule is invertible. Thus, invertibility is a special kind of permutability. Giving conditions for when two rules can be permuted over one another will then provide lemma8 as a corollary, for instance.
Permutability has been shown for specific logics, but not in a general setting. For in- stance, [Kleene, 1952] shows which inferences in LK and LJ can be permuted, which is extended in [Dyckhoff and Pinto,1999] and [Shankar, 1992]. In [Troelstra and Schwichten- berg,2000], the one-sided Gentzen-Sch¨utte systemGS1is analysed for permutability.
In a similar vein to permutability, albeit one further removed from the original area of study, one may investigate other meta-theoretical properties of sequent calculi. The prime candidate is that ofCut admissibility, which has been studied in some detail in [Ciabattoni and Terui, 2006a], [Curry, 1963], [Restall, 1999] etc. Indeed, looking at the conditions of [Restall, 1999] when compared with the defnition of a decomposable uniprincipal calculus (section5.5) one sees all that is missing for such a calculus to admitCut is the matching of principal constituents. In other words, should one provide conditions for when a cut, whose formula is principal inboth premisses, can be eliminated in such a calculus (subject to use of the induction hypothesis), then one would have provedCut admissibility in general.
Also of interest is Contraction. This is perhaps more accessible thanCut admissibility, given that only one rule at a time needs to be considered, rather than pairs of rules. However, as we saw in section4.5, based on [Dyckhoff and Negri,2000], a lot of non-standard lemmata are needed in some instances to proveContraction admissibility. Contraction is also, whilst interesting, not as important as Cut admissibility, nor indeed is it as important as proving
an interpolation result. As noted in section 4.3, interpolation results are useful in various branches of computer science. Given a logic, would it be possible to give conditions for when an interpolation theorem is provable?
8.2.4
Formalisation Problems
From the theorem prover side, there is one avenue which would be very interesting to ex- plore. The informal proofs given within this thesis often contain sequent calculus derivations, which the formal proof has to recreate line by line in a linear fashion. Ideally one would like to give a sequent calculus derivation inIsabelle using the same mechanism as one would in LATEX, with some notes to tell the system which rule to apply etc. Then, instead of get-
ting formally checked proofs, one would get formally checked proofs which contain formally checked sequent calculus derivations. This would, again, decrease the mismatch between the formal and informal approaches. Incidentally, this was one of the original ideas for the thesis which was never followed through; it is more a LATEX andIsabelle interface problem
than a proof theory problem.