# [FOM] Mismatch

Alessio Guglielmi Alessio.Guglielmi at inf.tu-dresden.de
Sat Jun 14 22:23:31 EDT 2003

```Hello,

a structural mismatch between meta and object level in deductive
systems. I'll use this argument as motivation and introduction for a
course on my work on deep inference.

In short, the argument goes as follows:
1   There is a mismatch between meta and object levels in many
calculi, like the sequent calculus and natural deduction.
2   The mismatch has undesirable proof theoretical consequences, the
most important being the inability to design deductive systems.
3   Since the object level is untouchable (it's the language you want
to deal with), one can fix the problem by `improving' the meta level;
4   or one can also sort of *merge* object and meta level by using
deep inference, and this is my favourite approach.

I'm not going to be particularly exhaustive or pedantic (I'll use
plenty of `quotes'), I just hope that people who see things
differently tell me so, and the same do people who know about ideas I
didn't consider here. I'm sure there are many imprecisions and
mistakes in what follows, please correct me and let me know of your
work if you think it's relevant.

-Alessio

1   THERE IS A MISMATCH BETWEEN META AND OBJECT LEVEL IN MANY CALCULI

I will here make more precise what I mean by `mismatch'. It's not
worth the effort to make this notion formal, because it has more of a
moral value than a technical one. To make my point, looking at the
sequent calculus suffices, because the reader can immediately make
the analogue with natural deduction, proof nets, etc.

1.1   Let us first see a case of perfect match in Gentzen's one-sided
sequent calculus, for propositional classical logic. Consider:

|- A, Gamma   |- B, Delta       |- A, B, Gamma
^ ------------------------- ,   V --------------- ,
|- A ^ B, Gamma, Delta         |- A V B, Gamma

where A and B are formulae and Gamma and Delta are multisets of
formulae. In the first inference, the object level ^ in the
conclusion corresponds to a meta level `and' between the two premises
(branches of the derivation). In the second inference, the object
level V corresponds to the meta level `or' expressed by commas in the
premise sequent.

We can test the correspondence by `flattening down to object' the
meta level and check whether inferences correspond to implications.
They do, because

((A V Gamma) ^ (B V Delta)) => ((A ^ B) V Gamma V Delta) ,

and trivially so for the V rule.

There is no mismatch, because, when going bottom-up, the connectives
at the object level (^, V) become corresponding structural relations
at the meta level (resp. `branching' and `being in a multiset').

1.2   Let us now see a case of mismatch, in Gentzen's one-sided
sequent calculus for linear logic. Consider:

|- A, Gamma   |- B, Delta
* ------------------------- ,
|- A * B, Gamma, Delta

where * is multiplicative conjunction (`times' or `tensor'). Can we
consider branching a meta-level *? We could, because

((A # Gamma) * (B # Delta)) -o ((A * B) # Gamma # Delta) ,

where # is multiplicative disjunction (`par') and -o is linear implication.

Now let's see additive conjunction & (`with'):

|- A, Gamma   |- B, Gamma
& ------------------------- .
|- A & B, Gamma

We assumed that branching is *, and we find that

((A # Gamma) * (B # Gamma)) /-o ((A & B) # Gamma) ,

so we have a mismatch. If we assume that branching is &, instead of *, then

((A # Gamma) & (B # Gamma)) -o ((A & B) # Gamma) ,

so the mismatch disappears here, but it reappears in the previous
case of *, because

((A # Gamma) & (B # Delta)) /-o ((A * B) # Gamma # Delta) .

We could go one step further by declaring that branching between
sequents (or derivations) S and T corresponds to !S * !T, where ! is
the `of course' modality. In this case, we have

(!(A # Gamma) * !(B # Delta)) -o ((A * B) # Gamma # Delta)   and
(!(A # Gamma) * !(B # Gamma)) -o ((A & B) # Gamma) .

But are we happy? No, because we still have a mismatch, since !(.) *
!(.), i.e., the branching relation, doesn't correspond neither to *,
nor to &, nor to any other connective in the language.

At this point you see how I could technically define my notion of
mismatch, in the sequent calculus. Of course, the notion would be
meaningful also in proof nets, but the definition for them would
become more delicate (given that branching is rather different), so
it's not worthy to get into this business for now.

What matters is that, for example, if one asks the question: `What
does branching correspond to?' then the answer cannot be given simply
and elegantly in terms of logical operators at the object level. I'm
going to argue that this difficulty causes trouble in structural
proof theory.

2   THE MISMATCH HAS UNDESIRABLE PROOF THEORETICAL CONSEQUENCES

My impression is that we can consider the sequent calculus perfect
for classical logic, since there's no mismatch, but it gets less and
less adequate the more we depart from classical logic.

2.1   If we move to linear logic, we see disturbing phenomena
occurring already, even more disturbing than the mismatch observed
above. Consider for example the promotion rule:

|- A, ?B1, ..., ?Bh
p -------------------- ,
|- !A, ?B1, ..., ?Bh

where ? ('why not') is the dual modality of !. Contrary to legitimate
expectation, applying the rule requires a global check on the entire
context of !A, for making sure that all formulae are modalised by ?.

If one is only interested in certain technical aspects of deductive
systems, like for example cut admissibility, then the sequent
calculus, in this case, does its job, and we only have to pay a price
in terms of elegance and conceptual clarity. In this sense, the rule
above is disturbing, for example, because the `meaning' of ! requires
a side condition on the context. As we will see later, pure
aesthetics of this rule can also be improved in a different approach.

The promotion rule is not `difficult' only in the sequent calculus:
proof nets for ! and ? are also unsatisfying, again for problems that
one can reconduct to the mismatch.

2.2   What I find the most serious drawback of the mismatch is the
impossibility, or at least the inability, or at the very least the
difficulty, of designing deductive systems for many logics that are
otherwise known:

- Impossibility: we know a logic, called BV, for which Alwen Tiu
proved that it's impossible to express it in Gentzen's sequent
calculus [1]. It's the only result I know of this kind. There is no
room here to give the details, suffices to say that BV is extremely
simple, and it's not a specially tailored pathological case.

- Inability and difficulty: probably the most common cases of
inability are found in modal logics. For several modal logics, like
S5, it is not known (if I'm not wrong) whether a (pure) Gentzen's
sequent calculus presentation is achievable. For other modal logics
one such presentation has been painfully found, often after
significant departures from the pure form of Gentzen's sequents.

Typically, the context structure of sequents is enriched: they are no
more just sets or multisets, but become order structures of varying,
and sometimes very high, complexity. All of this is done of course
for the sake of eliminating the mismatch.

2.3   There is a simple philosophy of Gentzen's sequent calculus, for
which, grossly speaking, the `meaning' of connectives is defined by
their rules, where connectives are `brought to the meta level' with
no reference to the context, according to some harmony-preserving
principles. Departing from pure sequents has a conceptual cost, which
is not entirely repaid, in my opinion, by achieving technical results
like cut elimination.

In fact, there is a value in having simple deductive systems, all
designed according to the same methodology. Having to tailor the
shape of sequents to each logic requires an effort, both for the
proof theorist and for the user of the deductive system, who has to
learn a new metalanguage for every system.

There is in addition a technological cost. `Exotic' logics are more
and more used in computer science, in verification, for example. If
one changes the metalanguage (of sequents, or of natural deduction,
typically), then the corresponding verification system must be
updated accordingly. In other words, it is difficult to design a
universal framework in which all sorts of metalanguages for
describing deductive systems are easily coded.

3   ONE CAN FIX THE PROBLEM BY `IMPROVING' THE META LEVEL

As I said above, the standard way of dealing with the mismatch is to
enrich the structure of the meta level. The idea is simple: if
branching and commas are not enough, let's supply some more structure.

Since doing this ad hoc, case by case, blurs the conceptual picture
and has costs for the users of deductive systems, a general solution
is desirable.

here) one strong bid in this direction comes from Belnap's display
calculus. The idea goes as follows:

3.1   Why does one need a meta level at all? Because, when looking
for a proof, one has to get inside formulae to access the information
lying there under several strata of connectives. To do so, one breaks
formulae into pieces, but has to keep track carefully of the
relations between pieces, whence the need of structure at the meta
level.

The display calculus does this: it defines a rich notion of structure
at the meta level, and makes sure that the various subformulae are
accessible (`on display'). Then a general cut elimination theorem
exists which is automatically inherited by all deductive systems
presented in the display calculus. Since the display calculus is
mainly studied by philosophers, I'm sure that a careful theory of
meaning has been elaborated (I'm not knowledgeable enough to be more
positive).

3.2   I'd be very much interested in listing all advantages and
disadvantages of the display calculus. My provisory list goes as
follows:

uniform treatment here, as opposed to ad hoc presentations in
variants of Gentzen's sequent calculus; generality, one cut
elimination theorem suffices for all systems.

- Disadvantages: complexity, I find the definition of display
calculus `very big'.

As a last remark, I have to say that I'm not entirely convinced of
the display calculus approach because it goes in the direction of
complexity instead of simplicity. There are two ways to reach
generality: 1) to take into account all possibilities and then study
them once and for all, 2) to try and find a simple principle
underlying a vast class of phenomena. My impression is that the
display calculus does 1, while I believe that in structural proof
theory something like 2 can be discovered.

4   ONE CAN `BRING THE META LEVEL DOWN TO THE OBJECT ONE' BY USING
DEEP INFERENCE

If the purpose of the meta level is to make subformulae accessible,
why don't we just get in and take them, by inferring directly inside
formulae, at any level of depth? Of course, one reason not to do so
could be being unable to do proof theory (for example proving cut
elimination)!

My colleagues and I, in the last few years, have developed a proof
theory for deep inference (called the Calculus of Structures), so we
claim to have a radical, good solution for the problem represented by
the mismatch. This is the content of my course, but, to complete this
email, I will explain here the main idea with a couple of examples,
and will make a summary of the advantages I see in my approach.

4.1   All inference rules are of the form

C{B}
---- ,
C{A}

meaning that subformula A inside formula C{A} is replaced inside
context C{ } by B, while going up in building a proof. There is no
branching, no commas, no meta-level structure. The formula itself
takes care of representing what used to be the meta-level structure.
Gien this, rule V of classical logic immediately disappears. Let us
just assume from now on that A and B, as above, do not occur in the
scope of an odd number of negations (otherwise we could use dual
rules to the ones we are going to see).

Rule ^ becomes what I call a `switch':

C{(A V D) ^ (B V E)}
s -------------------- ,
C{(A ^ B) V (D V E)}

which is good also for the linear logic case,

C{(A # D) * (B # E)}
s -------------------- .
C{(A * B) # (D # E)}

4.2   This was of course a triviality. The next step is looking at
cut and cut elimination. Identity and cut look like this:

C{true}              C{A ^ -A}
id ---------   and   cut --------- .
C{A V -A}             C{false}

It is easy to see how they correspond to their counterparts in the
sequent calculus. What is not easy is proving cut elimination,
because, in the presence of deep inference, the old methods are
basically useless. One can find in [2, 3, 4] proofs of cut
elimination for classical and linear logics. Many other papers are
available at [5], one of which also deals with a uniform theory for
modal logics [6].

4.3   Rules like promotion get important advantages from deep
inference. Here is promotion in the Calculus of Structures:

C{!(A # B)}
p ----------- ;
C{!A # ?B}

since there's more flexibility in managing the context than just
branching, this promotion rule can check its context incrementally,
one ?B at a time, this way removing the need for the global test that
afflicted promotion in the sequent calculus. (Why this is so is not
obvious, the reader can check [4] or [7] or [8] for details.)

More importantly, one can see that promotion is just a unary case of
switch. More in general, rules in the Calculus of Structures can be
derived from a more general, simple scheme, briefly described in [9].
(An even more dramatic simplification is speculated in [10].)

4.4   The biggest advantage that deep inference brings into the
picture is top-down symmetry for derivations. A derivation can be
flipped upside-down, and negated, and it remains a valid derivation.
One can see our new symmetry in the identity and cut rules shown
above. In the sequent calculus, this is impossible because there is a
top-down asymmetry: while going up, one has less and less of object
level and more and more of meta level.

The top-down symmetry has important proof theoretic consequences. For
example, the cut rule can be equivalently replaced by its atomic
version (the principal formula is an atom). This can be done without
having to prove full-blown cut elimination, but just by a very simple
inductive argument dual to the corresponding one for identity. This
contributes to completely different, and in certain cases very
simple, arguments for cut elimination (see [2]). In general, the
entire analysis of proofs benefits from the symmetry, for example
several interesting notions of normal forms can be studied which are
provably impossible in the sequent calculus.

4.5   In conclusion, given that the mismatch causes trouble, it seems
to me that the solution of eliminating the cause of the disease, by
deep inference, looks better than the alternative of living with it
by curing the symptoms. There are (at least) two serious objections
one can make:

1) There is no `theory of meaning' for what we do with deep
inference. My guess is that this might come one day, given that my
formalism is very clean and regular, and it possesses interesting
properties, that should stimulate the interest in studying it from a
philosophical perspective.

2) There is no general cut elimination theorem, like for display
logic. Again, I believe that this is a matter of time: I am
especially confident in the kind of research started in [10] and in
so-called splitting theorems (see [11] for an example), which appear
to have the right kind of encompassing regularity needed for this

REFERENCES

[1] Alwen Fernanto Tiu. Properties of a logical system in the
calculus of structures. Technical Report WV-01-06, Technische
Universitaet Dresden, 2001. URL:
<http://www.cse.psu.edu/~tiu/thesisc.pdf>.

[2] Kai Bruennler. Atomic cut elimination for classical logic.
Technical Report WV-02-11, Technische Universitaet Dresden, 2002.
URL:
<http://www.ki.inf.tu-dresden.de/~kai/AtomicCutElimination-short.pdf>,
accepted at CSL '03.

[3] Kai Bruennler. Deep Inference and Symmetry in Classical Proofs.
PhD thesis, Technische Universitaet Dresden, 2003. URL:
<http://www.ki.inf.tu-dresden.de/~kai/phd.pdf>.

[4] Lutz Strassburger. Linear Logic and Noncommutativity in the
Calculus of Structures. PhD thesis, Technische Universitaet Dresden,
2003. URL: <http://www.ki.inf.tu-dresden.de/~lutz/dissvonlutz.pdf>.

[5] Alessio Guglielmi, The Calculus of Structures. Web site. URL:
<http://www.ki.inf.tu-dresden.de/~guglielm/Research>.

[6] Charles Stewart and Phiniki Stouppa. A systematic proof theory
for several modal logics. Technical Report WV-03-08, Technische
Universitaet Dresden, 2003, URL:
<http://www.linearity.org/cas/papers/sysptf.pdf>.

[7] Alessio Guglielmi and Lutz Straßburger. Non-commutativity and
MELL in the Calculus of Structures. CSL 2001, LNCS 2142, pp. 54-68.
URL:
<http://www.ki.inf.tu-dresden.de/~guglielm/Research/GugStra/GugStra.pdf>

[8] Lutz Straßburger. MELL in the Calculus of Structures. Technical
Report WV-01-03, to appear in Theoretical Computer Science. URL:
<http://www.ki.inf.tu-dresden.de/~lutz/els.pdf>.

[9] Alessio Guglielmi. Recipe. Manuscript, 2002, URL:
<http://www.ki.inf.tu-dresden.de/~guglielm/Research/Notes/AG2.pdf>.

[10] Alessio Guglielmi. Subatomic logic. Manuscript, 2002, URL:
<http://www.ki.inf.tu-dresden.de/~guglielm/Research/Notes/AG8.pdf>.

[11] Alessio Guglielmi. A system of interaction and structure.
Technical Report WV-02-10, Technische Universitaet Dresden, 2002, URL:
<http://www.ki.inf.tu-dresden.de/~guglielm/Research/Gug/Gug.pdf>, submitted.

--

Alessio Guglielmi
Technische Universitaet Dresden
<http://www.ki.inf.tu-dresden.de/~guglielm/>

```