Sequent Calculus Sequents Natural Deduction Example exercise for Sequent Calculus and Natural Deduction Cut rule Asymmetrical representation Subformula properties Translation. Natural Deduction Sequent Calculus Cut Elimination theorem

1. Sequent Calculus - Natural
Deduction. Translation and Overview
Dostonbek Matyakubov

2. A little bit of history
Natural Deduction and Sequent calculus both were introduced by Gerhard
Gentzen, who first introduced Natural Deduction and later Sequent Calculus
which he found easier to work with(for consistency of proofs).
In Natural Deduction, the rules for logical connectives are introduction and
elimination rules (right).
In Sequent Calculus, there are no elimination rules: right introduction and left
introduction rules.
I.e. right elimination is traded against left introduction.

3. Sequent Calculus vs Natural Deduction
Differences:
● The sequent calculus is more suitable for automated proof search
● The cut rule is implicit in Natural Deduction and explicit in Sequent
Calculus.
● Natural Deduction gives more mathematical-like approach to reasoning
while the Sequent Calculus gives more structural and symmetrical
approach
Relations - one can be translated to another.

4. The Sequent Calculus
Sequent calculus gives principled answers to problems such as:
● How to soundly compile xxx?
● How to prove normalization of xxx?
● How should control operators and xxx interact?
● Deciding equivalence of normal forms

5. Sequent
A sequent is an expression A |- B where A and B are finite sequences of
formulae A1...An and B1...Bn.

6. Natural Deduction - Example
Let’s consider the following problem : P, P ⇒ Q ⊢ P ∧ Q
1. P Premise
2. P ⇒ Q Premise
3. Q Modus Ponens (2)
4. P ∧ Q Conjunction (3)
Note:
● Natural Deduction is solved from top-to-bottom unlike Sequent Calculus
● Natural Deduction is more mathematical-like approach.

7. Sequent Calculus Example
Consider |- A ⇒ (B ⇒ A).
Consider |- ∀x.Px ⇒ ∀y.Py.

8. Sequent Calculus - Structural rules (CL)
● The exchange rules:
● The weakening rules:
● The contraction rules:
The structural rules essentially say that A and B in the sequence A |- B are
multisets. These rules are the most important of the whole calculus.Symmetry

9. Intuitionistic Structural Rules
An intuitionistic sequent is a sequent A |- B where B has at most one formula.
The exchange and contraction rules:
The weakening rules:
Note that RX and RC rules are not possible. And the symmetry is broken.

10. Properties of Intuitionistic Sequent Calculus
Let’s consider a proof of |- A without cut. What could be the last rule?
● Structural rules cannot give us |- A.
● The identity axiom does not give us |- A.
● Left logical rules cannot do.
The last rule must be a right logical rule.

11. Disjunction Property vs Existence Property
If A = A’ ∨ A’’, the last rule must be R1∨ or R2∨. That is, we have |- A’ or |- A’’. If
|- A’ ∨ A’’, then |- A’ or A’’ . This is called the Disjunction Property.
I If A = ∃ξ.A’, the last rule must be R∃. That is, we have |- A’ [a/ξ]. If |- ∃ξ.A’ is
provable, then |- A’[a/ξ] for some term a. This is called the Existence
Property.

12. The Cut rule - General Idea
The cut rule is an inference rule of sequent calculus. It is a generalisation of
the classical modus ponens rule, meaning - if formula A appears as conclusion
in one proof as hypothesis in another, then another proof in which formula A
does not appear can be deduced.
Example - occurences of man are eliminated to deduce Socrates is
mortal: Every man is mortal, Socrates is a man - Socrates is mortal.
The cut rule is the subject of an important theorem - the cut elimination
theorem, which states that any judgement that possesses a proof in the
sequent calculus that makes use of the cut rule also possesses a cut-free
proof(a proof that does not make use of cut rule).

13. The Cut rule and Identity Axiom
For every formula C, we have the identity axiom.
The cut rule:
The cut rule can be seen as the symmetric rule to identity axiom. The identity
axiom states C-left is stronger than C-right. The cut rule states C-right is
stronger than C-left
The cut rule is not welcome in proof search. How can an algorithm guess C to
prove A, A` |- B, B` ?
For every proof for a sequent, there is a cut-free proof for the same sequent.
So, no need for cut rule

14. Unpredictable Cut rule
The cut rule is absolutely unpredictable, since an arbitrary formula C
disappears: it cannot be recovered from the conclusions. The cut rule is only
rule which behaves in this way, all other rules have the property that the
unspecified context part is preserved intact.
For example: for A / B as a conclusion, A and B must have been used as
premises.
All rules expect cut have the property that premises are made up of sub-
formula of the conclusions. In fact, a cut-free proof of sequent uses only
subformula of its formula.

15. Subformula Property
The cut rule is unpredictable. So, there is no way to guess the cut formula C.
● The immediate subformulae of A ∧ B, A ∨ B, and A ⇒ B are A and B;
● The immediate subformulae of ¬A is A;
● The immediate subformulae of ∀ξ.A and ∃ξ.A are A[a/ξ] with any term a.
Except the cut rule, all rules preserve “contexts” (written A, A’ , B, B’) and
change only one formula; moreover, the premises are immediate subformula
of the conclusion. This is called Subformula Property which is very useful in
automated deduction.

16. From Sequent Calculus to Natural Deduction
A proof of A|- B corresponds to deduction of B under parcels of hypotheses A.
Conversely, a deduction of B under parcels of hypotheses A can be
represented by a proof of A |- B.
Why not consider A |- B? - A deduction is for a formula, not formulae(s).

17. Identity Axiom and Cut rule - Translation
The identity group gives basic deductions.
For the identity axiom,
For the cut rule,

18. Structural rules Translation
Structural rules manage parcels. The rule LX is interpreted as the identity: the same
deduction before and after the rule.
For rule LW, add a new parcel. is interpreted as the creation of a mock parcel formed
from zero occurrences of C.
The rule LC merges two parcels.

19. Logical rules Translation
Right logical rules correspond to introduction. The rule R∧ will be interpreted by ∧I
The rule R⇒ will be interpreted by ⇒I.
The rule R∀ will be interpreted by ∀I:

20. From Sequent Calculus to Natural Deduction
Left logical rules correspond to elimination.
The rule L1∧ is interpreted by ∧1E:
The rule L⇒ is interpreted by ⇒E:
The rule L∀ is interpreted by ∀E

21. Properties of translation
The translation from sequent calculus into natural deduction is not 1–1: different proofs
give the same deduction, for example.
which differ only in the order of the rules, have the same translation:
Natural deductions reflect to our informal notion of “proofs” more closely.
Sequent calculus on the other hand manipulates such “proofs.” Hence, A |- B means a
“proof” of B from A.

22. Some Remarks on Translation
Right logical rules in sequent calculus correpond to introduction rules in natural
deduction. The translation expands the deduction downwards (to the root).
Left logical rules in sequent calculus correspond to elimination rules in natural
deduction. The translation expands the deduction upwards (to leaves).
We can make the translation expand downwards by the cut rule.

23. Automatic proof generator for SC
Sequent calculus proof generators.
● http://logitext.mit.edu/main
● http://bach.istc.kobe-u.ac.jp/seqprover/
● https://seqcalc.io/
Automatic ND to SC translator GAPT - General Architecture for proof theory.
● https://logic.at/gapt
● https://www.logic.at/gapt/release_archive.html
● https://github.com/gapt/gapt

24. Sequent Calculus - example
Let's consider how to build a CL proof for |- ((A ∧ B) ∧ C) -> (B ∧ A) is true.
First, we begin with the sequent as the goal and notice that the primary
connective exposed in the only proposition available is implication, so we can
apply the right implication rule:
Next, we may break down the conjunction in the consequence B ∧ A with the
right conjunction rule, splitting the proof into two parts

25. Sequent Calculus - example
At this point, the consequences of both our goals are generic, lacking any
specific connectives to work with. Therefore, we must shift our attention to
the left and begin breaking down the hypotheses. Since the hypothesis
(A∧B)∧C contains a superfluous C, we use the first left conjunction rule in both
branches of the proof to discard it:

26. Sequent Calculus - example
Now we may apply another left conjunction rule to select the appropriate
hypothesis needed for both sub-proofs:

27. Sequent Calculus - example
And finally, we can now close off both sub-proofs with the Axiom, finishing the
proof: