=Paper=
{{Paper
|id=Vol-1770/ARQNL2016_invited1
|storemode=property
|title=From Axioms to Proof Rules, then add
Quantifiers
|pdfUrl=https://ceur-ws.org/Vol-1770/ARQNL2016_invited1.pdf
|volume=Vol-1770
|authors=Revantha Ramanayake
}}
==From Axioms to Proof Rules, then add
Quantifiers==
https://ceur-ws.org/Vol-1770/ARQNL2016_invited1.pdf
FROM AXIOMS TO STRUCTURAL RULES, THEN ADD
QUANTIFIERS.
REVANTHA RAMANAYAKE
We survey recent developments in the program of generating proof calculi for
large classes of axiomatic extensions of a non-classical logics by translating each
axiom into a set of structural rules, starting from a base calculus. We will introduce
three proof formalisms: the sequent calculus, the hypersequent calculus and the
display calculus. The calculi that are obtained derive exactly the theorems of the
logic and satisfy a subformula property which ensures that a proof of a theorem only
contains statements that are ‘related to the conclusion’. These calculi can be used
as a starting point for developing automated reasoning systems and to facilitate a
proof-theoretic investigation of the logic (e.g. to prove interpolation, consistency,
decidability, complexity). In the final section we discuss how first-order quantifiers
may be added to these propositional calculi.
Much of the content here can be found in an extended form in the survey pa-
per [10]. The main purpose of this abstract is to provide a concise ‘hands-on’
description of the methods. We present a subjective selection of problems while
directing the reader to the references for a more exhaustive exposition.
1. Sequent calculus
Let L denote the intuitionistic language. The formulae from this language are
given by the grammar:
A, B := propositional variable p | ⊥ | > | A ∧ B | A ∨ B | A → B
The set of theorems of propositional intuitionistic logic is denoted by Int. Define
a sequent to be a tuple (denoted X ` A) where X (antecedent) is a multiset of
formulae and A (succedent) is a formula. A sequent calculus sInt for propositional
intuitionistic logic Int is given below.
X ` C (w) X, Y, Y ` C
⊥`C X`> p`p (c)
X, Y ` C X, Y ` C
X`A B, Y ` C A, X ` B A1 , A2 , X ` C
→l →r ∧l
A → B, X, Y ` C X`A→B A1 ∧ A2 , X ` C
X`A X ` B ∧r X ` Ai A, X ` C B, X ` C
∨r ∨l
X `A∧B X ` A1 ∨ A2 A ∨ B, X ` C
A derivation in the sequent calculus is defined recursively in the usual way as
either an initial sequent or the object obtained by applying a rule to the sequents
concluding a derivation. Given a multiset A1 , . . . , An , let ∧X denote A1 ∧ . . . ∧ An
if n > 0 else >. Then the relationship between the sequent calculus and the logic
can be described as follows.
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ARQNL 2016 1 CEUR-WS.org/Vol-1770
2 REVANTHA RAMANAYAKE
Theorem 1 (Gentzen [14]). Let X be a multiset and A a formula. Then X ` A is
derivable in sInt iff the formula ∧X → A is a theorem of Int. Also ` A is derivable
in sInt iff A is a theorem of Int.
The theorem reveals that sequents represent a certain normal form for formulae,
specifically conjunction of formulae → formula. The point of note here is not that
we have a proof calculus for the logic. After all, (axiomatic) Hilbert calculi for Int
are well-known (see e.g. [5]). Instead, the point is that this proof calculus has the
subformula property: every formula occurring in the premises occurs as a subfor-
mula of some formula in the conclusion. This means that any formula occurring in
the derivation must occur in the conclusion.
From an automated reasoning perspective, this immediately suggests a backward
proof-search procedure. The idea is to repeatedly applying proof-rules backwards.
If a derivation is obtained, then the sequent is derivable. The subformula property
tells us that the premise(s) are completely determined by the conclusion. Of course,
further pruning and optimisation on backward proof-search is required to obtain a
terminating efficient reasoning system.
Decidability is immediate: because of the weakening and contraction rules, we
can replace the multisets in the antecedent with sets. Then every sequent appearing
in backward proof search from ` A has the form Y ` B where Y ∈ P(Ω) and B ∈ Ω
where Ω is the set of all subformulae in X ` A and P is the powerset operator.
There are |P(Ω)| · |Ω| possibilities. Now enumerate the trees with nodes labelled
by such sequents (each node is permitted one or two children) such that the height
of the tree is ≤ |P(Ω)| · |Ω|. If one of these trees is a derivation of ` A then A is a
theorem. If not, there cannot possibly be a derivation of ` A so A is not a theorem.
Moreover, it is possible to amend the rules such that every premise antecedent is ⊇
the conclusion ancetedent. This corresponds to a pruning of the backward proof
search procedure and leads to the tight PSPACE upper-bound for Int.
Now suppose we wish to obtain a sequent calculus for the axiomatic extension
of Int with the axiom (p → q) ∨ (q → p) (denoted Int + (p → q) ∨ (q → p)). This
formula is not a theorem of Int (e.g. argue that ` (p → q)∨(q → p) is not derivable).
To obtain a sequent calculus for Int + (p → q) ∨ (q → p), it would be tempting
to add the sequent ` (p → q) ∨ (q → p) to sInt. Unfortunately not every theorem of
Int + (p → q) ∨ (q → p) is derivable using sInt + `(p → q) ∨ (q → p). The addition of
the (cut) rule below rectifies this: ` A derivable in sInt + (cut) + `(p → q) ∨ (q → p)
iff A is a theorem of Int + (p → q) ∨ (q → p).
X`A A, Y ` B
(cut)
X, Y ` B
Unfortunately the calculus sInt + (cut) + `(p → q) ∨ (q → p) lacks the subformula
property since the cut-formula A in (cut) need not appear in the conclusion. The
key to applications is having a calculus with the subformula property so we need
another solution.
2. Hypersequents
Theorem 1 reveals that every sequent can be read as a formula of the form
conjunction of formulae → formula. Let us call such a formula an implicational
formula. It turns out that the obstacle for constructing a calculus for Int + (p →
q) ∨ (q → p) is that the sequent-representation is too restrictive. A solution is to
move from an implicational formula to a finite disjunction of implicational formulae.
2
FROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS. 3
A hypersequent [1, 23] is a non-empty multiset of sequents denoted as below. In
particular, each Xi is a multiset of formulae and Ai is a formula.
(1) X1 ` A1 | X2 ` A2 | . . . | Xn+1 ` An+1
A sequent in the hypersequent is called a component. The notion of derivability
of a hypersequent is defined analogously to the sequent case. The rules of the
hypersequent calculus hInt are obtained from sInt by appending g | to each sequent
and adding the rules below left and centre. The g is a schematic variable that
can be omitted or instantiated with a hypersequent. Define the (cut) rule for the
hypersequent calculus as below right.
g|X ` B g|X ` B |X ` B g | X1 ` A h | A, Y ` B
ew ec (cut)
g|X ` B |Y ` A g|X ` B g | h | X, Y ` B
Theorem 2. The hypersequent h in (1) is derivable in hInt + (cut) iff the formula
(∧X1 → A1 ) ∨ . . . ∨ (∧Xn+1 → An+1 ) (the interpretation hI of h) is a theorem
of Int. Also ` A is derivable in hInt + (cut) iff A is a theorem of Int.
2.1. Translating suitable axioms into rules. The method introduced in [7]
transforms suitable axiom into a structural rule (i.e. a rule which contains no
logical connectives) and ultimately yields a calculus with the subformula property.
Starting from Theorem 2 it may be argued that h is derivable in hInt+(cut)+g | `
p → q | ` q → p iff hI is a theorem of Int + (p → q) ∨ (q → p) (replace top-level
disjunction symbols with | and add context g | ).
The new aim is thus to transform hInt + (cut) + g | ` p → q | ` q → p into a
hypersequent calculus with the subformula property which derives the same hyper-
sequents. Certainly we can simplify g | ` p → q | ` q → p to g | p ` q | q ` p by
repeated application of the invertible rules (→r). Invertible rules are rules
which preserve derivability upwards. In other words, the premises are derivable
whenever the conclusion is derivable. To deal with hInt + (cut) + g | p ` q | q ` p we
use the proof-theoretic form of Ackermann’s lemma [7, 11]:
Lemma 2.1 (Ackermann’s lemma). Let C be a hypersequent calculus extending sInt+
(cut). Let r1 , . . . , r4 be the rules defined below where S is a set of hypersequents
and the variables Y and Π do not appear other than where indicated. Then C + r1
and C + r2 (also C + r3 and C + r4 ) derive the same hypersequents.
S r1 S g | A, Y ` B
r2
g|X ` A g | X, Y ` B
S S g|Y ` A
l1 l2
g | A, X ` B g | X, Y ` B
Setting g | p ` q | q ` p as the zero-premise rule ρ0 , a single application of Ack-
ermann’s lemma tells us that hInt + (cut) + ρ0 and hInt + (cut) + ρ1 derive the
same hypersequents. Three further applications of Ackermann’s lemma yield that
hInt + (cut) + ρ4 also derives the same hypersequents (equivalent calculus).
g | X1 ` p g | X1 ` p g | X2 ` q
ρ1 ρ2
g | X1 ` q | q ` p g | X1 ` q | X2 ` p
g | X1 ` p g | X2 ` q g | Y1 , q ` B1 g | X1 ` p g | X2 ` q g | Y1 , q ` B1 Y2 , p ` B2
ρ3 ρ4
g | Y1 , X1 ` B1 | X2 ` p g | Y1 , X1 ` B1 | Y2 , X2 ` B2
3
4 REVANTHA RAMANAYAKE
The hypersequent calculus hInt + (cut) + ρ4 does not have the subformula property
because the propositional variables p and q in the premise do not appear in the
conclusion. To rectify this there is one final step: take the cut-closure on the
premises of the rule. In effect we delete those premises which contain propositional
variables which appear in either the antecedent or succedent but not both, and
we apply cut in all possible ways to the remaining the premises. This operation
may not terminate in general (consider the situation when the same propositional
variable appears in the same component of the antecedent and succedent). In the
case of ρ4 it does terminate to yield the rule
g | X1 , Y2 ` B2 g | X2 , Y1 ` B1
(com)
g | Y1 , X1 ` B1 | Y2 , X2 ` B2
When cut-closure terminates it can be shown that it yields an equivalent structural
rule. I.e. hInt + (cut) + (com) and hInt + (cut) + ρ4 are equivalent. It remains to
show that hInt + (cut) + (com) has cut-elimination i.e. hInt + (com) is an equivalent
calculus. In turns out that the structural rules obtained by the above procedure
satisfies sufficient conditions for cut-elimination (such rules are called analytic rules)
so we are done.
The four steps are summarised below.
1. Given the axiom ` A1 ∨ . . . ∨ An+1 apply all possible invertible rules backwards
to the hypersequent g | ` A1 | . . . | ` An+1
2. Use Ackermann’s lemma on each formula in the conclusion in order to obtain a
rule where the conclusion contains no formulae.
3. Apply all possible invertible rules backwards to the premises of the rule. (Fail
if one of the resulting hypersequents contain a compound formula which cannot
be made propositional by the invertible rules.)
4. Delete premises containing propositional variables appearing only on one side.
Apply all possible cuts to the remaining premises. (Fail if this step does not
terminate.)
As we would expect, not all axioms can be handled by this method. If the nesting
depth of logical connectives invertible in the antecedent/succedent is too great then
item 3 will fail. In the context of intuitionistic logic cut-closure always terminates
due to the presence of weakening and contraction. However this is not always the
case in substructural logics.
2.2. Some open problems. Hypersequents in the calculus hInt were built using
components X ` A where X is a multiset. If we take X as a list of formulae,
then the exchange rule (ex) below left, which states that formulae in the list can be
permuted, must be stated explicitly. Deleting structural properties of the comma
such as exchange (ex), weakening (w), contraction (c) and associativity lead to
hypersequent calculi for substructural logics. The substructural logics typically
contain additional language connectives. For example, in the absence of (w) and
(c), a commutative operator ⊗ distinct from ∧ can be defined using the rules below
centre and right. An identity 1 for the ⊗ operator may also be defined.
g | X, A, B, Y ` C g | X, A, B ` C g|X ` A h|Y ` B
(ex) ⊗l ⊗r
g | X, B, A, Y ` C g | X, A ⊗ B ` C g | h | X, Y ` A ⊗ B
Let hFLe denote the substructural hypersequent calculus lacking the weakening and
contraction properties (and containing the rules for ⊗). The corresponding logic is
the Full Lambek calculus with exchange (denoted FLe ).
4
FROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS. 5
• A hypersequent for the fuzzy logic MTL [13] (monoidal T-norm based logic)
can be constructed by the procedure above: add the (w) and (com) rules
to hFLe . This logic is known to be decidable via a semantic proof. It would
be interesting to obtain a proof by arguing directly on the hypersequent
calculus. This is turn would yield an upper bound on the logical complexity.
Meanwhile it is unknown if Involutive MTL is decidable. A hypersequent
for this calculus can be obtained by amending the rules of the calculus to use
a list of formulae in the succedent rather than a single formula. A syntactic
proof of decidability for MTL may provide a pathway for obtaining a proof
of decidability for IMTL.
• A hypersequent calculus for IUL [20] (involutive uninorm logic) can be ob-
tained by the addition of (com) to hFLe . A result of interest to the fuzzy
logic community is if this logic is standard complete [16]. This in turn would
follow by showing that whenever g | X ` Y, p | p, U ` V is derivable (propo-
sitional variable p occurs only where indicated) then so is g | X, U ` Y, V .
Such an argument is called density elimination [20, 8]. Some automated
solutions to this problem are described in [3].
See [21] for further details on these two problems. A more general open problem
concerns the handling of axioms which fail this methodology. A solution is to ven-
ture to a more expressive proof formalism as described in the following section. In
the context of modal logics, alternative approaches [19] to generating hypersequent
calculi from axioms have also been investigated.
3. Display Calculus
Since item 3 above fails when the invertible rules cannot reduce compound for-
mulae to propositional variables, it follows that the more invertible rules in the
calculus, the more axioms that can be presented. Essentially, the hypersequent cal-
culus was able to invert top-level disjunctions. The display calculus formalism [4]
extends [24] the hypersequent formalism: the formula corresponding to a display
sequent has a normal form which is broader and also more general in the sense that
the normal form is based on the algebraic semantics of the logic.
Below we introduce the display calculus δBiInt [28] for bi-intuitionistic logic BiInt.
The language of BiInt extends the intuitionistic language with the coimplication ←d .
This is forced because the display calculus formalism requires that the logical con-
nectives come in residuated pairs. To make the disjunction invertible on the right
we added the semicolon on the right. Residuation then necessitated the addition of
the structural connective < standing for ←d . An attractive feature of the display
calculus is the general sufficient conditions [4] for cut-elimination.
I`X X`I >r ⊥l
p`p >l ⊥r I`> ⊥`I
>`X X`⊥
A, B ` Y X ` A > B →r BY X < Y ` B ←d A
5
6 REVANTHA RAMANAYAKE
X`Y >Z XZ X”, “;” and “<” reflect the flexible normal form of the corresponding formulae.
Theorem 3 ([9]). Let C be a display calculus for the logic L satisfying the conditions
in [9]. Also suppose that {ri }i∈I are the analytic structural rules computed from the
finite set {αj }j∈J of axioms using the analogous procedure to the one in Section 2.
Then C + {ri }i∈I is a display calculus with the subformula property for L + {αj }j∈J .
For example, the logic BiInt + (p → ⊥) ∨ ((p → ⊥) → ⊥) can be presented
in this manner. Since BiInt + (p → ⊥) ∨ ((p → ⊥) → ⊥) is conservative over
Int + (p → ⊥) ∨ ((p → ⊥) → ⊥) (argue via the Kripke or algebraic semantics), we
can obtain a display calculus for the latter by deleting the logical rules (not the
structural rules!) introducing ←d . Incidentally, proving the conservativity directly
on the display calculus appears to be surprisingly difficult. The issue is with the
interaction of the display rules introducing < and contraction.
It should be noted that a version of the method of extracting analytic structural
rules from axioms appeared rather early on [17] in the context of display calculi
for tense logics. Recently, the tools of unified correspondence theory have been
applied [15] to extend that work in order to provide a new and uniform perspective
on the axioms to rules paradigm.
4. First-order quantifiers
A cut-free hypersequent calculus hIntfo for first-order intuitionistic logic Intf o is
obtained by adding to hInt the following rules for quantifiers [2, 6, 22]:
g | A(t), X ` B g | Γ ` A(a)
∀l ∀r
g | ∀xA(x), X ` B g | X ` ∀xA(x)
g | A(a), X ` B g | X ` A(t)
∃l ∃r
g | ∃xA(x), X ` B g | X ` ∃xA(x)
where the rules (∀r), (∃l) have an eigenvariable condition: the free variable a must
not occur in the lower hypersequent.
The addition of quantifiers to the hypersequent calculus can have some unex-
pected effects. For example, if we add the (com) rule to hIntfo then ` ∀x(A(x) ∨
B) → (∀xA(x) ∨ B) is derivable whenever x is not free in B. The corresponding
formula is known as the quantifier-shift or constant-domains formula and is not a
theorem of Intf o + (p → q) ∨ (q → p)! A non-standard hypersequent calculus for the
latter logic (known as Corsi’s logic [12] or Gödel-Dummett logic with non-constant
domains) has been introduced [25] where the eigenvariables are incorporated into
the hypersequent syntax. These hypersequents have no formula-interpretation in
the logic and it is not clear how to generalise the calculus to capture other logics.
6
FROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS. 7
4.1. Some open problems. The hypersequent calculus hIntfo + (com) presents
first-order Gödel logic. The question of interpolation for this logic is open. It is
conceivable to investigate this problem via the hypersequent proof-theory although
it has defied all such attempts thus far.
Meanwhile, the interaction between the (com) rule and the first-order quantifiers
witnessed in first-order Gödel logic can be generalised to a new question: let L be
a propositional axiomatic extension of Int and suppose that C is the hypersequent
calculus for it, obtained by the methods of the previous section. Let L1 be the
first-order axiomatic extension of L1 and let L2 consist of those formulae A such
that ` A is derivable in the extension of C by the first-order quantifier rules above.
What then is the relationship between L1 and L2 ?
The only investigation of first-quantifiers for display calculi appears in [27].
Adding the obvious correspondents of the quantifier rules above to δBiInt derives
the quantifier-shift formula. Meanwhile classical predicate logic may be treated as
a kind of propositional tense logic [26, 18], where the quantifiers ∃ and ∀ are treated
as a residuated pair analogous to _ and (see [17]). However the resulting dis-
play calculus fails cut-elimination due to the presence of certain rules. A decidable
minimal first-order system may be obtained by dropping these rules.
References
[1] A. Avron. A constructive analysis of RM. J. of Symbolic Logic, 52(4):939–951, 1987.
[2] M. Baaz and R. Zach. Hypersequents and the proof theory of intuitionistic fuzzy logic. In
Computer science logic (Fischbachau, 2000), volume 1862 of Lecture Notes in Comput. Sci.,
pages 187–201. Springer, Berlin, 2000.
[3] P. Baldi, A. Ciabattoni, and L. Spendier. Standard completeness for extensions of MTL: an
automated approach. In WOLLIC 2012, volume 7456 of LNCS, pages 154–167. Springer,
2012.
[4] N. D. Belnap, Jr. Display logic. J. Philos. Logic, 11(4):375–417, 1982.
[5] A. Chagrov and M. Zakharyashchev. Modal companions of intermediate propositional logics.
Studia Logica, 51(1):49–82, 1992.
[6] A. Ciabattoni. A proof-theoretical investigation of global intuitionistic (fuzzy) logic. Archive
of mathematical Logic, 44:435–457, 2005.
[7] A. Ciabattoni, N. Galatos, and K. Terui. From axioms to analytic rules in nonclassical logics.
In LICS 2008, pages 229 – 240, 2008.
[8] A. Ciabattoni and G. Metcalfe. Density elimination. Theor. Comput. Sci., 403(2-3):328–346,
2008.
[9] A. Ciabattoni and R. Ramanayake. Power and limits of structural display rules. ACM Trans.
Comput. Logic, 17(3):17:1–17:39, February 2016.
[10] A. Ciabattoni, R. Ramanayake, and H. Wansing. Hypersequent and display calculi – a unified
perspective. Studia Logica, 102(6):1245–1294, 2014.
[11] W. Conradie and A. Palmigiano. Algorithmic correspondence and canonicity for distributive
modal logic. Annals of Pure and Applied Logic, 163(3):338–376, 2012.
[12] G. Corsi. A cut-free calculus for Dummett’s LC quantified. Mathematical Logic Quarterly,
35(4):289–301, 1989.
[13] F. Esteva and L. Godo. Monoidal t-norm based logic: towards a logic for left-continuous
t-norms. Fuzzy Sets and Systems, 124:271–288, 2001.
[14] G. Gentzen. The collected papers of Gerhard Gentzen. Edited by M. E. Szabo. Studies in Logic
and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1969.
[15] G. Greco, M. Ma, A. Palmigiano, A. Tzimoulis, and Z. Zhao. Unified correspondence as a
proof-theoretic tool. to appear. Journal of Logic and Computation, 2016.
[16] P. Hájek. Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998.
[17] M. Kracht. Power and weakness of the modal display calculus. In Proof theory of modal logic
(Hamburg, 1993), volume 2 of Appl. Log. Ser., pages 93–121. Kluwer Acad. Publ., Dordrecht,
1996.
7
8 REVANTHA RAMANAYAKE
[18] D. Leivant. Quantifiers as modal operators. Studia Logica, 39:145–158, 1980.
[19] B. Lellmann. Hypersequent rules with restricted contexts for propositional modal logics.
Theoretical Computer Science, 656, Part A:76 – 105, 2016.
[20] G. Metcalfe and F. Montagna. Substructural fuzzy logics. J. of Symbolic Logic, 72(3):834–864,
2007.
[21] G. Metcalfe, N. Olivetti, and D. Gabbay. Proof Theory for Fuzzy Logics, volume 39 of Springer
Series in Applied Logic. Springer, 2009.
[22] G. Metcalfe, N. Olivetti, and D. Gabbay. Proof Theory for Fuzzy Logics, volume 39 of Springer
Series in Applied Logic. Springer, 2009.
[23] G. Pottinger. Uniform, cut-free formulations of T, S4 and S5 (abstract). J. of Symbolic Logic,
48(3):900, 1983.
[24] R. Ramanayake. Embedding the hypersequent calculus in the display calculus. Journal of
Logic and Computation, 25(3):921–942, 2015.
[25] A. Tiu. A hypersequent system for Gödel-Dummett logic with non-constant domains. In
Tableaux 2011. LNAI, pages 248–262. Springer, 2011.
[26] J. van Benthem. Modal foundations of predicate logic. Log. J. IGPL, 5:259–286, 1997.
[27] H. Wansing. Predicate logics on display. Studia Logica, 62(1):49–75, 1999.
[28] H. Wansing. Constructive negation, implication, and co-implication. J. Appl. Non-Classical
Logics, 18(2-3):341–364, 2008.
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