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Logic CL4

- The language of CL4
- The rules of CL4
- CL4 as a conservative extension of classical logic
- The soundness and completeness of CL4
- The decidability of the blind-quantifier-free fragment of CL4
- Other axiomatizations (affine and intuitionistic logics)

0

CL4 is by far the most expressive deductive system for computability logic known

to be sound and complete at present.

Its language is obtained from the full language described in Episode 14 by forbidding

only the parallel quantifiers and the recurrence operators (together with and ).

We refer to the formulas of this language as CL4-formulas.

The language of CL4To define CL4, we need to agree on some terminology.

(Here) understanding EF as an abbreviation of EF, a negative occurrence of a

subformula is one that is in the scope of an odd number of ’s. Otherwise, the

occurrence is positive.

A surface occurrence of a subformula means an occurrence that is not in the scope

of a choice operator.

A CL4-formula is said to be elementary iff it does not contain general letters and

choice operators. Note that elementary formulas = formulas of classical logic.

The elementarization of a CL4-formula is the result of replacing, in it, every surface

occurrence of the form E⊓F or ⊓xE by ⊤, every surface occurrence of the form E⊔For

⊔xE by ⊥, every negative surface occurrence of each general atom by ⊤, and every

positive surface occurrence of each general atom by ⊥.

A CL4-formula is said to be stable iff its elementarization is valid in classical logic,

i.e. provable in G3. Otherwise the formula is instable.

CL4 has the following four rules, with E,F standing for CL4-formulas and H for a

set of CL4-formulas:

The rules of CL4where F is stable and, whenever F has a positive (resp. negative)

surface occurrence of:

Rule A: , (i) G1⊓G2 (resp.G1⊔G2), for each i{1,2},

H contains the result of replacing in F that occurrence by Gi

(ii) ⊓xG(x) (resp. ⊔xG(x)),Hcontains the result of replacing

in F that occurrence by G(y) where y does not occur in F

H

F

where E is the result of replacing in F some negative (resp.

Rule B1: , positive) surface occurrence of G1⊓G2 (resp. G1⊔G2) by Gi

for some i{1,2}

E

F

where E is the result of replacing in F some negative (resp.

Rule B2: , positive) surface occurrence of ⊓xG(x) (resp. ⊔xG(x)) by G(t)

for some term t that has no bound occurrence in F.

E

F

where E is the result of replacing in F two – one positive and Rule C: , one negative – surface occurrences of some n-ary general letter

by an n-ary elementary letter that does not occur in F

E

F

CL4⊦F means “F is provable in CL4”, and CL4⊬F means “F is not provable in CL4”

Propositional examplesCL4 ⊦ PPP

Justification: The elementarization of this formula is

p⊤p.

1. pPp

And the latter is a classical tautology. So,pPp is stable.And it

does not contain any choice operators. Hence, it follows from the

empty set {} of premises by Rule A (is an “axiom”)

Justification: From 1 by Rule C

2. PPP

CL4 ⊬ PPP

Reason: The elementarization of the above formula is

which is not a

tautology. So, PPP isinstable and hence cannot be derived by Rule A. And it does

not contain choice operators, so it cannot be derived by Rules B1 or B2, either. Thus,

it could only be derived by Rule C. Then the premise should be pPp or p pP

for some elementary atom p. In either case we deal with an instable formula which

does not contain choice operators and contains only one occurrence of a general atom.

Such a formula cannot be derived by any of the four rules of CL4.

⊤⊥⊥,

This formula, as any other classically valid elementary formula,

follows from {} by Rule A (is an “axiom”).

CL4 ⊦ ppp

CL4 ⊦ p⊔(QR) (p⊔Q)(p⊔R)

More propositional examples1. p pp

from {} by Rule A

2. p p(p⊔R)

from 1 by Rule B1

3. p (p⊔Q)(p⊔R)

from 2 by Rule B1

4. qr qr

from {} by Rule A

5. qR qR

from 4 by Rule C

6. QR QR

from 5 by Rule C

7. QR Q(p⊔R)

from 6 by Rule B1

8. QR (p⊔Q)(p⊔R)

from 7 by Rule B1

9. p⊔(QR) (p⊔Q)(p⊔R)

from {3,8} by Rule A

On the other hand, one can show that

CL4 ⊬ P⊔(QR) (P⊔Q)(P⊔R)

1. p(z) p(z)

from {} by Rule A

Examples with quantifiersCL4 ⊦⊓x⊔y(P(x)P(y))

2. P(z) P(z)

from 1 by Rule C

3.⊔y(P(z)P(y))

from 2 by Rule B2

4. ⊓x⊔y(P(x)P(y))

from {3} by Rule A

Indeed, this instable formula cannot be the

conclusion of any rule but B2. If it is derived by this

rule, the premise should be

CL4 ⊬⊔y⊓x(P(x)P(y))

⊓x(P(x)P(t))

for some term t different from x. In turn, ⊓x(P(x)P(t)) could only be derived by

Rule A where, for some variable z different from t, P(z)P(t) is a (the) premise. The

latter is an instable formula and does not contain choice operators, so the only rule by

which it can be derived is C, where the premise is p(z)p(t) for some elementary letter

p. Now we deal with a classically non-valid and hence instable elementary formula, and

it cannot be derived by any of the four rules of CL4.

1. yx(p(x)p(y)) from {} by Rule A

(see Slide 5.16)

CL4 ⊦yx(P(x)P(y))

2. yx(P(x)P(y)) from 1 by Rule C

in contrast to the previous example.

Show that:

Propositional exercises(1) CL4 ⊦PP

(2) CL4 ⊬P⊔P

(3) CL4 ⊦ P⊔PP

(4) CL4 ⊬ PPP

(5) CL4 ⊦ PQP⊓Q

(6) CL4 ⊬ P⊓QPQ

(7) CL4 ⊦(P⊔Q)(P⊔R) P⊔(QR) [Hint: see Slide 9.4]

(8) CL4 ⊬ P⊔(QR) (P⊔Q)(P⊔R)

(9) CL4 ⊦(PP)(PP) (PP)(PP) [Hint: see Slide 8.6]

(10) CL4 ⊦(((P(R⊓S))⊓((Q(R⊓S)))⊓(((P⊓Q)R)⊓((P⊓Q)S)) (P⊓Q)(R⊓S)

Below CL4⊦EF means “Both CL4⊦EF and CL4⊦FE”. Show that:

Exercises with quantifiers(1) CL4 ⊦xP(x)⊓xP(x)

(2) CL4 ⊬⊓xP(x)xP(x)

(3) CL4 ⊦x(P(x)Q(x)) xP(x)xQ(x)

(4) CL4 ⊬⊓x(P(x)Q(x)) ⊓xP(x)⊓xQ(x)

(5) CL4 ⊦⊓xyP(x,y) y⊓xP(x,y) [Similarly for ⊔ instead of ⊓,

and/or instead of ]

(6) CL4 ⊦xP(x)⊓xQ(x) x(P(x)⊓Q(x))[Similarly for ⊔instead of ⊓,

and/or instead of ]

(7)CL4 ⊦ ⊓x⊔y(q(x)p(y))(⊓x(p(x)⊔p(x)) ⊓x(q(x)⊔q(x)))

[See Slide 9.11]

(8) CL4 ⊬(⊓x(p(x)⊔p(x))⊓x(q(x)⊔q(x))) ⊓x⊔y(q(x)p(y))

[See Slide 9.12]

(9)CL4 ⊦ ⊓x((P(x)⊓xQ(x)) ⊓ (⊓xP(x)Q(x)))⊓xP(x) ⊓xQ(x)

Theorem 16.1. For any CL4-formula F, we have

CL4 ⊦ F iff F is valid iff F is uniformly valid.

Furthermore:

Uniform-constructive soundness: There is an effective

procedure that takes any CL4-proof of any formula F and constructs

a uniform solution for F.

Strong completeness: If CL4⊬F, then F* is not computable for

some interpretation * which interprets every elementary letter as a

(,,)- combination of 1-predicates, and interprets every general

letter as a (⊓,⊔)-combination of such combinations.

The soundness and completeness of CL4Here “1-predicate” means a predicate of the form yp(y,x1,...,xn),

where p(y,x1,...,xn) is a decidable predicate.

Fact 16.2.CL4 is a conservative extension of classical first-order logic. That is

in the sense that an elementary formula (formula of classical logic) is provable

in CL4 if and only if it is valid in the classical sense (i.e. provable in G3).

The decidability of the ,-free fragment of CL4The above fact is established by the simple observation that, when F is elementary,

it can only be derived from {} by Rule A, which, in turn, means that F is stable; but

stability for an elementary formula simply means its validity in the classical sense.

Thus, Gödel’s completeness theorem for classical logic is a relatively simple special

case of our Theorem 16.1. Specifically, it is Theorem 16.1 restricted to only the

elementary fragment of CL4.

Classical first-order logic, as we know, is undecidable, which implies that CL4 is

not decidable, either. Yet, as it turns out, it is only the blind (rather than the choice)

quantifiers that make trouble:

Theorem 16.3. For the formulas that do not contain blind quantifiers (but may

contain choice quantifiers), the question on provability in CL4 and hence the

question on validity or uniform validity is decidable.

Thus, not only does CL4 provide a systematic, never-failing effective tool for telling

what is valid, but --- in the case of ,-free formulas --- also for telling what is not valid.

Both CL5 and CL4 are new and proof-theoretically rather unusual logics, created

within the program of finding sound and complete axiomatizations for various fragments

of computability logic.

Other axiomatizationsHere we very briefly survey two other, well known logics that existed long before

computability logic was introduced.

One is affine logic, a variation of Girard’s linear logic. Affine logic, in its full

first-order language, turns out to be sound but incomplete with respect to the semantics

of computability logic (validity or uniform validity), with the additives understood as

choice operators, the multiplicatives as parallel connectives, and exponentials as either

sort of our recurrence operators.

The other one is intuitionistic logic. It, in the full first-order language, has been shown

to be sound with respect to the semantics of computability logic when the intuitionistic

“absurd” is understood as ⊥, implication as , and the other intuitionistic operators

(including quantifiers) as choice operators. At the same time, just as in the case of affine

logic, there is no completeness. Yet, the positive (⊥-free) propositional fragment of

intuitionistic logic turns out to be complete. Furthermore, completeness extends to

the full propositional fragment if we understand “absurd” not as ⊥ but rather as $,

a “problem of universal strength”. The implicative fragment of intuitionistic logic

also turns out to be sound and complete with implication seen as .

The soundness theorems for both affine and intuitionistic logics hold in the strong,

uniform-constructive sense.

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