Yin-Yang of Programs and Three-Valued Logics

Jerzy TyszkiewiczInstytut Informatyki UW

Programs

Three-valued logics

Three-Valued Logics for Programs

• Explicito SQLo Datalog with negation

• Implicito Every regular programming languageo Conditional expressions

SQL, Heyting-Kleene-Łukasiewicz,Goodman-Nguyen-Walker

OR 0 1 NULL

0 0 1 NULL

1 1 1 1

NULL NULL 1 NULL

AND 0 1 NULL

0 0 0 0

1 0 1 NULL

NULL 0 NULL NULL

Sobociński, Schay-Adams-Calabrese

OR 0 1 $

0 0 1 0

1 1 1 1

$ 0 1 $

AND 0 1 $

0 0 0 0

1 0 1 1

$ 0 1 $

Strict evaluation, Bochvar

OR 0 1 $

0 0 1 $

1 1 1 $

$ $ $ $

AND 0 1 $

0 0 0 $

1 0 1 $

$ $ $ $

Lazy evaluation

OR 0 1 $

0 0 1 $

1 1 1 1

$ $ $ $

AND 0 1 $

0 0 0 0

1 0 1 $

$ $ $ $

Conditionals

• An object can red r, yellow y or blue b• Pr(r)=Pr(y)=Pr(b)=1/3

• What is the probability, that it is b, given that if it is not r, then it is b?

• Pr(b |(b |¬ r))• Not a well-defined expression

Conditionals

Conditionals G-N-W

COND 0 1 $

0 $ 0 0

1 $ 1 $

$ $ $ $

Conditional S-A-C

COND 0 1 $

0 $ 0 0

1 $ 1 1

$ $ $ $

Datalog

q(X, Y) ← edge(X, Y)q(X, Z) ← edge(X, Y), q(Y, Z)

Computation stabilises eventually, if started from empty intentional relation q.

The resulting q is two-valued.

Datalog with negation

q(X, Y) ← r(X, Y, Z), ¬q(Y, X)q(X, Y) ← s(X, Y, Z), r(X, Y, Z), q(X, Y), ¬q(Y, Y)

Relation q need not stabilise during computation.

Datalog with negation

• Some tuples are almost always in q• Some tuples are almost always not in q• Some tuples continue changing status forever

Datalog with negation

The resulting relation is three-valued.

•Some tuples are in the computed q•Some tuples are not in the computed q•Some tuples are undecided

Mutual definability of conditional algebras

•Are the connectives of SAC and GNW algebras definable in each other?

Mutual definability of conditional algebras

•The main question is equivalent to defibability of certain three-valued functions by composition of other functions.

•We use programs to enumerate all possible two-argument compositions of the basic connectives.

Mutual definability of conditional algebras

Theorem 1. An n-argument connective f : {0,1,$} →{0,1,$} is definable in SAC iff f($,...,$) =$.There are 6561 of binary ones.In particular, all connectives of GNW aredefinable in SAC.

Mutual definability of conditional algebras

I could pretend not using computer in the proof of Theorem 1.

Mutual definability of conditional algebras

Theorem 2. The conjunction and disjunction of SACare not definable in GNW.COND of SAC is definable in GNW.There are only 849 two-argument connectives definable in GNW.

Mutual definability of conditional algebras

At present I could not give a proof of Theorem 2 without reference to the computer.

Internal structure of conditional algebras

Theorem 3. SAC an GNW have Sheffer-like definable connectives: ones that are definable and suffice alone to define all other connectives.

My co-authors

Piotr Chrząstowski-WachtelAchim HoffmannArthur Ramer

Thank

you!