Negation and choice operations

1
Negation and choice operations
Episode 7

• About the operations studied in computability
  logic
• Negation
• The double negation principle
• Choice conjunction and disjunction
• Choice quantifiers
• DeMorgans laws for choice operations
• The constructive character of choice operations
• Failure of the principle of the excluded middle
  for choice disjunction

0
2
About the operations studied in computability
logic
7.1
Logical operators in computability logic
stand for operations on games. There is an
open-ended pool of operations of potential
interest, and which of those to study may depend
on particular needs and taste. Yet, there is
a core collection of the most basic and natural
game operations, to the definitions of which the
present section is devoted the propositional ---
more precisely, constant-game --- connectives
?,
?, ?, ?, ?, , , , (together with the
defined impication-style connectives ?, ,
) and the quantifiers
?, ?, ?, ?, ?, ?.
Among these we see all operators of classical
logic, and our choice of the classical notation
for them is no accident. Classical logic is
nothing but the elementary fragment of
computability logic (the fragment that deals only
with predicates elementary games). And each of
the classically-shaped operations, when
restricted to elementary games, turns out to be
can be virtually the same as the corresponding
operator of classical logic. For instance, if A
and B are elementary games, then so is A?B, and
the latter is exactly the classical conjunction
of A and B understood as an (elementary) game.
3
Negation ?
7.2.a
For a run ?, by ?? we mean the negative
image of ? (green and red interchanged).
Definition 7.1. The game ?A is defined by

• Lre?A ? ???LreA
• Wne?A ??? ? iff WneA ???? ?

A
?A
0
1
0
1
0
1
0
1
0
1
0
1
4
Negation ?
7.2.b
For a run ?, by ?? we mean the negative
image of ? (green and red interchanged).
Definition 7.1. The game ?A is defined by

• Lre?A ? ???LreA
• Wne?A ??? ? iff WneA ???? ?

? Chess
5
Negation ?
7.2.b
For a run ?, by ?? we mean the negative
image of ? (green and red interchanged).
Definition 7.1. The game ?A is defined by

• Lre?A ? ???LreA
• Wne?A ??? ? iff WneA ???? ?

? Chess
Chess
6
Negation ?
7.2.b
For a run ?, by ?? we mean the negative
image of ? (green and red interchanged).
Definition 7.1. The game ?A is defined by

• Lre?A ? ???LreA
• Wne?A ??? ? iff WneA ???? ?

Chess
? Chess
Chess
7
Negation ?
7.2.b
For a run ?, by ?? we mean the negative
image of ? (green and red interchanged).
Definition 7.1. The game ?A is defined by

• Lre?A ? ???LreA
• Wne?A ??? ? iff WneA ???? ?

Chess
? Chess
Chess
8
Negation ?
7.2.b
For a run ?, by ?? we mean the negative
image of ? (green and red interchanged).
Definition 7.1. The game ?A is defined by

• Lre?A ? ???LreA
• Wne?A ??? ? iff WneA ???? ?

Chess
? Chess
Chess
9
The double negation principle holds
7.3
? ? A A
A
10
The double negation principle holds
7.3
? ? A A
A
11
The double negation principle holds
7.3
? ? A A
A
12
The double negation principle holds
7.3
? ? A A
A
13
The double negation principle holds
7.3
? ? A A
A
14
Choice conjunction ? and disjunction ?
7.4
Choice conjunction ?
A0 ? A1
A0
A1
Choice disjunction ?
A0 ? A1
A0
A1
15
Choice universal quantifier ? and existential
quantifier ?
7.5
Choice universal quantifier ?
?xA(x)
A(0) ? A(1) ? A(2) ? ...
0
1
2
. . .
A(0)
A(1)
A(2)
Choice existential quantifier ?
?xA(x)
A(0) ? A(1) ? A(2) ? ...
0
1
2
. . .
A(0)
A(1)
A(2)
16
Representing the problem of computing a function
7.6
A
0
1
2
...
...
...
...
1
2
0
3
1
2
0
3
1
2
0
3
?y?z (zy1)
This game can be written as
One of the legal runs, won by the machine, is
?1,2?
17
Representing the problem of computing a function
7.6
A
0
1
2
...
...
...
...
1
2
0
3
1
2
0
3
1
2
0
3
?y?z (zy1)
This game can be written as
One of the legal runs, won by the machine, is
?1,2?
18
Representing the problem of computing a function
7.6
?1?A
...
1
2
0
3
?y?z (zy1)
This game can be written as
One of the legal runs, won by the machine, is
?1,2?
19
Representing the problem of computing a function
7.6
?1?A
...
1
2
0
3
?y?z (zy1)
This game can be written as
One of the legal runs, won by the machine, is
?1,2?
20
Representing the problem of computing a function
7.6
?1,2?A
?y?z (zy1)
This game can be written as
One of the legal runs, won by the machine, is
?1,2?
Generally, the problem of computing a function f
can be written as
?y?z (zf(y))
21
Another example
7.7
?
0
1
...
?
?
...
...
1
2
0
1
2
0
00x
10x
20x
01x
11x
21x
?y?z (zyx)
This game can be written as
22
Representing the problem of deciding a predicate
7.8
...
2
1
0
3
4
5
0
1
0
1
0
1
0
1
0
1
0
1
This game is about deciding what predicate?
Even(x)
?x(?Even(x) ? Even(x))
How can it be written?
Generally, the problem of deciding a predicate
p(x) can be written as
?x(?p(x) ? p(x))
23
Representing the problem of deciding a predicate
7.8
Position
?x(?Even(x) ? Even(x))
? ?
...
2
1
0
3
4
5
0
1
0
1
0
1
0
1
0
1
0
1
24
Representing the problem of deciding a predicate
7.8
Position
?x(?Even(x) ? Even(x))
? ?
...
2
1
0
3
4
5
0
1
0
1
0
1
0
1
0
1
0
1
Making move 4 means asking the machine the
question Is 4 even?
25
Representing the problem of deciding a predicate
7.8
Position
?x(?Even(x) ? Even(x))
? ?
?4?
?Even(4)?Even(4)
0
1
Making move 4 means asking the machine the
question Is 4 even?
This move brings the game down to ?Even(4) ?
Even(4), in the sense that
?4? ?x(?Even(x)?Even(x)) ?Even(4)?Even(4)
26
Representing the problem of deciding a predicate
7.8
Position
?x(?Even(x) ? Even(x))
? ?
?4?
?Even(4)?Even(4)
0
1
Making move 4 means asking the machine the
question Is 4 even?
This move brings the game down to ?Even(4) ?
Even(4), in the sense that
?4? ?x(?Even(x)?Even(x)) ?Even(4)?Even(4)
Making move 1 in this position means answering
Yes.
27
Representing the problem of deciding a predicate
7.8
Position
?x(?Even(x) ? Even(x))
? ?
?4?
?Even(4)?Even(4)
Even(4)
?4,1?
Making move 4 means asking the machine the
question Is 4 even?
This move brings the game down to ?Even(4) ?
Even(4), in the sense that
?4? ?x(?Even(x)?Even(x)) ?Even(4)?Even(4)
Making move 1 in this position means answering
Yes.
This move brings the game down to Even(4), in the
sense that
?1? ?Even(4)?Even(4) Even(4) ( ?)
The play hits ?, so the machine is the winner.
28
Representing the problem of deciding a predicate
7.8
Position
?x(?Even(x) ? Even(x))
? ?
?4?
?Even(4)?Even(4)
?Even(4)
?4,0?
Making move 4 means asking the machine the
question Is 4 even?
This move brings the game down to ?Even(4) ?
Even(4), in the sense that
?4? ?x(?Even(x)?Even(x)) ?Even(4)?Even(4)
Making move 1 in this position means answering
Yes.
This move brings the game down to Even(4), in the
sense that
?1? ?Even(4)?Even(4) Even(4) ( ?)
The machine would have lost if it had made move 0
(answered No) instead.
29
Formal definitions
7.9
Definition 7.2. (a) The game A0?A1 is defined
by (b) The game A0?A1 is defined by (c)
The game ?xA(x) is defined by (d) The game
?xA(x) is defined by

• LreA0?A1 ?? ? ?i, ?? i?0,1,
  ??LreAi.
• WneA0?A1?? ? WneA0?A1 ?i, ?? WneAi???.

• LreA0?A1 ?? ? ?i, ?? i?0,1,
  ??LreAi.
• WneA0?A1?? ? WneA0?A1 ?i, ?? WneAi???.

• Lre?xA(x) ?? ? ?c, ?? c?0,1,2,...,
  ??LreA(c).
• Wne?xA(x) ?? ? Wne?xA(x) ?c, ?? WneA(c)
  ???.

• Lre?xA(x) ?? ? ?c, ?? c?0,1,2,...,
  ??LreA(c).
• Wne?xA(x) ?? ? Wne?xA(x) ?c, ?? WneA(c)
  ???.

30
DeMorgans laws hold
7.10
?(A ? B) ?A ? ?B
A ? B ?(?A ? ?B)
?(A ? B) ?A ? ?B
A ? B ?(?A ? ?B)
??xA ?x?A
?xA ??x?A
??xA ?x?A
?xA ??x?A
?(A ? B)
?A ? ?B
B ? A
31
The constructive character of choice operations
7.11
As noted in Episode 1, computability logic
revises traditional logic through replacing
truth by computability. And computability of a
problem means existence of a machine (
algorithmic strategy) that wins the corresponding
game.
Correspondingly, while classical logic
defines validity as being always true,
computability logic understands it as being
always computable.
The operators of classical logic are not
constructive. Consider, for example,
?x?y(yf(x)).
It is true in the classical sense as long as f is
a (total) function. Yet its truth has little (if
any) practical import, as ?y merely signifies
existence of y, without implying that such a y
can actually be found. And, indeed, if f is an
incomputable function such as Kolmogorov
complexity, there is no method for finding y.
On the other hand, the choice operations of
computability logic are constructive.
Computability (truth) of
?x?y(yf(x))
means more than just existence of y it means the
possibility to actually find (compute, construct)
a corresponding y for every x.
32
Failure of the principle of the excluded middle
7.12
Similarly, let H(x) be the predicate Turing
machine x halts on input 0. Consider
?x(?H(x)?H(x)).
It is true in classical logic, yet not in a
constructive sense. Its truth means that, for
every x, either ?H(x) or H(x) is true, but it
does not imply existence of an actual way to
tell which of these two is true after all.
And such a way does not really exist, as the
halting problem is undecidable. This means that
?x(?H(x)?H(x))
is not computable.
Generally, the principle of the excluded
middle A OR NOT A, validated by classical
logic and causing the indignation of the
constructivistically-minded, is not valid in
computability logic with OR understood as choice
disjunction. The following is an example of
a constant game of the form A??A with no
algorithmic solution (why?)
?x(?H(x)?H(x)) ? ??x(?H(x)?H(x))
33
Chess ? ?Chess a really hard game
7.13
?
To win this game means to choose between
playing white or black, and then win the chosen
game. No human and no modern computer can
handle this task with a full guarantee of
success. Most probably, no future machines can
succeed either, even though Chess is a finite
game and, theoretically, there should be an
algorithmic winning strategy here (excluding the
possibility of draw outcomes, of course they can
be considered wins for the black player, for
example).