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Formalizing Alpha:
Soundness and Completeness
Bram van Heuveln
Dept. of Cognitive Science
RPI
Overview
• This presentation discusses:
–
–
–
–
Formal Syntax of Alpha
Formal Semantics of Alpha
Soundness of Alpha
Completeness of Alpha
Formalizing Alpha
Formal Syntax
Alpha Graphs are recursively defined as follows:
1.
2.
is an Alpha Graph.
P
is an Alpha Graph, with P an atomic statement.
3. is an Alpha Graph, with and Alpha Graphs.
4.
is an Alpha Graph, with an Alpha Graph.
Formalizing Alpha
Formal Semantics
A truth-assignment h is a function that assigns either True
or False to any Alpha Graph in accordance with:
1. h(
2. h(
) = True.
P
) = True or False.
3. h( ) = True iff h( ) = True and h( ) = True.
4. h(
) = True iff is h( ) = False.
Formalizing Alpha
Consequence
• Let us use the following notation:
– h |= iff h() = True
– |= iff for any truth-assignment h: h |=
• Let us define:
– is a truth-functional consequence of iff for any
truth-assignment h: if h |= then h |= .
Formalizing Alpha
Soundness and Completeness
• Let us use the symbol TF to indicate truthfunctional consequence:
– TF iff is a truth-functional consequence of .
• Let us use the symbol EG to indicate truthfunctional provability in Alpha:
– EG iff there exists a formal proof from to .
• We will show two things:
– Alpha is truth-functionally sound:
• For any and : if EG then TF
– Alpha is truth-functionally complete:
• For any and : if TF then EG
Soundness
Proving Soundness
• A straightforward way to demonstrate the soundness of
some system S is to demonstrate that each of the inference
rules of S is sound.
• If the inference rules deal with subproofs, such a proof can
actually become rather technical, even if the soundness of
each inference rules is intuitive clear (see e.g. proof of
soundness of F in “Language, Proof, and Logic”).
• The good news is that Alpha does not deal with subproofs.
• However, the bad news is that, except for double cut, the
soundness of Alpha’s inference rules is not intuitive.
• To make the soundness of the inference rules more
intuitive, we use the notion of a recursive conditional
reading as well as 3 less commonly used inference rules.
Soundness
2 Rules of Inference and 1 Rule of Equivalence
Strengthening the
Antecedent
pr
(p q) r
Absorption Case 1
Weakening the
Consequent
p (q r)
pr
Absorption Case 2
p and q r
p and (q p) r
p and q r
p and q (r p)
Soundness
Recursive Conditional Reading
One can see graphs with multiple cuts inside each other as
expressing recursively conditioned conditionals. For example:
P
Q
R
S
This graph can be read as: ‘if P is true then Q is true, but if P is
true, then it is also true that if R is true then S true’. Hence the
conditional ‘if R then S’ is conditionally true. Thus, as more
and more conditions get added, the truth of more and more
statements can be asserted.
Soundness
Relative Recursive Conditional Reading I
0
1
2k-1
2k
Relative to any subgraph existing at an odd level 2k-1,
one can see the graph as a whole in the above manner (note:
if there is no cut next to , then one has to add an empty
double cut next to to make it work).
Soundness
Relative Recursive Conditional Reading II
0
1
2k-1
2k
The Recursive Conditional Reading (RCR) of a graph relative to
a subgraph existing at odd level 2k – 1 is defined as follows:
0 &
1 2 &
(1 3) 4
(1 2k-1 ) 2k
Soundness
Relative Recursive Conditional Reading III
0
1
2k-1
2k
Relative to any subgraph existing at an even level 2k, one
can see the graph as a whole in the above manner.
Soundness
Relative Recursive Conditional Reading IV
0
1
2k-1
2k
The Recursive Conditional Reading (RCR) of a graph relative to
a subgraph existing at even level 2k is defined as follows:
0 &
1 2 &
(1 3) 4
(1 2k-1) 2k
Soundness
Soundness of Insertion
0
1
2k-1
2k
0 &
1 2 &
(1 2k-1) 2k
? IN
0
1
2k-1
2k
pr
(p q) r
0 &
1 2 &
(1 2k-1 ) 2k
Insertion corresponds to Strengthening the Antecedent
Soundness
Soundness of Erasure
0
1
2k-1
2k
0 &
1 2 &
(1 2k-1) (2k )
? E
0
1
2k-1
2k
p (q r)
pr
0 &
1 2 &
(1 2k-1) 2k
Erasure corresponds to Weakening the Consequent
Soundness
Soundness of Iteration/Deiteration
? IT/DE
• To demonstrate the
soundness of iteration
and deiteration, it
suffices to demonstrate
that the subgraph that
has the original at level
0 is equivalent to the
corresponding
subgraph in the result
of the iteration or
deiteration.
Soundness
Soundness of Iteration/Deiteration Case 1
1
2k-1
2k
&
1 2 &
(1 2k-1) 2k
? IT/DE
1
2k-1
2k
p and q r
p and (q p) r
&
1 2 &
(1 2k-1 ) 2k
Iteration/Deiteration (Case 1) corresponds to Absorption (Case 1)
Soundness
Soundness of Iteration/Deiteration Case 2
1
2k-1
2k
&
1 2 &
(1 2k-1) 2k
? IT/DE
1
2k-1
2k
p and q r
p and q (r p)
&
1 2 &
(1 2k-1) (2k )
Iteration/Deiteration (Case 2) corresponds to Absorption (Case 2)
Completeness
Proving Completeness
• On the next slides, we will provide a direct proof
of the completeness of Alpha that very much
follows the strategy used in “Language, Proof, and
Logic” to prove that system F is complete.
Completeness
Alpha Consistency and Alpha Completeness
A graph is Alpha inconsistent iff
EG
A graph is Alpha complete iff for any graph :
EG or EG
Completeness
Either-Or Lemma
For any graph : if is Alpha consistent and
Alpha complete, then for any graph :
EG or EG
but not both EG and EG
Proof: Suppose is Alpha consistent and Alpha complete. Then:
By Alpha completeness: EG or EG
Now suppose EG and EG . Then:
EG
i.e. is Alpha inconsistent.
Contradiction, so not both EG and EG
Completeness
Deduction Theorem
For any graphs and :
EG
iff
EG
Proof:
‘if’:
EG
‘only if’:
EG
Completeness
Transposition Lemma
For any graphs and :
EG
iff
EG
Proof:
‘only if’:
So:
‘if’:
Suppose EG . Then (Ded. Thm.): EG
Suppose
So:
EG
EG . Then (‘only if’):
EG
EG
Completeness
Expansion Theorem
For any graph : if is Alpha consistent, then there exists a
’ = for some such that ’ is Alpha consistent and
Alpha complete.
Proof: Obtain ’ from the following routine:
As = Set of all atomic statements
While As :
Select and remove some A from As
if EG A or EG A
’ =
A otherwise
{
= ’
Next two slides: ’ is Alpha consistent and Alpha complete.
Completeness
Expansion Theorem (Alpha Consistency of ’)
To prove: ’ as obtained by the routine is Alpha consistent.
Proof: By induction we’ll show that at any point ’ is Alpha consistent
Base: is Alpha consistent
Step: Suppose is Alpha consistent, and suppose that adding A As
to makes ’ = A Alpha inconsistent, i.e: A EG
Then:
A
A EG
So (Transposition Theorem): EG A
Hence (Ded. Thm): EG A
A
So, A would not be added to .
Contradiction, so adding A keeps Alpha consistent.
Completeness
Expansion Theorem (Alpha Completeness of ’)
To prove: ’ as obtained by the routine is Alpha complete.
Proof: By induction on the composition of any statement .
Base: For every atomic statement A, ’ EG A or ’ EG A
Step: Case 1: = 1 2
By inductive assumption: ’ EG i or ’ EG i
If ’ EG 1 and ’ EG 2 then ’ ’ ’ EG 1 2
If ’ EG 1 or ’ EG 2 then ’ EG 1 2 by IN
Case 2: = ’
By inductive assumption: ’ EG ’ or ’ EG ’
Hence, ’ EG ’ ’ or ’ EG
Completeness
Consistency Lemma
For any graph : if is Alpha consistent and Alpha complete,
then is consistent.
Proof: Suppose is Alpha consistent and Alpha complete.
Define truth-value assignment h as follows:
For any atomic statement A: h(A) = True iff EG A
(the Either-Or Lemma guarantees that h is well-defined).
On the next slide we’ll show that for any graph :
h() = True iff EG
Finally, because EG , h() = True. Hence, is
consistent.
Completeness
Consistency Lemma (Continued)
To prove: for any graph : h() = True iff EG
Proof: By induction on the composition of any statement .
Base: For every atomic statement A, h(A) = True iff EG A (def. h)
Step: Case 1: = 1 2
By inductive assumption: h(i) = True iff EG i
So, h(1 2) = True iff h(1) = True and h(1) = True iff
EG 1 and EG 2 iff (trivial) EG 1 2 iff EG
Case 2: = ’ By ind. assumption: h(’) = true iff EG ’
Hence, h() = True iff h(’) = false iff not EG ’ iff
(Either-Or Lemma) EG ’ iff EG
Completeness
The Central Theorem
For any graph :
if is Alpha consistent, then is consistent.
Proof:
Suppose is Alpha consistent.
By the Expansion Theorem there exists a ’ = for
some such that ’ is Alpha consistent and Alpha complete.
By the Consistency Lemma, this ’ is consistent.
So, is consistent.
Therefore, is consistent.
Completeness
Basic Completeness Theorem
For any : if TF then EG
Proof:
Suppose TF
Then
is inconsistent.
Hence (Central Theorem):
Therefore
is Alpha inconsistent.
EG
So (Transposition Theorem): EG
Completeness
Completeness Theorem
For any and : if TF then EG
Proof:
Suppose TF
Then TF
Hence (Basic Completeness Theorem): EG
Therefore (Deduction Theorem): EG
