1. Properties of Relational Logic
Computational Logic Lecture 8
Properties of Relational Logic
Michael Genesereth Autumn 2011

2. Programme Expressiveness What we can say in First-Order Logic
And what we cannot
Semidecidability and Decidability
Using Godel’s Completeness Theorem
Complexity of Arithmetic
Godel’s Incompleteness Theorem

3. Interpretation: Signature Structure
Structures
A structure is a vector consisting of a universe of discourse and values for the items in the signature of a language (when the signature is ordered).
Interpretation: Signature Structure
Note that there is no additional information in a structure. It is just a different (but useful) way of thinking about an interpretation.

4. Example Signature: a, b, f, r Interpretation: |i| = {1, 2} i(a) = 1
i(b) = 2
i(f) = {12, 21}
i(r) = {1,2, 1,1, 2,2}
Structure:
{1, 2}, 1, 2, {12, 21}, {1,2, 1,1, 2,2}

5. Definability
One of the roles of logic is to define classes of structures, distinguishing those that are in the class from those that are not.
Example - Open Partial Orders:
¬r(x,x)
r(x,y)  ¬r(y,x)
r(x,y)  r(y,z)  r(x,z)
Examples: Non-Examples:
{{a,b,c}, {a,b,b,c,a,c} {{a,b,c}, {a,a,a,b,a,c}
{{a,b,c}, {a,b,a,c} {{a,b,c}, {a,b,b,a}
{{a,b,c}, {} {{a,b,c}, {a,b,b,c}

6. Example Definition of Open Partial Orders: ¬r(x,x) r(x,y)  ¬r(y,x)
r(x,y)  r(y,z)  r(x,z)
Examples:
{{a,b,c}, {a,b,b,c,a,c}
{{a,b,c}, {a,b,a,c}
{{a,b,c}, {}
Non-Examples:
{{a,b,c}, {a,a,a,b,a,c}
{{a,b,c}, {a,b,b,a}
{{a,b,c}, {a,b,b,c}

7. Elementary Equivalence
NB: There are pairs of structures that cannot be distinguished from each other in Relational Logic.
Two structures are elementarily equivalent if and only if they satisfy the same set of sentences for all signatures.

8. Examples {1,2}, 1, 2, {1,2, 2,1} {1,2}, 2, 1, {1,2, 2,1}
{3,4}, 3, 4, {3,4, 4,3}
{,}, , , {,, ,}
Note, however, that these structures are isomorphic - they have the same structure.
Q, <
R, <

9. Transitivity Theorem
It is not possible in first-order logic to define transitive closure in first-order logic.
More precisely, it is not possible characterize the set of structures U,p,r consisting of an arbitrary universe U, an arbitrary binary relation p, and the transitive closure r of that relation.
NB: This is similar to the open partial orders problem earlier except that (1) we do not care about reflexivity and antisymmetry and (2) we care about the relationship between two relations (p and r).

10. Counterargument and Rebuttal
Really? What about this definition?
r(x,z)  p(x,z)  y.(r(x,y)  r(y,z))
Counterexample 1:
Counterexample 2:
In other words, there is a point between every pair of points between 3 and 4. Require infinite universe. 1 3 2 1 2 3 4

11. x.z.(r(x,z)  p(x,z)  y.(r(x,y)  r(y,z)))
Size of the Universe
Models with universes of at least size 2:
x.y.(p(x)  ¬p(y))
x.y.(xy)
Models with universes of at most size 2:
x.y.z.(zx  zy)
Models with infinite universes:
x.z.(r(x,z)  p(x,z)  y.(r(x,y)  r(y,z)))
x.y.(p(x,y)  ¬r(x,z))

12. Lowenheim Skolem Tarski Theorem
If there is a model of a set of first-order sentences of any infinite cardinality, then there is a model of every infinite cardinality.

13. Programme Expressiveness What we can say in First-Order Logic
And what we cannot
Semidecidability and Decidability
Using Godel’s Completeness Theorem
Complexity of Arithmetic
Godel’s Incompleteness Theorem

14. Logical Entailment
A set of premises logically entails a conclusion if and only if every interpretation that satisfies the premises also satisfies the conclusion.

15. Formal Proofs
A formal proof of  from  is a sequence of sentences terminating in  in which each item is either:
1. a premise (a member of )
2. an instance of an axiom schema
3. the result of applying a rule of inference to earlier items in the sequence.
A sentence  is provable from a set of sentences  if and only if there is a finite formal proof of  from  using only Modus Ponens, Universal Generalization, and the Mendelson axiom schemata.

16. Soundness and Completeness
Soundness Theorem: If  is provable from , then  logically entails .
Completeness Theorem (Godel): If  logically entails , then  is provable from .

17. Decidability
A class of questions is decidable if and only if there is a procedure such that, when given as input any question in the class, the procedure halts and says yes if the answer is positive and no if the answer is negative.
Example: For any natural number n, determining whether n is prime.

18. Semidecidability
A class of questions is semidecidable if and only if there is a procedure that halts and says yes if the answer is positive.
Obvious Fact: If a class of questions is decidable, it is semidecidable.

19. Semidecidability of Logical Entailment

20. Decidability Not Proved
Note that we have not shown that logical entailment for Relational Logic is decidable.
The procedure may not halt.
p(x)  p(f(x))
p(f(f(a)))
p(f(b))?
We cannot just run procedure on negated sentence because that may not be logically entailed either!
p(f(b))?

21. Undecidability of Logical Entailment
Metatheorem: Logical Entailment for Relational Logic is not decidable.
Proof: Suppose there is a machine p that decides the question of logical entailment. Its inputs are  and .
We can encode the behavior of this machine and its inputs as sentences and ask whether the machine halts as a conclusion.
What happens if we give this description and question to p? It says yes.
Yes  p f No

22. Undecidability (continued)
It is possible to construct a larger machine p’ that enters an infinite loop if p says yes and halts if p says no.
We can also encode a description of this machine as a set of sentences and ask whether the machine halts as a conclusion.
What happens if we give this description and question to p’? If p says yes, then p’ runs forever, contradicting the hypothesis that p computes correctly. If p says no, then p’ halts, once again leading to contradiction. QED No p
Halts 

23. Closure
The closure S* of a set S of sentences is the set of all sentences logically entailed by S.
S*={ | S|=}
Set of Sentences: Closure:
p(a) p(a)
p(x)p(f(x)) p(f(a))
p(f(f(a)))
p(a)  p(f(a))
p(x)p(f(x)) …

24. Theories
A theory is a set of sentences closed under logical entailment, i.e. T is a theory if and only if T*=T.

25. Finite Axiomatizability
A theory T is finitely axiomatizable if and only if there is a finite set  of sentences such that T=*.

26. Theory Completeness
A theory T is complete if and only, for all , either T or T.
Note: Not every theory is complete. Consider the theory consisting of all consequences of p(a,b). Does this include p(b,a)? Does it include p(b,a)?
Note: There is one and only inconsistent theory, viz. the set of all sentences in the language.

27. Relationships on Theories
Decidable
Semidecidable Finitely Axiomatizable

28. Programme Expressiveness What we can say in First-Order Logic
And what we cannot
Semidecidability and Decidability
Using Godel’s Completeness Theorem
Complexity of Arithmetic
Godel’s Incompleteness Theorem

29. Arithmetization of Logical Entailment
The theory of arithmetic is the set of all sentences true of the natural numbers, 0, 1, +, *, and <.
Fact: It is possible to assign numbers to sentences such that
(1) Every sentence  is assigned a unique number n.
(2) The question of logical entailment |= can be expressed as a numerical condition r(n,n).
Conclusion: The theory of arithmetic is not decidable.

30. Incompleteness Theorem
Metatheorem (Godel): If  is a finite subset of the theory of arithmetic, then * is not complete.
Variant: Arithmetic is not finitely axiomatizable.
Proof: If there were a finite axiomatization, then the theory would be decidable. However, arithmetic is not decidable. Therefore, there is no finite axiomatization.

31. Summary Logical Entailment for Relational Logic is semidecidable.
Logical Entailment for Relational Logic is not decidable.
Arithmetic is not finitely axiomatizable in Relational Logic.