1. CIS341 Artificial Intelligence
Weeks 8 – 11
Predicate calculus and tableau proofs
Natural language syntax and parsing
Search and problem solving
Parsing as proof, proof as a search problem
Coursework 2 proposed dates:
Given out Friday March 13, due in Friday April 3rd
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

2. Week 8: logic and inference revisited
Revision of propositional logic
Brief overview of proof techniques
Introduction to predicate calculus
Demonstration of tableau proofs on whiteboard
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

3. Automated reasoning and formal logic
Logic is concerned with validity of inference
A central concern of 20th-century mathematical logic was to develop systematic, mechanical techniques for proving logical consequence
The AI subfields of logic programming and automated reasoning have built on these efforts to develop systems to perform logical inferences.
Considerations of speed and efficiency put limitations on the kinds of inference that can be implemented.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

4. Overview of “classical” logic
Just as many people understand the word music to refer to European classical music composed between the 17th and 20th centuries, so the word logic is often shorthand for a variant called classical logic.
The remainder of these slides provide an overview of “classical” logic, comprising
Boolean or propositional logic
First-order predicate calculus (the words first-order are often omitted)
There are many non-classical logics but they do not come into this course.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

5. Some important terminology
A proposition is something that can be true or false
A proposition is consistent if it is possible for it to be true
A proposition is inconsistent or contradictory if it is not possible for it to be true
A proposition is a necessary truth or a tautology if it is not possible for it to be false.
Class exercise 1: think up examples in each of these categories.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

6. Boolean or propositional logic
Boolean logic is so-called because logical formulas are interpreted in terms of Boolean “truth values”
True or False
T or F
1 or 0
The semantics of Boolean logic tells us how to calculate the truth values of complex formulas as a function of the truth values of their constituent “literals” using truth tables
Simple propositions generally written P, Q etc.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

7. Syntax of propositional logic
Formulas of Boolean logic are made up of propositional letters P, Q, R etc and the symbols
& or  : conjunction, read as “and”
 : inclusive disjunction, read as “or”
 or ~ : negation, read as “not”
 or  : implication, read as “implies” or “if … then”
 or  : biconditional, read as “if and only if” or “iff”
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

8. CIS341 Artificial Intelligence - Goldsmiths Dept of Computing
Truth tables P Q
P  Q
True
False P P
True
False
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

9. CIS341 Artificial Intelligence - Goldsmiths Dept of Computing
Truth table for & P Q
P  Q
True
False
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

10. Truth table for implicationQ
P  Q
True
False
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

11. Inference involving implication
“If the solution is acid, the paper will turn red”.
True  True: the solution is acid, the paper turns red. The implication is verified.
True  False: the solution is acid, the paper does not turn red. The implication is falsified.
False  True: the solution is not acid, the paper still turns red.
False  False: the solution is not acid, the paper does not turn red.
Is the implication verified or falsified in cases 3 and 4?
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

12. Class/self-study exercise 2
Construct truth tables for:
P  Q
(P & Q)
(P  Q)
(P &  Q)
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

13. CIS341 Artificial Intelligence - Goldsmiths Dept of Computing
Validity of inference
An inference is valid if it is not possible for the premises to be true and the conclusion false.
Valid:
P, P Q / Q
(P  Q) / P
Invalid:
(P & Q) / P
Q, P Q / P
Validity in propositional logic can be checked using truth tables.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

14. CIS341 Artificial Intelligence - Goldsmiths Dept of Computing
Some proof techniques
Modus ponens
P  Q; P / Q
Modus tollens
P  Q; Q / P
Provide a “short cut” for calculating inferences without constructing truth tables.
A proof theory consists of a set of such rules
There are proof theories for propositional logic which allow all valid inferences to be proved.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

15. CIS341 Artificial Intelligence - Goldsmiths Dept of Computing
Resolution rule
Equivalent of modus ponens
P  Q; P / Q
Basis of logic programming
Requires disjunctions with at most one positive literal, for example:
(P & Q)  R =
 (P & Q)  R =
P   Q  R
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

16. CIS341 Artificial Intelligence - Goldsmiths Dept of Computing
Tableau proof
A “graphic” proof technique based on tree diagrams called semantic tableaux
Will be covered in later slides
Good tutorial in Wilfred Hodges, Logic
Useful online tutorial at designed for Oxford philosophy students
Key point: there are many proof techniques which are all equally sound and complete, i.e. they will prove all and only valid arguments.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

17. Shortcomings of propositional logic
A simple proof:
Fred is ill.
If Fred is ill, he should not drive.
So, Fred should not drive.
In symbolic form:
P = “Fred is ill''
Q = “Fred should not drive”
Modus ponens:
P  Q, P / Q
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

18. Shortcomings of propositional logic (cont.)
So far, so good. But what if Charlie is ill?
Do we construct a separate rule for every individual?
P2 = ``Charlie is ill''
Q2 = ``Charlie should not drive”
P2  Q2
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

19. From propositional logic to predicate calculus
Solution is to decompose statements into predicates and arguments:
P(x) = “x is ill''
Q(x) = “x should not drive”
P(x)  Q(x)
Proof will look more like this:
P(fred)
P(fred)  Q(fred)
 Q(fred)
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

20. From propositional logic to predicate calculus (cont.)
Boolean logic is the logic of propositions.
Predicate calculus introduces predicates and arguments.
Arguments can be constant terms or variables.
Predicates have to be constant terms in the first-order predicate calculus.
There are higher-order logics with predicate variables but we will not be concerned with them.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

21. Quantifiers in predicate calculus I
Implicational rules are generally written like this:
x(P(x)  Q(x))
The upside–down A is called the universal quantifier: this means that the implication is true for all values of x (i.e. for any individual).
Read as “for all x, P(x) implies Q(x)”
Can also be read as “all Ps are Q” or “all Ps Q”:
x(fish(x)  swim(x))
x(cow(x)  mammal(x))
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

22. Quantifiers in predicate calculus II
The existential quantifier  means that statement that follows it is true of some individual, though we may not know which.
x(Px & Qx)
Read as:
“there exists an x such that P(x) and Q(x)”
“some P is a Q”
“some Ps Q”
x(mammal(x) & lays-eggs(x))
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

23. Quantifiers in predicate calculus III
You may have noticed that  generally goes with the conjunction “&” and  goes with the implication “”
This is very important. What would the following mean? Are they true?
x(fish(x)  sings(x))
x(cow(x) & mammal(x))
This is probably what beginning logic students most often get wrong.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

24. Quantifiers in predicate calculus IV
Multiple quantifiers can be combined in the same formula:
x(student(x)  y (tutor(y,x)))
Every student has a tutor
x(professor(x) & y(course(y) & teach(x,y)))
There is a professor who does not teach any courses
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

25. Duality of first-order quantifiers
The quantifiers  and  can be defined in terms of each other, using Boolean equivalences:
x(P(x)  Q(x)) =
x(P(x)  Q(x)) =
x(P(x) & Q(x))
x(P(x) & Q(x)) =
x(P(x) & Q(x)) =
x(P(x)  Q(x))
Class/self-study exercise 3:
Convince yourself that the above equivalences are correct.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

26. Class/self-study exercise 3
Convert the following to predicate calculus:
All mammals are vertebrates.
Not all vertebrates are mammals.
Only fish swim.
Some primates have tails.
No primates have feathers.
Rhinos eat either leaves or grass.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

27. Inference in predicate calculus
Techniques such as modus ponens and tableau proof are applicable to predicate calculus, with extra complications caused by the use of variables.
Example – given the premises:
x(P(x)  Q(x))
y(Q(y)  R(y))
P(fred)
we can infer via modus ponens or using tableaux (several steps omitted):
Q(fred); R(fred)
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

28. Beginning tableau proofs
The Tableau method uses a standard technique of proof by contradiction:
To show whether an argument is valid, attempt to show that the premises are inconsistent with the negation of the conclusion
To show whether a complex proposition is a tautology, attempt to show that its negation is inconsistent
(see whiteboard and handouts)
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing

29. Useful further reading
Books:
A very short introduction to logic, Graham Priest, OUP. A clear and stimulating guide to basic concepts of logic.
Logic, Wilfred Hodges, Penguin. Goes into a lot more detail than Priest, useful for reference/revision.
Website
Introduction to logic,
Tutorial on propositional and predicate logic aimed at Oxford philosophy students. May not display correctly in all browsers. Seems to work best in Internet Explorer.
11/11/2018
CIS341 Artificial Intelligence - Goldsmiths Dept of Computing