1. First-Order Logic (FOL) aka. predicate calculus
Tuomas Sandholm
Carnegie Mellon University
Computer Science Department

2. First-Order Logic (FOL) Syntax
User defines these primitives:
Constant symbols (i.e., the "individuals" in the world) E.g., Mary, 3
Function symbols (mapping individuals to individuals) E.g., father-of(Mary) = John, color-of(Sky) = Blue
Predicate symbols (mapping from individuals to truth values) E.g., greater(5,3), green(Grass), color(Grass, Green)

3. First-Order Logic (FOL) Syntax…
FOL supplies these primitives:
Variable symbols. E.g., x,y
Connectives. Same as in PL: not (~), and (^), or (v), implies (=>), if and only if (<=>)
Quantifiers: Universal (A) and Existential (E)

4. Quantifiers
Universal quantification corresponds to conjunction ("and") in that (Ax)P(x) means that P holds for all values of x in the domain associated with that variable.
E.g., (Ax) dolphin(x) => mammal(x)
Existential quantification corresponds to disjunction ("or") in that (Ex)P(x) means that P holds for some value of x in the domain associated with that variable.
E.g., (Ex) mammal(x) ^ lays-eggs(x)
Universal quantifiers are usually used with "implies" to form "if-then rules."
E.g., (Ax) cs student(x) => smart(x) means "All cs students are smart."
You rarely use universal quantification to make blanket statements about every individual in the world: (Ax)cs student(x) ^ smart(x) meaning that everyone in the world is a cs student and is smart.

5. Quantifiers …
Existential quantifiers are usually used with "and" to specify a list of properties or facts about an individual.
E.g., (Ex) cs student(x) ^ smart(x) means "there is a cs student who is smart."
A common mistake is to represent this English sentence as the FOL sentence: (Ex) cs student(x) => smart(x)
Switching the order of universal quantifiers does not change the meaning: (Ax)(Ay)P(x,y) is logically equivalent to (Ay)(Ax)P(x,y). Similarly, you can switch the order of existential quantifiers.
Switching the order of universals and existentials does change meaning:
Everyone likes someone: (Ax)(Ey)likes(x,y)
Someone is liked by everyone: (Ey)(Ax)likes(x,y)

6. First-Order Logic (FOL) Syntax…
Sentences are built up of terms and atoms:
A term (denoting a real-world object) is a constant symbol, a variable symbol, or a function e.g. left-leg-of ( ). For example, x and f(x1, ..., xn) are terms, where each xi is a term.
An atom (which has value true or false) is either an n-place predicate of n terms, or, if P and Q are atoms, then ~P, P V Q, P ^ Q, P => Q, P <=> Q are atoms
A sentence is an atom, or, if P is a sentence and x is a variable, then (Ax)P and (Ex)P are sentences
A well-formed formula (wff) is a sentence containing no "free" variables. I.e., all variables are "bound" by universal or existential quantifiers.
E.g., (Ax)P(x,y) has x bound as a universally quantified variable, but y is free.

7. Translating English to FOL
Every gardener likes the sun. (Ax) gardener(x) => likes(x,Sun)
You can fool some of the people all of the time. (Ex)(At) (person(x) ^ time(t)) => can-fool(x,t)
You can fool all of the people some of the time. (Ax)(Et) (person(x) ^ time(t) => can-fool(x,t)
All purple mushrooms are poisonous. (Ax) (mushroom(x) ^ purple(x)) => poisonous(x)

8. Translating English to FOL…
No purple mushroom is poisonous. ~(Ex) purple(x) ^ mushroom(x) ^ poisonous(x) or, equivalently, (Ax) (mushroom(x) ^ purple(x)) => ~poisonous(x)
There are exactly two purple mushrooms. (Ex)(Ey) mushroom(x) ^ purple(x) ^ mushroom(y) ^ purple(y) ^ ~(x=y) ^ (Az) (mushroom(z) ^ purple(z)) => ((x=z) v (y=z))
Deb is not tall. ~tall(Deb)
X is above Y if X is on directly on top of Y or else there is a pile of one or more other objects directly on top of one another starting with X and ending with Y. (Ax)(Ay) above(x,y) <=> (on(x,y) v (Ez) (on(x,z) ^ above(z,y)))

9. Inference Rules for FOL
Inference rules for PL apply to FOL as well. For example, Modus Ponens, And-Introduction, And-Elimination, etc.
New sound inference rules for use with quantifiers:
Universal Elimination If (Ax)P(x) is true, then P(c) is true, where c is a constant in the domain of x. For example, from (Ax)eats(Ziggy, x) we can infer eats(Ziggy, IceCream).
The variable symbol can be replaced by any ground term, i.e., any constant symbol or function symbol applied to ground terms only.
Existential Introduction If P(c) is true, then (Ex)P(x) is inferred.
For example, from eats(Ziggy, IceCream) we can infer (Ex)eats(Ziggy, x).
All instances of the given constant symbol are replaced by the new variable symbol. Note that the variable symbol cannot already exist anywhere in the expression.
Existential Elimination From (Ex)P(x) infer P(c).
For example, from (Ex)eats(Ziggy, x) infer eats(Ziggy, Cheese).
Note that the variable is replaced by a brand new constant that does not occur in this or any other sentence in the Knowledge Base. In other words, we don't want to accidentally draw other inferences about it by introducing the constant. All we know is there must be some constant that makes this true, so we can introduce a brand new one to stand in for that (unknown) constant.

10. Inference Rules for FOL
Paramodulation
Given two sentences (P1 v ... v PN) and (t=s v Q1 v ... v QM) where each Pi and Qi is a literal and Pj contains a term t, derive new sentence (P1 v ... v Pj-1 v Pj[s] v Pj+1 v ... v PN v Q1 v ... v QM) where Pj[s] means a single occurrence of the term t is replaced by the term s in Pj
Example: From P(a) and a=b derive P(b)

11. Inference Rules for FOL
Generalized Modus Ponens (GMP)
Combines And-Introduction, Universal-Elimination, and Modus Ponens
E.g.: from P(c), Q(c), and (Ax)(P(x) ^ Q(x)) => R(x), derive R(c)
In general, given atomic sentences P1, P2, ..., PN, and implication sentence (Q1 ^ Q2 ^ ... ^ QN) => R, where Q1, ..., QN and R are atomic sentences, and subst(Theta, Pi) = subst(Theta, Qi) for i=1,...,N, derive new sentence: subst(Theta, R)
subst(Theta, alpha) denotes the result of applying a set of substitutions defined by Theta to the sentence alpha
A substitution list Theta = {v1/t1, v2/t2, ..., vn/tn} means to replace all occurrences of variable symbol vi by term ti.
Substitutions are made in left-to-right order in the list.
E.g.: subst({x/IceCream, y/Ziggy}, eats(y,x)) = eats(Ziggy, IceCream)

12. Automated Inference in FOL
Automated inference in FOL is harder than in PL because variables can take on potentially an infinite number of possible values from their domain. Hence there are potentially an infinite number of ways to apply the Universal-Elimination rule of inference
Goedel's Completeness Theorem says that FOL entailment is semidecidable. That is, if a sentence is true given a set of axioms, there is a procedure that will determine this. However, if the sentence is false, then there is no guarantee that a procedure will ever determine this. In other words, the procedure may never halt in this case.
Goedel's Incompleteness Theorem says that in a slightly extended language (that enables mathematical induction), there are entailed sentences that cannot be proved
The truth table method of inference is not complete for FOL because the truth table size may be infinite
Natural deduction is complete for FOL but is not practical for automated inference because the "branching factor" in a search is too large, caused by the fact that we would have to potentially try every inference rule in every possible way using the set of known sentences

13. Incompleteness of the generalized modus ponens inference rule
Backward chaining
Forward chaining
x P(x) => Q(x)
x P(x) => Q(x)
x Q(x) => S(x)
x R(x) => S(x)
 Cannot be converted to Horn
Want to conclude S(A)
S(A) is true if Q(A) or R(A) is true, and one of those must be true because either P(A) or P(A)
Incomplete: there are entailed sentences that the procedure cannot infer.

14. Generalized Modus Ponens in Horn FOL
Generalized Modus Ponens (GMP) is complete for KBs containing only Horn clauses
A Horn clause is a sentence of the form: (Ax) (P1(x) ^ P2(x) ^ ... ^ Pn(x)) => Q(x) where there are 0 or more Pi's, and the Pi's and Q are positive (i.e., un-negated) literals
Horn clauses represent a subset of the set of sentences representable in FOL. For example, P(a) v Q(a) is a sentence in FOL but is not a Horn clause.
Natural deduction using GMP is complete for KBs containing only Horn clauses. Proofs start with the given axioms/premises in KB, deriving new sentences using GMP until the goal/query sentence is derived. This defines a forward chaining inference procedure because it moves "forward" from the KB to the goal.

15. Example of forward chaining
Example: KB = All cats like fish, cats eat everything they like, and Ziggy is a cat. In FOL, KB =
(Ax) cat(x) => likes(x, Fish)
(Ax)(Ay) (cat(x) ^ likes(x,y)) => eats(x,y)
cat(Ziggy)
Goal query: Does Ziggy eat fish?
Proof:
Use GMP with (1) and (3) to derive: 4. likes(Ziggy, Fish)
Use GMP with (3), (4) and (2) to derive eats(Ziggy, Fish)
So, Yes, Ziggy eats fish.
Data-driven

16. Backward chaining
Backward-chaining deduction using GMP is complete for KBs containing only Horn clauses. Proofs start with the goal query, find implications that would allow you to prove it, and then prove each of the antecedents in the implication, continuing to work "backwards" until we get to the axioms, which we know are true.
Example: Does Ziggy eat fish?
To prove eats(Ziggy, Fish), first see if this is known from one of the axioms directly. Here it is not known, so see if there is a Horn clause that has the consequent (i.e., right-hand side) of the implication matching the goal.
Proof: Goal Driven
Goal matches RHS of Horn clause (2), so try and prove new sub-goals cat(Ziggy) and likes(Ziggy, Fish) that correspond to the LHS of (2)
cat(Ziggy) matches axiom (3), so we've "solved" that sub-goal
likes(Ziggy, Fish) matches the RHS of (1), so try and prove cat(Ziggy)
cat(Ziggy) matches (as it did earlier) axiom (3), so we've solved this sub-goal
There are no unsolved sub-goals, so we're done. Yes, Ziggy eats fish.

17. Resolution Procedure (aka Resolution Refutation Procedure)
Resolution procedure is a sound and complete inference procedure for FOL (BFS is complete for Horn, DFS may go into infinite loops)
Resolution procedure uses a single rule of inference: the Resolution Rule (RR), which is a generalization of the same rule used in PL
Resolution Rule for PL: From sentence P1 v P2 v ... v Pn and sentence ~P1 v Q2 v ... v Qm derive resolvent sentence: P2 v ... v Pn v Q2 v ... v Qm
Examples
From P and ~P v Q, derive Q (Modus Ponens)
From (~P v Q) and (~Q v R), derive ~P v R
From P and ~P, derive False
From (P v Q) and (~P v ~Q), derive True

18. Resolution Procedure…
Resolution Rule for FOL:
Given sentence P1 v ... v Pn
and sentence Q1 v ... v Qm
where each Pi and Qi is a literal, i.e., a positive or negated predicate symbol with its terms,
if Pj and ~Qk unify with substitution list Theta,
then derive the resolvent sentence: subst(Theta, P1 v ... v Pj-1 v Pj+1 v ... v Pn v Q1 v ... Qk-1 v Qk+1 v ... v Qm)
Example
From clause P(x, f(a)) v P(x, f(y)) v Q(y)
and clause ~P(z, f(a)) v ~Q(z),
derive resolvent clause P(z, f(y)) v Q(y) v ~Q(z) using Theta = {x/z}

19. Unification
Unification is a "pattern matching" procedure that takes two atomic sentences, called literals, as input, and returns "failure" if they do not match and a substitution list, Theta, if they do match.
That is, unify(p,q) = Theta means subst(Theta, p) = subst(Theta, q) for two atomic sentences p and q.
Theta is called the most general unifier (mgu)
All variables in the given two literals are implicitly universally quantified
To make literals match, replace (universally-quantified) variables by terms

20. Unification Algorithm
procedure unify(p, q, theta)
Scan p and q left-to-right and find the first corresponding terms where p and q "disagree" ; where p and q not equal
If there is no disagreement, return theta ; success
Let r and s be the terms in p and q, respectively, where disagreement first occurs
If variable(r) then theta = union(theta, {r/s}) unify(subst(theta, p), subst(theta, q), theta)
else if variable(s) then theta = union(theta, {s/r}) unify(subst(theta, p), subst(theta, q), theta)
else return "failure" end

21. Unification… Examples Literal 1 Literal 2 Literal 3
parents(x, father(x), mother(Bill))
parents(x, father(x), mother(Jane))
parents(Bill, father(Bill), y)
parents(Bill, father(y), z)
parents(Bill, father(y), mother(y))
{x/Bill, y/mother(Bill)}
{x/Bill, y/Bill, z/mother(Bill)}
Failure

22. Unification…
Unify is a linear time algorithm that returns the most general unifier (mgu), i.e., a shortest length substitution list that makes the two literals match.
(In general, there is not a unique minimum length substitution list, but unify returns one of them.)
A variable can never be replaced by a term containing that variable. For example, x/f(x) is illegal. This "occurs check" should be done in the above pseudo-code before making the recursive calls.

23. Resolution Procedure…
Proof by contradiction method
Given a consistent set of axioms KB and goal sentence Q, we want to show that KB |= Q. This means that every interpretation I that satisfies KB, satisfies Q. But we know that any interpretation I satisfies either Q or ~Q, but not both. Therefore if in fact KB |= Q, an interpretation that satisfies KB, satisfies Q and does not satisfy ~Q. Hence KB union {~Q} is unsatisfiable, i.e., that it's false under all interpretations.
In other words, (KB |- Q) <=> (KB ^ ~Q |- False)
What's the gain? If KB union ~Q is unsatisfiable, then some finite subset is unsatisfiable
Resolution procedure can be used to establish that a given sentence Q is entailed by KB; however, it cannot, in general, be used to generate all logical consequences of a set sentences. Also, the resolution procedure cannot be used to prove that Q is not entailed by KB.
Resolution procedure won't always give an answer since entailment is only semidecidable. And you can't just run two proofs in parallel, one trying to prove Q and the other trying to prove ~Q, since KB might not entail either one.

24. Resolution Algorithm procedure resolution-refutation(KB, Q)
;; KB is a set of consistent, true FOL sentences
;; Q is a goal sentence that we want to derive
;; return success if KB |- Q, and failure otherwise
KB = union(KB, ~Q)
while false not in KB do
pick 2 sentences, S1 and S2, in KB that contain literals that unify (if none, return "failure“)
resolvent = resolution-rule(S1, S2)
KB = union(KB, resolvent)
return "success"

25. Resolution example (using PL sentences)
From “Heads I win, tails you lose” prove that “I win”
First, define the axioms in KB:
"Heads I win, tails you lose." (Heads => IWin) or, equivalently, (~Heads v IWin) (Tails => YouLose) or, equivalently, (~Tails v YouLose)
Add some general knowledge axioms about coins, winning, and losing: (Heads v Tails) (YouLose => IWin) or, equivalently, (~YouLose v IWin) (IWin => YouLose) or, equivalently, (~IWin v YouLose)
Goal: IWin

26. Resolution example (using PL sentences)…
Resolvent
~IWin
~Heads
Tails
YouLose
IWin
~Heads v IWin
Heads v Tails
~Tails v YouLose
~YouLose v Iwin
False

27. Problems yet to be addressed
Resolution rule of inference is only applicable with sentences that are in the form P1 v P2 v ... v Pn, where each Pi is a negated or nonnegated predicate and contains functions, constants, and universally quantified variables, so can we convert every FOL sentence into this form?
Resolution strategy
How to pick which pair of sentences to resolve?
How to pick which pair of literals, one from each sentence, to unify?

28. Converting FOL sentences to clause form
Every FOL sentence can be converted to a logically equivalent sentence that is in a "normal form" called clause form
Steps to convert a sentence to clause form:
Eliminate all <=> connectives by replacing each instance of the form (P <=> Q) by expression ((P => Q) ^ (Q => P))
Eliminate all => connectives by replacing each instance of the form (P => Q) by (~P v Q)
Reduce the scope of each negation symbol to a single predicate by applying equivalences such as converting
~~P to P
~(P v Q) to ~P ^ ~Q
~(P ^ Q) to ~P v ~Q
~(Ax)P to (Ex)~P
~(Ex)P to (Ax)~P

29. Converting FOL sentences to clause form…
Standardize variables: rename all variables so that each quantifier has its own unique variable name. For example, convert (Ax)P(x) to (Ay)P(y) if there is another place where variable x is already used.
Eliminate existential quantification by introducing Skolem functions. For example, convert (Ex)P(x) to P(c) where c is a brand new constant symbol that is not used in any other sentence. c is called a Skolem constant. More generally, if the existential quantifier is within the scope of a universal quantified variable, then introduce a Skolem function that depends on the universally quantified variable.
For example, (Ax)(Ey)P(x,y) is converted to (Ax)P(x, f(x)). f is called a Skolem function, and must be a brand new function name that does not occur in any other sentence in the entire KB.
Example: (Ax)(Ey)loves(x,y) is converted to (Ax)loves(x,f(x)) where in this case f(x) specifies the person that x loves.
(If we knew that everyone loved their mother, then f could stand for the mother-of function.)

30. Converting FOL sentences to clause form…
Remove universal quantification symbols by first moving them all to the left end and making the scope of each the entire sentence, and then just dropping the "prefix" part. E.g., convert (Ax)P(x) to P(x)
Distribute "and" over "or" to get a conjunction of disjunctions called conjunctive normal form.
convert (P ^ Q) v R to (P v R) ^ (Q v R)
convert (P v Q) v R to (P v Q v R)
Split each conjunct into a separate clause, which is just a disjunction ("or") of negated and nonnegated predicates, called literals
Standardize variables apart again so that each clause contains variable names that do not occur in any other clause

31. Converting FOL sentences to clause form…
Examples:
Convert the sentence
(Ax)(P(x) => ((Ay)(P(y) => P(f(x,y))) ^ ~(Ay)(Q(x,y) => P(y))))
Eliminate <=> Nothing to do here.
Eliminate => (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ ~(Ay)(~Q(x,y) v P(y))))
Reduce scope of negation (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ (Ey)(Q(x,y) ^ ~P(y))))

32. Converting FOL sentences to clause form…
Standardize variables (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ (Ez)(Q(x,z) ^ ~P(z))))
Eliminate existential quantification (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ (Q(x,g(x)) ^ ~P(g(x)))))
Drop universal quantification symbols (~P(x) v ((~P(y) v P(f(x,y))) ^ (Q(x,g(x)) ^ ~P(g(x)))))
Convert to conjunction of disjunctions (~P(x) v ~P(y) v P(f(x,y))) ^ (~P(x) v Q(x,g(x))) ^ (~P(x) v ~P(g(x)))

33. Converting FOL sentences to clause form…
Create separate clauses
~P(x) v ~P(y) v P(f(x,y))
~P(x) v Q(x,g(x))
~P(x) v ~P(g(x))
Standardize variables
~P(z) v Q(z,g(z))
~P(w) v ~P(g(w))

34. Example: Hoofers Club [example from Charles Dyer]
Problem Statement: Tony, Shi-Kuo and Ellen belong to the Hoofers Club. Every member of the Hoofers Club is either a skier or a mountain climber or both. No mountain climber likes rain, and all skiers like snow. Ellen dislikes whatever Tony likes and likes whatever Tony dislikes. Tony likes rain and snow.
Query: Is there a member of the Hoofers Club who is a mountain climber but not a skier?

35. Example: Hoofers Club…
Translation into FOL Sentences
Let S(x) mean x is a skier, M(x) mean x is a mountain climber, and L(x,y) mean x likes y, where the domain of the first variable is Hoofers Club members, and the domain of the second variable is snow and rain. We can now translate the above English sentences into the following FOL wffs:
(Ax) S(x) v M(x)
~(Ex) M(x) ^ L(x, Rain)
(Ax) S(x) => L(x, Snow)
(Ay) L(Ellen, y) <=> ~L(Tony, y)
L(Tony, Rain)
L(Tony, Snow)
Query: (Ex) M(x) ^ ~S(x)
Negation of the Query: ~(Ex) M(x) ^ ~S(x)

36. Example: Hoofers Club…
Conversion to Clause Form
S(x1) v M(x1)
~M(x2) v ~L(x2, Rain)
~S(x3) v L(x3, Snow)
~L(Tony, x4) v ~L(Ellen, x4)
L(Tony, x5) v L(Ellen, x5)
L(Tony, Rain)
L(Tony, Snow)
Negation of the Query: ~M(x7) v S(x7)
Write these on blackboard

37. Example: Hoofers Club…
Resolution Refutation Proof
Clause 1
Clause 2
Resolvent
MGU (i.e., Theta) 8 9 10 11 1 3 4 7
9. S(x1)
10. L(x1, Snow)
11. ~L(Tony, Snow)
12. False
{x7/x1}
{x3/x1}
{x4/Snow, x1/Ellen} {}

38. Example: Hoofers Club…
Answer Extraction
Clause 1
Clause 2
Resolvent
MGU (i.e., Theta)
~M(x7) v S(x7) v (M(x7) ^ ~S(x7)) 9 10 11 1 3 4 7
9. S(x1) v (M(x1) ^ ~S(x1))
10. L(x1, Snow) v (M(x1) ^ ~S(x1))
11. ~L(Tony, Snow) v (M(Ellen) ^ ~S(Ellen))
12. M(Ellen) ^ ~S(Ellen)
{x7/x1}
{x3/x1}
{x4/Snow, x1/Ellen} {}
Answer to the query: Ellen!

39. Resolution procedure as search
Resolution procedure can be thought of as the bottom-up construction of a search tree, where the leaves are the clauses produced by KB and the negation of the goal. When a pair of clauses generates a new resolvent clause, add a new node to the tree with arcs directed from the resolvent to the two parent clauses. The resolution procedure succeeds when a node containing the False clause is produced, becoming the root node of the tree.
A strategy is complete if its use guarantees that the empty clause (i.e., false) can be derived whenever it is entailed

40. Some strategies for controlling resolution's search
Breadth-First
Level 0 clauses are those from the original axioms and the negation of the goal. Level k clauses are the resolvents computed from two clauses, one of which must be from level k-1 and the other from any earlier level.
Compute all level 1 clauses possible, then all possible level 2 clauses, etc.
Complete, but very inefficient.
Set-of-Support
At least one parent clause must be from the negation of the goal or one of the "descendents" of such a goal clause (i.e., derived from a goal clause)
Complete (assuming all possible set-of-support clauses are derived)
Unit Resolution
At least one parent clause must be a "unit clause," (clause containing a single literal)
Not complete in general, but complete for Horn clause KBs
Input Resolution
At least one parent from the original KB (axioms and the negation of the goal)
Linear Resolution
P and Q can be resolved together if either P is in the original KB or if P is an ancestor of Q in the proof tree
Complete

41. Subsumption Motivation: Helps keep the KB small => less search
Never keep more specific (“subsumed”) sentences in the KB than more general ones
E.g., if the KB has P(x), then eliminate
P(c)
P(a) v P(b)