First Order Logic in Discrete Math Presentation

First-Order Logic
Chapter 8.1-8.3
CMSC 471
Adapted from slides by
Tim Finin and
Marie desJardins.
Some material adopted from notes
by Andreas Geyer-Schulz

Outline
• First-order logic
– Properties, relations, functions, quantifiers, …
– Terms, sentences, axioms, theories, proofs, …
• Extensions to first-order logic
• Logical agents
– Reflex agents
– Representing change: situation calculus, frame problem
– Preferences on actions
– Goal-based agents

First-order logic
• First-order logic (FOL) models the world in terms of
– Objects, which are things with individual identities
– Properties of objects that distinguish them from other objects
– Relations that hold among sets of objects
– Functions, which are a subset of relations where there is only one
“value” for any given “input”
• Examples:
– Objects: Students, lectures, companies, cars ...
– Relations: Brother-of, bigger-than, outside, part-of, has-color,
occurs-after, owns, visits, precedes, ...
– Properties: blue, oval, even, large, ...
– Functions: father-of, best-friend, second-half, one-more-than ...

User provides
• Constant symbols, which represent individuals in the world
– Mary
– 3
– Green
• Function symbols, which map individuals to individuals
– father-of(Mary) = John
– color-of(Sky) = Blue
• Predicate symbols, which map individuals to truth values
– greater(5,3)
– green(Grass)
– color(Grass, Green)

FOL Provides
• Variable symbols
– E.g., x, y, foo
• Connectives
– Same as in PL: not (), and (), or (), implies (), if
and only if (biconditional )
• Quantifiers
– Universal x or (Ax)
– Existential x or (Ex)

Sentences are built from terms and atoms
• A term (denoting a real-world individual) is a constant symbol, a
variable symbol, or an n-place function of n terms.
x and f(x1, ..., xn) are terms, where each xi is a term.
A term with no variables is a ground term
• An atomic sentence (which has value true or false) is an n-place
predicate of n terms
• A complex sentence is formed from atomic sentences connected
by the logical connectives:
P, PQ, PQ, PQ, PQ where P and Q are sentences
• A quantified sentence adds quantifiers  and 
• A well-formed formula (wff) is a sentence containing no “free”
variables. That is, all variables are “bound” by universal or
existential quantifiers.
(x)P(x,y) has x bound as a universally quantified variable, but y is free.

Quantifiers
• Universal quantification
– (x)P(x) means that P holds for all values of x in the
domain associated with that variable
– E.g., (x) dolphin(x)  mammal(x)
• Existential quantification
– ( x)P(x) means that P holds for some value of x in the
domain associated with that variable
– E.g., ( x) mammal(x)  lays-eggs(x)
– Permits one to make a statement about some object
without naming it

Quantifiers
• Universal quantifiers are often used with “implies” to form “rules”:
(x) student(x)  smart(x) means “All students are smart”
• Universal quantification is rarely used to make blanket statements
about every individual in the world:
(x)student(x)smart(x) means “Everyone in the world is a student and is smart”
• Existential quantifiers are usually used with “and” to specify a list of
properties about an individual:
(x) student(x)  smart(x) means “There is a student who is smart”
• A common mistake is to represent this English sentence as the FOL
sentence:
(x) student(x)  smart(x)
– But what happens when there is a person who is not a student?

Quantifier Scope
• Switching the order of universal quantifiers does not change
the meaning:
– (x)(y)P(x,y) ↔ (y)(x) P(x,y)
• Similarly, you can switch the order of existential
quantifiers:
– (x)(y)P(x,y) ↔ (y)(x) P(x,y)
• Switching the order of universals and existentials does
change meaning:
– Everyone likes someone: (x)(y) likes(x,y)
– Someone is liked by everyone: (y)(x) likes(x,y)

Connections between All and Exists
We can relate sentences involving  and 
using De Morgan’s laws:
(x) P(x) ↔ (x) P(x)
(x) P ↔ (x) P(x)
(x) P(x) ↔  (x) P(x)
(x) P(x) ↔ (x) P(x)

Quantified inference rules
• Universal instantiation
– x P(x)  P(A)
• Universal generalization
– P(A)  P(B) …  x P(x)
• Existential instantiation
– x P(x) P(F)  skolem constant F
• Existential generalization
– P(A)  x P(x)

Universal instantiation
(a.k.a. universal elimination)
• If (x) P(x) is true, then P(C) is true, where C is any
constant in the domain of x
• Example:
(x) eats(Ziggy, x)  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 instantiation
(a.k.a. existential elimination)
• From (x) P(x) infer P(c)
• Example:
– (x) eats(Ziggy, x)  eats(Ziggy, Stuff)
• Note that the variable is replaced by a brand-new constant
not occurring in this or any other sentence in the KB
• Also known as skolemization; constant is a skolem
constant
• In other words, we don’t want to accidentally draw other
inferences about it by introducing the constant
• Convenient to use this to reason about the unknown object,
rather than constantly manipulating the existential quantifier

Existential generalization
(a.k.a. existential introduction)
• If P(c) is true, then (x) P(x) is inferred.
• Example
eats(Ziggy, IceCream)  (x) 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

Translating English to FOL
Every gardener likes the sun.
x gardener(x)  likes(x,Sun)
You can fool some of the people all of the time.
x t person(x) time(t)  can-fool(x,t)
You can fool all of the people some of the time.
x t (person(x)  time(t) can-fool(x,t))
x (person(x)  t (time(t) can-fool(x,t))
All purple mushrooms are poisonous.
x (mushroom(x)  purple(x))  poisonous(x)
No purple mushroom is poisonous.
x purple(x)  mushroom(x)  poisonous(x)
x (mushroom(x)  purple(x))  poisonous(x)
There are exactly two purple mushrooms.
x y mushroom(x)  purple(x)  mushroom(y)  purple(y) ^ (x=y)  z
(mushroom(z)  purple(z))  ((x=z)  (y=z))
Clinton is not tall.
tall(Clinton)
X is above Y iff X is on directly on top of Y or there is a pile of one or more other
objects directly on top of one another starting with X and ending with Y.
x y above(x,y) ↔ (on(x,y)  z (on(x,z)  above(z,y)))
Equivalent
Equivalent

Monty Python and The Art of Fallacy
Cast
–Sir Bedevere the Wise, master of (odd) logic
–King Arthur
–Villager 1, witch-hunter
–Villager 2, ex-newt
–Villager 3, one-line wonder
–All, the rest of you scoundrels, mongrels, and
nere-do-wells.

An example from Monty Python
by way of Russell & Norvig
• FIRST VILLAGER: We have found a witch. May we burn
her?
• ALL: A witch! Burn her!
• BEDEVERE: Why do you think she is a witch?
• SECOND VILLAGER: She turned me into a newt.
• B: A newt?
• V2 (after looking at himself for some time): I got better.
• ALL: Burn her anyway.
• B: Quiet! Quiet! There are ways of telling whether she is a
witch.

Monty Python cont.
• B: Tell me… what do you do with witches?
• ALL: Burn them!
• B: And what do you burn, apart from witches?
• Third Villager: …wood?
• B: So why do witches burn?
• V2 (after a beat): because they’re made of wood?
• B: Good.
• ALL: I see. Yes, of course.

Monty Python cont.
• B: So how can we tell if she is made of wood?
• V1: Make a bridge out of her.
• B: Ah… but can you not also make bridges out of stone?
• ALL: Yes, of course… um… er…
• B: Does wood sink in water?
• ALL: No, no, it floats. Throw her in the pond.
• B: Wait. Wait… tell me, what also floats on water?
• ALL: Bread? No, no no. Apples… gravy… very small
rocks…
• B: No, no, no,

Monty Python cont.
• KING ARTHUR: A duck!
• (They all turn and look at Arthur. Bedevere looks up, very
impressed.)
• B: Exactly. So… logically…
• V1 (beginning to pick up the thread): If she… weighs the
same as a duck… she’s made of wood.
• B: And therefore?
• ALL: A witch!

Monty Python Fallacy #1
• x witch(x)  burns(x)
• x wood(x)  burns(x)
• -------------------------------
•  z witch(x)  wood(x)
• p  q
• r  q
• ---------
• p  r Fallacy: Affirming the conclusion

Monty Python Near-Fallacy #2
• wood(x)  can-build-bridge(x)
• -----------------------------------------
•  can-build-bridge(x)  wood(x)
• B: Ah… but can you not also make bridges out of stone?

• x wood(x)  floats(x)
• x duck-weight (x)  floats(x)
• -------------------------------
•  x duck-weight(x)  wood(x)
• p  q
• r  q
• -----------
•  r  p

• z light(z)  wood(z)
• light(W)
• ------------------------------
•  wood(W) ok…………..
• witch(W)  wood(W) applying universal instan.
to fallacious conclusion #1
• wood(W)
• ---------------------------------
•  witch(z)

Example: A simple genealogy KB by FOL
• Build a small genealogy knowledge base using FOL that
– contains facts of immediate family relations (spouses, parents, etc.)
– contains definitions of more complex relations (ancestors, relatives)
– is able to answer queries about relationships between people
• Predicates:
– parent(x, y), child(x, y), father(x, y), daughter(x, y), etc.
– spouse(x, y), husband(x, y), wife(x,y)
– ancestor(x, y), descendant(x, y)
– male(x), female(y)
– relative(x, y)
• Facts:
– husband(Joe, Mary), son(Fred, Joe)
– spouse(John, Nancy), male(John), son(Mark, Nancy)
– father(Jack, Nancy), daughter(Linda, Jack)
– daughter(Liz, Linda)
– etc.

• Rules for genealogical relations
– (x,y) parent(x, y) ↔ child (y, x)
(x,y) father(x, y) ↔ parent(x, y)  male(x) (similarly for mother(x, y))
(x,y) daughter(x, y) ↔ child(x, y)  female(x) (similarly for son(x, y))
– (x,y) husband(x, y) ↔ spouse(x, y)  male(x) (similarly for wife(x, y))
(x,y) spouse(x, y) ↔ spouse(y, x) (spouse relation is symmetric)
– (x,y) parent(x, y)  ancestor(x, y)
(x,y)(z) parent(x, z)  ancestor(z, y)  ancestor(x, y)
– (x,y) descendant(x, y) ↔ ancestor(y, x)
– (x,y)(z) ancestor(z, x)  ancestor(z, y)  relative(x, y)
(related by common ancestry)
(x,y) spouse(x, y)  relative(x, y) (related by marriage)
(x,y)(z) relative(z, x)  relative(z, y)  relative(x, y) (transitive)
(x,y) relative(x, y) ↔ relative(y, x) (symmetric)
• Queries
– ancestor(Jack, Fred) /* the answer is yes */
– relative(Liz, Joe) /* the answer is yes */
– relative(Nancy, Matthew)
/* no answer in general, no if under closed world assumption */
– (z) ancestor(z, Fred)  ancestor(z, Liz)

Semantics of FOL
• Domain M: the set of all objects in the world (of interest)
• Interpretation I: includes
– Assign each constant to an object in M
– Define each function of n arguments as a mapping Mn => M
– Define each predicate of n arguments as a mapping Mn => {T, F}
– Therefore, every ground predicate with any instantiation will have a
truth value
– In general there is an infinite number of interpretations because |M| is
infinite
• Define logical connectives: ~, ^, , =>, <=> as in PL
• Define semantics of (x) and (x)
– (x) P(x) is true iff P(x) is true under all interpretations
– (x) P(x) is true iff P(x) is true under some interpretation

• Model: an interpretation of a set of sentences such that every
sentence is True
• A sentence is
– satisfiable if it is true under some interpretation
– valid if it is true under all possible interpretations
– inconsistent if there does not exist any interpretation under which the
sentence is true
• Logical consequence: S |= X if all models of S are also
models of X

Axioms, definitions and theorems
•Axioms are facts and rules that attempt to capture all of the
(important) facts and concepts about a domain; axioms can
be used to prove theorems
–Mathematicians don’t want any unnecessary (dependent) axioms –ones
that can be derived from other axioms
–Dependent axioms can make reasoning faster, however
–Choosing a good set of axioms for a domain is a kind of design
problem
•A definition of a predicate is of the form “p(X) ↔ …” and
can be decomposed into two parts
–Necessary description: “p(x)  …”
–Sufficient description “p(x)  …”
–Some concepts don’t have complete definitions (e.g., person(x))

More on definitions
• A necessary condition must be satisfied for a statement to be true.
• A sufficient condition, if satisfied, assures the statement’s truth.
• Duality: “P is sufficient for Q” is the same as “Q is necessary for P.”
• Examples: define father(x, y) by parent(x, y) and male(x)
– parent(x, y) is a necessary (but not sufficient) description of
father(x, y)
• father(x, y)  parent(x, y)
– parent(x, y) ^ male(x) ^ age(x, 35) is a sufficient (but not necessary)
description of father(x, y):
father(x, y)  parent(x, y) ^ male(x) ^ age(x, 35)
– parent(x, y) ^ male(x) is a necessary and sufficient description of
father(x, y)
parent(x, y) ^ male(x) ↔ father(x, y)

More on definitions
P(x)
S(x)
S(x) is a
necessary
condition of P(x)
(x) P(x) => S(x)
S(x)
P(x)
S(x) is a
sufficient
condition of P(x)
(x) P(x) <= S(x)
P(x)
S(x)
S(x) is a
necessary and
sufficient
condition of P(x)
(x) P(x) <=> S(x)

Higher-order logic
• FOL only allows to quantify over variables, and variables
can only range over objects.
• HOL allows us to quantify over relations
• Example: (quantify over functions)
“two functions are equal iff they produce the same value for all
arguments”
f g (f = g)  (x f(x) = g(x))
• Example: (quantify over predicates)
r transitive( r )  (xyz) r(x,y)  r(y,z)  r(x,z))
• More expressive, but undecidable. (there isn’t an effective
algorithm to decide whether all sentences are valid)
– First-order logic is decidable only when it uses predicates with only one
argument.

Expressing uniqueness
• Sometimes we want to say that there is a single, unique
object that satisfies a certain condition
• “There exists a unique x such that king(x) is true”
– x king(x)  y (king(y)  x=y)
– x king(x)  y (king(y)  xy)
– ! x king(x)
• “Every country has exactly one ruler”
– c country(c)  ! r ruler(c,r)
• Iota operator: “ x P(x)” means “the unique x such that p(x)
is true”
– “The unique ruler of Freedonia is dead”
– dead( x ruler(freedonia,x))

Notational differences
• Different symbols for and, or, not, implies, ...
–         
– p v (q ^ r)
– p + (q * r)
– etc
• Prolog
cat(X) :- furry(X), meows (X), has(X, claws)
• Lispy notations
(forall ?x (implies (and (furry ?x)
(meows ?x)
(has ?x claws))
(cat ?x)))

Logical agents for the Wumpus World
Three (non-exclusive) agent architectures:
–Reflex agents
• Have rules that classify situations, specifying how to
react to each possible situation
–Model-based agents
• Construct an internal model of their world
–Goal-based agents
• Form goals and try to achieve them

A simple reflex agent
• Rules to map percepts into observations:
b,g,u,c,t Percept([Stench, b, g, u, c], t)  Stench(t)
s,g,u,c,t Percept([s, Breeze, g, u, c], t)  Breeze(t)
s,b,u,c,t Percept([s, b, Glitter, u, c], t)  AtGold(t)
• Rules to select an action given observations:
t AtGold(t)  Action(Grab, t);
• Some difficulties:
– Consider Climb. There is no percept that indicates the agent should
climb out – position and holding gold are not part of the percept
sequence
– Loops – the percept will be repeated when you return to a square,
which should cause the same response (unless we maintain some
internal model of the world)

Representing change
• Representing change in the world in logic can be
tricky.
• One way is just to change the KB
– Add and delete sentences from the KB to reflect changes
– How do we remember the past, or reason about changes?
• Situation calculus is another way
• A situation is a snapshot of the world at some
instant in time
• When the agent performs an action A in
situation S1, the result is a new situation
S2.

Situation calculus
• A situation is a snapshot of the world at an interval of time during which
nothing changes
• Every true or false statement is made with respect to a particular situation.
– Add situation variables to every predicate.
– at(Agent,1,1) becomes at(Agent,1,1,s0): at(Agent,1,1) is true in situation (i.e., state)
s0.
– Alternatively, add a special 2nd-order predicate, holds(f,s), that means “f is true in
situation s.” E.g., holds(at(Agent,1,1),s0)
• Add a new function, result(a,s), that maps a situation s into a new situation as a
result of performing action a. For example, result(forward, s) is a function that
returns the successor state (situation) to s
• Example: The action agent-walks-to-location-y could be represented by
– (x)(y)(s) (at(Agent,x,s)  onbox(s))  at(Agent,y,result(walk(y),s))

Deducing hidden properties
• From the perceptual information we obtain in situations, we
can infer properties of locations
l,s at(Agent,l,s)  Breeze(s)  Breezy(l)
l,s at(Agent,l,s)  Stench(s)  Smelly(l)
• Neither Breezy nor Smelly need situation arguments
because pits and Wumpuses do not move around

Deducing hidden properties II
• We need to write some rules that relate various aspects of a
single world state (as opposed to across states)
• There are two main kinds of such rules:
– Causal rules reflect the assumed direction of causality in the world:
(l1,l2,s) At(Wumpus,l1,s)  Adjacent(l1,l2)  Smelly(l2)
( l1,l2,s) At(Pit,l1,s)  Adjacent(l1,l2)  Breezy(l2)
Systems that reason with causal rules are called model-based
reasoning systems
– Diagnostic rules infer the presence of hidden properties directly
from the percept-derived information. We have already seen two
diagnostic rules:
( l,s) At(Agent,l,s)  Breeze(s)  Breezy(l)
( l,s) At(Agent,l,s)  Stench(s)  Smelly(l)

Representing change:
The frame problem
• Frame axioms: If property x doesn’t change as a result of
applying action a in state s, then it stays the same.
– On (x, z, s)  Clear (x, s) 
On (x, table, Result(Move(x, table), s)) 
On(x, z, Result (Move (x, table), s))
– On (y, z, s)  y x  On (y, z, Result (Move (x, table), s))
– The proliferation of frame axioms becomes very cumbersome in
complex domains

The frame problem II
• Successor-state axiom: General statement that
characterizes every way in which a particular predicate can
become true:
– Either it can be made true, or it can already be true and not be
changed:
– On (x, table, Result(a,s)) 
[On (x, z, s)  Clear (x, s)  a = Move(x, table)] 
[On (x, table, s)  a  Move (x, z)]
• In complex worlds, where you want to reason about longer
chains of action, even these types of axioms are too
cumbersome
– Planning systems use special-purpose inference methods to reason
about the expected state of the world at any point in time during a
multi-step plan

Qualification problem
• Qualification problem:
– How can you possibly characterize every single effect of an action,
or every single exception that might occur?
– When I put my bread into the toaster, and push the button, it will
become toasted after two minutes, unless…
• The toaster is broken, or…
• The power is out, or…
• I blow a fuse, or…
• A neutron bomb explodes nearby and fries all electrical components,
or…
• A meteor strikes the earth, and the world we know it ceases to exist,
or…

Ramification problem
• Similarly, it’s just about impossible to characterize every side effect of
every action, at every possible level of detail:
– When I put my bread into the toaster, and push the button, the bread will
become toasted after two minutes, and…
• The crumbs that fall off the bread onto the bottom of the toaster over tray will
also become toasted, and…
• Some of the aforementioned crumbs will become burnt, and…
• The outside molecules of the bread will become “toasted,” and…
• The inside molecules of the bread will remain more “breadlike,” and…
• The toasting process will release a small amount of humidity into the air because
of evaporation, and…
• The heating elements will become a tiny fraction more likely to burn out the next
time I use the toaster, and…
• The electricity meter in the house will move up slightly, and…

Knowledge engineering!
• Modeling the “right” conditions and the “right” effects at
the “right” level of abstraction is very difficult
• Knowledge engineering (creating and maintaining
knowledge bases for intelligent reasoning) is an entire field
of investigation
• Many researchers hope that automated knowledge
acquisition and machine learning tools can fill the gap:
– Our intelligent systems should be able to learn about the conditions
and effects, just like we do!
– Our intelligent systems should be able to learn when to pay attention
to, or reason about, certain aspects of processes, depending on the
context!

Preferences among actions
• A problem with the Wumpus world knowledge base that we
have built so far is that it is difficult to decide which action
is best among a number of possibilities.
• For example, to decide between a forward and a grab,
axioms describing when it is OK to move to a square would
have to mention glitter.
• This is not modular!
• We can solve this problem by separating facts about
actions from facts about goals. This way our agent can be
reprogrammed just by asking it to achieve different
goals.

• The first step is to describe the desirability of actions
independent of each other.
• In doing this we will use a simple scale: actions can be
Great, Good, Medium, Risky, or Deadly.
• Obviously, the agent should always do the best action it can
find:
(a,s) Great(a,s)  Action(a,s)
(a,s) Good(a,s)  (b) Great(b,s)  Action(a,s)
(a,s) Medium(a,s)  ((b) Great(b,s)  Good(b,s))  Action(a,s)
...

• We use this action quality scale in the following way.
• Until it finds the gold, the basic strategy for our agent is:
– Great actions include picking up the gold when found and climbing
out of the cave with the gold.
– Good actions include moving to a square that’s OK and hasn't been
visited yet.
– Medium actions include moving to a square that is OK and has
already been visited.
– Risky actions include moving to a square that is not known to be
deadly or OK.
– Deadly actions are moving into a square that is known to have a pit
or a Wumpus.

Goal-based agents
• Once the gold is found, it is necessary to change strategies.
So now we need a new set of action values.
• We could encode this as a rule:
– (s) Holding(Gold,s)  GoalLocation([1,1]),s)
• We must now decide how the agent will work out a
sequence of actions to accomplish the goal.
• Three possible approaches are:
– Inference: good versus wasteful solutions
– Search: make a problem with operators and set of states
– Planning: to be discussed later

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