Lecture 10: Logic

# Artificial Intelligence

### Logic
  
---

# Logic

---

# Why Logic?

* Remember: Everything in AI is either representation or search 
  
  * Logical formulas are often a very convenient representation
  
  * Why convenient?
     
      - Can be generated automatically
      - Can be processed automatically 
      - Are very expressive

* Logic is used extensively in AI!
  
---

# Propositional Logic 
  
  * `$a \wedge b$`: a **and** b
  
  * `$a \vee b$`: a **or** b (or both!)

*  `$\neg a$`: **not** a 
  
  * `$a \rightarrow b$`: a **implies** b, which means: if `a` is true, `b` also has to be true (but if `a` is false, `b` can be true or false). Logically equivalent to `$\neg a \vee b$`
  
  * When you see **iff** in text it is not a typo, it means "if and only if", a bidirectional implication ("a iff b" means that if `a` is true `b` is true, but if `a` is false, `b` is also false)
  
---

# Propositional Logic

* How do we tell if a formula is true?
  
  * We need to know if a and b (and all other constants) are "true"
  
  * In other words, if we have some "interpretation" `W` of `a` and `b`, we can determine truth values
  
  * `$ W \models  a \wedge b $` iff `a` and `b` are both true in `W`, i.e. `$ W \models a $` **and** `$W \models b $`
  
  * So what is W?
  
---

# Interpretations

* W is an *interpretation* (also called *state* or *world*), which assigns truth values to every constant

* Sometimes it is convenient to just represent W as a set that contains everything that is true 
  
  * Conversely, everything that is not in the set is false ("closed world assumption")
  
  * A *model* is an interpretation under which a formula is satisfied 
  
  * We say W *models* a formula, if the formula is satisfied under that interpretation
  
  * A formula is a tautology if *all* interpretations are models
  
---

# Interpretations as sets

* Why might it be convenient to represent `W` as a set?

* Determining the truth value of a formula reduces to set-operations

$$
W \models (\neg a) \equiv a \not\in W\\\\
W \models (a \wedge b) \equiv a \in W, b \in W\\\\
W \models (a \vee b) \equiv a \in W \text{or} b \in W\\\\
W \models (a \rightarrow b) \equiv a \not\in W \text{or} b \in W
$$

More complex formulas need to be split up until we reach atoms.
  
---

# Interpretation

Given an interpretation `$W = \{ a, b, c \}$`. For each of the following formulas determine if `$W$` is a model for that formula.

$$
(a \wedge b)\\\\
(b \rightarrow d)\\\\
(d \rightarrow b)\\\\
b \wedge ((c \rightarrow e) \rightarrow d)\\\\
e \vee (c \wedge \neg d)\\\\
\neg d \rightarrow \neg e
$$

---

# Interpretation

$$
W = \\{a,b,c\\}
$$

$$
W \models a \wedge b
$$

### because `a` and `b` are both elements of `W`.

---

# Interpretation

$$
W = \\{a,b,c\\}
$$

$$
W \not\models (b \rightarrow d)
$$

### because `b` is an element of `W`, but `d` is not.

---

# Interpretation

$$
W = \\{a,b,c\\}
$$

$$
W \models b \wedge ((c \rightarrow e) \rightarrow d)
$$

* `b` is an element of `W`

* Next we need to interpret `$((c \rightarrow e) \rightarrow d)$`

* For **that** we need to interpret `$c \rightarrow e$`

* `c` is an element of `W`, but `e` is not, so `W` does not model `$c \rightarrow e$`

* However, this means that `W` **does** model `$((c \rightarrow e) \rightarrow d)$`

* And therefore `W` models our entire formula `$b \wedge ((c \rightarrow e) \rightarrow d)$`

---

# Interpretation

$$
W = \\{a,b,c\\}
$$

$$
W \models e \vee (c \wedge \neg d)
$$

* `e` is not an element of `W`

* But maybe we have more luck with the right side `$c \wedge \neg d$`

* `c` is an element of `W`, `d` is not, so `$c \wedge \neg d$` is true

---

# Interpretation

$$
W = \\{a,b,c\\}
$$

$$
W \models \neg d \rightarrow \neg e
$$

* `d` is not an element of `W`

* Neither is `e`

* This means both sides of our implication are true

* And therefore `W` is a model of this formula

---

# Predicate Logic

---

# Predicate Logic

* Representing any non-trivial state in propositional logic is tedious (imagine chess: there has to be one variable for each possible location of each piece)
  
  * *Predicate Logic*, on the other hand, represents the world as objects and relations between them 
  
  * Objects: constants
  
  * Relations: Sets of n-tuples of objects, called predicates (n is the *arity* of the predicate)
  
  * (Functors: Assignments of n-tuples of objects to objects)
  
  * We also have quantifiers `$ \forall, \exists $`
  
---

# Predicate Logic: Example

$$
\mathit{human}(\mathit{socrates})\\\\
\forall h: \mathit{human}(h) \rightarrow \mathit{mortal}(h)\\\\
\mathit{friends}(\mathit{socrates}, \mathit{plato})
$$
--
`$\mathit{human}(h)$` is an unary relation, i.e. a set of 1-tuples/single elements. It is often more convenient to write:

$$
\forall h \in \mathit{human}: \mathit{mortal}(h)
$$
--
Using set-operators, we could also write `$(\mathit{socrates}, \mathit{plato}) \in \mathit{friends}$`

In fact, as we will see, that's how our interpretations work.

---

# Predicate Logic: Syntax

* A predicate name with parameters is called an *Atom* or *atomic formula*, e.g. `$\mathit{human}(h)$`
  
  * A *literal* is an atom of the negation of an atom, e.g. `$\neg \mathit{human}(h)$`
  
  * The `h` can either be a constant or a variable 
  
  * If `h` is part of our *domain*, it is a constant 
  
  * If `h` does not refer to any concrete object, it is a variable. A variable can be *bound* by a quantifier, or it can be *unbound*/*free*
  
  * A formula with no free variables is called *ground*
  
---

# Predicate Logic: Syntax

* Given two predicate logic formulas `$\phi$` and `$\psi$`, we can use the usual logical connectives `$\wedge, \vee, \rightarrow, \neg $` to construct more complex formulas

* Conceptually, this will result in a (syntax) tree, where each interior node is an operator, and the leaves are atoms
  
  * If we have a formula `$\phi$` that contains a free variable `x`, we can *substitute* that variable with a value t, written `$\phi[t/x]$`
  
---

# Predicate Logic: Semantics

* Given a (ground) formula `$\phi$` and an interpretation/world `$W$`, we want to know if the formula holds in that world, i.e. if the world is a model for the formula
  
  * We write this as `$W \models \phi$` ("W models phi")
  
  * How do we determine that? 
  
     - If `$\phi$` is an atom, it is true iff `$\phi \in W$`
     - If `$\phi$` is a more complex formula, use the syntax tree to determine the truth value
     
---

# Predicate Logic: Semantics

$$
W \models \neg \phi \:\:\text{iff}\:\:W \not\models \phi\\\\
W \models \phi \wedge \psi\:\:\text{iff}\:\:W \models \phi\:\:\text{and}\:\:W \models \psi\\\\
W \models \phi \vee \psi\:\:\text{iff}\:\:W \models \phi\:\:\text{or}\:\:W \models \psi\\\\
W \models \phi \rightarrow \psi\:\:\text{iff}\:\:W \not\models \phi\:\:\text{or}\:\:W \models \psi\\\\
W \models \forall x \in X: \phi\:\:\text{iff}\:\:W \models \phi[t/x]\:\:\text{for all}\:t \in X\\\\
W \models \exists x \in X: \phi\:\:\text{iff}\:\:W \models \phi[t/x]\:\:\text{for any}\:t \in X
$$

---

# Predicate Logic: Interpretations

* We start with a *domain*, the set of all objects we can talk about

* Technically, an interpretation has to assign objects to all our constants, since Socrates and Plato could be the same person in an interpretation! We 
  will use the *unique-names assumption* to avoid such problems and will just assume that the constants are the same as the objects.
  
  * Then, the interpretation has to contain definitions for every predicate, in the form of a set of tuples.
  
  * The *closed world assumption* states that everything we don't know to be true (i.e. everything not in the set representing a predicate) is in fact false.
  
  * Using these assumptions, we can represent our interpretation as a set of true atoms, or as a collection of sets, one for each predicate. 
  
---

# Predicate Logic: Interpretations

W:
$$
\mathit{human} = \\{\mathit{socrates}, \mathit{plato}, \mathit{aristotle}\\}\\\\
\mathit{mortal} = \\{\mathit{socrates}, \mathit{plato}, \mathit{aristotle}, \mathit{DonaldDuck}\\}\\\\
\mathit{friends} = \\{(\mathit{socrates}, \mathit{plato}), (\mathit{socrates}, \mathit{DonaldDuck})\\}\\\\
$$

$$
W \models \mathit{human}(\mathit{socrates}) \\\\
W \models \forall h: \mathit{human}(h) \rightarrow \mathit{mortal}(h) \\\\
W \models \mathit{friends}(\mathit{socrates}, \mathit{plato})\\\\
W \not\models \mathit{friends}(\mathit{socrates}, \mathit{socrates})\\\\
\ldots
$$

---

# Predicate Logic: Interpretations

$$
W = \\{\mathit{human}(\mathit{socrates}), \mathit{human}(\mathit{plato}), \mathit{human}(\mathit{aristotle}), \\\\
\mathit{mortal}(\mathit{socrates}), \mathit{mortal}(\mathit{plato}), \mathit{mortal}(\mathit{aristotle}), \mathit{mortal}(\mathit{DonaldDuck}),\\\\
\mathit{friends}(\mathit{socrates}, \mathit{plato}), \mathit{friends}(\mathit{socrates}, \mathit{DonaldDuck})\\}\\\\
$$

---

# Example

You are given an interpretation

$$
W = \\{ \mathit{Lannister}(\mathit{Tywin}), \mathit{Lannister}(\mathit{Tyrion}),
       \mathit{Lannister}(\mathit{Cersei}), \\\\
       \mathit{Lannister}(\mathit{Jaime}), \mathit{owes}(\mathit{Tyrion}, \mathit{Bronn}), \mathit{owes}(\mathit{Jaime},\mathit{Brienne}), \\\\
       \mathit{owes}(\mathit{Brienne}, \mathit{Catelyn}),
       \mathit{paidDebt}(\mathit{Tyrion}, \mathit{Bronn}),
       \mathit{paidDebt}(\mathit{Jaime}, \mathit{Brienne}) \\}
$$

Over the domain

$$
D= \\{\mathit{Tywin},  \mathit{Tyrion}, \mathit{Cersei}, \mathit{Jaime}, \mathit{Bronn}, \mathit{Brienne}, \mathit{Catelyn}\\}
$$

For each of the following formulas, determine if $W$ is a model of that formula over $D$.

$$
\forall x \forall y \in Lannister: \mathit{owes}(y,x) \rightarrow \mathit{paidDebt}(y,x)\\\\
\exists x \forall y: \neg \mathit{owes}(x,y)\\\\
\exists x \forall y: \neg \mathit{owes}(y,x)\\\\
\forall y \exists x: \mathit{owes}(y,x) \vee \mathit{owes}(x,y)\\\\
\forall x: \mathit{owes}(x, \mathit{Catelyn}) \rightarrow \neg \mathit{Lannister}(x)
$$

---

# Example

$$
W \models \forall x \forall y \in Lannister: \mathit{owes}(y,x) \rightarrow \mathit{paidDebt}(y,x)\\\\
$$

* We need to consider all `x` and `y` (for which `y` is in the set Lannister).

* If `y` owes `x`, they also have to pay their debt.

* Tyrion owes Bronn, and Jaime owes Brienne, which are the only Lannisters to owe anyone, and both paid their debt.

**A Lannister always pays his debt**

---

# Example

$$
W \models \exists x \forall y: \neg \mathit{owes}(x,y)
$$

* "There is someone for whom it is false that they owe them, for everyone else" ("There is someone that does not owe anyone").

* Tywin, in our interpretation, does not owe anyone.

---

# Example

$$
W \models \exists x \forall y: \neg \mathit{owes}(x,y)
$$

* "There is someone for whom it is false that they are owed something, for everyone else" ("There is someone that is not owed by anyone").

* Once again, we can use Tywin as `x`, because no one owes Tywin.

---

# Example

$$
W \not\models \forall y \exists x: \mathit{owes}(y,x) \vee \mathit{owes}(x,y)
$$

* "For everyone there is someone that they owe to or that owes them".

* We need to look at each character and see if they have any outgoing or incoming debts.

* However, Tywin has (as we discovered in formula 2) no one that he owes, nor (as we discovered in formula 3) anyone that owed him, and therefore `W` is not a model for this formula.

---

# Example

$$
W \models \exists x \forall y: \neg \mathit{owes}(x,y)
$$

* "Everyone that owes Catelyn is not a Lannister".

* For every `x` we first need to determine if they owe Catelyn, which is only true for Brienne.

* For every other `x` the antecendent is false, and therefore the implication is true.

* For Brienne, the consequence is true, since she is not a Lannister. Therefore, the implication holds for all `x`.

---

# WHY?

* You might say: "But we've already done logic in CS1300"

* Or: "This seems more complicated than necessary"

* So, why are we doing this?

* AI: Representation and Search; logic is our representation

* But that's not enough, we also need to implement it!

---

# Implementing Logic

### What do we need to actually use this?

* Data structure for interpretations and formulas

* Methods to determine truth values ("models")

### What about changes in the world and/or actions our agent can perform?

### Next time we will look into these topics