Logic: logic.py; Chapters 6-8

This notebook describes the logic.py module, which covers Chapters 6 (Logical Agents), 7 (First-Order Logic) and 8 (Inference in First-Order Logic) of Artificial Intelligence: A Modern Approach. See the intro notebook for instructions.

We'll start by looking at Expr, the data type for logical sentences, and the convenience function expr. We'll be covering two types of knowledge bases, PropKB - Propositional logic knowledge base and FolKB - First order logic knowledge base. We will construct a propositional knowledge base of a specific situation in the Wumpus World. We will next go through the tt_entails function and experiment with it a bit. The pl_resolution and pl_fc_entails functions will come next. We'll study forward chaining and backward chaining algorithms for FolKB and use them on crime_kb knowledge base.

But the first step is to load the code:

Logical Sentences

The Expr class is designed to represent any kind of mathematical expression. The simplest type of Expr is a symbol, which can be defined with the function Symbol:

Or we can define multiple symbols at the same time with the function symbols:

We can combine Exprs with the regular Python infix and prefix operators. Here's how we would form the logical sentence "P and not Q":

This works because the Expr class overloads the & operator with this definition:

def __and__(self, other): return Expr('&',  self, other)

and does similar overloads for the other operators. An Expr has two fields: op for the operator, which is always a string, and args for the arguments, which is a tuple of 0 or more expressions. By "expression," I mean either an instance of Expr, or a number. Let's take a look at the fields for some Expr examples:

It is important to note that the Expr class does not define the logic of Propositional Logic sentences; it just gives you a way to represent expressions. Think of an Expr as an abstract syntax tree. Each of the args in an Expr can be either a symbol, a number, or a nested Expr. We can nest these trees to any depth. Here is a deply nested Expr:

Operators for Constructing Logical Sentences

Here is a table of the operators that can be used to form sentences. Note that we have a problem: we want to use Python operators to make sentences, so that our programs (and our interactive sessions like the one here) will show simple code. But Python does not allow implication arrows as operators, so for now we have to use a more verbose notation that Python does allow: |'==>'| instead of just ==>. Alternately, you can always use the more verbose Expr constructor forms:

Here's an example of defining a sentence with an implication arrow:

expr: a Shortcut for Constructing Sentences

If the |'==>'| notation looks ugly to you, you can use the function expr instead:

expr takes a string as input, and parses it into an Expr. The string can contain arrow operators: ==>, <==, or <=>, which are handled as if they were regular Python infix operators. And expr automatically defines any symbols, so you don't need to pre-define them:

For now that's all you need to know about expr. If you are interested, we explain the messy details of how expr is implemented and how |'==>'| is handled in the appendix.

Propositional Knowledge Bases: PropKB

The class PropKB can be used to represent a knowledge base of propositional logic sentences.

We see that the class KB has four methods, apart from __init__. A point to note here: the ask method simply calls the ask_generator method. Thus, this one has already been implemented and what you'll have to actually implement when you create your own knowledge base class (if you want to, though I doubt you'll ever need to; just use the ones we've created for you), will be the ask_generator function and not the ask function itself.

The class PropKB now.

• __init__(self, sentence=None) : The constructor __init__ creates a single field clauses which will be a list of all the sentences of the knowledge base. Note that each one of these sentences will be a 'clause' i.e. a sentence which is made up of only literals and ors.
• tell(self, sentence) : When you want to add a sentence to the KB, you use the tell method. This method takes a sentence, converts it to its CNF, extracts all the clauses, and adds all these clauses to the clauses field. So, you need not worry about telling only clauses to the knowledge base. You can tell the knowledge base a sentence in any form that you wish; converting it to CNF and adding the resulting clauses will be handled by the tell method.
• ask_generator(self, query) : The ask_generator function is used by the ask function. It calls the tt_entails function, which in turn returns True if the knowledge base entails query and False otherwise. The ask_generator itself returns an empty dict {} if the knowledge base entails query and None otherwise. This might seem a little bit weird to you. After all, it makes more sense just to return a True or a False instead of the {} or None But this is done to maintain consistency with the way things are in First-Order Logic, where, an ask_generator function, is supposed to return all the substitutions that make the query true. Hence the dict, to return all these substitutions. I will be mostly be using the ask function which returns a {} or a False, but if you don't like this, you can always use the ask_if_true function which returns a True or a False.
• retract(self, sentence) : This function removes all the clauses of the sentence given, from the knowledge base. Like the tell function, you don't have to pass clauses to remove them from the knowledge base; any sentence will do fine. The function will take care of converting that sentence to clauses and then remove those.

Wumpus World KB

Let us create a PropKB for the wumpus world with the sentences mentioned in section 7.4.3.

We define the symbols we use in our clauses.
$P_{x, y}$ is true if there is a pit in [x, y].
$B_{x, y}$ is true if the agent senses breeze in [x, y].

Now we tell sentences based on section 7.4.3.
There is no pit in [1,1].

A square is breezy if and only if there is a pit in a neighboring square. This has to be stated for each square but for now, we include just the relevant squares.

Now we include the breeze percepts for the first two squares leading up to the situation in Figure 7.3(b)

We can check the clauses stored in a KB by accessing its clauses variable

We see that the equivalence $B_{1, 1} \iff (P_{1, 2} \lor P_{2, 1})$ was automatically converted to two implications which were inturn converted to CNF which is stored in the KB.
$B_{1, 1} \iff (P_{1, 2} \lor P_{2, 1})$ was split into $B_{1, 1} \implies (P_{1, 2} \lor P_{2, 1})$ and $B_{1, 1} \Longleftarrow (P_{1, 2} \lor P_{2, 1})$.
$B_{1, 1} \implies (P_{1, 2} \lor P_{2, 1})$ was converted to $P_{1, 2} \lor P_{2, 1} \lor \neg B_{1, 1}$.
$B_{1, 1} \Longleftarrow (P_{1, 2} \lor P_{2, 1})$ was converted to $\neg (P_{1, 2} \lor P_{2, 1}) \lor B_{1, 1}$ which becomes $(\neg P_{1, 2} \lor B_{1, 1}) \land (\neg P_{2, 1} \lor B_{1, 1})$ after applying De Morgan's laws and distributing the disjunction.
$B_{2, 1} \iff (P_{1, 1} \lor P_{2, 2} \lor P_{3, 2})$ is converted in similar manner.

Inference in Propositional Knowlwdge Base

In this section we will look at two algorithms to check if a sentence is entailed by the KB. Our goal is to decide whether $\text{KB} \vDash \alpha$ for some sentence $\alpha$.

Truth Table Enumeration

It is a model-checking approach which, as the name suggests, enumerates all possible models in which the KB is true and checks if $\alpha$ is also true in these models. We list the $n$ symbols in the KB and enumerate the $2^{n}$ models in a depth-first manner and check the truth of KB and $\alpha$.

Note that tt_entails() takes an Expr which is a conjunction of clauses as the input instead of the KB itself. You can use the ask_if_true() method of PropKB which does all the required conversions. Let's check what wumpus_kb tells us about $P_{1, 1}$.

Looking at Figure 7.9 we see that in all models in which the knowledge base is True, $P_{1, 1}$ is False. It makes sense that ask_if_true() returns True for $\alpha = \neg P_{1, 1}$ and False for $\alpha = P_{1, 1}$. This begs the question, what if $\alpha$ is True in only a portion of all models. Do we return True or False? This doesn't rule out the possibility of $\alpha$ being True but it is not entailed by the KB so we return False in such cases. We can see this is the case for $P_{2, 2}$ and $P_{3, 1}$.

Proof by Resolution

Recall that our goal is to check whether $\text{KB} \vDash \alpha$ i.e. is $\text{KB} \implies \alpha$ true in every model. Suppose we wanted to check if $P \implies Q$ is valid. We check the satisfiability of $\neg (P \implies Q)$, which can be rewritten as $P \land \neg Q$. If $P \land \neg Q$ is unsatisfiable, then $P \implies Q$ must be true in all models. This gives us the result "$\text{KB} \vDash \alpha$ if and only if $\text{KB} \land \neg \alpha$ is unsatisfiable".
This technique corresponds to proof by contradiction, a standard mathematical proof technique. We assume $\alpha$ to be false and show that this leads to a contradiction with known axioms in $\text{KB}$. We obtain a contradiction by making valid inferences using inference rules. In this proof we use a single inference rule, resolution which states $(l_1 \lor \dots \lor l_k) \land (m_1 \lor \dots \lor m_n) \land (l_i \iff \neg m_j) \implies l_1 \lor \dots \lor l_{i - 1} \lor l_{i + 1} \lor \dots \lor l_k \lor m_1 \lor \dots \lor m_{j - 1} \lor m_{j + 1} \lor \dots \lor m_n$. Applying the resolution yeilds us a clause which we add to the KB. We keep doing this until:

First-Order Logic Knowledge Bases: FolKB

The class FolKB can be used to represent a knowledge base of First-order logic sentences. You would initialize and use it the same way as you would for PropKB except that the clauses are first-order definite clauses. We will see how to write such clauses to create a database and query them in the following sections.

Criminal KB

In this section we create a FolKB based on the following paragraph.
The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.
The first step is to extract the facts and convert them into first-order definite clauses. Extracting the facts from data alone is a challenging task. Fortnately we have a small paragraph and can do extraction and conversion manually. We'll store the clauses in list aptly named clauses.

“... it is a crime for an American to sell weapons to hostile nations”
The keywords to look for here are 'crime', 'American', 'sell', 'weapon' and 'hostile'. We use predicate symbols to make meaning of them.

$\text{American}(x) \land \text{Weapon}(y) \land \text{Sells}(x, y, z) \land \text{Hostile}(z) \implies \text{Criminal} (x)$

"The country Nono, an enemy of America"
We now know that Nono is an enemy of America. We represent these nations using the constant symbols Nono and America. the enemy relation is show using the predicate symbol Enemy.

$\text{Enemy}(\text{Nono}, \text{America})$

"Nono ... has some missiles"
This states the existance of some missile which is owned by Nono. $\exists x \text{Owns}(\text{Nono}, x) \land \text{Missile}(x)$. We invoke existential instantiation to introduce a new constant M1 which is the missile owned by Nono.

$\text{Owns}(\text{Nono}, \text{M1}), \text{Missile}(\text{M1})$

"All of its missiles were sold to it by Colonel West"
If Nono owns something and it classifies as a missile, then it was sold to Nono by West.

$\text{Missile}(x) \land \text{Owns}(\text{Nono}, x) \implies \text{Sells}(\text{West}, x, \text{Nono})$

"West, who is American"
West is an American.

$\text{American}(\text{West})$

We also know, from our understanding of language, that missiles are weapons and that an enemy of America counts as “hostile”.

$\text{Missile}(x) \implies \text{Weapon}(x), \text{Enemy}(x, \text{America}) \implies \text{Hostile}(x)$

Now that we have converted the information into first-order definite clauses we can create our first-order logic knowledge base.

Inference in First-Order Logic

In this section we look at a forward chaining and a backward chaining algorithm for FolKB. Both the aforementioned algorithms rely on a process called unification, a key component of all first-order inference algorithms.

Unification

We sometimes require finding substitutions that make different logical expressions look identical. This process, called unification, is done by the unify algorithm. It takes as input two sentences and returns a unifier for them if one exists. A unifier is a dictionary which stores the substitutions required to make the two sentences identical. It does so by recursively unifying the components of a sentence, where the unification of a variable symbol var with a constant symbol Const is the mapping {var: Const}. Let's look at a few examples.

In cases where there is no possible substitution that unifies the two sentences the function return None.

We also need to take care we do not unintentionally use same variable name. Unify treats them as a single variable which prevents it from taking multiple value.

Forward Chaining Algorithm

We consider the simple forward-chaining algorithm presented in Figure 9.3. We look at each rule in the knoweldge base and see if the premises can be satisfied. This is done by finding a substitution which unifies the each of the premise with a clause in the KB. If we are able to unify the premises the conclusion (with the corresponding substitution) is added to the KB. This inferencing process is repeated until either the query can be answered or till no new sentences can be aded. We test if the newly added clause unifies with the query in which case the substitution yielded by unify is an answer to the query. If we run out of sentences to infer, this means the query was a failure.

The function fol_fc_ask is a generator which yields all substitutions which validate the query.

Let's find out all the hostile nations. Note that we only told the KB that Nono was an enemy of America, not that it was hostile.

The generator returned a single substitution which says that Nono is a hostile nation. See how after adding another enemy nation the generator returns two substitutions.

Note: fol_fc_ask makes changes to the KB by adding sentences to it.

Backward Chaining Algorithm

This algorithm works backward from the goal, chaining through rules to find known facts that support the proof. Suppose goal is the query we want to find the substitution for. We find rules of the form $\text{lhs} \implies \text{goal}$ in the KB and try to prove lhs. There may be multiple clauses in the KB which give multiple lhs. It is sufficient to prove only one of these. But to prove a lhs all the conjuncts in the lhs of the clause must be proved. This makes it similar to And/Or search.

OR

The OR part of the algorithm comes from our choice to select any clause of the form $\text{lhs} \implies \text{goal}$. Looking at all rules's lhs whose rhs unify with the goal, we yield a substitution which proves all the conjuncts in the lhs. We use parse_definite_clause to attain lhs and rhs from a clause of the form $\text{lhs} \implies \text{rhs}$. For atomic facts the lhs is an empty list.

AND

The AND corresponds to proving all the conjuncts in the lhs. We need to find a substitution which proves each and every clause in the list of conjuncts.

Now the main function fl_bc_ask calls fol_bc_or with substitution initialized as empty. The ask method of FolKB uses fol_bc_ask and fetches the first substitution returned by the generator to answer query. Let's query the knowledge base we created from clauses to find hostile nations.

You may notice some new variables in the substitution. They are introduced to standardize the variable names to prevent naming problems as discussed in the Unification section

Appendix: The Implementation of |'==>'|

Consider the Expr formed by this syntax:

What is the funny |'==>'| syntax? The trick is that "|" is just the regular Python or-operator, and so is exactly equivalent to this:

In other words, there are two applications of or-operators. Here's the first one:

What is going on here is that the __or__ method of Expr serves a dual purpose. If the right-hand-side is another Expr (or a number), then the result is an Expr, as in (P | Q). But if the right-hand-side is a string, then the string is taken to be an operator, and we create a node in the abstract syntax tree corresponding to a partially-filled Expr, one where we know the left-hand-side is P and the operator is ==>, but we don't yet know the right-hand-side.

The PartialExpr class has an __or__ method that says to create an Expr node with the right-hand-side filled in. Here we can see the combination of the PartialExpr with Q to create a complete Expr:

This trick is due to Ferdinand Jamitzky, with a modification by C. G. Vedant, who suggested using a string inside the or-bars.

Appendix: The Implementation of expr

How does expr parse a string into an Expr? It turns out there are two tricks (besides the Jamitzky/Vedant trick):

1. We do a string substitution, replacing "==>" with "|'==>'|" (and likewise for other operators).
2. We eval the resulting string in an environment in which every identifier is bound to a symbol with that identifier as the op.

In other words,

is equivalent to doing:

One thing to beware of: this puts ==> at the same precedence level as "|", which is not quite right. For example, we get this:

which is probably not what we meant; when in doubt, put in extra parens:

Examples

Authors

This notebook by Chirag Vartak and Peter Norvig.