Propositional Logic

Let’s take a look at the syntax (how sentances are structured)

and semantics (how truth of sentances is determined)

of Propositional Logic

Then we’ll look at a simple algorithm of logical inference that implements entailment. From there, we’ll reason about the treacherous Wumpus World…

Syntax

Syntax defines our allowable sentences.

Atomic sentences consist of a single proposition symbol.

Each proposition symbol can either be true or false. Example proposition symbols: \[ \mathbf{P,Q,R,W_{1,3}},FacingEast \]

The symbols could be anything, but are often mnemonic, e.g. \(W_{1,3}\) could mean that the Wumpus is in \([1,3]\).

Important: Symbols are atomic, the \([1,3]\) might as well be anything.

There are two fixed-meaning propositions: \(True\) and \(False\).

Complex sentences are built-up from from other sentences using \((\ )\) and logical connectives, of which we’ll introduce five:

  • \(\lnot\) … not
  • \(\land\) … and
    • A sentences which uses \(\land\) as the primary connective, e.g. \(W_{1,3}\land P_{3,1}\), is called a conjunction
  • \(\lor\) … or (not-exclusive)
    • similar to the previous connective, disjunction
  • \(\implies\) … implies
    • Consider (\(W_{1,3}\land P_{3,1} \implies \lnot W_{2,2}\))… called an implication (or conditional), the premise or antecedent is (\(W_{1,3}\land P_{3,1}\)), the conclusion or consequent is \(\lnot W_{2,2}\)
    • We can also think of them as rules or if-then statements. (some other books may write this as \(\supset\), or \(\rightarrow\))
  • \(\iff\) … iff (if and only if), a biconditional

Here is a formal grammar of propositional logic presented in BNF (augmented with operator precedence, to resolve ambiguity).

Bracus-Naur Form

Semantics

With the syntax out of the way, we can now focus on meaning.

In propositional logic, a model just sets a truth value (true or false) for every proposition symbol. Example: \[ m_1 = \{P_{1,2}=false,P_{2,2}=false,P_{3,1}=true\} \]

With three proposition symbols, there are how many possible models? ::: {.incremental} - \(2^3=8\) ::: Exactly as seen in: Possible Models

It’s important to note that these modes are purely mathematical objects that don’t necessarily relate to wumpus worlds. \(P_{1,2}\) is just a symbol which could just as well mean “There’s a pit in [1,2]” as I am pogchamp today and tomorrow.

Given a model, our semantics must specify how to compute the truth value of any sentence, given a model, (done recursively).

This requires us to specify how to compute atomic sentences and sentences formed with the five connectives.

First, atomic sentences: ::: {.incremental} - \(True\) is true in every model and \(False\) is false in every model. - The truth of every other proposition must be specified directly in the model. ::: For complex sentences, we have five rules, which hold for any subsentences \(P,Q\) in \(m\): ::: {.incremental} - \(\lnot P\) is true iff \(P\) is false in \(m\) - \(P \land Q\) is true iff both \(P\) or \(Q\) is true in \(m\) - \(P \lor Q\) is true iff either \(P\) or \(Q\) is true in \(m\) - \(P \implies Q\) is true unless \(P\) is true and \(Q\) is false in \(m\) - \(P \iff Q\) is true iff \(P\) and \(Q\) are both true or both false in \(m\) ::: The rules can be expressed by truth tables that specify the truth values to its components.

From these, the truth value of any sentence \(s\) may be computed with respect to any model \(m\) by recursive evaluation.

Let’s try!: not, (\(P\land Q\))

How about: \(\lnot P_{1,2}\land(P_{2,2}\lor P_{3,1})\)

Complete truth table for the five connectives:

Truth Table

One subtle point: \(P\lor Q\) is true if \(P\) is true or \(Q\) is true or both. “Exclusive or” or “xor” is another connective, which yields false when both disjuncts are true. You may see xor as: \(\lor,\neq,or\ \oplus\)

Another odd point is the fact that the following sentence is true under propositional logic:

"5 is odd implies Tokyo is the capital of Japan"

Does this strike anyone as odd :::{.incremental} - implication is not the same as causation or even relevance :::

Let’s think of “\(P\implies Q\)” as “If \(P\) is true, then I am claiming that \(Q\) is true; otherwise I am making no claim”.

A simple knowledge base

Let’s start with the immutable aspects of the wumpus world (we’ll cover mutable aspects later). We need some symbols for each \([x,y]\) location:

  • \(P_{x,y}\) is true if there is a pit in \([x,y]\)
  • \(W_{x,y}\) is true if there is a wumpus in \([x,y]\), dead or alive
  • \(B_{x,y}\) is true if there is a breeze in \([x,y]\)
  • \(S_{x,y}\) is true if there is a stench in \([x,y]\)
  • \(L_{x,y}\) is true if the agent is in location \([x,y]\)

These will be sufficient to derive \(\lnot P_{1,2}\) as was done before (informally). We’ll lable our sentences \(R_i\) so we can refer to them

  • There is no pit in [1,1]:
    • \(R_1: \lnot P_{1,1}\)
  • A square is breezy iff there is a pit in a neighboring square (This is stated for every square)
    • \(R_2: B_{1,1} \iff (P_{1,2}\lor P_{2,1})\)
    • \(R_3: B_{1,1} \iff (P_{1,1}\lor P_{2,2} \lor P_{3,1})\)

These are true in all wumpus worlds!

Now… to include the breeze percepts for the first two squares visited in our specific world:

$R_4: \lnot B_{1,1}$
$R_5: B_{2,1}

A simple inference procedure

Our goal is to decide whether \(KB\models \alpha\) (KB models alpha) for some sentence alpha.

Models are assignments of \(true\) or \(false\) to every proposition symbol.

In our Wumpus World, the relevant propositions are \(B_{1,1},B_{2,1},P_{1,1},P_{1,2},P_{2,1},P_{2,2},P_{3,1}\) With seven symbols we have \(128\) possible models… in three of them, \(KB\) is true.

In those three, \(\lnot P_{1,2}\) is true, therefore there is no pit in \([x,y]\). However, \(P_{2,2}\) is true in two, but false in one… so we can’t tell yet.

Our large truth table: Truth Table

A general algorithm for deciding entailment in propositional logic is below.

Like our \(BacktrackingSearch\), \(TTEntails\) searches recursively over a finite space of assignments to symbols. It is both sound and complete… why?

Truth Table Enumeration

That said, “finite” is not always the same as “small”.

If \(KB\) and \(\alpha\) contain \(n\) symbols, then there are \(2^n\) models. Thus, the time complexity is \(O(2^n)\) (the space complexity is \(O(n)\) due to it being DFS).

Unfortunately, propositional entailment is co-NP-complete (probably no easier than NP-complete).

Lab time!