I had previously mentioned that the classic AI textbook by Russell and Norvig (2010) was fairly easy reading compared to most computer science textbooks. Now that I’ve recently gone through the first half of the book (which is about 500 pages) in the span of two weeks, I stand by my claim. Reading all these pages, however, does not necessarily mean that I would sufficiently absorb the material to the extent I wish, so in this post, I’ll give a brief overview of what’s covered in the first half of the book. The first two chapters serve as an introduction to AI, as a review of how the field came to be, and how we wish to design AI agents that are rational, which means that they make decisions that “make sense” according to some utility definition. There isn’t that much to see here.

Part II: Problem Solving

This part encompasses chapters 3 through 6 and is about problem-solving. Yes! Now we’re onto something that’s interesting, and something that’s also covered in every AI course. And every algorithms course, because what’s in chapter 3? Search algorithms on graphs!

Chapter 3: Solving Problems by Searching

The following list outlines the most important search algorithms to know:

  • Breadth-First Search (BFS), a strategy where we start from a root node, expand it to generate its children, and then put those children in a queue (i.e, FIFO) to expand then later. This means all nodes at some depth level \(d\) of the tree get expanded before any node at depth level \(d+1\) gets expanded. The goal test is applied when nodes are immediately detected (i.e., before adding it to the queue) because there’s no benefit to continue checking nodes. BFS is complete and optimal, but it also suffers from horrible space and time complexity.

  • Uniform Cost Search (UCS) is like BFS, except that it orders the nodes to expanded in a priority queue based on some path cost function \(g(n)\). One would want to use UCS rather than BFS in cases when a step cost (i.e., the cost of traversing from one node to another) is not uniform. Technically, this means without a uniform step function, BFS isn’t optimal, but usually we are smart enough to not apply BFS in those situations. UCS applies the goal test when a node is expanded, i.e., when it is pulled off the queue. This is later than when BFS would check. UCS is complete and optimal (so long as edge costs are nonzero, to prevent infinite loops) but can suffer from the same complexity issues as BFS.

  • Depth First Search (DFS) expands the deepest node in the search frontier, so it stores the frontier as a LIFO stack. Unfortunately, this means that DFS can traverse one really long path forever, without stopping to check back at other unexpanded noes near the beginning, so it’s clearly non-optimal. The real savings for DFS comes with space complexity, because once a node has been expanded, it can be removed from memory once all its descendants have been fully explored.

  • Depth-Limited Search is like DFS, except that nodes at a depth limit \(\ell\) are treated as if they had no children. This can avoid DFS spiraling off in wild directions, but it also means that we will never reach the goal if the shallowest goal is beyond the depth limit.

  • Iterative Deepening Search (IDS) is another version of depth-first search, and here we run depth-limited search multiple times, increasing the depth limit by one each time so as to gradually get close to the goal. It’s not as slow as one might think, because we would be repeating most of the initial node expansions, which have a lower branching factor.

  • Bidirectional Search means that we run two searches, one from the starting state and another from the ending state. The real challenge is how to combine the two search paths in the middle, and how to backtrack if necessary. In the \(n\)-queens problem, it’s not clear how to backtrack.

  • Greedy Best-First Search uses a heuristic function (explained later) to choose which node to expand. This means that nodes are stored in a priority queue according to \(h(n)\). This may seem a lot like UCS (and it is), but here, \(h(n)\) is the estimated cost of reaching the goal from node \(n\), not the overall path cost we have seen so far.

  • A-Star Search fixes problems with greedy best-first search by supplementing \(h(n)\) with the path cost seen so far, \(g(n)\). So here, the nodes would be stored in a priority queue based on \(g(n)+h(n)\), where the first term indicates a known cost so far, and the second indicates our estimate of the future cost. A-Star Search is probably the most widely used form of best-first search.

For problems that use heuristic functions (i.e., \(h(n)\) in the above notation), one would like heuristics that are admissible and consistent, because then that would make A-Star search complete and optimal. Admissible means that the function never overestimates the cost of reaching to the goal, and consistent means that the function obeys the triangle inequality1. Every consistent heuristic is also admissible, but the reverse is not true. Probably the canonical example of a consistent heuristic in traveling salesman-like problems is when we use the straight-line distance from one city to another (well, assuming that our travel speed would be uniform across all possible routes).

If one has multiple admissible heuristics \(h_1,\ldots,h_m\) for a problem, and none dominates the other, then we should take the max of those for each node, \(h(n) = \max\{h_1(n), \ldots, h_m(n)\}\). Do not take the sum if we want an admissible heuristic!

Amusingly enough, this chapter is about trying to extend the previous one to bring it closer to the “real world.” Though admittedly, yes it is more like what most people would use. The first part of this chapter gives a very brief introduction to the field of optimization. Hill-climbing search is a search algorithm that attempts to move in the direction of increasing utility value, and is a more general version of the commonly-used gradient-descent algorithm, which is only applicable in continuous domains. The main problem with these local algorithms is that they can get stuck in local minima (or maxima, depending on whichever is most convenient for the problem description), so one should run the algorithm multiple times with random starting positions. Alternatively (or in addition), one can use simulated annealing. The way to think about how that works is to imagine a local minima problem where we have a small ping-pong ball on a curvy, bumpy surface and are trying to get it to rest in the deepest crevice. The ball would obviously get stuck in a local minima easily due to gravity, so simulated annealing is like “shaking” the surface enough to shoot the ball out of a small valley, but not out of the actual global minima, which would be like a deep, giant pit. I find this analogy a lot easier to understand than most descriptions of simulated annealing, by the way.

The fact that we bring up gradient descent is interesting, because the search problems in Chapter 3 cannot handle continuous domains due to the infinite branching factor. Said another way, a human has an infinite number of ways to walk in a specific direction along the 360 degree circle; how would we design DFS to help a robot do that? One way to find optimal solutions in continuous spaces is to discretize the problem, so going back to my “human walking” example, we might limit the search directions to be anywhere from 0 up to 30 degrees, any where from 30 up to 60 degrees, etc. Another, of course, is to use the gradient and update the current state according to \(X \leftarrow X + \alpha \nabla f(X)\). We could also have constrained optimization problems, of which the best known and easily solvable problems are of the linear programming variety.

Remember that in Chapter 3, the environment is assumed to be fully deterministic and observable. It is also interesting to see how we would design an agent to search in spaces that have non-deterministic actions and if the agent can only have partial or even no (!) observations about the world. The key idea here is that we have to make use of an agent’s percepts that will help inform it what state it is in, so that we can say things like “if X happens, do Y, else do Z.” We can still design search trees and traverse them to reach the goal, but the trees have different flavors. In the nondeterministic environment case, we would need to have edges between a parent node and all its possible children nodes that could result. In the partial observation case, we would have the tree’s nodes be belief states, so each node is actually a set that contains the possible states the agent could be in. Gradually moving around in the search space might narrow down the set of possible belief states. The book’s vacuum cleaner example helps to explain how even an agent with no observations can still tell which state it is in given that it executes a specific sequence of actions and knows the consequences. In fact, having a sensor-less agent can be advantageous in situations when it would be expensive to pin down an exact state, which is why doctors tend to prescribe a broad spectrum antibiotic rather than perform detailed analysis of a patient to decide on an incredibly specific drug.

As an interesting side note, I was reading sections 4.3 and 4.4 and noticed the similarities between the graphs provided here and the finite automata I learned from my undergraduate theoretical computer science course. We have notions of a state, a transition function, actions, and final (i.e., goal) states. Section 4.5, discussing online search, also uses these graphs. The nice thing about the book’s organization is that the search algorithms from Chapter 3 can be applied to the graphs on Chapter 4, along with additional problem-specific restrictions.

We spent a lot of time in my undergraduate AI course on adversarial search. This is like what we consider in Chapters 3 and 4, except that the agent is no longer alone, and its actions are in conflict with other agents. The simplest abstraction of adversarial search is a two-player game involving one overall game score. The players are named MAX and MIN because they wish to maximize and minimize, respectively, the overall game score. In normal search algorithms, a player named MAX (who by convention tends to start these games) would just search for and form a sequence of moves that would reach a terminal state. Unfortunate, MIN can stop this in some cases. So the best algorithm for MAX to pursue is the minimax algorithm, because it minimizes the worst-case scenario2. The easiest way to view this algorithm is to draw a graph of the game tree with various scores. Levels would alternate between a MAX player and a MIN player, so the MAX player should trace through the entire game tree and recursively backtrack to check and see the best score it can get on each node assuming that MIN plays optimally.

The problem with minimiax search is that nontrivial games have too many possible states; it’s actually exponential in the depth of the tree. By using alpha-beta pruning to prune away branches that cannot possibly affect the final decision, we can cut the exponent in about half and still return the same solution used by minimiax. For any given state at any time in the search algorithm, alpha is the value of the best choice found along the path for MAX, and beta is the value of the best choice found along the path for MIN. Alpha starts as negative infinity and tries to go up, whereas beta starts as positive infinity and tries to go down. It’s easiest to see how this works by tracing through some game trees.

I remember these algorithms well because my undergraduate AI courses used the Berkeley Pacman assignments3, which involved heavy use of minimiax search and alpha-beta pruning. I remember that our problem involved four agents: Pacman (us) and three ghosts who wanted to eat him. With more than two players, we can associate a vector of values, and I think that’s what we did in the assignment, since the description says:

Now you will write an adversarial search agent in the provided MinimaxAgent class stub in multiAgents.py. Your minimax agent should work with any number of ghosts, so you’ll have to write an algorithm that is slightly more general than what you’ve previously seen in lecture. In particular, your minimax tree will have multiple min layers (one for each ghost) for every max layer.

If you’ve checked the AI project description, you’ll also see that we only run minimax algorithms to a limited depth, sometimes as small as just two layers. This is necessary due to the exponential explosion in the number of states in the Pacman maze. Another way to speed up the searching (but again, at the cost of optimality) is to treat nonterminal nodes at a given level as terminal nodes, and create a heuristic evaluation function for their values. (Yes, this is very similar to Chapter 3 material!) After all, this is what humans do when they play games. I can’t remember 20 moves ahead in a chess game, but I can reason that moving my queen to capture an opponent’s queen, while not threatening any of my pieces in the process, will have a higher utility for me.

One can also use minimax algorithms with games involving chance, which means that game nodes have chance nodes in addition to the normal MAX and MIN nodes. To make correct decisions here, we have to change our analysis to consider expected values.

Chapter 6: Constraint Satisfaction Problems

Now we’ll switch gears and focus on problems that have more sophisticated notions of a “state.” The reason for doing this is that algorithms like DFS, BFS, etc., assume that states are just black boxes. There is no domain-specific part of those search algorithm to those problems4. With constraint satisfaction problems (CSPs), we represent each state as a set of variables \((X_1,\ldots,X_m)^{(i)}\), and a problem is solved when each variable has a value that satisfies all the constraints imposed from the states and problem formulation. The example used in the book is about coloring the seven regions of Australia so that no two bordering regions have the same color. To formulate it as a CSP, we

  • define seven variables to be the seven regions
  • define the domain for each variable, which consists of three colors for us
  • define the constraints, which means listing all the color inequalities from bordering regions

Just to be clear, why do we want to use CSPs? Here are a few reasons:

  • It is nice to have a single solver for a CSP. We can then solve a problem by converting it to a CSP, and then running our CSP solver. This is what a lot of theorists do when they reduce problems to known ones.
  • There is no need to develop a detailed, problem-specific heuristic.
  • CSPs can eliminate large portions of the search space all at once by quickly identifying variable assignments that violate constraints.

It’s worth discussing that last point in more detail. In the search problem of previous chapters, our search algorithm can search. With a CSP, we can perform inference called constraint propagation, which uses the constraints to reduce the number of legal values for variables. As the book delightfully points out, Sudoku is a problem that has “introduced millions of people to constraint satisfaction problems.” Constraint propagation is obvious here: if I see a column of variables that has all values other than 3 and 7 filled in (i.e., two empty squares), I can identify one of those spots that I want to fill in and constrain the number of possible variables from nine to two. If I then see that the square coincides with a row that already has 3 in it (but not 7, unless something went wacky), then that further constraints the choice of my variable to be 7 … and I will obviously put 7 there.

But while Sudoku problems can be solved by inference over constraints, sometimes we just have to search for a solution, and here is where backtracking search comes into play. This is a depth-first search algorithm that goes down the tree assigning values to variables, and if it reaches a point where a variable has no legal values left to assign, then there is clearly no solution, so it backtracks to previous variables to try and perform different assignments. The order that we assign the variables does not matter, which helps to cut down on the branching factor.

To make backtracking search more efficient without using problem-specific knowledge, we should decide on solid heuristics for the following:

  • What is the order in which we should choose to assign variables, and in what order should the possible assignments be done?
  • What inferences should be performed at each step in the search?
  • When the search arrives at an assignment that violates a constraint, how can we avoid repeating this failure?

For choosing the variable ordering, one way is the minimum-remaining-values (MRV) heuristic, or choosing the variable that has the fewest legal values, because then we can detect failure quicker. When we do choose a variable, but have to assign it from the list of possible values, it actually makes sense to follow least-constraining-value heuristic, or choose the variable that rules out the fewest choices for the neighboring variables in the constraint graph (this is a graphical representation of the CSP) because it allows the possibility of more solutions down the road (ideally). So, most constrained when choosing a variable, and least constrained when assigning that chosen variable.

To perform inference, we can do forward checking after we assign a variable. This establishes arc consistency among adjacent variables in the constraint graph by iteratively updating constraints on those variables. However, this will fail in simple cases such as after we have assigned a variable a color in a problem where we have to two-color the \(\mathcal{K}_3\) graph, because forward checking can’t reason about arcs that don’t directly include the currently assigned variable.

When we violate a constraint, we can backtrack one step up in the DFS tree like normal DFS, but that tends to work poorly because if we have an inconsistency, then we may have made a mistake much earlier in our sequence of variable assignments, so we want to backtrack far up in the tree beyond the most recent decision point. We can design a backjumping method by maintaining conflict sets for each variable, or in other words, a set of assignments that are in conflict with a variable assignment. Then the backtracking process would go back to the most recent variable assignment in the conflict set. However, forward checking already supplies the conflict set (check this yourself!), and so “simple” backjumping as previously described is redundant in a forward checking search or a search that utilizes arc consistency measures.

Instead, we can use the more sophisticated conflict-directed backjumping. Instead of backjumping once we detect a failure based on conflict sets, we should backtrack all the way before that to the point where the branch “gets doomed.” Clearly, this is a more challenging task, and we do this by redefining what it means to be a conflict set: for a variable, its conflict set is the set of preceding variables that caused this one, together with any subsequent variables, to have no consistent solution. These conflict sets are computed by an ingenious method of “absorbing” other nodes’ conflict sets.

Once we have our constraint graph, we can also apply some local search techniques from Chapter 4 (e.g., simulated annealing) to CSPs.

The previous stuff is very general, but honestly, if you look at the problem and can figure out something from it that is “obviously” going to make the problem easier, do it! In the Australian color mapping, Tasmania was not connected to the mainland, so it’s obvious that it shouldn’t have been part of the original coloring problem at all! Thus, splitting the graph into connected components would have been a smart tactic. Another way to make a problem easier is to reduce their constraint graphs to trees, because any tree-structured CSP can be solved in time linear in the number of variables. We can assign values to variables so that the remaining ones form a tree, or we can do a more sophisticated tree-decomposition, where nodes are now a set of variables, and variables can be part of multiple nodes. Each node represents its own subproblem.

Part III: Knowledge, Reasoning, and Planning

This part of the book is a little dry, and is about how one can design “languages” or various formalisms for agents to help them reason and plan about the world by extracting from a knowledge base. In my undergraduate AI course, we barely covered this part at all, and it was only towards the last week of class, when attendance was half the normal level because we didn’t have a final exam. I’m not sure how important this part is to AI research nowadays, since AI tends to be synonymous with machine learning these days. But maybe in some parts of robot motion planning?

Nevertheless, I still decided to read the entirety of it as there might be some important stuff here.

Chapter 7: Logical Agents

In which we design agents that can form representations of a complex world, use a process of inference to derive new representations about the world, and use these new representations to deduce what to do.

This somewhat boring chapter5 introduces the class of logic known as propositional logic, which lets agents represent the world through a series of statements and provides inference techniques to make conclusions. This is an upgrade over the agents in previous chapters. Why? When we tell Pacman to perform DFS to determine where it should move in the game, Pacman doesn’t really know anything about the game. A human can deduce a number of facts from the world, such as that Pacman should avoid going towards dead ends if a ghost is behind it and there is no power-up available there, but to the point of view of the DFS search agent, that knowledge is irrelevant. To say it another way, search agents only know the world in a very narrow, inflexible sense, and they can’t make real conclusions. They cannot reason. Constraint Satisfaction Problems alleviate this knowledge block by changing the representation of states from atomic to a set of variables, which allows for more efficient inference techniques (arc consistency, etc.), so there is a little bit of reasoning going on. Here, in Chapter 7, we take another step by representing the world not through a set of states and variables, but through logic. Remember that throughout this chapter (and the subsequent chapters), the overall theme is representation.

A few terms are in order to review:

  • A knowledge base (KB) is the central component of our agents and will contain all the set of sentences (each represented with a specific syntax) that are known to the agent. A knowledge base is monotonic if the set of entailed sentences only increases as new information is added. Otherwise, it would be like the model is changing its mind.
  • But wait, what is entailment? It is the idea that a sentence follows logically from another sentence. By writing \(\alpha \models \beta\), we state that \(\alpha\) entails \(\beta\), so that in every model in which \(\alpha\) is true, \(\beta\) is also true. The relation implies that \(\alpha\) is a stronger assertion than \(\beta\) (check this yourself).
  • An inference algorithm is one that uses existing logical sentences and derives logically valid conclusions from them. If we have in our knowledge base that \(x=0\) and \(y=1\) then we should be able to make the conclusion somehow that \(x+y=1\). Algorithms that are sound derive only entailed sentences (this is a good thing), and algorithms that are complete can derive any sentence that is entailed from the KB, so complete algorithms are also sound. The slowest complete inference algorithm (assuming finite spaces) is model-checking because it enumerates all models to check for entailment. This is not a scalable solution.

Propositional logic includes the following:

  • atomic sentences, which consist of one symbol
  • not connectives (\(\neg\)) to negate an expression
  • and connectives (\(\wedge\)) to join two expressions together (producing a conjunction)
  • or connectives (\(\vee\)) to join two expressions together (producing a disjunction)
  • implies relationships: \(\alpha \Rightarrow \beta\)
  • if and only if relationships: \(\alpha \iff \beta\)

The semantics of these relationships are what one would expect, i.e., directly based on your discrete math or math logic class.

How do we use these facts to perform inference, and to do it efficiently6? In other words, the ultimate question is that we want to decide if \(KB \models \alpha\) (logically equivalent to \(KB \Rightarrow \alpha\)) for some sentence \(\alpha\). For this, we do some theorem-proving. An important rule is Modus Ponens, which states that whenever \(\alpha \Rightarrow \beta\) is true, and if \(\alpha\) is true, then \(\beta\) has to be true. (This makes sense because \(\alpha \Rightarrow \beta\) is only false if \(\alpha\) were true but \(\beta\) false.) There are a few others, such as and-elimination, which reduces \(\alpha \wedge \beta\) to (without loss of generality) \(\alpha\), but I generally apply these rules by directly appealing to what I remember about logic, rather than trying to remember rule names and their exact syntax. This must be why I hate resolving logic by hand.

A rule like Modus Ponens is sound, but incomplete. For a complete rule, we want to use resolution7, which simplifies our problem by eliminating clauses that resolve with each other and don’t contribute to the resulting truth values. If we have \((P_1 \vee P_2) \wedge (\neg P_1 \vee P_3)\), then we can simplify the sentence to be \(P_2 \vee P_3\) because \(P_1 \vee \neg P_1\) are complimentary, so they cannot both force their respective clauses to be true. This is resolution. It applies to pairs of arbitrarily long clauses ANDed together. The key fact is that a resolution-based theorem prover can, for any sentences \(\alpha\) and \(\beta\) in propositional logic, decide whether \(\alpha \models \beta\). Why?

  • Every sentence of propositional logic is logically equivalent to a conjunction of clauses, or said another way, every sentence can be converted into CNF form. This is important for resolution because it relies on there being a disjunction of literals. (Again, a sentence is in CNF form if it is an AND of ORs, and a clause is a disjunction of symbols.)
  • Resolution-based theorem provers work by using contradiction. To show that \(KB \models \alpha\), we show \((KB \wedge \neg \alpha)\) is unsatisfiable. Starting with a sentence in CNF form, we apply the resolution rule to pairs of clauses to produce (potentially) new clauses. We continue with this until there are no new clauses that can be added (\(KB\) does not entail \(\alpha\)) or if any two clauses resolve to yield the empty clause (\(KB \models \alpha\)).
  • Termination of the above algorithm follows due to the finite amount of symbols in the knowledge base, so long as useless literals such as \(A \vee \neg A\) are removed throughout the clause formation process. The proof of completeness for resolution is the ground resolution theorem.

While resolution is complete, sometimes we do not need its full power, or it is too slow. A Horn clause is a disjunction of literals of which at most one is positive, and if our clauses are of this form, we can perform more efficient inference using forward-chaining and backward-chaining algorithms. Forward-chaining means we start with the known facts and try to draw conclusions (e.g., using Modus Ponens) and propagate inference through the AND-OR graph. Backward-chaining does this in reverse. These algorithms decide entailment in linear time. And they are easy to describe to humans. Yay. Of course, we need Horn clauses for these to apply.

Moving away from resolution and back to model checking (remember how inefficient that is?), we can devise several heuristics to improve model checking, such as backtracking search and hill-climbing search. Backtracking search is a depth-first enumeration of possible models with early termination, pure symbol heuristics, and unit clause heuristics. Hill-climbing search is a seemingly crazy way of doing inference. It randomly picks an unsatisfied clause and flips a symbol in it. Obviously, this may go on forever if we get unlucky in our draws, but if we do get a solution, then we know for sure that a solution actually exists!

Chapter 8: First-Order Logic

To design an agent based on propositional logic, as in the Wumpus world, one has to perform cumbersome steps to take care of variables representing the same world object, but at different times. The next upgrade of logic into what is known as first-order logic will alleviate us of this nuisance because of existential (\(\exists\)) and universal (\(\forall\)) quantifiers. The following rule:

\[\forall x\quad King(x) \Rightarrow Person(x)\]

means that “For all \(x\), if \(x\) is a king, then \(x\) is a person.” More formally, first-order logic assumes that the world now consists of facts, objects, relations, and functions, while propositional logic only assumes the world consists of facts (or propositions). The syntax terms are that constant symbols represent objects, predicate symbols represent relations, and function symbols stand for functions. Functions are a special type of relation where there is only one value for an input (which is the standard way we think of functions). In the above rule, \(x\) is a variable; terms with no variables are called ground terms.

A model in first order logic consists of not only objects, but also various interpretations of each predicate and function. If our world consists of three terms \(A, B,\) and \(C\), and two objects, there are multiple ways we could map those terms: all of them could mean the first object, or \(A\) could mean the first and \(B\) and \(C\) could both mean the second, and so on. Due to the amount of ways one could assign symbols to various objects or change the definition of a relation, model-checking for entailment (which must apply to all possible models) in first-order logic is much slower than it is for propositional logic.

Going back to universal quantification, we say that \(\forall x\: P\) is true in a given model if \(P\) is true in all possible extended interpretations constructed from the interpretation given in the model. This is a fancy way of saying that if our model has three objects (e.g., Richard the Lionheart, King John, and Richard’s Left Leg), then we better be able to plug in all three of those objects in as the variable \(x\) in the rule \(P\) and have those statements be true. For existential quantifiers, we just need at least one statement true in the extended interpretation for \(\exists x\: P\) to be true.

First-order logic also includes equality. This is convenient when we need two variables to be unequal. Consider the following rule:

\[\exists x, y\quad Brother(x,Richard) \wedge Brother(y, Richard) \wedge \neg(x = y)\]

This is stating that Richard has at least two brothers. If we removed the \(\neg(x = y)\) part, then we could assign \(x\) and \(y\) to be the same person. But even if \(x\) and \(y\) referred to two different names (e.g., Daniel and Darius) then they could still refer to the same symbol/object. Thus, to make things easier for our brains, we will follow the unique-name assumption. We can also invoke the closed-world assumption in which atomic sentences not known to be true are false.

Chapter 9: Inference in First-Order Logic

One way to perform inference for first-order logic is to convert a first-order knowledge base to a propositional one, and then apply the propositional inference algorithms from Chapter 7. (Yes, I know you can tell that this will be crazily inefficient, but it might be useful to see how that works.) There are two techniques that help:

  • Universal instantiation means we can substitute a ground term for a variable in a universally quantified rule. The rule is that if \(\forall v\: \alpha\) is true, then so is \(Subst(\{v/g\}, \alpha)\), where \(g\) is the ground term8.
  • Existential instantiation means that in an existentially quantified rule, we can create a single new constant symbol that does not appear in the knowledge base. If \(\exists v\: \alpha\) is true, then so is \(Subst(\{v/k\},\alpha)\), where \(k\) is that new symbol. This new symbol is a Skolem constant and is part of a general process called skolemization.

These two methods help us to discard universal and existential quantifiers, respectively. (The former would require us to make many new rules, the latter requires only one new rule.) There’s more to this technique of propositionalization, but the point is that we can transform first-order inference queries into propositional form while preserving entailment. Unfortunately, the question of entailment is semidecidable. We will be able to prove entailment for every entailed sentence, but we cannot refute entailment (in layman’s terms, “say no”) to every non-entailed sentence. The reason for this is that functions can construct infinitely-many ground-term substitutions, and we found out earlier that propositional inference algorithms (i.e., resolution) terminate precisely because we are guaranteed to have finitely many terms.

Despite this somewhat sorrowing news, there is better news to be had with regards to how efficiently we can “propositionalize.” For this, there are two techniques we can draw from: Generalized Modus Ponens and Unification.

  • Generalized Modus Ponens is an inference rule which states that for atomic sentences \(p_i, p_i'\) and \(q\), if there is a substitution \(\theta\) such that \(Subst(\theta,p_i') = Subst(\theta,p_i)\), then if the following are true:

    \[p_1', \ldots, p_n'\] \[(p_1 \wedge \cdots \wedge p_n) \Rightarrow q\]

    then the conclusion is that \(Subst(\theta,q)\) is true. So what does this mean in English? The conclusion is the sentence \(q\) after we have applied the substitution \(\theta\) that created equivalence between \(p_i\) and \(p_i'\). This is helpful in cases when the \(p_i\)s are variable rules (e.g., \(King(x)\)) and the \(p_i'\)s are knowledge-base sentences (e.g., \(King(John)\)). By applying Generalized Modus Ponens with appropriate substitutions (like \(\{x/John\}\)), we can avoid the unnecessary extended interpretations of \((p_1 \wedge \cdots \wedge p_n) \Rightarrow q\).

    Before moving on, it’s worth connecting this rule to Modus Ponens from Chapter 7. It’s obvious that there are some similarities: we make the conclusion \(q\), which is the result of an implication \(\alpha \Rightarrow q\). But why is this called the generalized version? It’s because we “generalize” this rule from propositional logic to first-order logic by introducing variables and substitutions, which we know are not present in propositional logic. The book uses the term lifted for this, but it seems a bit arbitrary to me.

  • Our next rule relates to the previous one. Remember that we have to make sure that \(Subst(\theta,p_i') = Subst(\theta,p_i)\), but this will require a lot of comparisons. Unification is the hugely-important process of making different logical expressions have identical meanings. For two sentences \(p\) and \(q\), unification returns \(\theta\), a set of substitutions for their variables to make them identical, if one exists.

    Unification may require standardizing apart variables to avoid name clashes. Also, more than one unifications may be possible for a set of statements, so it is logical to pick the one that places fewest restrictions on the variables.

    A naive (but sound) algorithm for unification recursively explores the expressions and builds up a unifier along the way, but has complexity quadratic in the size of the expressions being unified. More complicated unification algorithms can run in linear time.

Let us now briefly discuss three families of first-order inference algorithms: forward chaining, backward chaining, and resolution. These should be familiar from propositional logic, since all we are doing here is extending them to fit in the framework of a first-order logic system, but it is important to understand where exactly the extensions occur. We will clearly have to use rules like Generalized Modus Ponens and Unification here.

Forward chaining in first-order logic applies Generalized Modus Ponens repeatedly to add more atomic sentences to the knowledge base until no further inferences are possible. This is similar to propositional logic, where forward chaining would repeatedly apply Modus Ponens. But remember how propositional forward chaining required Horn clauses (a generalized version of propositional definite clauses)? In first-order logic, forward chaining requires first-order definite clauses, which are disjunctions of literals of which exactly one is positive. Many (but all not all) knowledge bases can be converted into a set of definite clauses, which acts as a preprocessing step. Then, as stated earlier, we apply Generalized Modus Ponens, ideally until we’ve solved our query or reached a fixed point. Again, it’s similar to the propositional logic version, except here we include universally quantified atomic sentences. It is sound and complete, but entailment with definite clauses is semidecidable.

There are three sources of inefficiency in the naive forward chaining algorithm: (1) that unification involves searching through too many sets of facts on the knowledge base, (2) that the algorithm rechecks each rule on every iteration to see whether its premises are satisfied, and (3) that the algorithm generates facts that may be irrelevant to the goal.

Backward chaining in first-order logic means we work backward from the goal, searching for substitutions and unifications to satisfy \(lhs \Rightarrow rhs\) where the \(rhs\) expression is already known. To find suitable substitutions for \(lhs\), which is a list of conjuncts which must all be positive, we may have to perform additional backtracking. The naive backward chaining algorithm is depth-first search, so it suffers from some standard problems with DFS (e.g, lack of completeness) that forward chaining avoids.

Backward chaining is used in logic programming, a technology where systems are constructed and make conclusions using processes similar to what happens in first-order logic. Prolog is an example of a logic programming language, but it is incomplete as a theorem prover for definite clauses. To avoid redundant computations, backward chaining should memoize solutions to sub-problems.

We can extend resolution from propositional logic to create a complete inference procedure for first-order logic. As before, the first step is to convert first-order sentences into inferentially equivalent CNF sentences, which is always possible, and forms the basis for future proofs-by-contradiction resolution procedures. This conversion process isn’t too difficult, though we need to eliminate existential quantifiers via Skolemization (briefly mentioned earlier). The process might involve creating Skolem functions to clarify variable dependencies.

The resolution inference rule is a generalization of the propositional reference rule to handle variables. Two clauses standardized apart (i.e., no shared variables) can be resolved (and therefore removed from proof as they don’t affect the outcome) if they contain complementary literals. In first order logic, complimentary literals are those in which one unifies with the negation of the other; remember that in propositional logic, we just had to worry about straightforward negations.

Resolution is refutation-complete in the following sense: if \(S\) is an unsatisfiable set of clauses, then the application of a finite number of resolution steps to \(S\) will yield a contradiction. Resolution cannot generate all logical consequences of a set of sentences.

Chapter 10: Classical Planning

This chapter introduces a representation for planning problems in single-agent, deterministic, observable environments, which scales far better than the earlier search agents of Chapter 3 and the logical agents of Chapter 7. As a starting step to analyze and standardize language, AI researchers have introduced the PDDL language, the Planning Domain Definition Language. It can describe the things we need to define a search problem:

  • the initial state, or states in general, which are conjunctions of ground (i.e., no variables or functions) atoms called fluents.
  • the actions, which have variables, preconditions, and effects. They are only applicable in a state if the state satisfies the preconditions.
  • the list of actions at each state, and the result of applying each action.
  • the goal test, which is again a conjunction of literals, though these may contain variables. The goal is to find a sequence of actions that lead to a goal state.

There are several straightforward examples of PDDL in the book. They are intuitive descriptions of various problems written in a structured framework, though there is some trickery involved (e.g., with inequalities).

PDDL maps planning problems to search problems, and we can solve these with forward searching or (you must know what’s coming…) backward searching through states. Forward searching needs heuristics, because as stated earlier, the branching factor is too large to apply one of the Chapter 3 or 4 search algorithms directly. Backward searching avoids many irrelevant states, and PDDL makes it easy to represent the backtracking process, but requires sets of states and does not lend itself to easy heuristics.

To get an admissible heuristic, we can relax the problem to make it easier and apply the resulting solution to the original one. The corresponding search graph has nodes as states and actions as edges. Some ideas for heuristics:

  • Ignore preconditions, so every action becomes applicable in every state.
  • Ignore delete lists, applicable if all goals and preconditions are positive literals. This removes all negative literals from all actions, and the problem is easier now because no action will undo progress made by another action.
  • To reduce the number of states, ignore some of the fluents.
  • Assume subgoal independence, so if the goal is \(G_1 \wedge \cdots \wedge G_n\), take as the estimate the maximum cost over all \(n\), or sum up the estimated costs for each state (note: this is inadmissible).

A special data structure called a planning graph can provide better admissible heuristics than the ones previously suggested; we build the graph, and then search over it (this is called “GraphPlan” in the book). The planning graph is a directed graph of states, and is a polynomial-sized approximation to the exponential-sized tree that consists of all the possible paths taken from the starting state, with the goal of finding a path to the goal state. It consists of alternating state levels \(S_i\) and action levels \(A_i\). The state levels contain the literals that might be true (since it’s an approximation) at a given state9, and the action levels contain the actions whose preconditions are satisfied by those literals in the previous state level. Literals that show up later in the tree (i.e., farther away from the starting state) are “harder” to achieve, so the true states that contain those literals are “harder” to reach.

A key property of state and action layers is that they contain mutual exclusion or mutex links between literals and actions, respectively. A mutex relation holds between two actions if they have inconsistent effects, interference, or competing needs. A mutex relation holds between two literals if one is the negation of the other, or if the pairs of actions that could lead to those literals are mutex.

Now that we’ve constructed a planning graph, how do we use it? As stated earlier, we can estimate the cost of achieving any goal literal \(g_i\) as the level in which it first appears in the planning graph, i.e., the level cost. If the goal state is a conjunction of literals, which is the normal case anyway, then some ideas are:

  • Take the maximum level cost over all literals. This is admissible.
  • Take the sum over all literals. This is not admissible, though it might work if the individual literals are reasonably independent of each other.
  • Take the level cost of the level in which all literals are in the planning graph, without there being any mutex relations between the two. This is admissible and also clearly dominates the maximum level cost heuristic because the level we return will have all the literals.

As an alternative, we can run the GraphPlan algorithm to search directly on the planning graph. The algorithm repeatedly adds levels to the planning graph, and finishes once all the goals show up as non-mutex in some possible state for a state layer. If the action and state levels do not increase, then the algorithm returns failure.

The downside of planning graphs is that they only work for propositional planning problems, though if we wanted, we could obviously convert a first-order logic encoding of a plan into propositional logic. They also fail to detect unsolvable problems with three-way mutex relations but no two-way mutex relations.

There are a few other classical planning paradigms:

  • We can treat this as a theorem-proving problem by transforming the PDDL description into a form that can be processed by a SAT solver. This step involves propositionalizing the actions and the goal, as well as adding in more axioms to handle successor states and mutual exclusions.
  • We can use first-order logical deduction rather than PDDL. Rather than tie time directly to fluents, we can use situation calculus and create new rules to apply to our states. The downside of this approach is that it’s hard to get good heuristics.
  • We can transform the problem of finding a plan of length \(k\) as a constraint satisfaction problem (from Chapter 6), similar to the encoding for a SAT problem.
  • We can also create partially ordered plans, which is useful with independent subproblems. We can create such plans by searching through the space of plans, rather than the state space. Unfortunately, it doesn’t represent states easily, and these fell out of favor (after the 1990s) in place of plans that search through states with strong heuristics.

Chapter 11: Planning and Acting in the Real World

This rather amusingly-titled chapter now brings us into the “real world” by requiring that our agent representation must handle not only planning, but also handle a changing environment. Here are some of the new assumptions we make in our world:

  • We may have to deal with time and resource constraints.
  • We may have to organize plans in a hierarchical fashion.
  • We may have to handle nondeterminism and uncertainty in our environment.
  • We may have to deal with multiple, competing agents.

Let’s briefly discuss how we design an agent to handle these four cases.

To deal with time and resource constraints, we augment the language of the problem formulation to include amounts for certain resources (e.g., \(Inspectors(9)\) means there are nine inspectors available in a car inspection problem), as well as Consume and Use keywords to indicate whether the resource is gone or available again after usage (e.g., inspectors would usually be available again after their usage). We can represent the problem with a directed graph that obeys the time/resource constraints, and then find the critical path through the graph, which is the path with the longest duration and thus is the “limiting factor” in the schedule. A heuristic to find the minimum cost path: for each iteration, choose the action with all predecessors satisfied, and which has the least amount of slack.

In order for AI systems to think like humans do, AI systems will need to make plans at a higher level of abstraction by forming hierarchies. The classic example is when we organize a plane trip to Hawaii. The high-level abstraction is: take the BART to the airport, search for the gate, etc. Humans do not think: first, open the door carefully, then take seven paces to the right, then walk down this way for 500 steps, then put the ticket inside the BART gate, etc. That kind of detail would lead to too much combinatorial explosion in AI systems. So AI agents must use high-level actions (HLA) that can possibly be refined later. One way to solve such hierarchy problems is to start with one or multiple HLAs that solve the problem, and then refine it (e.g., in a BFS fashion) until we get a sequence of primitive actions that accomplishes the goal. This can be substantially faster than normal BFS over the space of primitives. To get an even greater — potentially exponential — speedup in search, what we would like to do is only search through the space of HLAs, and once we find a sequence of HLAs that work, then we can refine that one plan into primitives, since we know it works. To do this, we set up preconditions and effects for each HLA, and the state space will be a set of fluents (as it was before in many areas of Chapter 10). We can define a search problem known as angelic search that utilizes reachable sets of HLAs. Pessimistic and optimistic reachable sets can prune away refinements that have no chance of reaching the goal.

Our agents will have to deal with nondeterminism and uncertainty in the environment. With no observations, we can perform sensorless planning. With partial observations, we can perform contingency planning, and for unknown environments, we can do online learning. Some material is similar to what was presented in Chapter 4, and the main difference here is that we have a far richer state representation (fluents rather than atomic) and thus we can represent belief states easily using \(O(n)\)-length conjunctions (well, assuming that our belief states are 1-CNF). It can be tricky to update belief states after an action. An example problem with fluents might be that we have to paint two chairs to be the same color, and a sensorless agent could solve the problem by just dumping a can of paint on both chairs, without knowing the color of the paint at all. A contingent agent can also solve the problem, and often does so more efficiently.

Finally, we can consider the multiagent case, either when one “super” agent controls multiple smaller agents, or when there are multiple, separate agents who only control themselves (and whose goals may be in competition with one another). If the agents are loosely coupled, it makes sense to decompose the transition model into independent subproblems to avoid an exponential branching factor.

Chapter 12: Knowledge Representation

In which we show how to use first-order logic to represent the most important aspects of the real world, such as action, space, time, thoughts, and shopping.

Wait, shopping? All right all right, let’s see what’s in store here in the final chapter of the Knowledge, Reasoning, and Planning aspect of AI. The previous chapters have come up with the technology (e.g., first-order logic and its various inference methods) for knowledge-based agents, but now, we have to figure out the content to put in an agent’s knowledge base.

I don’t think there is much material in this chapter that I need to know, and a lot of it is common sense. But here are the highlights regardless:

  • Ontological engineering is the process of representing abstract concepts of events, time, physical objects, and beliefs in various domains. We clearly have to do something like this to design an agent! One way is to describe an upper ontology of the world by listing some general things first, and then moving to more specific items down the tree (this is what we do in object oriented programming).
  • We need to represent the following: categories, objects, and events. We can represent categories by using straightforward predicates (e.g., the category of basketballs could be \(Basketball(b)\)) or by reification10, a.k.a. thingification, which means representing it as an object \(Basketballs\). To reason about categories, we can use the graphically appealing semantic network framework, or appeal to formalism with description logic11. Objects should be arranged in a hierarchy of categories with subclassing and inheritance, kind of like (again) how we do it in object-oriented programming. For events, rather than the situational calculus we saw earlier, we should use event calculus to deal with continuity. Event calculus reifies fluents and events.
  • Sometimes, we may wish to represent mental beliefs by model logic rather than first-order logic, because the former lets us take sentences as arguments, and allows us to represent a set of possible worlds of beliefs. The notation \(K_AP\) means that agent \(A\) knows \(P\).
  • At the end of the chapter, there is a shopping mall example, which I find to be amusing but not that educational.

Whew! Reading the above ten chapters, as well as Chapters 1 and 2 in the book, and Chapter 13 (which is about basic probability, nothing to see here!) brings me to the 500-page mark out of 1000 pages. Now I only have 500 pages to go!

  1. Said another way, it never overestimates the cost of reaching a given state. 

  2. It is also possible that the MIN player is dumb and doesn’t play optimally, but the MAX player following the minimax strategy would perform even better if that were the case. 

  3. These assignments are a great way to see more complicated applications of algorithms from the textbook. 

  4. Various evaluation functions (e.g., heuristics) are of course domain-specific, but the point is that the search algorithms are not domain-specific. It’s worth thinking about this often. 

  5. There’s also a nice Wumpus example in this chapter that makes it twice as much fun to read. That part is not dry. 

  6. The book states that propositional entailment is co-NP-complete, so every known inference algorithm for propositional logic has a worst-case complexity exponential in the size of the input. Still, this is “only” worst case complexity. 

  7. Technically, resoultion is complete if it is coupled with a complete search algorithm. 

  8. The notation \(Subst(\theta, \alpha)\) denotes the result of applying the substitution \(\theta\) to sentence \(\alpha\). The substitution rule should be of the form \(\{x/y\}\) where \(x\) is a variable and \(y\) is a real term that we want to plug in. 

  9. A literal cannot appear too late in a state level, because that would be like over-estimating the cost of that literal and the states it belongs to, resulting in inadmissible heuristics. 

  10. One of the advantages of reification is that by doing so, we represent what we want in terms of objects; then we can add an arbitrary amount of information about them. For instance, refiying events means we can add in as many descriptions about the event as we want by adding in more conjuncts. 

  11. Representing categories these ways also helps us to establish default values for categories. Unlike in previous forms of knowledge representation, semantic networks and description logics make it easy for us to have exceptions for objects. For example, most tomatoes are red, but some can be purple. Thus, the category of tomatoes should have a default color attribute of red, with individual objects potentially overriding those values. The connections between this and programming languages is once again obvious. Also, note that if we allow overriding, then this violates monotonicity of the logic. Monotonicity means that if \(KB \models \alpha\), then for any \(\beta\), \(KB \wedge \beta \models \alpha\).