You're walking down the street and need to pass someone going the opposite way. You take a step left, but they're thinking the same thing and take a step to their right, aka your left. You're still blocking each other. Then you take a step to the right, and they take a step to their left, and you're back to where you started. I've heard this called "walkwarding" Let's model this in TLA+. TLA+ is a formal methods tool for finding bugs in complex software designs, most often involving concurrency. Two people trying to get past each other just also happens to be a concurrent system. A gentler introduction to TLA+'s capabilities is here, an in-depth guide teaching the language is here. The spec ---- MODULE walkward ---- EXTENDS Integers VARIABLES pos vars == <<pos>> Double equals defines a new operator, single equals is an equality check. <<pos>> is a sequence, aka array. you == "you" me == "me" People == {you, me} MaxPlace == 4 left == 0 right == 1 I've gotten into the habit of assigning string "symbols" to operators so that the compiler complains if I misspelled something. left and right are numbers so we can shift position with right - pos. direction == [you |-> 1, me |-> -1] goal == [you |-> MaxPlace, me |-> 1] Init == \* left-right, forward-backward pos = [you |-> [lr |-> left, fb |-> 1], me |-> [lr |-> left, fb |-> MaxPlace]] direction, goal, and pos are "records", or hash tables with string keys. I can get my left-right position with pos.me.lr or pos["me"]["lr"] (or pos[me].lr, as me == "me"). Juke(person) == pos' = [pos EXCEPT ![person].lr = right - @] TLA+ breaks the world into a sequence of steps. In each step, pos is the value of pos in the current step and pos' is the value in the next step. The main outcome of this semantics is that we "assign" a new value to pos by declaring pos' equal to something. But the semantics also open up lots of cool tricks, like swapping two values with x' = y /\ y' = x. TLA+ is a little weird about updating functions. To set f[x] = 3, you gotta write f' = [f EXCEPT ![x] = 3]. To make things a little easier, the rhs of a function update can contain @ for the old value. ![me].lr = right - @ is the same as right - pos[me].lr, so it swaps left and right. ("Juke" comes from here) Move(person) == LET new_pos == [pos[person] EXCEPT !.fb = @ + direction[person]] IN /\ pos[person].fb # goal[person] /\ \A p \in People: pos[p] # new_pos /\ pos' = [pos EXCEPT ![person] = new_pos] The EXCEPT syntax can be used in regular definitions, too. This lets someone move one step in their goal direction unless they are at the goal or someone is already in that space. /\ means "and". Next == \E p \in People: \/ Move(p) \/ Juke(p) I really like how TLA+ represents concurrency: "In each step, there is a person who either moves or jukes." It can take a few uses to really wrap your head around but it can express extraordinarily complicated distributed systems. Spec == Init /\ [][Next]_vars Liveness == <>(pos[me].fb = goal[me]) ==== Spec is our specification: we start at Init and take a Next step every step. Liveness is the generic term for "something good is guaranteed to happen", see here for more. <> means "eventually", so Liveness means "eventually my forward-backward position will be my goal". I could extend it to "both of us eventually reach out goal" but I think this is good enough for a demo. Checking the spec Four years ago, everybody in TLA+ used the toolbox. Now the community has collectively shifted over to using the VSCode extension.1 VSCode requires we write a configuration file, which I will call walkward.cfg. SPECIFICATION Spec PROPERTY Liveness I then check the model with the VSCode command TLA+: Check model with TLC. Unsurprisingly, it finds an error: The reason it fails is "stuttering": I can get one step away from my goal and then just stop moving forever. We say the spec is unfair: it does not guarantee that if progress is always possible, progress will be made. If I want the spec to always make progress, I have to make some of the steps weakly fair. + Fairness == WF_vars(Next) - Spec == Init /\ [][Next]_vars + Spec == Init /\ [][Next]_vars /\ Fairness Now the spec is weakly fair, so someone will always do something. New error: \* First six steps cut 7: <Move("me")> pos = [you |-> [lr |-> 0, fb |-> 4], me |-> [lr |-> 1, fb |-> 2]] 8: <Juke("me")> pos = [you |-> [lr |-> 0, fb |-> 4], me |-> [lr |-> 0, fb |-> 2]] 9: <Juke("me")> (back to state 7) In this failure, I've successfully gotten past you, and then spend the rest of my life endlessly juking back and forth. The Next step keeps happening, so weak fairness is satisfied. What I actually want is for both my Move and my Juke to both be weakly fair independently of each other. - Fairness == WF_vars(Next) + Fairness == WF_vars(Move(me)) /\ WF_vars(Juke(me)) If my liveness property also specified that you reached your goal, I could instead write \A p \in People: WF_vars(Move(p)) etc. I could also swap the \A with a \E to mean at least one of us is guaranteed to have fair actions, but not necessarily both of us. New error: 3: <Move("me")> pos = [you |-> [lr |-> 0, fb |-> 2], me |-> [lr |-> 0, fb |-> 3]] 4: <Juke("you")> pos = [you |-> [lr |-> 1, fb |-> 2], me |-> [lr |-> 0, fb |-> 3]] 5: <Juke("me")> pos = [you |-> [lr |-> 1, fb |-> 2], me |-> [lr |-> 1, fb |-> 3]] 6: <Juke("me")> pos = [you |-> [lr |-> 1, fb |-> 2], me |-> [lr |-> 0, fb |-> 3]] 7: <Juke("you")> (back to state 3) Now we're getting somewhere! This is the original walkwarding situation we wanted to capture. We're in each others way, then you juke, but before either of us can move you juke, then we both juke back. We can repeat this forever, trapped in a social hell. Wait, but doesn't WF(Move(me)) guarantee I will eventually move? Yes, but only if a move is permanently available. In this case, it's not permanently available, because every couple of steps it's made temporarily unavailable. How do I fix this? I can't add a rule saying that we only juke if we're blocked, because the whole point of walkwarding is that we're not coordinated. In the real world, walkwarding can go on for agonizing seconds. What I can do instead is say that Liveness holds as long as Move is strongly fair. Unlike weak fairness, strong fairness guarantees something happens if it keeps becoming possible, even with interruptions. Liveness == + SF_vars(Move(me)) => <>(pos[me].fb = goal[me]) This makes the spec pass. Even if we weave back and forth for five minutes, as long as we eventually pass each other, I will reach my goal. Note we could also by making Move in Fairness strongly fair, which is preferable if we have a lot of different liveness properties to check. A small exercise for the reader There is a presumed invariant that is violated. Identify what it is, write it as a property in TLA+, and show the spec violates it. Then fix it. Answer (in rot13): Gur vainevnag vf "ab gjb crbcyr ner va gur rknpg fnzr ybpngvba". Zbir thnenagrrf guvf ohg Whxr qbrf abg. More TLA+ Exercises I've started work on an exercises repo. There's only a handful of specific problems now but I'm planning on adding more over the summer. learntla is still on the toolbox, but I'm hoping to get it all moved over this summer. ↩

I started writing this early last week but Real Life Stuff happened and now you're getting the first-draft late this week. Warning, unedited thoughts ahead! New Logic for Programmers release! v0.9 is out! This is a big release, with a new cover design, several rewritten chapters, online code samples and much more. See the full release notes at the changelog page, and get the book here! Write the cleverest code you possibly can There are millions of articles online about how programmers should not write "clever" code, and instead write simple, maintainable code that everybody understands. Sometimes the example of "clever" code looks like this (src): # Python p=n=1 exec("p*=n*n;n+=1;"*~-int(input())) print(p%n) This is code-golfing, the sport of writing the most concise code possible. Obviously you shouldn't run this in production for the same reason you shouldn't eat dinner off a Rembrandt. Other times the example looks like this: def is_prime(x): if x == 1: return True return all([x%n != 0 for n in range(2, x)] This is "clever" because it uses a single list comprehension, as opposed to a "simple" for loop. Yes, "list comprehensions are too clever" is something I've read in one of these articles. I've also talked to people who think that datatypes besides lists and hashmaps are too clever to use, that most optimizations are too clever to bother with, and even that functions and classes are too clever and code should be a linear script.1. Clever code is anything using features or domain concepts we don't understand. Something that seems unbearably clever to me might be utterly mundane for you, and vice versa. How do we make something utterly mundane? By using it and working at the boundaries of our skills. Almost everything I'm "good at" comes from banging my head against it more than is healthy. That suggests a really good reason to write clever code: it's an excellent form of purposeful practice. Writing clever code forces us to code outside of our comfort zone, developing our skills as software engineers. Debugging is twice as hard as writing the code in the first place. Therefore, if you write the code as cleverly as possible, you [will get excellent debugging practice at exactly the right level required to push your skills as a software engineer] — Brian Kernighan, probably There are other benefits, too, but first let's kill the elephant in the room:2 Don't commit clever code I am proposing writing clever code as a means of practice. Being at work is a job with coworkers who will not appreciate if your code is too clever. Similarly, don't use too many innovative technologies. Don't put anything in production you are uncomfortable with. We can still responsibly write clever code at work, though: Solve a problem in both a simple and a clever way, and then only commit the simple way. This works well for small scale problems where trying the "clever way" only takes a few minutes. Write our personal tools cleverly. I'm a big believer of the idea that most programmers would benefit from writing more scripts and support code customized to their particular work environment. This is a great place to practice new techniques, languages, etc. If clever code is absolutely the best way to solve a problem, then commit it with extensive documentation explaining how it works and why it's preferable to simpler solutions. Bonus: this potentially helps the whole team upskill. Writing clever code... ...teaches simple solutions Usually, code that's called too clever composes several powerful features together — the "not a single list comprehension or function" people are the exception. Josh Comeau's "don't write clever code" article gives this example of "too clever": const extractDataFromResponse = (response) => { const [Component, props] = response; const resultsEntries = Object.entries({ Component, props }); const assignIfValueTruthy = (o, [k, v]) => (v ? { ...o, [k]: v } : o ); return resultsEntries.reduce(assignIfValueTruthy, {}); } What makes this "clever"? I count eight language features composed together: entries, argument unpacking, implicit objects, splats, ternaries, higher-order functions, and reductions. Would code that used only one or two of these features still be "clever"? I don't think so. These features exist for a reason, and oftentimes they make code simpler than not using them. We can, of course, learn these features one at a time. Writing the clever version (but not committing it) gives us practice with all eight at once and also with how they compose together. That knowledge comes in handy when we want to apply a single one of the ideas. I've recently had to do a bit of pandas for a project. Whenever I have to do a new analysis, I try to write it as a single chain of transformations, and then as a more balanced set of updates. ...helps us master concepts Even if the composite parts of a "clever" solution aren't by themselves useful, it still makes us better at the overall language, and that's inherently valuable. A few years ago I wrote Crimes with Python's Pattern Matching. It involves writing horrible code like this: from abc import ABC class NotIterable(ABC): @classmethod def __subclasshook__(cls, C): return not hasattr(C, "__iter__") def f(x): match x: case NotIterable(): print(f"{x} is not iterable") case _: print(f"{x} is iterable") if __name__ == "__main__": f(10) f("string") f([1, 2, 3]) This composes Python match statements, which are broadly useful, and abstract base classes, which are incredibly niche. But even if I never use ABCs in real production code, it helped me understand Python's match semantics and Method Resolution Order better. ...prepares us for necessity Sometimes the clever way is the only way. Maybe we need something faster than the simplest solution. Maybe we are working with constrained tools or frameworks that demand cleverness. Peter Norvig argued that design patterns compensate for missing language features. I'd argue that cleverness is another means of compensating: if our tools don't have an easy way to do something, we need to find a clever way. You see this a lot in formal methods like TLA+. Need to check a hyperproperty? Cast your state space to a directed graph. Need to compose ten specifications together? Combine refinements with state machines. Most difficult problems have a "clever" solution. The real problem is that clever solutions have a skill floor. If normal use of the tool is at difficult 3 out of 10, then basic clever solutions are at 5 out of 10, and it's hard to jump those two steps in the moment you need the cleverness. But if you've practiced with writing overly clever code, you're used to working at a 7 out of 10 level in short bursts, and then you can "drop down" to 5/10. I don't know if that makes too much sense, but I see it happen a lot in practice. ...builds comradery On a few occasions, after getting a pull request merged, I pulled the reviewer over and said "check out this horrible way of doing the same thing". I find that as long as people know they're not going to be subjected to a clever solution in production, they enjoy seeing it! Next week's newsletter will probably also be late, after that we should be back to a regular schedule for the rest of the summer. Mostly grad students outside of CS who have to write scripts to do research. And in more than one data scientist. I think it's correlated with using Jupyter. ↩ If I don't put this at the beginning, I'll get a bajillion responses like "your team will hate you" ↩

Recently I got a question on formal methods1: how does it help to mathematically model systems when the system requirements are constantly changing? It doesn't make sense to spend a lot of time proving a design works, and then deliver the product and find out it's not at all what the client needs. As the saying goes, the hard part is "building the right thing", not "building the thing right". One possible response: "why write tests"? You shouldn't write tests, especially lots of unit tests ahead of time, if you might just throw them all away when the requirements change. This is a bad response because we all know the difference between writing tests and formal methods: testing is easy and FM is hard. Testing requires low cost for moderate correctness, FM requires high(ish) cost for high correctness. And when requirements are constantly changing, "high(ish) cost" isn't affordable and "high correctness" isn't worthwhile, because a kinda-okay solution that solves a customer's problem is infinitely better than a solid solution that doesn't. But eventually you get something that solves the problem, and what then? Most of us don't work for Google, we can't axe features and products on a whim. If the client is happy with your solution, you are expected to support it. It should work when your customers run into new edge cases, or migrate all their computers to the next OS version, or expand into a market with shoddy internet. It should work when 10x as many customers are using 10x as many features. It should work when you add new features that come into conflict. And just as importantly, it should never stop solving their problem. Canonical example: your feature involves processing requested tasks synchronously. At scale, this doesn't work, so to improve latency you make it asynchronous. Now it's eventually consistent, but your customers were depending on it being always consistent. Now it no longer does what they need, and has stopped solving their problems. Every successful requirement met spawns a new requirement: "keep this working". That requirement is permanent, or close enough to decide our long-term strategy. It takes active investment to keep a feature behaving the same as the world around it changes. (Is this all a pretentious of way of saying "software maintenance is hard?" Maybe!) Phase changes In physics there's a concept of a phase transition. To raise the temperature of a gram of liquid water by 1° C, you have to add 4.184 joules of energy.2 This continues until you raise it to 100°C, then it stops. After you've added two thousand joules to that gram, it suddenly turns into steam. The energy of the system changes continuously but the form, or phase, changes discretely. Software isn't physics but the idea works as a metaphor. A certain architecture handles a certain level of load, and past that you need a new architecture. Or a bunch of similar features are independently hardcoded until the system becomes too messy to understand, you remodel the internals into something unified and extendable. etc etc etc. It's doesn't have to be totally discrete phase transition, but there's definitely a "before" and "after" in the system form. Phase changes tend to lead to more intricacy/complexity in the system, meaning it's likely that a phase change will introduce new bugs into existing behaviors. Take the synchronous vs asynchronous case. A very simple toy model of synchronous updates would be Set(key, val), which updates data[key] to val.3 A model of asynchronous updates would be AsyncSet(key, val, priority) adds a (key, val, priority, server_time()) tuple to a tasks set, and then another process asynchronously pulls a tuple (ordered by highest priority, then earliest time) and calls Set(key, val). Here are some properties the client may need preserved as a requirement: If AsyncSet(key, val, _, _) is called, then eventually db[key] = val (possibly violated if higher-priority tasks keep coming in) If someone calls AsyncSet(key1, val1, low) and then AsyncSet(key2, val2, low), they should see the first update and then the second (linearizability, possibly violated if the requests go to different servers with different clock times) If someone calls AsyncSet(key, val, _) and immediately reads db[key] they should get val (obviously violated, though the client may accept a slightly weaker property) If the new system doesn't satisfy an existing customer requirement, it's prudent to fix the bug before releasing the new system. The customer doesn't notice or care that your system underwent a phase change. They'll just see that one day your product solves their problems, and the next day it suddenly doesn't. This is one of the most common applications of formal methods. Both of those systems, and every one of those properties, is formally specifiable in a specification language. We can then automatically check that the new system satisfies the existing properties, and from there do things like automatically generate test suites. This does take a lot of work, so if your requirements are constantly changing, FM may not be worth the investment. But eventually requirements stop changing, and then you're stuck with them forever. That's where models shine. As always, I'm using formal methods to mean the subdiscipline of formal specification of designs, leaving out the formal verification of code. Mostly because "formal specification" is really awkward to say. ↩ Also called a "calorie". The US "dietary Calorie" is actually a kilocalorie. ↩ This is all directly translatable to a TLA+ specification, I'm just describing it in English to avoid paying the syntax tax ↩

Short one this time because I have a lot going on this week. In computation complexity, NP is the class of all decision problems (yes/no) where a potential proof (or "witness") for "yes" can be verified in polynomial time. For example, "does this set of numbers have a subset that sums to zero" is in NP. If the answer is "yes", you can prove it by presenting a set of numbers. We would then verify the witness by 1) checking that all the numbers are present in the set (~linear time) and 2) adding up all the numbers (also linear). NP-complete is the class of "hardest possible" NP problems. Subset sum is NP-complete. NP-hard is the set all problems at least as hard as NP-complete. Notably, NP-hard is not a subset of NP, as it contains problems that are harder than NP-complete. A natural question to ask is "like what?" And the canonical example of "NP-harder" is the halting problem (HALT): does program P halt on input C? As the argument goes, it's undecidable, so obviously not in NP. I think this is a bad example for two reasons: All NP requires is that witnesses for "yes" can be verified in polynomial time. It does not require anything for the "no" case! And even though HP is undecidable, there is a decidable way to verify a "yes": let the witness be "it halts in N steps", then run the program for that many steps and see if it halted by then. To prove HALT is not in NP, you have to show that this verification process grows faster than polynomially. It does (as busy beaver is uncomputable), but this all makes the example needlessly confusing.1 "What's bigger than a dog? THE MOON" Really (2) bothers me a lot more than (1) because it's just so inelegant. It suggests that NP-complete is the upper bound of "solvable" problems, and after that you're in full-on undecidability. I'd rather show intuitive problems that are harder than NP but not that much harder. But in looking for a "slightly harder" problem, I ran into an, ah, problem. It seems like the next-hardest class would be EXPTIME, except we don't know for sure that NP != EXPTIME. We know for sure that NP != NEXPTIME, but NEXPTIME doesn't have any intuitive, easily explainable problems. Most "definitely harder than NP" problems require a nontrivial background in theoretical computer science or mathematics to understand. There is one problem, though, that I find easily explainable. Place a token at the bottom left corner of a grid that extends infinitely up and right, call that point (0, 0). You're given list of valid displacement moves for the token, like (+1, +0), (-20, +13), (-5, -6), etc, and a target point like (700, 1). You may make any sequence of moves in any order, as long as no move ever puts the token off the grid. Does any sequence of moves bring you to the target? This is PSPACE-complete, I think, which still isn't proven to be harder than NP-complete (though it's widely believed). But what if you increase the number of dimensions of the grid? Past a certain number of dimensions the problem jumps to being EXPSPACE-complete, and then TOWER-complete (grows tetrationally), and then it keeps going. Some point might recognize this as looking a lot like the Ackermann function, and in fact this problem is ACKERMANN-complete on the number of available dimensions. A friend wrote a Quanta article about the whole mess, you should read it. This problem is ludicrously bigger than NP ("Chicago" instead of "The Moon"), but at least it's clearly decidable, easily explainable, and definitely not in NP. It's less confusing if you're taught the alternate (and original!) definition of NP, "the class of problems solvable in polynomial time by a nondeterministic Turing machine". Then HALT can't be in NP because otherwise runtime would be bounded by an exponential function. ↩

I have a lot in the works for the this month's Logic for Programmers release. Among other things, I'm completely rewriting the chapter on Logic Programming Languages. I originally showcased the paradigm with puzzle solvers, like eight queens or four-coloring. Lots of other demos do this too! It takes creativity and insight for humans to solve them, so a program doing it feels magical. But I'm trying to write a book about practical techniques and I want everything I talk about to be useful. So in v0.9 I'll be replacing these examples with a couple of new programs that might get people thinking that Prolog could help them in their day-to-day work. On the other hand, for a newsletter, showcasing a puzzle solver is pretty cool. And recently I stumbled into this post by my friend Pablo Meier, where he solves a videogame puzzle with Prolog:1 Summary for the text-only readers: We have a test with 10 true/false questions (denoted a/b) and four student attempts. Given the scores of the first three students, we have to figure out the fourth student's score. bbababbabb = 7 baaababaaa = 5 baaabbbaba = 3 bbaaabbaaa = ??? You can see Pablo's solution here, and try it in SWI-prolog here. Pretty cool! But after way too long studying Prolog just to write this dang book chapter, I wanted to see if I could do it more elegantly than him. Code and puzzle spoilers to follow. (Normally here's where I'd link to a gentler introduction I wrote but I think this is my first time writing about Prolog online? Uh here's a Picat intro instead) The Program You can try this all online at SWISH or just jump to my final version here. :- use_module(library(dif)). % Sound inequality :- use_module(library(clpfd)). % Finite domain constraints First some imports. dif lets us write dif(A, B), which is true if A and B are not equal. clpfd lets us write A #= B + 1 to say "A is 1 more than B".2 We'll say both the student submission and the key will be lists, where each value is a or b. In Prolog, lowercase identifiers are atoms (like symbols in other languages) and identifiers that start with a capital are variables. Prolog finds values for variables that match equations (unification). The pattern matching is real real good. % ?- means query ?- L = [a,B,c], [Y|X] = [1,2|L], B + 1 #= 7. B = 6, L = [a, 6, c], X = [2, a, 6, c], Y = 1 Next, we define score/33 recursively. % The student's test score % score(student answers, answer key, score) score([], [], 0). score([A|As], [A|Ks], N) :- N #= M + 1, score(As, Ks, M). score([A|As], [K|Ks], N) :- dif(A, K), score(As, Ks, N). First key is the student's answers, second is the answer key, third is the final score. The base case is the empty test, which has score 0. Otherwise, we take the head values of each list and compare them. If they're the same, we add one to the score, otherwise we keep the same score. Notice we couldn't write if x then y else z, we instead used pattern matching to effectively express (x && y) || (!x && z). Prolog does have a conditional operator, but it prevents backtracking so what's the point??? A quick break about bidirectionality One of the coolest things about Prolog: all purely logical predicates are bidirectional. We can use score to check if our expected score is correct: ?- score([a, b, b], [b, b, b], 2). true But we can also give it answers and a key and ask it for the score: ?- score([a, b, b], [b, b, b], X). X = 2 Or we could give it a key and a score and ask "what test answers would have this score?" ?- score(X, [b, b, b], 2). X = [b, b, _A], dif(_A,b) X = [b, _A, b], dif(_A,b) X = [_A, b, b], dif(_A,b) The different value is written _A because we never told Prolog that the array can only contain a and b. We'll fix this later. Okay back to the program Now that we have a way of computing scores, we want to find a possible answer key that matches all of our observations, ie gives everybody the correct scores. key(Key) :- % Figure it out score([b, b, a, b, a, b, b, a, b, b], Key, 7), score([b, a, a, a, b, a, b, a, a, a], Key, 5), score([b, a, a, a, b, b, b, a, b, a], Key, 3). So far we haven't explicitly said that the Key length matches the student answer lengths. This is implicitly verified by score (both lists need to be empty at the same time) but it's a good idea to explicitly add length(Key, 10) as a clause of key/1. We should also explicitly say that every element of Key is either a or b.4 Now we could write a second predicate saying Key had the right 'type': keytype([]). keytype([K|Ks]) :- member(K, [a, b]), keytype(Ks). But "generating lists that match a constraint" is a thing that comes up often enough that we don't want to write a separate predicate for each constraint! So after some digging, I found a more elegant solution: maplist. Let L=[l1, l2]. Then maplist(p, L) is equivalent to the clause p(l1), p(l2). It also accepts partial predicates: maplist(p(x), L) is equivalent to p(x, l1), p(x, l2). So we could write5 contains(L, X) :- member(X, L). key(Key) :- length(Key, 10), maplist(contains([a,b]), L), % the score stuff Now, let's query for the Key: ?- key(Key) Key = [a, b, a, b, a, a, b, a, a, b] Key = [b, b, a, b, a, a, a, a, a, b] Key = [b, b, a, b, a, a, b, b, a, b] Key = [b, b, b, b, a, a, b, a, a, b] So there are actually four different keys that all explain our data. Does this mean the puzzle is broken and has multiple different answers? Nope The puzzle wasn't to find out what the answer key was, the point was to find the fourth student's score. And if we query for it, we see all four solutions give him the same score: ?- key(Key), score([b, b, a, a, a, b, b, a, a, a], Key, X). X = 6 X = 6 X = 6 X = 6 Huh! I really like it when puzzles look like they're broken, but every "alternate" solution still gives the same puzzle answer. Total program length: 15 lines of code, compared to the original's 80 lines. Suck it, Pablo. (Incidentally, you can get all of the answer at once by writing findall(X, (key(Key), score($answer-array, Key, X)), L).) I still don't like puzzles for teaching The actual examples I'm using in the book are "analyzing a version control commit graph" and "planning a sequence of infrastructure changes", which are somewhat more likely to occur at work than needing to solve a puzzle. You'll see them in the next release! I found it because he wrote Gamer Games for Lite Gamers as a response to my Gamer Games for Non-Gamers. ↩ These are better versions of the core Prolog expressions \+ (A = B) and A is B + 1, because they can defer unification. ↩ Prolog-descendants have a convention of writing the arity of the function after its name, so score/3 means "score has three parameters". I think they do this because you can overload predicates with multiple different arities. Also Joe Armstrong used Prolog for prototyping, so Erlang and Elixir follow the same convention. ↩ It still gets the right answers without this type restriction, but I had no idea it did until I checked for myself. Probably better not to rely on this! ↩ We could make this even more compact by using a lambda function. First import module yall, then write maplist([X]>>member(X, [a,b]), Key). But (1) it's not a shorter program because you replace the extra definition with an extra module import, and (2) yall is SWI-Prolog specific and not an ISO-standard prolog module. Using contains is more portable. ↩