#623 - Lambda Count
The lambda-calculus is a universal model of computation at the core of functional programming languages. It is based on lambda-terms, a minimal programming language featuring only function definitions, function calls and variables. Lambda-terms are built according to the following rules:
- Any variable \(x\) (single letter, from some infinite alphabet) is a lambda-term.
- If \(M\) and \(N\) are lambda-terms, then \((M N)\) is a lambda-term, called the application of \(M\) to \(N\).
- If \(x\) is a variable and \(M\) is a term, then \((\lambda x. M)\) is a lambda-term, called an abstraction. An abstraction defines an anonymous function, taking \(x\) as parameter and sending back \(M\).
A lambda-term \(T\) is said to be closed if for all variables \(x\), all occurrences of \(x\) within \(T\) are contained within some abstraction \((\lambda x. M)\) in \(T\). The smallest such abstraction is said to bind the occurrence of the variable \(x\). In other words, a lambda-term is closed if all its variables are bound to parameters of enclosing functions definitions. For example, the term \((\lambda x. x)\) is closed, while the term \((\lambda x. (x y))\) is not because \(y\) is not bound.
Also, we can rename variables as long as no binding abstraction changes. This means that \((\lambda x. x)\) and \((\lambda y. y)\) should be considered equivalent since we merely renamed a parameter. Two terms equivalent modulo such renaming are called \(\alpha\)-equivalent. Note that \((\lambda x. (\lambda y. (x y)))\) and \((\lambda x. (\lambda x. (x x)))\) are not \(\alpha\)-equivalent, since the abstraction binding the first variable was the outer one and becomes the inner one. However, \((\lambda x. (\lambda y. (x y)))\) and \((\lambda y. (\lambda x. (y x)))\) are \(\alpha\)-equivalent.
The following table regroups the lambda-terms that can be written with at most \(15\) symbols, symbols being parenthesis, \(\lambda\), dot and variables.
\[\begin{array}{|c|c|c|c|} \hline (\lambda x.x) & (\lambda x.(x x)) & (\lambda x.(\lambda y.x)) & (\lambda x.(\lambda y.y)) \\ \hline (\lambda x.(x (x x))) & (\lambda x.((x x) x)) & (\lambda x.(\lambda y.(x x))) & (\lambda x.(\lambda y.(x y))) \\ \hline (\lambda x.(\lambda y.(y x))) & (\lambda x.(\lambda y.(y y))) & (\lambda x.(x (\lambda y.x))) & (\lambda x.(x (\lambda y.y))) \\ \hline (\lambda x.((\lambda y.x) x)) & (\lambda x.((\lambda y.y) x)) & ((\lambda x.x) (\lambda x.x)) & (\lambda x.(x (x (x x)))) \\ \hline (\lambda x.(x ((x x) x))) & (\lambda x.((x x) (x x))) & (\lambda x.((x (x x)) x)) & (\lambda x.(((x x) x) x)) \\ \hline \end{array}\]Let be \(\Lambda(n)\) the number of distinct closed lambda-terms that can be written using at most \(n\) symbols, where terms that are \(\alpha\)-equivalent to one another should be counted only once. You are given that \(\Lambda(6) = 1\), \(\Lambda(9) = 2\), \(\Lambda(15) = 20\) and \(\Lambda(35) = 3166438\).
Find \(\Lambda(2000)\). Give the answer modulo \(1\,000\,000\,007\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=623. Published Sunday, 25th March 2018, 10:00 am. Solved by 394 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact integer reasoning and algorithmic efficiency. Foundational rigour and computational discipline rather than a direct trading application.
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 19.2 Primes, Sieves, and Integer Factorization
Recommended stepping-stone problems: #230 · #473 · #220
Concepts: computational-geometry number-theory string-processing brute-force-reduction
Likely techniques: hashing
Learning mode
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Understand the problem
- What exactly is the input to problem 623? Is it a bound (1000000007), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single residue (the answer is reduced modulo a given number, so keep everything in modular arithmetic from the start). Required format: answer modulo 1000000007.
- Which objects exactly are in scope, and which are excluded by the wording (strict vs non-strict inequality, 'distinct', 'proper', 'below' vs 'up to')?
- What constraint does the bound 1000000007 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 1000000007?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1000000007 and the cost of testing one.
- Which string-processing fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says string-processing / hashing - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1000000007, and the wall-clock time you expect. Write both down now.
- Predict the key data structure: what is stored, keyed by what, and how large will it get at full scale?
- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it (the statement says: "For example, the term ( x.") - then run it. A surprise here is worth more than an hour of debugging later.
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Optimization
You have a correct answer. That is the start of the learning, not the end.
- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
- Reduce memory. Can you stream, or keep only the last k states?
- Replace brute force with a closed form, a sieve, a recurrence, or a symmetry argument.
- Prove the optimized version computes the same thing.
- Compare two implementations and time them.
Explain it
Which step of your solution were you least confident about, and what evidence would settle it?
What did you try first, and what specifically made you abandon it - a proof, a timing, or a wrong small-case answer?
Where did the string-processing structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the hashing idea faster? Which words in the statement were pointing at it, and did you notice them?
What was the bug that cost you the most time, and what CLASS of bug was it (off-by-one, definition misread, precision, state under-specified)?
How would your solution change if the bound 1000000007 were multiplied by 1000? Does it survive, or does it need a different idea?
What is the honest complexity of what you wrote (not what you intended), and where is the remaining slack?
Which problem you have already solved is this most similar to, and what is the shared skeleton - is it really 'hashing' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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Mastery check
- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: #230 · #473 · #220
- Transfer: where else does this technique appear? Name a lesson and a real computational setting.
- Spaced re-attempt: come back after the review interval and re-solve it with no hints.