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I’ve taken some graduate courses in programming languages where I’ve learned about the lambda calculus and about type theory, and I really appreciate this post

I’ve taken some graduate courses in programming languages where I’ve learned about the lambda calculus and about type theory, and I really appreciate this post since this provides a nice timeline of how programming language theory has evolved. One topic I’m very interested in is how computation can be expressed in many different ways and the tradeoffs behind various expressions. I apologize if this is off-topic, but I wonder if there has been work done on algorithm analysis using the lambda calculus? The texts that I’ve read on algorithms describe the algorithms using procedural code, and then the analysis is done on the amount of procedures that get called. However, I would imagine that the description would be different for non-procedural styles of programming. I’ve heard of a book called “Functional Data Structures” but I haven’t read it yet. I’m wondering if there has been work done on algorithm analysis at the lambda calculus level.

yaseer 7y ago
It's an interesting question that poses interesting problems. The main problem is that beta reduction is an awkward operation from the perspective of computational complexity, which is a fairly fundamental concept in algorithm analysis. There's a discussion of that here: https://cstheory.stackexchange.com/q/41776 https://cstheory.stackexchange.com/q/41776 P.S: Purely Functional Data Structures is a great book. It's focus is more on the implementation details of algorithms in a functional language (ML), and the necessary abstractions. It's not an analysis of algorithms within a formal model, like lambda calculus.
- olooney 7y ago
  Right. The naive way to do complexity analysis of a lambda calculus algorithm is to count β-reductions; but it's immediately clear that the cost of a β-reduction (which is basically a string substitution) is proportional to the length of the program. Here is a simple lambda calculus program: ((λ x . (x x)) (λ y . y)) If we apply the β-reduction rule once, we obtain: ((λ y . y) (λ y . y)) However, in order to perform this operation, we have to imagine the program as a string and stepping through the entire program symbol by symbol, either copying it unchanged (if it's not "x") or replacing it by "(λ y . y)" (if it is "x".) This in an O(m) algorithm, where m is the symbol length of the program. That is how a Turing machine emulating the lambda calculus would have to do it, for example. Surely we should prefer to analyze our algorithms in terms of atomic operations that can be implemented in O(1) on a Turing machine, no? Space complexity is also an issue; if our program expands to m symbols, surely space complexity is at least O(m) as well? Worse yet, β-"reductions" don't necessarily make the program shorter; and it can expand to arbitrary size before it starts to get smaller. It's not usually obvious how m (length of the program) is related to n (number of elements acted on, in the usual O(n) sense.) An "awkward operation from the perspective of computational complexity" indeed!
  - whatshisface 7y ago
    What if there is a way to write lambda calculus programs that makes beta reduction "easier" in that sense? We are used to reading things like ((λ y . y) (λ y . y)), but there may be another notation with better properties. If there was, we wouldn't even have to use it, because simply the fact that it exists would justify using beta reduction counts as a measure of complexity.
    - jcora 7y ago
      But this is clearly not a property of the particular notation, but lambda-calculus itself (and the way beta-reduction is defined). If you were to make it a constant-time operation in length it literally wouldn't be lambda-calculus. Which actually makes me curious: what sort of change did you have in mind?
      - whatshisface 7y ago
        If you could quickly translate a lambda-calculus expression into a program written in some other model of computation (that was easier to simulate on a Turing machine) and back, then you could do the complete reduction in the other model, only using lambda-calculus as your input and output. To perform the entire reduction you would still have to scan the expression in O(n), but it wouldn't necessarily be O(n * number of reduction steps). There could potentially be a relationship between the number of beta reductions and the number of steps required to execute the reduction in the other model, in which case beta reduction count would be justified as a measure of time.
  - vilhelm_s 7y ago
    That's not immediately clear at all! A β-reduction is basically the same as a function call, we know how to implement those, and they take constant time. Your example could be directly interpreted as a program in e.g. Lisp or ML, and there the function call is definitely not implemented by stepping through the entire program symbol by symbol. I think the issue in the stackoverflow post above is that the lambda calculus is nondeterministic, so if you try to define the cost as the smallest number of β-reductions to reduce a term to a value, then you're in trouble because it's in general very hard to figure out what the optimal way of reducing a program would be. But that also shows that it's a completely unrealistic model. Surely what we should do is pick an evaluation order, e.g. call-by-value, and then count the number of reductions there. This corresponds to what actually happens when you run your program on a real computer. In fact, CBV evaluation is really well-behaved, and you can given an extremely intuitive cost semantics to such a lambda calculus: see section 7.4 in [1], and also see [2] to show that this semantics is sound in the sense that you can compile it down to an abstract machine respecting the cost. (Incidentally, this is something that Bob Harper has been talking about for a long time, see e.g. his blog post [3] for some polemics.) If anything, I think what this argument shows is that Turing machines are bad for complexity-theoretic purposes. We do not want to "analyze our algorithms in terms of atomic operations that can be implemented in O(1) on a Turing machine", we want to count atomic operations that can be executed in O(1) time on the actual computers we run stuff on. So Turing machines are ok if you just want to distinguish polynomial from non-polynomial time, but for finer-grained distinctions you need a more accurate model, like a RAM, or... like the lambda calculus! [1] https://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf https://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf [2] http://www.cs.cmu.edu/~blelloch/papers/BG95.pdf http://www.cs.cmu.edu/~blelloch/papers/BG95.pdf [3] https://existentialtype.wordpress.com/2011/03/16/languages-and-machines/ https://existentialtype.wordpress.com/2011/03/16/languages-a...
bodhi 7y ago
It’s not exactly what you were after, but there’s a paper called “The Expressive Power of Programming Languages” that might be worth a look: https://www.sciencedirect.com/science/article/pii/016764239190036W https://www.sciencedirect.com/science/article/pii/0167642391...
jdyb 7y ago
Perhaps the work on binary lambda calculus and Kolmogorov complexity by John Tromp would be interesting to you? http://tromp.github.io/cl/cl.html http://tromp.github.io/cl/cl.html "Functional Bits: Lambda Calculus based Algorithmic Information Theory" http://tromp.github.io/cl/LC.pdf http://tromp.github.io/cl/LC.pdf
pwm 7y ago
(On my phone walking so very quick and dirty post) purely functional DSs in general will cost you an extra O(log n) cause of their underlying tree representation for immutability (immutability achieved by copying paths in the tree to retain old data). Also some DSs are very difficult to express in a purely functional way.
- adjkant 7y ago
  While true on the first part, these days many cases don't have insane primary efficiency needs that would be affected by that log(n). As always though, you choose the tools based on the job at hand, not the tools. In my experience, functional philosophy often is about the gained mental and consistency advantages achieved by the features offered.
  - pwm 7y ago
    Don’t get me wrong, you are preaching to the choir (I write Haskell at work), just thought it’s worth mentioning this to the OP. In practice sophisticated compilers like GHC produce insanely fast binaries and is getting better and better.
- continuational 7y ago
  Array indexing is not O(1) on any modern machine due to the cache hierachy.
reuben364 7y ago
I found the OPLSS lectures on parallel cost semantics using lambda calculus quite enjoyable, but I don't really know much about it. https://www.youtube.com/watch?v=lSc0EtF6_Zk https://www.youtube.com/watch?v=lSc0EtF6_Zk

yaseer 7y ago

It's an interesting question that poses interesting problems. The main problem is that beta reduction is an awkward operation from the perspective of computational complexity, which is a fairly fundamental concept in algorithm analysis. There's a discussion of that here: https://cstheory.stackexchange.com/q/41776 https://cstheory.stackexchange.com/q/41776 P.S: Purely Functional Data Structures is a great book. It's focus is more on the implementation details of algorithms in a functional language (ML), and the necessary abstractions. It's not an analysis of algorithms within a formal model, like lambda calculus.

olooney 7y ago
Right. The naive way to do complexity analysis of a lambda calculus algorithm is to count β-reductions; but it's immediately clear that the cost of a β-reduction (which is basically a string substitution) is proportional to the length of the program. Here is a simple lambda calculus program: ((λ x . (x x)) (λ y . y)) If we apply the β-reduction rule once, we obtain: ((λ y . y) (λ y . y)) However, in order to perform this operation, we have to imagine the program as a string and stepping through the entire program symbol by symbol, either copying it unchanged (if it's not "x") or replacing it by "(λ y . y)" (if it is "x".) This in an O(m) algorithm, where m is the symbol length of the program. That is how a Turing machine emulating the lambda calculus would have to do it, for example. Surely we should prefer to analyze our algorithms in terms of atomic operations that can be implemented in O(1) on a Turing machine, no? Space complexity is also an issue; if our program expands to m symbols, surely space complexity is at least O(m) as well? Worse yet, β-"reductions" don't necessarily make the program shorter; and it can expand to arbitrary size before it starts to get smaller. It's not usually obvious how m (length of the program) is related to n (number of elements acted on, in the usual O(n) sense.) An "awkward operation from the perspective of computational complexity" indeed!
- whatshisface 7y ago
  What if there is a way to write lambda calculus programs that makes beta reduction "easier" in that sense? We are used to reading things like ((λ y . y) (λ y . y)), but there may be another notation with better properties. If there was, we wouldn't even have to use it, because simply the fact that it exists would justify using beta reduction counts as a measure of complexity.
  - jcora 7y ago
    But this is clearly not a property of the particular notation, but lambda-calculus itself (and the way beta-reduction is defined). If you were to make it a constant-time operation in length it literally wouldn't be lambda-calculus. Which actually makes me curious: what sort of change did you have in mind?
    - whatshisface 7y ago
      If you could quickly translate a lambda-calculus expression into a program written in some other model of computation (that was easier to simulate on a Turing machine) and back, then you could do the complete reduction in the other model, only using lambda-calculus as your input and output. To perform the entire reduction you would still have to scan the expression in O(n), but it wouldn't necessarily be O(n * number of reduction steps). There could potentially be a relationship between the number of beta reductions and the number of steps required to execute the reduction in the other model, in which case beta reduction count would be justified as a measure of time.
- vilhelm_s 7y ago
  That's not immediately clear at all! A β-reduction is basically the same as a function call, we know how to implement those, and they take constant time. Your example could be directly interpreted as a program in e.g. Lisp or ML, and there the function call is definitely not implemented by stepping through the entire program symbol by symbol. I think the issue in the stackoverflow post above is that the lambda calculus is nondeterministic, so if you try to define the cost as the smallest number of β-reductions to reduce a term to a value, then you're in trouble because it's in general very hard to figure out what the optimal way of reducing a program would be. But that also shows that it's a completely unrealistic model. Surely what we should do is pick an evaluation order, e.g. call-by-value, and then count the number of reductions there. This corresponds to what actually happens when you run your program on a real computer. In fact, CBV evaluation is really well-behaved, and you can given an extremely intuitive cost semantics to such a lambda calculus: see section 7.4 in [1], and also see [2] to show that this semantics is sound in the sense that you can compile it down to an abstract machine respecting the cost. (Incidentally, this is something that Bob Harper has been talking about for a long time, see e.g. his blog post [3] for some polemics.) If anything, I think what this argument shows is that Turing machines are bad for complexity-theoretic purposes. We do not want to "analyze our algorithms in terms of atomic operations that can be implemented in O(1) on a Turing machine", we want to count atomic operations that can be executed in O(1) time on the actual computers we run stuff on. So Turing machines are ok if you just want to distinguish polynomial from non-polynomial time, but for finer-grained distinctions you need a more accurate model, like a RAM, or... like the lambda calculus! [1] https://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf https://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf [2] http://www.cs.cmu.edu/~blelloch/papers/BG95.pdf http://www.cs.cmu.edu/~blelloch/papers/BG95.pdf [3] https://existentialtype.wordpress.com/2011/03/16/languages-and-machines/ https://existentialtype.wordpress.com/2011/03/16/languages-a...

bodhi 7y ago

It’s not exactly what you were after, but there’s a paper called “The Expressive Power of Programming Languages” that might be worth a look: https://www.sciencedirect.com/science/article/pii/016764239190036W https://www.sciencedirect.com/science/article/pii/0167642391...

jdyb 7y ago

Perhaps the work on binary lambda calculus and Kolmogorov complexity by John Tromp would be interesting to you? http://tromp.github.io/cl/cl.html http://tromp.github.io/cl/cl.html "Functional Bits: Lambda Calculus based Algorithmic Information Theory" http://tromp.github.io/cl/LC.pdf http://tromp.github.io/cl/LC.pdf

pwm 7y ago

(On my phone walking so very quick and dirty post) purely functional DSs in general will cost you an extra O(log n) cause of their underlying tree representation for immutability (immutability achieved by copying paths in the tree to retain old data). Also some DSs are very difficult to express in a purely functional way.

adjkant 7y ago
While true on the first part, these days many cases don't have insane primary efficiency needs that would be affected by that log(n). As always though, you choose the tools based on the job at hand, not the tools. In my experience, functional philosophy often is about the gained mental and consistency advantages achieved by the features offered.
- pwm 7y ago
  Don’t get me wrong, you are preaching to the choir (I write Haskell at work), just thought it’s worth mentioning this to the OP. In practice sophisticated compilers like GHC produce insanely fast binaries and is getting better and better.
continuational 7y ago
Array indexing is not O(1) on any modern machine due to the cache hierachy.

reuben364 7y ago

I found the OPLSS lectures on parallel cost semantics using lambda calculus quite enjoyable, but I don't really know much about it. https://www.youtube.com/watch?v=lSc0EtF6_Zk https://www.youtube.com/watch?v=lSc0EtF6_Zk