> Formal languages are basically laboratory-sized versions, or models, of natural languages.
I can understand why a hundred years ago explaining what formal is (in the context of formal languages) could have been difficult. You had to say that it means something whose form can be manipulated without "understanding", or by rules that pertain to form rather than meaning. But since the late 1930s explaining what formal means has become much simpler: it means mechanical. A formal language is one that can be precisely and accurately interpreted and manipulated by a machine.
When we talk about "formal proofs" we don't mean precise proofs, official proofs, or proofs written by a mathematician. We mean proofs written in a language, and following a procedure, that can be mechanically checked (and by a fairly basic algorithm).
While it is still a little colloquial, these days we can say that formal languages are those languages that can always be correctly interpreted by a computer. I think this captures the meaning of "formal" much more than saying these are "models of natural language".
Undecidable languages are formal languages, too, even though there's no Turing machine that can accurately determine for any string whether it is part of the language or not.
A formal language is a set of finite-length sequences (called "words") of symbols from another set (called the "alphabet"). It's essentially a very crude approximation of some strings of letters in an alphabetic writing system forming words in a natural language, while other combinations are just nonsense.
For a given formal language, there don't necessarily have to be any rules governing the words of the language, though the languages used for writing formal proofs are typically more well-behaved.
You're talking about formal languages in the context of computer science. Formal languages in the context of logic predate computer science (or could be said to be a direct precursor to computer science). These logic languages are also trivially decidable in the computer-science sense of formal languages, i.e. their set of strings is easily decidable. When we talk of decidability in those languages we ususally mean the decidability of whether a statement is provable or not (using the language's inference rules).
While my explanation of "formal" is meant to be introductory and not entirely precise, that some problem tackled by an algorithm is undecidable does not mean that that problem isn't precisely interpretable by the computer. A Python interpreter doesn't terminate for all inputs (and therefore doesn't decide halting), yet it does interpret all of its inputs precisely.
It does get worse in the sense that there could be languages whose description is incompressible (we can simulate this, assuming hash functions approximate random oracles, by saying "choose a secret key; now the language is 'every string whose HMAC value under that secret key is even'").
If you accept some axiomatic assumptions about infinite sets (that are common in mathematics; I'm not sure exactly what the weakest required axiom is for this), then you can even believe that there are infinite languages that have no finite description at all, very much akin to the commonplace claim that there are real numbers that have no finite description at all. There are formulations of mathematics in which this is not true, but most mathematicians seemingly work in formulations in which it is true.
I even expect that we can probably prove this directly using the powerset operation and diagonalization, which doesn't require particularly strong assumptions about infinities.
Use of the word "mechanical" to describe formal reasoning predates computers.
Here's the first sentence of Godel's 1931 On formally undecidable propositions...
"The development of mathematics in the direction of greater exactness has—as is well known—led to large tracts of it becoming formalized, so that proofs can be carried out according to a few mechanical rules."
Leibniz had discussed calculating machines (and even thought about binary arithmetic being the most appropriate implementation), so the general idea probably goes back quite far
Edit: Oh, I guess by "late 1930s" you're referring to Turing's 1936 paper where he defines Turing machines, rather than actual electronic computers. Still, understanding "formal" as "mechanical" predates it.
Yes, by Godel's time the notion of "calculability" was already at least intuitively grasped, and it was then that "formal" was understood to mean mechanical. Turing made the connection rigorous.
Leibniz spoke of "automatons" and dreamt of some sort of "thoughtless" reasoning, but I don't know if he had the right building blocks to even think of mechanisation as we could since the 19th century. E.g. here's how Leibniz tries to justify the utility of formal reasoning: "Our thoughts are for the most part what I call ‘blind thoughts’. I mean that they are empty of perception and sensibility, and consist in the wholly unaided use of symbols... We often reason in words, with the object itself virtually absent from our mind."
So he definitely had the right concept - which is why formal logic is so old - but not the right language that most people would intuitively understand today.
It's not a "belief"; that's what computability is. This definition is the whole point of the work by Church and Turing that resulted in the lambda calculus and the Turing machine, respectively.
Also i highly recommend everybody to read the great logician Alfred Tarski's classic book Introduction to Logic: And to the Methodology of Deductive Sciences to really understand "Logic" which is what Formal Reasoning is based on.
Agreed. Also, I feel strongly that logic should be part of the core curriculum in liberal arts colleges if not high school. I took a Logic class as an undergrad, in a course that covered Sentential, Predicate, and Aristotelian (syllogistic) Logic, then became a paid tutor the next semester. It was profoundly useful, and applicable to nearly every other field of study. So many otherwise well-educated people frequently fall prey to common logical fallacies, likely because their grasp of logic is strictly intuitive and implicit.
Logic should be taught from the intuitive aspects first before turning it formal; With lots of domain examples to show how sentences often have a common Form (i.e. structure) which can then be generalized as Symbolic Logic ranging over a Domain of Discourse/Domain of Interpretation. Once this distinction between "Structure/Form"(i.e. Syntax) and "Meanings"(i.e. Semantics) is grasped then mechanical manipulation strictly following rules, starts to make sense.
Instead, what is taught is starting with truth tables for logical operators and thus students simply don't learn to think/understand. For example, almost everybody is thrown off by the "implication" logical operator (evaluates to True when the antecedent is False irrespective of the consequent). But if you use some actual domain examples then it becomes clear why it is so.
Alfred Tarski's classic book mentioned above does a good job of building up logic from the first principles and hence is a great companion to any standard textbook.
Well, they can't always be correctly interpreted by computers. Computers misinterpret formal languages all the time! And since GPT-2, computers are reasonably frequently able to interpret informal languages correctly too.
Since LLMs are great at coding but bad at logic, maybe an approach like this can bridge the gap? So first let it translate natural language to a formal language, from there allow it to use a logic engine to make verifiable transformations (correctness-preserving), and finally translate back to natural language.
3) User "bytebach" gives a nice example of using Prolog as an intermediate DSL in the prompt to an LLM so as to transform English declarative -> Imperative code - https://news.ycombinator.com/item?id=41549823
There's also [1], containing further bibliography references along with practical applications in discrete planning.
Prolog is quite popular and successful as a target for LLMs.
And it's no accident considering Prolog was introduced to represent natural language statements in (predicate) logic.
> So first let it translate natural language to a formal language, from there allow it to use a logic engine to make verifiable transformations (correctness-preserving), and finally translate back to natural language.
Linguists in the Richard Montague tradition have indeed attempted to use tools like formal logic, lambda calculus, continuations, monads, modalities etc. to try and understand the semantics of natural language in a way that's both logical/formal and compositional - i.e. accounting at least partially for the "deep" syntax of natural language itself, such that a fragment can be said to have a semantics of its own and the global semantics of a broader construction arises from "composing" these narrower semantics in a reasonably straightforward way.
This is pretty much the same as trying to take the "let's translate natural language into formal logic" proof-of-concept exercises from a text like OP (or from your average logic textbook) seriously and extending them to natural language as a whole. It turns out that this is really, really hard, because natural language mixes multiple "modalities" together in what looks like a very ad-hoc way. We only barely have the tools in formal logic to try and replicate this, such as continuations, modalities and monads. (Linguists actually talk about many phenomena of this kind, talking about "modalities" is just one example that's both general enough to give a broad idea and happens to be straightforward enough on the logical side. You have quantification, intensionality, anaphora, scope, presupposition, modality proper, discourse-level inference, pragmatics, ellipsis, indexicals, speech acts, etc. etc. etc.)
And because the semantics of natural language is both so general and so hard to pin down, it doesn't seem useful to "reason" about the logical semantics of natural languages so directly. You can of course use logical/mathematical modeling to address all sorts of problems, but this doesn't occur via a verbatim "translation" from some specific language utterance.
I've been strapping different LLM based setups to Lean 4 with a variety of different prompting methods. My biggest conclusion here is that LLMs are worse at formalizing than humans are. Additionally, for Lean 4 specifically, I don't think there's enough training data.
I'm of the opinion that formalization is the biggest bottleneck of current generation LLMs. However, I don't think that this necessarily suggests that LLMs don't benefit from formal methods. Given existing abstractions, Lean4's exceptional tooling allows for more efficient iteration with LLMs and requires less human supervision since Lean's language server provides specific and actionable feedback whenever the LLM makes a mistake. I've also noticed that LLMs can reason about code written in Lean4 far more effectively than in Python, despite Python having orders of magnitude more training data than Lean.
Nonetheless, I concur that LLMs don't yet know how to translate a request stated in a prompt to a complete Lean4 interpretation. My practice so far has usually required me to first choose an existing reference file that is similar to my desired goals, and use this reference as "inspiration" for how the LLM should go about formalization.
Yeah, we really need LLMs to work swimmingly with Lean 4. It is currently hot garbage and it does not understand proof composition, exploring proof extensions, lemma search, etc. It does not explore an open-ended node to a mathematical knowledge graph by substituting various options.
I'd happily work with someone on a conversational theorem prover, if anyone's up for it.
I think a big issue with this approach is that the initial and last steps are prone to sycophancy: the machine wants you to believe it's getting the job done, which may lead it to do something correct-looking over something correct. The middle steps (correct-by-construction transformations) do not need an LLM at all. It's what a certified compiler does.
I think the way forward, for the immediate future, is to feed AI agents with a mixture of (hand-written) natural language and formal blueprints, then use as many mechanized analysis methods as possible on the generated code (from unit/regression testing to static analysis, and possibly more powerful software verification procedures). Potentially feed the output of these analyses back to the agents.
You should check out Math Academy. It'll give you as much math background as any engineering student and they aim to provide the equivalent of a full undergrad math degree in the next few years.
I've also had a lot of success with the Art Of Problem Solving text-books, the regular ones not the competition ones. As someone who's starting from the ground up with arithmetic.
I think of Math Academy as ideal for efficiently mastering subjects at a level slightly deeper than a typical education in a shorter amount of time, and AoPS is for taking your time to go much deeper into the topics.
Doing them both is probably best, but doing one or the other would still work.
>I think of Math Academy as ideal for efficiently mastering subjects at a level slightly deeper than a typical education in a shorter amount of time, and AoPS is for taking your time to go much deeper into the topics.
I've been searching hi and lo for the ultimate beginner resource and nothing comes close to AoPS for the exact reason you mentioned. I started with principia mathematica by bertrand russel because I thought math worked like a big tree all stemming from one trunk. Boy was that a bad move lol. I'd be reading some textbook and I have an infinite amount of why's that would block me on the most basic of things, tons of assumed knowledge. AoPS treats you like a caveman who first discovered fire, and then discovered their books, in that order.
I'll give MathAcademy a go too, I heard lots of great things.
> Formal languages are basically laboratory-sized versions, or models, of natural languages.
I can understand why a hundred years ago explaining what formal is (in the context of formal languages) could have been difficult. You had to say that it means something whose form can be manipulated without "understanding", or by rules that pertain to form rather than meaning. But since the late 1930s explaining what formal means has become much simpler: it means mechanical. A formal language is one that can be precisely and accurately interpreted and manipulated by a machine.
When we talk about "formal proofs" we don't mean precise proofs, official proofs, or proofs written by a mathematician. We mean proofs written in a language, and following a procedure, that can be mechanically checked (and by a fairly basic algorithm).
While it is still a little colloquial, these days we can say that formal languages are those languages that can always be correctly interpreted by a computer. I think this captures the meaning of "formal" much more than saying these are "models of natural language".
Undecidable languages are formal languages, too, even though there's no Turing machine that can accurately determine for any string whether it is part of the language or not.
A formal language is a set of finite-length sequences (called "words") of symbols from another set (called the "alphabet"). It's essentially a very crude approximation of some strings of letters in an alphabetic writing system forming words in a natural language, while other combinations are just nonsense.
For a given formal language, there don't necessarily have to be any rules governing the words of the language, though the languages used for writing formal proofs are typically more well-behaved.
You're talking about formal languages in the context of computer science. Formal languages in the context of logic predate computer science (or could be said to be a direct precursor to computer science). These logic languages are also trivially decidable in the computer-science sense of formal languages, i.e. their set of strings is easily decidable. When we talk of decidability in those languages we ususally mean the decidability of whether a statement is provable or not (using the language's inference rules).
While my explanation of "formal" is meant to be introductory and not entirely precise, that some problem tackled by an algorithm is undecidable does not mean that that problem isn't precisely interpretable by the computer. A Python interpreter doesn't terminate for all inputs (and therefore doesn't decide halting), yet it does interpret all of its inputs precisely.
It does get worse in the sense that there could be languages whose description is incompressible (we can simulate this, assuming hash functions approximate random oracles, by saying "choose a secret key; now the language is 'every string whose HMAC value under that secret key is even'").
If you accept some axiomatic assumptions about infinite sets (that are common in mathematics; I'm not sure exactly what the weakest required axiom is for this), then you can even believe that there are infinite languages that have no finite description at all, very much akin to the commonplace claim that there are real numbers that have no finite description at all. There are formulations of mathematics in which this is not true, but most mathematicians seemingly work in formulations in which it is true.
I even expect that we can probably prove this directly using the powerset operation and diagonalization, which doesn't require particularly strong assumptions about infinities.
Use of the word "mechanical" to describe formal reasoning predates computers.
Here's the first sentence of Godel's 1931 On formally undecidable propositions...
"The development of mathematics in the direction of greater exactness has—as is well known—led to large tracts of it becoming formalized, so that proofs can be carried out according to a few mechanical rules."
Leibniz had discussed calculating machines (and even thought about binary arithmetic being the most appropriate implementation), so the general idea probably goes back quite far
Edit: Oh, I guess by "late 1930s" you're referring to Turing's 1936 paper where he defines Turing machines, rather than actual electronic computers. Still, understanding "formal" as "mechanical" predates it.
Yes, by Godel's time the notion of "calculability" was already at least intuitively grasped, and it was then that "formal" was understood to mean mechanical. Turing made the connection rigorous.
Leibniz spoke of "automatons" and dreamt of some sort of "thoughtless" reasoning, but I don't know if he had the right building blocks to even think of mechanisation as we could since the 19th century. E.g. here's how Leibniz tries to justify the utility of formal reasoning: "Our thoughts are for the most part what I call ‘blind thoughts’. I mean that they are empty of perception and sensibility, and consist in the wholly unaided use of symbols... We often reason in words, with the object itself virtually absent from our mind."
So he definitely had the right concept - which is why formal logic is so old - but not the right language that most people would intuitively understand today.
Leibniz even invented a calculating machine. I didn't know he'd actually built one
https://en.wikipedia.org/wiki/Calculus_ratiocinator
Perhaps it has to be that way, the motivation to build a mechanical computer is based on the belief that computation can be mechanised.
It's not a "belief"; that's what computability is. This definition is the whole point of the work by Church and Turing that resulted in the lambda calculus and the Turing machine, respectively.
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Well said.
Also i highly recommend everybody to read the great logician Alfred Tarski's classic book Introduction to Logic: And to the Methodology of Deductive Sciences to really understand "Logic" which is what Formal Reasoning is based on.
Agreed. Also, I feel strongly that logic should be part of the core curriculum in liberal arts colleges if not high school. I took a Logic class as an undergrad, in a course that covered Sentential, Predicate, and Aristotelian (syllogistic) Logic, then became a paid tutor the next semester. It was profoundly useful, and applicable to nearly every other field of study. So many otherwise well-educated people frequently fall prey to common logical fallacies, likely because their grasp of logic is strictly intuitive and implicit.
Logic should be taught from the intuitive aspects first before turning it formal; With lots of domain examples to show how sentences often have a common Form (i.e. structure) which can then be generalized as Symbolic Logic ranging over a Domain of Discourse/Domain of Interpretation. Once this distinction between "Structure/Form"(i.e. Syntax) and "Meanings"(i.e. Semantics) is grasped then mechanical manipulation strictly following rules, starts to make sense.
Instead, what is taught is starting with truth tables for logical operators and thus students simply don't learn to think/understand. For example, almost everybody is thrown off by the "implication" logical operator (evaluates to True when the antecedent is False irrespective of the consequent). But if you use some actual domain examples then it becomes clear why it is so.
Alfred Tarski's classic book mentioned above does a good job of building up logic from the first principles and hence is a great companion to any standard textbook.
Well, they can't always be correctly interpreted by computers. Computers misinterpret formal languages all the time! And since GPT-2, computers are reasonably frequently able to interpret informal languages correctly too.
Since LLMs are great at coding but bad at logic, maybe an approach like this can bridge the gap? So first let it translate natural language to a formal language, from there allow it to use a logic engine to make verifiable transformations (correctness-preserving), and finally translate back to natural language.
People are already using Prolog for this;
1) A series of excellent and detailed blog posts by Eugene Asahara Prolog in the LLM Era - https://eugeneasahara.com/category/prolog-in-the-llm-era/
2) Previous HN discussion Use Prolog to improve LLM's reasoning - https://news.ycombinator.com/item?id=41831735
3) User "bytebach" gives a nice example of using Prolog as an intermediate DSL in the prompt to an LLM so as to transform English declarative -> Imperative code - https://news.ycombinator.com/item?id=41549823
There's also [1], containing further bibliography references along with practical applications in discrete planning.
Prolog is quite popular and successful as a target for LLMs. And it's no accident considering Prolog was introduced to represent natural language statements in (predicate) logic.
[1]: https://quantumprolog.sgml.net/llm-demo/part1.html
That is pretty neat!
As big Prolog fan, thanks for sharing those resources.
> So first let it translate natural language to a formal language, from there allow it to use a logic engine to make verifiable transformations (correctness-preserving), and finally translate back to natural language.
Linguists in the Richard Montague tradition have indeed attempted to use tools like formal logic, lambda calculus, continuations, monads, modalities etc. to try and understand the semantics of natural language in a way that's both logical/formal and compositional - i.e. accounting at least partially for the "deep" syntax of natural language itself, such that a fragment can be said to have a semantics of its own and the global semantics of a broader construction arises from "composing" these narrower semantics in a reasonably straightforward way.
This is pretty much the same as trying to take the "let's translate natural language into formal logic" proof-of-concept exercises from a text like OP (or from your average logic textbook) seriously and extending them to natural language as a whole. It turns out that this is really, really hard, because natural language mixes multiple "modalities" together in what looks like a very ad-hoc way. We only barely have the tools in formal logic to try and replicate this, such as continuations, modalities and monads. (Linguists actually talk about many phenomena of this kind, talking about "modalities" is just one example that's both general enough to give a broad idea and happens to be straightforward enough on the logical side. You have quantification, intensionality, anaphora, scope, presupposition, modality proper, discourse-level inference, pragmatics, ellipsis, indexicals, speech acts, etc. etc. etc.)
And because the semantics of natural language is both so general and so hard to pin down, it doesn't seem useful to "reason" about the logical semantics of natural languages so directly. You can of course use logical/mathematical modeling to address all sorts of problems, but this doesn't occur via a verbatim "translation" from some specific language utterance.
I've been strapping different LLM based setups to Lean 4 with a variety of different prompting methods. My biggest conclusion here is that LLMs are worse at formalizing than humans are. Additionally, for Lean 4 specifically, I don't think there's enough training data.
I'm of the opinion that formalization is the biggest bottleneck of current generation LLMs. However, I don't think that this necessarily suggests that LLMs don't benefit from formal methods. Given existing abstractions, Lean4's exceptional tooling allows for more efficient iteration with LLMs and requires less human supervision since Lean's language server provides specific and actionable feedback whenever the LLM makes a mistake. I've also noticed that LLMs can reason about code written in Lean4 far more effectively than in Python, despite Python having orders of magnitude more training data than Lean.
Nonetheless, I concur that LLMs don't yet know how to translate a request stated in a prompt to a complete Lean4 interpretation. My practice so far has usually required me to first choose an existing reference file that is similar to my desired goals, and use this reference as "inspiration" for how the LLM should go about formalization.
Yeah, we really need LLMs to work swimmingly with Lean 4. It is currently hot garbage and it does not understand proof composition, exploring proof extensions, lemma search, etc. It does not explore an open-ended node to a mathematical knowledge graph by substituting various options.
I'd happily work with someone on a conversational theorem prover, if anyone's up for it.
Join the Lean Zulip. There are many people interested in this.
https://leanprover.zulipchat.com/
I think a big issue with this approach is that the initial and last steps are prone to sycophancy: the machine wants you to believe it's getting the job done, which may lead it to do something correct-looking over something correct. The middle steps (correct-by-construction transformations) do not need an LLM at all. It's what a certified compiler does.
I think the way forward, for the immediate future, is to feed AI agents with a mixture of (hand-written) natural language and formal blueprints, then use as many mechanized analysis methods as possible on the generated code (from unit/regression testing to static analysis, and possibly more powerful software verification procedures). Potentially feed the output of these analyses back to the agents.
That's the approach we're taking to verify LLM-generated SQL code at http://sql.ai.
Essentially there is growing interest in the "formal" math community (combinatorics, mining, etc ..) to do exactly this.
This is great reading and a great supplement to my limited education in math, comp sci, and formal logic.
You should check out Math Academy. It'll give you as much math background as any engineering student and they aim to provide the equivalent of a full undergrad math degree in the next few years.
I've also had a lot of success with the Art Of Problem Solving text-books, the regular ones not the competition ones. As someone who's starting from the ground up with arithmetic.
Yeah they're great too!
I think of Math Academy as ideal for efficiently mastering subjects at a level slightly deeper than a typical education in a shorter amount of time, and AoPS is for taking your time to go much deeper into the topics.
Doing them both is probably best, but doing one or the other would still work.
>I think of Math Academy as ideal for efficiently mastering subjects at a level slightly deeper than a typical education in a shorter amount of time, and AoPS is for taking your time to go much deeper into the topics.
I've been searching hi and lo for the ultimate beginner resource and nothing comes close to AoPS for the exact reason you mentioned. I started with principia mathematica by bertrand russel because I thought math worked like a big tree all stemming from one trunk. Boy was that a bad move lol. I'd be reading some textbook and I have an infinite amount of why's that would block me on the most basic of things, tons of assumed knowledge. AoPS treats you like a caveman who first discovered fire, and then discovered their books, in that order.
I'll give MathAcademy a go too, I heard lots of great things.
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