Formal Reasoning [pdf]
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Formal Reasoning [pdf]
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Re: Formal Reasoning [pdf]
#2Since 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.
Re: Formal Reasoning [pdf]
#3Since 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.
Essentially there is growing interest in the "formal" math community (combinatorics, mining, etc ..) to do exactly this.
Re: Formal Reasoning [pdf]
#4This is great reading and a great supplement to my limited education in math, comp sci, and formal logic.
Re: Formal Reasoning [pdf]
#5> 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".
Re: Formal Reasoning [pdf]
#6Since 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
Re: Formal Reasoning [pdf]
#7Since 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 trans…
As big Prolog fan, thanks for sharing those resources.
Re: Formal Reasoning [pdf]
#8> 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…
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.
Re: Formal Reasoning [pdf]
#9> 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…
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.
Re: Formal Reasoning [pdf]
#10> 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…
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.