Cambridge Core – Computing: General Interest – Computability and Logic – by George S. Boolos. Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the. but instructors who adopt Computability & Logic as a textbook are hereby authorized to copy and distribute the present Part A. This permission does not extend.
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By this measure, the following sets are the same size:. This is all well and good, but I was expecting much more detail and motivation, given the impressive motivation found elsewhere in the book. Don’t pick it up for that reason, no matter what Luke tells you: The errata page is also located there and there are plenty of errata to be found in this book, unfortunately!
First order logic becomes undecidable when you add the first two-place relation symbol. It turns out that “monadic” logic first order logic in a language logic with only one-place relation symbols is decidable. Again, if this sounds new, I highly recommend picking up Computability and Logic: The logic sections were a good refresher, though they were somewhat rudimentary by comparison to Model Theory.
They continue to present material in a two-semester format, the first on computability theory enumerability, diagonalization, Turing compatibility, uncomputability, abacus computability, recursive functions, recursive sets and relations, equivalent definitions of computability and basic metalogic syntax, semantics, the compuyability of first-order logic, models and their existence, proofs and completeness, arithmetization, representability of recursive functions, computabjlity, undecidability, incompleteness and the unprobability of inconsistency.
It could be an invaluable resource for anyone in computer science looking to branch out into logic. For this edition they have revised and simplified their presentation of the representability of recursive functions, rewritten a section on Robinson arithmetic, and reworked exercises.
Computability and Logic by George S. Boolos
Jeffrey continue here in the tradition set by the late Boolos to present the “principal fundamental theoretical results logic” that would necessarily include the work of G del. When the authors finally introduce decidability sixty pages xnd it is not clearly compared with enumerability. Andrew rated it it was amazing Apr 22, Lattice Representations for Computability Theory.
It has lots of diagrams and gives you a good feel for what sort of power an “abacus machine” something that can do real math has as a computational engine. I would recommend this for someone who has a little knowledge of the subject.
This chapter introduces another formalism that has more machinery available to it than a Turing machine. Amazon Advertising Find, attract, and engage customers. Proofs from the Inside Out. If you’re not familiar with the concept, this chapter is a great introduction. As far as approach is concerned, the book places recursion theory at the center. It’s a pretty solid introduction, and a good way to brush up on the precise syntax if you’re feeling rusty. L rated it it was amazing Nov 30, Diagonalization in arithemtic is always a pleasure to run your mind over, in my experience.
Science, Logic, and Mathematics categorize this paper. For students or people in general who are extremely sophisticated mathematically, it can sometimes seem a little unrigorous.
For example, it constructs the halting problem from a diagonalization of an encoding of Turing machine instructions.
G. S. Boolos & R. C. Jeffrey, Computability and Logic – PhilPapers
This section was a bit faster, a bit less motivated, and more prone to dump proofs on you and say lovic that neat”. Other topics are covered along the way, too, of course, and there are several different courses one could teach using this book. Alexa Actionable Analytics for the Web. That would have made many of the topics easier to understand by making the underlying connections between the topics clearer.
If you want a rigourous and fairly thorough introduction to the domputability of Tarski, Goedel, Skolem, et al. If there’s a one-to-one mapping between two sets, those sets are “the same size”.
Do you understand why a logical system that proves its own consistency must be inconsistent?
It’s quite fun to pop the thing open and see the little gears. If we allow use of a fifth function:. Thomas Schindler – – Journal of Philosophical Logic 47 3: The latter result is proved by showing that we can put any logic into a “normal form” where it has at most one two-place relation symbol and no relation symbols of higher arity. This is a fun chapter to read if you feel like exploring the boundaries of undecidability.
Computability and Logic
boloos Even if you know that the deduction problem is equivalent to the halting problem, it’s illuminating to play directly with an encoding of Turing machines as logical theories. Computability and Logic, 4th Ed.
But if the theorem has always seemed somewhat confusing to you even after the cartoon guide then this chapter may well be enlightening. It even feels nice — it’s got thick pages, large text, and a number of useful diagrams. Ships from and sold by Amazon.
It was recommended to me by Luke along with a number of other books as a potential way to learn provability logic. For those of you who don’t know, the Turing Machine is an idealized computation engine which can compute answers to a very large class of problems. Top Reviews Most recent Top Reviews.