Questions tagged [lo.logic]

Computational and mathematical logic.

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Solid applications of category theory in TCS?

I've been learning a few bits of category theory. It certainly is a different way of looking at things. (Very rough summary for those who haven't seen it: category theory gives ways of expressing all ...
Ryan Williams's user avatar
83 votes
5 answers
5k views

Techniques for Reversing the Order of Quantifiers

It is well-known that in general, the order of universal and existential quantifiers cannot be reversed. In other words, for a general logical formula $\phi(\cdot,\cdot)$, $(\forall x)(\exists y) \...
Sadeq Dousti's user avatar
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71 votes
7 answers
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Which interesting theorems in TCS rely on the Axiom of Choice? (Or alternatively, the Axiom of Determinacy?)

Mathematicians sometimes worry about the Axiom of Choice (AC) and Axiom of Determinancy (AD). Axiom of Choice: Given any collection ${\cal C}$ of nonempty sets, there is a function $f$ that, given a ...
Ryan Williams's user avatar
52 votes
3 answers
6k views

Shallow versus Deep Embeddings

When encoding a logic into a proof assistant such as Coq or Isabelle, a choice needs to be made between using a shallow and a deep embedding. In a shallow embedding logical formulas are written ...
Dave Clarke's user avatar
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51 votes
8 answers
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Are there non-constructive algorithm existence proofs?

I remember I might have encountered references to problems that have been proven to be solvable with a particular complexity, but with no known algorithm to actually reach this complexity. I struggle ...
jkff's user avatar
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49 votes
3 answers
4k views

How do 'tactics' work in proof assistants?

Question: How do 'tactics' work in proof assistants? They seem to be ways of specifying how to rewrite a term into an equivalent term (for some definition of 'equivalent'). Presumably there are formal ...
John Salvatier's user avatar
48 votes
5 answers
7k views

What is the most intuitive dependent type theory I could learn?

I am interested in getting a really solid grasp on dependent typing. I've read most of TaPL and read (if not fully absorbed) 'Dependent Types' in ATTaPL. I've also read and skimmed a bunch of articles ...
John Salvatier's user avatar
48 votes
2 answers
6k views

Explaining Applicative functor in categorical terms - monoidal functors

I'd like to understand Applicative in terms of category theory. The documentation for Applicative says that it's a strong lax ...
Petr's user avatar
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46 votes
4 answers
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Are there any proofs the undecidability of the halting problem that does not depend on self-referencing or diagonalization ?

This is a question related to this one. Putting it again in a much simpler form after a lot of discussion there, that it felt like a totally different question. The classical proof of the ...
Mohammad Alaggan's user avatar
44 votes
4 answers
5k views

How would I go about learning the underlying theory of the Coq proof assistant?

I'm going over the course notes at CIS 500: Software Foundations and the exercises are a lot of fun. I'm only at the third exercise set but I would like to know more about what's happening when I use ...
user avatar
40 votes
5 answers
1k views

Results in Theoretical CS independent of ZFC

I'm going to ask a quite vague question, since the borderline between theoretical computer science and math is not always easy to distinguish. QUESTION: Are you aware of any interesting result in CS ...
OldFella's user avatar
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39 votes
5 answers
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Is there a logic without induction that captures much of P?

The Immerman-Vardi theorem states that PTIME (or P) is precisely the class of languages that can be described by a sentence of First-Order Logic together with a fixed-point operator, over the class of ...
András Salamon's user avatar
38 votes
3 answers
6k views

Extended Church-Turing Thesis

One of the most discussed questions on the site has been What it Would Mean to Disprove the Church-Turing Thesis. This is partly because Dershowitz and Gurevich published a proof of the Church-Turing ...
Aaron Sterling's user avatar
38 votes
2 answers
4k views

Axioms necessary for theoretical computer science

This question is inspired by a similar question about applied mathematics on mathoverflow, and that nagging thought that important questions of TCS such as P vs. NP might be independent of ZFC (or ...
Artem Kaznatcheev's user avatar
35 votes
4 answers
1k views

Correspondence between complexity classes and logic

I took a class once on Computability and Logic. The material included a correlation between complexity / computability classes (R, RE, co-RE, P, NP, Logspace, ...) and Logics (Predicate calculus, ...
ripper234's user avatar
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33 votes
4 answers
3k views

What are the differences between logical relations and simulations?

I'm a beginner working on methods proving program equivalence. I've read a few papers about defining logical relations or simulations to prove two programs are equivalent. But I am quite confused ...
Hongjin Liang's user avatar
32 votes
6 answers
12k views

What is the difference between propositions and judgments?

I get confused by the subtle difference between propositions and judgments when exposed to intuitionistic type theory. Can any one explain to me what is the point to distinguish them and what ...
day's user avatar
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30 votes
4 answers
3k views

Why do we need formal semantics for predicate logic?

Consider this question solved. I will not pick a best answer as all of them have contributed to my understanding of the topic. Im unsure what benefit we have by formally defining the semantics of ...
Martin's user avatar
  • 309
30 votes
6 answers
1k views

Well known classes of boolean formulas that require exponentially long resolution proofs

You might often find cutting plane methods, variable propagation, branch and bound, clause learning, intelligent backtracking or even handwoven human heuristics in SAT solvers. Yet for decades the ...
30 votes
3 answers
2k views

Curry-Howard and programs from non-constructive proofs

This is a follow up question to What is the difference between proofs and programs (or between propositions and types)? What program would correspond to a non-constructive (classical) proof of the ...
Kaveh's user avatar
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30 votes
3 answers
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Translating SAT to HornSAT

Is it possible to translate a boolean formula B into an equivalent conjunction of Horn clauses? The Wikipedia article about HornSAT seems to imply that it is, but I have not been able to chase down ...
Evgenij Thorstensen's user avatar
29 votes
1 answer
936 views

Is there a reasonable automated proof system for TCS theorems?

Suppose I wanted to formalize Turing's proof regarding the halting problem so that a machine could check it. Some of the well-known automated theorem proving systems include Mizar, Coq, and HOL4. I ...
user avatar
29 votes
1 answer
985 views

Inductive types for large countable ordinal notations.

I'm looking to build notations for large countable ordinals in a "natural way". By "natural way" I mean that given an inductive data type X, that equality should be the usual recursive equality (the ...
Russell O'Connor's user avatar
28 votes
6 answers
3k views

How should I think about proof nets?

In his answer to this question, Stephane Gimenez pointed me to a polynomial-time normalization algorithm for proofs in linear logic. The proof in Girard's paper uses proof nets, which are an aspect of ...
Neel Krishnaswami's user avatar
28 votes
4 answers
6k views

Church's Theorem and Gödel's Incompleteness Theorems

I have recently been reading up on some of the ideas and history of the ground-breaking work done by various logicians and mathematicians regarding computability. While the individual concepts are ...
Noldorin's user avatar
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27 votes
2 answers
5k views

What's the difference between ADTs, GADTs, and inductive types?

Might anyone be able to explain the difference between: Algebraic Datatypes (which I am fairly familiar with) Generalized Algebraic Datatypes (what makes them generalized?) Inductive Types (e.g. Coq) ...
ninjagecko's user avatar
27 votes
2 answers
4k views

What is the logarithm or root operation in type-space?

I was recently reading The Two Dualities of Computation: Negative and Fractional Types. The paper expands on sum-types and product-types, giving semantics to the types ...
efrey's user avatar
  • 373
26 votes
5 answers
1k views

What is the difference between proofs and programs (or between propositions and types)?

Given that the Curry-Howard Correspondence is so widely spread/extended, is there any difference between proofs and programs (or between propositions and types)? Can we really identify them?
day's user avatar
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26 votes
1 answer
1k views

Are types propositions? (What are types exactly?)

I've been reading a lot on type systems and such and I understand roughly why they were introduced (in order to resolve Russel's paradox). I also understand roughly their practical relevance in ...
Rehno Lindeque's user avatar
26 votes
4 answers
2k views

When does (or should) Theoretical CS care about intuitionistic proofs?

From what I understand (which is very little, so please correct me where I err!), theory of programming languages is often concerned with "intuitionistic" proofs. In my own interpretation, the ...
usul's user avatar
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26 votes
1 answer
700 views

Unification and Gaussian Elimination

Does anyone knows of references that precisely spell out the connection between the unification algorithm and Gaussian elimination? I'm particularly interested in the relationship between triangular ...
Neel Krishnaswami's user avatar
26 votes
1 answer
594 views

Interesting algorithms in the formalization of the Feit-Thompson theorem?

It looks like George Gonthier and his collaborators have finished formalizing the Odd Order Theorem. In his earlier work on the Four Color Theorem, Gonthier invented a bunch of new algorithms (...
Neel Krishnaswami's user avatar
25 votes
5 answers
2k views

Are there any annotated formal verification systems for pure functional programming languages?

ACSL (Ansi C Specification Language), is a specification for C code, annotated with special comments, that allows C code to be formally verified. I have not looked into it, but I imagine that the ...
Nathan BeDell's user avatar
25 votes
2 answers
2k views

Do dependent types give you everything subtyping does?

Types and Programming Languages focuses quite a bit on subtyping, but as far as I can tell, subtyping doesn't seem especially fundamental. Does subtyping give you anything more than dependent types do?...
John Salvatier's user avatar
25 votes
2 answers
2k views

Sum-of-squares proof system

Recently I have seen several articles on arxiv that refer to a proof system called sum-of-squares. Can someone explain what is a sum-of-squares proof and why such proofs are important/interesting? ...
Anonymous's user avatar
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25 votes
1 answer
2k views

What's the expressive power of Simply Typed Lambda calculus?

The standard approach to simply typed lambda calculus considers computations over Church numerals. If input and outputs are Church numerals always typed as $Int$, where $Int = (\tau \rightarrow \tau) ...
Mycol's user avatar
  • 411
24 votes
1 answer
2k views

Where is the proof that Coq + Excluded Middle is consistent

I've seen (and heard) it claimed that it is safe to add the classical axiom of excluded middle to Coq, but I can not seem to find a paper supporting this claim. The papers I see listed on the Coq wiki ...
Mark Reitblatt's user avatar
23 votes
4 answers
2k views

Starting SAT solver papers

I want to make a first SAT solver. I know the SAT competition and the SAT conference, and there are just so many papers on this subject. I'm a starter, an overwhelmed starter. Where should I begin? ...
Zirui Wang's user avatar
23 votes
4 answers
4k views

What's the point of $\eta$-conversion in lambda calculus?

I think I'm not understanding it, but $\eta$-conversion looks to me as a $\beta$-conversion that does nothing, a special case of $\beta$-conversion where the result is just the term in the lambda ...
Trylks's user avatar
  • 604
23 votes
6 answers
818 views

Curriculum: Logical/Formal Methods in Security

At present I teach a small course (Four two hour lectures at the Masters level) on Logical Methods in Security, though the title Formal Methods in Security might be more apt. It covers briefly the ...
Dave Clarke's user avatar
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23 votes
1 answer
5k views

Prove proof irrelevance in Coq?

Is there a way to prove the following theorem in Coq? Theorem bool_pirrel : forall (b : bool) (p1 p2 : b = true), p1 = p2. EDIT: An attempt to give a brief ...
day's user avatar
  • 2,805
23 votes
2 answers
2k views

What was the original intent for the creation of Lambda calculus?

I've read that initially Church proposed the $\lambda$-calculus as part of his Postulates of Logic paper (which is a dense read). But Kleene proved his "system" inconsistent after which, Church ...
PhD's user avatar
  • 5,305
23 votes
2 answers
1k views

Circuit lower bounds and kolmogorov complexity

Consider the following reasoning: Let $K(x)$ denote the Kolmogorov complexity of the string $x$. Chaitin's incompleteness theorem says that for any consistent and sufficiently strong formal ...
Magnus Find's user avatar
22 votes
7 answers
1k views

How can we know that formal methods work?

An important goal of formal methods is to prove the correctness of systems, either by automated or human-directed means. However, it seems that even if you can give a correctness proof, you may NOT be ...
Jenny's user avatar
  • 337
21 votes
3 answers
2k views

Why can Lambda Calculus not represent some combinators?

This paper suggests that there are combinators (representing symbolic computations) that can not be represented by the Lambda calculus (if I understand things correctly):
hawkeye's user avatar
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21 votes
2 answers
1k views

Decidability of diophantine equations over {=, +, gcd}

It is well-known that polynomial diophantine equations are undecidable (Hilbert's 10th problem): that is, given a quantifier-free formula over the language $\{=, +, \cdot, 1\}$ (of equality, addition, ...
Caleb Stanford's user avatar
21 votes
3 answers
543 views

Are recursive forms of Godel's statement possible?

The self-referentiality of the P/NP problem has sometimes been highlighted as a barrier to its resolution, see, for instance, Scott Aaronson's paper, is P vs. NP formally independent? One of the many ...
Anand Kulkarni's user avatar
21 votes
3 answers
2k views

Smallest possible universal combinator

I am looking for the smallest possible universal combinator, measured by the number of abstractions and applications required to specify such a combinator in the lambda calculus. Examples of universal ...
user76284's user avatar
  • 662
21 votes
1 answer
871 views

Combinators for the Primitive Recursive Functions

It is well-known that the S and K combinators are Turing Complete. Are there combinators that suffice to yield (only) the primitive recursive functions?
NietzscheanAI's user avatar
20 votes
3 answers
4k views

Symbolic Execution is a case of Abstract Interpretation?

This is written in the wiki entry of Symbolic Execution, but I can't find any reference for it. Can anyone show me a pointer? Thank you.
sean's user avatar
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