Questions tagged [logic]

Questions related to mathematical logic and its use in computer science

58 questions
8k views

Learning Automated Theorem Proving

I am learning Automated Theorem Proving / SMT solvers / Proof Assistants by myself and post a series of questions about the process, starting here. Note that these topics are not easily digested ...
5k views

DNF to CNF conversion: Easy or Hard

In relation to the thread Proving that the conversion from CNF to DNF is NP-Hard (and a related Math thread): How about the other direction, from DNF to CNF? Is it easy or hard? On Page 2 of this ...
7k views

What is coinduction?

I've heard of (structural) induction. It allows you to build up finite structures from smaller ones and gives you proof principles for reasoning about such structures. The idea is clear enough. But ...
231 views

Creating a logical circuit

Task: Design a 2 bit comparator. Input: 2x 2 bit (I take it as 2 2-bit values, let them be unsigned for simplicity) Output: 1 if result input1>input2 is true, 0 otherwise Develop truth table and ...
92 views

First Order interpretation of arbitrary structures as a graph

I am currently trying to get some intuition on the concept of First Order reductions, and have come across this exercise question by Immerman, dubbed "Everything is a Graph". Given some arbitrary ...
12k views

Is there any concrete relation between Gödel's incompleteness theorem, the halting problem and universal Turing machines?

I've always thought vaguely that the answer to the above question was affirmative along the following lines. Gödel's incompleteness theorem and the undecidability of the halting problem both being ...
825 views

Does the Y combinator contradict the Curry-Howard correspondence?

The Y combinator has the type $(a \rightarrow a) \rightarrow a$. By the Curry-Howard Correspondence, because the type $(a \rightarrow a) \rightarrow a$ is inhabited, it must correspond to a true ...
921 views

Equivalence of Büchi automata and linear $\mu$-calculus

It's a known fact that every LTL formula can be expressed by a Büchi $\omega$-automaton. But, apparently, Büchi automata are a more powerful, expressive model. I've heard somewhere that Büchi automata ...
565 views

210 views

Difference between fully-reduced BDD and quasi-reduced BDD

I am trying to figure out difference between fully- and quasi-reduced BDDs. I have read a lot of material but still it is not very clear. As I am trying to figure out the quasi reduced version for ...
4k views

719 views

Why doesn't Godel's Second Incompleteness Theorem rule out a formalizable proof of P!=NP?

I'm sure there must be something wrong with the following reasoning because otherwise a lot of P vs. NP research would be curtailed but I cannot determine my error: For any fixed integer $k>0$ ...
951 views

Types of Automated Theorem Provers

I am learning Automated Theorem Proving / SMT solvers / Proof Assistants by myself and post a series of questions about the process, starting here. Which are the relevant automated theorem provers? I ...
358 views

Why do some inference engines need human assistance while others don't?

I am learning Automated Theorem Proving / SMT solvers / Proof Assistants by myself and post a series of questions about the process, starting here. Why is it that automated theorem provers, i.e. ACL2,...
191 views

Is there a relationship between “sound and complete” in logic and “type safety” in PLs?

I've been wondering if there's a connection between "good logics" and "good programming languages". It seems that logics are shown to be "locally sound and complete" whereas programming languages are ...
1k views

Do undecidable languages exist in constructivist logic?

Constructivist logic is a system which removes the Law of the Excluded Middle, as well as Double Negation, as axioms. It's described on Wikipedia here and here. In particular, the system doesn't ...
2k views

What is beta equivalence?

In the script I am currently reading on the lambda calculus, beta equivalence is defined as this: The $\beta$-equivalence $\equiv_\beta$ is the smallest equivalence that contains $\rightarrow_\beta$...
4k views

Understanding DPLL algorithm

I'm trying to understand DPLL algorithm for solving SAT problem. And here it is: ...
2k views

Consistency and completeness imply soundness?

I understand that soundness implies consistency. Also, I understand that consistency alone does not imply soundness. But shouldn't consistency + completeness imply soundness? Scott Aaronson in his ...
2k views

Recursive definitions over an inductive type with nested components

Consider an inductive type which has some recursive occurrences in a nested, but strictly positive location. For example, trees with finite branching with nodes using a generic list data structure to ...
3k views

Example of Soundness & Completeness of Inference

Is the following example correct about whether an inference algorithm is sound and complete? Suppose we have needles a, b, c in a haystack, and have also an inference algorithm that is designed to ...
4k views

Is resolution complete or only refutation-complete?

Going through some knowledge representation tutorials on resolution at the moment, and I came across slide 05.KR, no77. There it is mentioned that "the procedure is also complete". I think this ...
526 views

Algorithm for deciding alpha-equivalence of terms in languages with bindings

I am interested in the alpha equivalence relation in languages with variable bindings, such as: ...
365 views

What is wrong with this seeming contradiction with a paper about AND-compression of SAT?

Got a simple construction seemingly contradicting a paper assuming plausible conjecture. Since it is unlikely the conjecture to be false, what is wrong with the argument? From a paper An AND-...
88 views

Developing invariants for comparing two strings

The following algorithm is supposed to compare two strings $S_1$ and $S_2$ ("/\" for empty string): ...
1k views

Propositional formula in DNF can be decided in polynomial time?

For a given propositional formula f in DNF, one can decide in polynomial time, if the formula is satisfiable: Just walk through all subformulas (l_1 and ... and l_k) and check, wheter it has NO ...
281 views

Normal order sequencing vs applicative order sequencing

I'm trying to understand this lecture, section 2.7. Why would the normal order sequencing print out "hello" "world" and not ...
144 views

Equivalence preserving operator from CTL* to LTL

The question is about an operator that transforms any CTL* formula ${\psi}$ into a (not necessarily equivalent) LTL formula ${A\psi^d}$, where $d$ means syntactically removing all $A,E$ quantifiers ...
48 views

What is the connection between the logic and the logic programming?

As I understand, then logic (i.e. particular theory of logic with some theory's axioms and some possible assignments of the variables) describes some specific world (in the non-modal case) or some set ...
364 views

Is this possible to solve boolean satisfiablility by using karnaugh maps to simplify the whole given boolean formula by simplifying subformulas?

Building karnaugh map for the whole given boolean formula always costs Θ(2n) both time and space complexities, where $n$ is the number of boolean variables in the given boolean formula. It is ...
34 views

What is the largest possible minimal 3CNF formula as function of the number of variables?

I have already defined here what is minimal 3CNF formula. In the answer to the question, D.W. answered: What you are thinking is wrong. A minimal, unsatisfiable formula can have more than 8 clauses. ...
285 views

Convert conjunctive normal form to equivalent boolean formula with only NAND gates

Let $\varphi$ be a boolean formula in 3-CNF form (conjunctive normal form with three literals at most per clause). I want to convert it to an equivalent boolean formula that uses only NAND gates with ...
68 views

Can a propositional threshold connective be expressed by standard connectives?

We are given a finite set of propositional atoms $\{x_1, \dots, x_n\}$ and an integer $k$. Can we capture through a propositional formula $\varphi$ (built from the standard connectives \$\neg, \wedge, \...
104 views

What does “refuting random 3CNF” formulas mean?

Intuitively, recall what 3CNF formulas mean: Its a boolean formula with conjunctive normal form (i.e. formula of ANDs of clauses with ORs) with no more than three variables per conjunct. I was ...
156 views

on `On the cruelty of really teaching computing science' [closed]

Dijkstra, in his essay On the cruelty of really teaching computing science, makes the following proposal for an introductory programming course: On the one hand, we teach what looks like the ...
166 views

Someone explain the venn diagram for the logic equation A*(B+C)

So, I am studying logic circuits and how to prove them with Venn diagrams. When drawing a Venn diagram for the equation A*(B+C) I figured it would look something like this: But according to the ...
379 views

Transition systems that satisfy LTL but not CTL, and vice versa

I am learning about temporal logic and model checking systems. One conceptual exercise that I am struggling with is how to create a transition system which satisfies only one of two given properties, ...