Questions tagged [logic]

Questions related to mathematical logic and its use in computer science

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54 views

Number of substrings possible with even characters

Consider a string 'ABBAA' Possible substrings with even number of characters are $4$ 'ABBA' : Count of 'A' is even and 'B' is even 'AA' : Count of 'A' is even and 'B' is even - ($0$) Similarly 'BB' ...
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How do computers *really* work? (at the most basic level)

While learning about computers I will read about RAM and Storage and the CPU, and while these explain the architecture of a computer and how parts of a computer work together, I still don't understand ...
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1answer
18 views

Proving using a SAT solver that KB entails D

Suppose you're given this KB: $$KB = (A, A ⇒ B, A ⇒ C, B ∧ C ⇒ D)$$ How would you show using a SAT solver that $KB \vDash D$?
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35 views

Oracle that can only definitively say if an instance is unsatisfiable

Assuming I have an Oracle that takes as input a strictly 3SAT Boolean instance and states whether the instance is satisfiable or not. If it says instance is unsatisfiable then the instance is ...
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1answer
29 views

Number of substrings with exactly k distinct characters

Given a string s and an int k, return an int representing the number of substrings (not unique) of s with exactly k distinct characters. If the given string doesn't have k distinct characters, return ...
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1answer
25 views

Showing that a bijection preserves homomorphism

I have the vocabulary $\tau = \{+, 0 \}$ and the structure $R = (\mathbb{R}, +^R, 0^R)$ where $+^R$ is a binary function symbol representing the normal addition of numbers and $0^R$ is the constant $0$...
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Two Statements and Conclusions: Which Choice is Logically Correct? [closed]

This is a question asked in GATE 2021. The problem is in C and D options. If one interpret question in different ways both C and D can be correct. Can someone clarify? Two statements and conclusions ...
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2answers
78 views

Encoding the theorem on friends and strangers in predicate logic

In every group chat of six (distinct) people on Instagram, there is a group of three (distinct) people such that either: everyone in the group are friends everyone in the group are strangers (i.e not ...
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CTL and the modal mu-calculus

Is there any literature / results to which fragment of the modal mu-calculus the logic CTL corresponds? I know that in order to tranlsate the until operators of CTL, it suffices to use a single mu-...
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1answer
27 views

Converting into CNF

I am still a bit confused by how to convert into CNF even though I have the rules written down. How do I convert the following sentence into CNF? $(I \lor M) \Rightarrow H$ I know I need to get rid of ...
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28 views

Multiple parameter predicates and the Relational Model

I've got a very general question about the relational model and it's relationship to 1st order predicate calculus. It will probably seem very basic to most, I'm afraid, a consequence of me grappling ...
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2answers
31 views

Propositional Logic: Entailment

Given the sentences, S1, S2, if S1 |= S2 then all models that satisfy S1 also satisfy S2, how is the following statement correct? A ∧ ¬A |= B How can something and ...
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In propositional logic how is the negation equivalent?

I am unable to understand how the negation of a sentence into CNF actually makes sense, hopefully somebody can explain it to me in understandable terms! Given the sentence: If the battery is charged (...
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A necessary condition for a relation to be in 2NF but not in 3NF is that some non-prime attribute must be determined by a non-prime attribute

I will state the complete question now, since it did not fit in the title. Is the statement given below correct? A necessary condition for a relation to be in 2NF but not in 3NF is that some non-...
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37 views

Is there a diagram of related logics?

Proving software is based on logics and there any many logics. AFAIK all of the logics can be shown as a tree diagram of how they are related; for example, a tree of how types of number are releated ...
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50 views

Proving equivalence of two substitutions by induction

I'm trying to prove the following reduction: $$ t\{x:=u\}\{y:=v\} = t\{y:=v\}\{x:=u\{y:=v\}\} $$ under the following assumptions: $x \neq y$ $x$ is not a free variable of $v$ (in symbols, $x \...
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1answer
56 views

What's the average number of transistor switches needed to do an N-bit x N-bit multiply?

I want to know how switch-efficient a multiplier can be. If I need to do many $N$-bit by $N$-bit multiplies, and each bit is determined by flipping a coin, what's the average number of transistor ...
2
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1answer
38 views

Clarifying the definition of reduction with regards to NP-complete problems

In my logic class we started learning about the different complexity classes. In particular, we focused on the NP complexity class. A problem is in NP if it is solvable in polynomial time using a ...
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0answers
338 views

Prove that PRE-WORK-POST is Turing Complete

This is a homework question and I am trying to fully understand what I have to do and how to go about it. Therefore, I don't want full answers to the question. The programming language PRE-WORK-POST ...
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0answers
36 views

Do FOL theorem provers accept axiom schemata?

Axiom schemata (such as ZFC) are, in a sense, infinite sets of axioms. Do the ATPs designed to work with FOL (such as Vampire) accept axiom schemata? I looked in the Vampire "manual" briefly,...
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1answer
23 views

Consider 4 positive integers a,b,c,d having exactly 10^11 bits (considering the leading zeroes) in the binary representations

Consider 4 positive integers a,b,c,d having exactly 10^11 bits (considering the leading zeroes) in the binary representations. Positions are numbered from 1 to 10^11. Every 3rd bit of a is equal to 1 ...
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21 views

Comparing Registers with 16-bits

I am given with this information: "CMP OP1,OP2 will compare registers OP1 and OP2 if they are equal, flag values will be ZF=1, CF=0, if the first operands value is greater flag values will be ZF=...
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2answers
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A basic question about predicate logic

Suppose we have a statement: "Tom has a car" and we have an auxiliary clause that is has(x, y): x has y then to write the statement in logic I am guessing ...
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1answer
37 views

An equivalent of $\neg ((a \wedge b) \Leftrightarrow c)$ based on a specific requirement

I am solving a problem that is asking to provide an equivalent for $\neg ((a \wedge b) \Leftrightarrow c)$ and the equivalent shouldn't contain $\wedge$ (and), $\Rightarrow$ (implication) or $\...
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42 views

What is the difference between type theory and logic programming (in terms of declarative programming and specification)

How is does type theory (coq, lean, agda), and logic programming (prolog, datalog) differ from each other. Logic programming is a way of declarative specifying an algorithm, using classical 1st order ...
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1answer
26 views

Which of these properties hold for all FO theories? (but not regarding fragments thereof)

Which of these properties hold for all FO theories? (but not regarding fragments thereof) a. Decidable b. At least expressive as propositional logic c. NP-complete a) Decidable: no, some first order ...
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1answer
31 views

Create an Finite Deterministic Automata for a regular expression

I want to create a finite state machine that accepts the following language: $$ L=\{w\in\{a,b\}^* | w \text{ contains abb but not on the first position}\} $$ So I began by writing a regular expression ...
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1answer
40 views

Is Conjunctive Normal Form or not?

I have one formula that I do not understand why it is CNF and one that is not CNF, namely. p && !q (NOT NCF) and !!p(CNF). According to the exercise where I found these examples, 1 is not ...
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19 views

Is my UML Use Case Diagram valid?

I'm preparing an UML use case diagram for my following school project: ...
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0answers
35 views

Help me understand whether these critical pairs are joinable

I have the following TRS $R$: $$ l_1 = f(g(x)) \to f(x) = r_1 \\ l_2 = g(f(y)) \to g(y) = r_2 $$ I want to know if $R$ is confluent, and whether $g(f(f(x))) \leftrightarrow_R^* g(g(g(x)))$. I have ...
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Why is most research in logic and verification conducted in Europe rather than the US?

This is the CSRankings output for logic and formal verification for the last 10 years, worldwide: http://csrankings.org/#/index?log&world For the International Conference on Computer Aided ...
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How to combine multiple Boolean Functions in CDNF efficiently for implementation on a CPU?

I am trying to see how fast I can implement a 6-bits to 6-bites lookup table. Generally, I am attempting to do this by using the common method mentioned in CS textbooks of using the Quine-McCluskey ...
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20 views

Prove the boolean function E = F + G contains of the sum of the minterms of F and G

I'm given a boolean function E which is result of the sum of F and G (Where both F and G are boolean functions for sure). By using K-Map I can easily understand why E would have the sum of the ...
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1answer
32 views

is duality principle in boolean algebra is true for every expression

Let say A = 1 and B = 1 and then A+B = 1 now by using duality(replacing or gate by and gate and 1 by 0) we can say that, A.B = 0 but this is not 0, because 1.1 = 1, so please anyone clear my ...
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22 views

Automated algorithm design

There is a huge field of study dedicated to automated theorem proving. Is there any similar research dedicated to automated algorithm design? And is there any form of "duality" between these ...
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36 views

Check if given safety properties are regular, and if so construct NFAs

Let $\mathit{AP} = \{a, b, c\}$. Consider the following LT properties: Between two neighboring occurrences of $a$, $b$ always holds. Between two neighboring occurrences of $a$, $b$ occurs more often ...
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33 views

Name for a function on formulae that does not break the structure of given formula

I have two logics, $L_1$ and $L_2$, and a mapping $f: Formulas_{L1} \rightarrow Formulas_{L2}$, and want to argue that the mapping is "nice", in the sense that it is structure-preserving. ...
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0answers
202 views

Can every sentence of first-order logic be converted into an equisatisfiable equation in Boolean algebra?

There may be some theoretical literature, unknown to me, that addresses this question. If possible, I would like a practical approach to this problem. My attempt involves the use of an equational ...
3
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2answers
89 views

How expensive is it to verify that a gate is universal?

I know that a NAND gate is considered universal because we can use it to implement an AND, OR, and NOT gate. Given an arbitrary gate, or set of gates, how many operations would it take to verify if ...
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1answer
41 views

For a logic gate to be universal, must it necessarily be able to perform duplication?

It is said that a gate that can simulate AND and NOT is universal and able to recreate any classical circuit. I was looking at some of the circuits simulated by NAND, and for some of them, we need to ...
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1answer
25 views

first order logic to normal form order of operations

∃y∀x [A(x) ∧ B(y) -> C(x,y)] ∃y∀x [¬(A(x) ∧ B(y)) v C(x,y)] ∃y∀x [¬A(x) v ¬B(y) v C(x,y)] I need to convert the above to conjunctive normal form. I'm a ...
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1answer
56 views
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1answer
40 views

Universal Quantifiers in QBFs

I've been looking into reductions to/from the TQBF language and have managed to get stuck on something that is almost certainly not true (or, if it is true I'm missing a significant computational cost ...
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1answer
37 views

Lambda calculus simplification excercise

Below is the lambda expression which I am finding difficult to reduce i.e. I am not able to understand how to go about this problem. (λx.λy.yx)z (λw.w) I am lost with this. if anyone could lead me in ...
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23 views

Two variants of Fredkin gate

Sometimes two outpus of Fredkin gate are swapped when constant bit is 0, sometimes when it is 1. What variant is right or more modern? If it was conscious decision to change behavior what was ...
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25 views

In theory, should neuromorphic computers be more efficient than traditional computers when performing logic?

There is this general sentiment about neuromorphic computers that they are simply "more efficient" than von Neuman. I've heard a lot of talk about neuromorphic computers in the context of ...
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1answer
26 views

Represent a DNF formula as a multivariate linear formula?

Lets say I have the following DNF: (x or y) and (z or i) / $(x\lor y)\land(z\lor i)$ How do I convert that into a polynomial form?
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1answer
40 views

Test two LTL expression trees for equivalence

Is there an algorithm on how to check if two LTL expressions (represented as binary trees) are semantically equivalent? Since there are many smaller equivalences such as $a\Rightarrow b \equiv \neg a \...
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46 views

Building an ALU on nandgame's website

I'm working on nandgame's website found here. I'm working on the ALU and here is an image of my implementation: My Implementation: And I compared it to this website's solution: Solution However when ...
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Can HOL be simulated in the CiC?

I was wondering if HOL (higher-order logic) can be simulated in the Calculus of Inductive constructions (CiC)

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