Questions related to mathematical logic and its use in computer science

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Digital Logic Settle Time

I have an exam tonight and I'm reviewing my midterm exam. I got this question completely wrong, with no solution given. ...
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Efficiently decidable logics

So propositional logic (PL) is efficiently (in P) decidable because I can convert formulas to an equisatisifiable CNF-formula, negate and convert (efficiently, by De Morgans laws) to DNF. I can then ...
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2answers
35 views

The use of modal logic in computer science [closed]

I have a tentative understanding of modal logic. Can anyone explain modal logic as it is used in computer science?
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1answer
250 views

DNF to CNF conversion: Easy or Hard

In relation to the thread CNF to DNF — conversion 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 paper, they seem to ...
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1answer
66 views

A logic function that is true iff the first operand is less than the second operand

In my computer organization class I have been given a series of problems. One I'm stuck on currently is below: Assume that $X$ consists of 4 bits, $x_3 x_2 x_1 x_0$, and $Y$ consists of 4 bits, ...
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1answer
48 views

If $F$ is valid then $F \cup \{res(C_1,C_2,A_i)\}$ is valid

I have to prove the following problem in propositional logic: Let $F$ be a set of clauses and let $F' = F \cup \{res(C_1,C_2,A_i)\}$ be the extension of $F$ by a resolvent of some clauses $C_1,C_2 ...
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1answer
71 views

Abstract algebra and programming languages

Quite often, I stumble upon abstract algebra concepts like initial algebra, free algebra, and similar while reading papers on programming languages. For instance, in papers on algebraic data types, ...
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1answer
58 views

Finding a Hoare logic correctness proof for a Repeat-Until loop

How can we prove a program in repeat until using Hoare Logic? I've found a rule like this: {P} S {R}, {R ^ ~B -> P}, {R ^ B -> Q} for ...
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2answers
77 views

Skolemization with multiple arguments — how to unify

Edit: answerers keep finding (valid!) problems with my example. I'll try again. The older version is below the horizontal line. Thanks to Klaus below for pointing out the last problem. My ...
6
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1answer
79 views

What do we gain by having “dependent types”?

I thought I understood dependent typing (DT) properly, but the answer to this question: Why was there a need for Martin-Löf to create intuitionistic type theory? has had me thinking otherwise. ...
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0answers
27 views

Solving Subproblem in Logic (first-order, propositional, pddl)

One sentence question Is there any algorithm able to prove (solve) a logic problem (first-order, propositional, pddl) by finite induction? Background I am researching Hierarchical planning solvers ...
3
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1answer
186 views

What's the purpose of ANDing a bit mask with all bits set to a value?

I'm adapting an algorithm for calculating a CRC 16-bit CCITT (XModem) value from an ASCII input. I've found some code here. I'm using the function the poster has in his question, but I've noticed at ...
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1answer
19 views

Unification — removing equations and updating the solution

This question is concerned with the first order unification. Suppose I have a set $D$ of equations and a solution to these equations. Let this solution be a set $S$ of substitutions. Now, suppose I ...
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3answers
147 views

Combinational Logic - Need help with proof by rewrite

In my Computer Logic class we were assigned the following problem: Complete a truth table that has $3$ inputs $(A, B,C)$ and one output $(F)$. $F$ is asserted whenever $B$ or $C$ are ...
3
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1answer
46 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, ...
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2answers
51 views

Logical Reduction

Reducing one computable problem to another by providing an algorithm which transforms an instance of one problem to one of the other (and limiting the time or space of that algorithm) is clear to me. ...
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1answer
54 views

CTL - model checking for formula $A [a \cup b]$

I'm trying to verify if the following model satisfy $A [a \cup b]$: The algorithm I'm using is taken from "Concepts, Algorithms, and Tools for Model Checking", Joost-Pieter Katoen. In particular I ...
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0answers
34 views

What are the fundamental principles/algorithms on the process of equation solving?

I have seen a lot of solvers that are capable of, for example, getting an equation such as x ^ 2 + x = 12 and finding x = [3, -4]. I know some of them are implemented by hardcoding special cases. For ...
2
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1answer
49 views

CTL vs LTL - when a formula satisfy a model

I'm trying to understand the difference between LTL and CTL. In particular, i'm trying to understand when a model (a transition system eg. Kripke structure) satisfy a formula. This is my point of ...
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2answers
99 views

Show this is a tautology without using a truth table

I am doing a homework assignment and I've been stuck on this question for a long time now. The question says to prove this equation is a tautology without using a truth table. I assume we have to use ...
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1answer
51 views

Defining a new operator in CTL

Lets consider a new operator $B$ where $a B b$ means "in every execution, if $b$ holds some time, then $a$ does so before it" and we're asked to define it in CTL. My working: the system can only ...
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0answers
43 views

Formulating an arrangement problem with STRIPS

The problem is rearranging furniture in a flat. We are given rectangular rooms of natural width and height with doors between them, (the walls have no width) and rectangular furniture at starting ...
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1answer
53 views

Defining a new (informal) operator in CTL

If you were given a "new operator" Wh and a formula a Wh b meaning that a holds for at least as long as b does (in all executions). How would you define this operator in CTL? This is an exercise ...
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3answers
51 views

Simplifying a boolean expression

I need to prove: XY+~XZ+YZ=XY+~XZ I cannot think how to do this. I have tried factorising, but I just don't know of any rule that removes one of the terms like above. I start with the LHS, ...
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2answers
95 views

Some slight confusion with the UNTIL operator in CTL (e.g. a U b)

I've sketched a very small transition system in paint that I'll use as an example. I want to see if $A(aUb)$ holds for this transition system. From my understanding, this CTL formula is asking if ...
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2answers
108 views

Is Karnaugh Map possible for Maxterms?

I read about Minterms i.e. sums of products, simplification using Karnaugh Graph. Can this graph be used for Maxterms, i.e. products of sums, as well? If yes, then how? If not, then is there some ...
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1answer
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Given this transition system, for which states are these (very basic) LTL formulas fulfilled?

I missed a lot of lectures for this module due to surgery so I'm trying to teach it to myself now. This is the question I've been working on: First of all, would I be correct in saying that the LTL ...
2
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3answers
79 views

Can someone clarify this unification algorithm?

I've been having trouble understanding a unification algorithm for first order logic, as I don't know what a compound expression is. I googled it, but found nothing relevant. I also don't know what a ...
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3answers
95 views

Turing Machine That Accepts Machines With Undecidable Languages

So I'm reviewing my Computability notes for my final, and I understand how reduction arguments work, but I'm having trouble framing one for the following Turing machine: Undecidable TM = { ⟨M⟩ | L(M) ...
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0answers
29 views

Open Standard Prolog Knowledge Bases [closed]

Are there any standard Prolog knowledge bases available anywhere that have the same purpose as Cyc, namely to encode generally accepted common sense and human knowledge? Typically containing ...
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1answer
37 views

Resolution refutation [closed]

This is an assignment so I don't want the answer, just maybe a push in the right direction. The question is: Suppose you are given an algorithm $R$ that can test for any propositional formula in ...
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0answers
17 views

SMT solves with functions for free varibles

So this sounds like this might lead to an undecidable theory but I thought I would give it a try and ask about it after I found nothing on the subject. I am somewhat interested in finding functions ...
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1answer
45 views

Is the extension of every undecidable theory undecidable?

While it is not the case that the extension of every decidable theory is decidable, is it true that: the extension of every undecidable theory undecidable? In other words, given an undecidable ...
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2answers
64 views

Underlying language to specify various types of logic

There exist several different types of logic -- 1st order, 2nd and higher order with many different sets of inference rules possible. What I'm having trouble understanding is what's the "underlying ...
2
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1answer
139 views

Is There a Programming Language That Embraces Globals?

All programming languages have globally defined symbols. While best practices invariably abjure their use as mutable entities the philosophy of what is mutable and what is not mutable is highly ...
3
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2answers
155 views

Free and bound variables

I am familiar with free and bound variables theory , but while learning I somewhere saw this lambda expression ((lambda var ((fn1 var) & (fn2 var))) argument) From what I have learned it ...
2
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0answers
30 views

Understanding a paper on polynomial recursion in all finite types

So I wasn't sure weather or not this counted as "research level" or not but I figured it wasn't so I decided to post it here. There is a paper by S. Bellantoni et al. called "Higher Type Recursion, ...
2
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1answer
232 views

Proving Linear Time Temporal Logic formula □ ◊ f ⇔ ◊ □ f

I am new to this topic, Linear Time Temporal Logic and I am trying to prove this equivalence -- $\Box\Diamond f \Leftrightarrow \Diamond\Box f$ This is my take -- Basic definitions: $(\sigma, j) ...
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1answer
176 views

Hamming Weight to find the sum of 1 bits in the range between A and B inclusive [closed]

I am trying to find the sum of 1 bits in the range between A and B inclusive, where -2^31 <= A <= B <= 2^31 - 1 Input Format: The first line contains the number of test cases T ...
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1answer
33 views

Type checking relation to logic

I am learning logic and functional programming and there is a topic called Type checking and their relationship to logic , I am unable to find any source that can help me in understanding this topic. ...
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2answers
50 views

Working out the connectives (And, Or, Not) in a Truth Table that has the outputs [duplicate]

I don't understand how to work backwards to work out a truth table that has been filled out already (I don't know the logical operators). E.g P | Q | Output 1 | 1 | 1 1 | 0 | 0 0 | 0 | 0 0 | 1 | 0 ...
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1answer
17 views

Prove the existence of a proposition logical formula so that following conditions are fulfilled

For two proposition logical formulas $\phi$ and $\chi$ so that $\phi\implies\chi$ is generally valid. How can I prove that there is a formular $\psi$ with $var(\psi )\subseteq var(\phi )\cap var(\chi ...
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0answers
52 views

Creating branch less than equals zero MIPS instruction in single cycle circuit

As a project I'm trying to implement a MIPS processor. I have a circuit similar to the one below. My question is what would be a way to implement a ...
6
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2answers
146 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 ...
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1answer
35 views

Karnaugh Map is this the simplest solution possible?

I'm learning to use a Karnaugh map but I'm not sure if I obtained the simplest expression possible. Have a look at this example Truth table (where A B C yield F): ...
0
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1answer
48 views

Difference between equivalence and implication

In terms of CTL formulae, what is the difference between equivalence and implication? (prop = some proposition, && = conjunction, AG = CTL syntax for "globally holds") E.g. AG (prop1 ...
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1answer
49 views

How can I translate this quantified logical expression into english

I was reading chapter-1 The Foundations: Logic and Proofs from this book. The chapter gives example of translating English sentence : "There is a woman who has taken a flight on every airline in the ...
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1answer
35 views

Meaning of empty clause

Why does the empty clause is logically equivalent to a contraddiction: $\square \cup \square \Longleftrightarrow \perp$ and why the empty cube is logically equivalent to a tautology: $\square \cap ...
2
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1answer
24 views

Terminology about the word Function : General vs Computable

I've seen two different concepts referred to by the term "function": A small part of a program specified by the composition of constants and other functions as paramaters, such as the "functions" in ...
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2answers
34 views

Using the rules of inferences

I know the rules of inferences and logical equivalence but I cannot seem to validate this argument. I rewrote the first premise as $\neg p\vee q$ other from that I am stuck. Any help will be ...