Questions related to logic and its use in computer science

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3
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Does a logical system have semantics?

From Wikipedia: A logical system or, for short, logic, is a formal system together with a form of semantics, usually in the form of model-theoretic interpretation, which assigns truth values to ...
2
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1answer
31 views

Does Herbrand's theorem mean any first-order logic formula can be expressed in CNF?

Herbrand's theorem shows that any formula of first-order logic can be expressed as a disjunction of quantifier-free formulas of first-order logic. Is this equivalent to saying that Herbrand's theorem ...
4
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1answer
30 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 ...
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2answers
88 views

Example for an unsatisfiable formula that can be made satisfiable by reordering quantifiers [closed]

Please give me an example of an unsatisfiable quantified 2 CNF formula. I need it in my proof and I am unable to think of one. I am looking for such an unsatisfiable quantified 2 CNF formula which ...
3
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1answer
57 views

Can quantified renamable Horn formulas be identified using the same procedure as unquantified formulas?

Definition: A renamable Horn formula is a Boolean formula that can be transformed into a Horn formula by flipping the polarity of every instance of one of more of its variables. Example: $\qquad ...
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0answers
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How to ascertain that the result of 1's complement arithmetic is invalid/valid

Trying to do (5)$_{10}$-(-5)$_{10}$ in 4 bit 1's complement system: 0101-1010 = 1011 with borrow 1 Subtracting back borrow 1011 - 1 = 1010 So sign bit is 1 in final answer, thus in 1's ...
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1answer
39 views

Why is Mixed Quantified Horn SAT in PSPACE?

I want to prove that Mixed Quantified Horn SAT is a PSPACE-complete problem. I have proved that it is PSPACE-hard. How can I prove that it is in PSPACE? My study: To prove QSAT to be in PSPACE: ...
0
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1answer
104 views

Concept used in the proof [closed]

In the paper "Resolution for Quantified Boolean Formulas", I am unable to understand the proof of theorem 3.4. Please help me with the basic concept used on page 4: The concept that I am referring ...
3
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2answers
78 views

Composition of combinators with arities greater than one

In combinatory logic, the axiom of composibility asserts that for any two combinators, $A$ and $B$, there exists a combinator $C$ that composes $A$ and $B$. That is, for all $A,B,x$ there exists a ...
0
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1answer
33 views

How to design xml schema for digital circuits? [closed]

how can i design XML Schema for logical and digital circuits? i cant find any help or manual for this work for example i have a digital circuits with AND OR NOR ,... gates now i want design xml ...
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0answers
41 views

Translation of natural language to logic [closed]

Given a statement in natural language, what can be said about how many possible translations there are in first-order-logic? What happen if we take a more complex logic like second-order logic and ...
2
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1answer
70 views

Solving SAT using tableau calculus

I've learned about tableau calculus which is a decision procedure solving the problem of satisfiability of a first order logic formula. Now I'm wondering why this technique can't be used to solve the ...
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1answer
79 views

Quantifier String Placement [closed]

This is the edited question: Suppose I have $(x_1 \vee y_1 \vee y_2)$. x is existential and y is universal. Then it should be like this in the quantifier string: $\forall y_1 \forall y_2 \exists x_1$ ...
3
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3answers
165 views

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

EDIT 3: Might be wrong, but I am still confused by the answer's claim "It does not have to output an instance that preserves all satisfying assignments for all the input instances". This appears to ...
7
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3answers
903 views

Gödels (first) incompleteness Theorem and the Halting Problem - How limiting is it?

When I first heard of these things I was very fascinated as I thought it sets really a limit to mathematics and science in general. But how practically relevant are these things? For the Halting ...
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5answers
183 views

Real life description for (~A->A)->A

It can be shown that the logical preposition [ :- (~A->A)->A ] is a theorem (always true). I want to know if anybody knows a real life description for the preposition above? I mean an expression in ...
4
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1answer
62 views

Showing that a Particular Word Problem is Decidable

I need to give an algorithm to show that the word problem in the group $\langle x,y \mid \mid x^{1984} = y^{2014} = 1 \rangle$ is decidable. How do I show this? I'm not too sure where to start.
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1answer
71 views

Hilbert's 10th Problem and Chaitin's Diophantine Equation “Computer”?

In Chaitin's Meta Math! The Quest For Omega, he briefly talks about Hilbert's 10th Problem. He then says that any Diophantine Equation $p=0$ can be changed into two equal polynomials with positive ...
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0answers
36 views

Introductory book on Logic and Computation

Can you give me some suggestions about a good introductory (but comprehensive) bookabout Logic and Computation? Some fuzzy topics that I have in mind are: Presburger artihm., PA, ZF, ZFC, HOL Set ...
2
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1answer
288 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 ...
1
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1answer
54 views

Logic applications and security protocols references request

I'm a student of logic and I'm interested in looking at logic applications, of which I am told there are many, in computer science and allied areas. I've seen some applications. More specifically, I ...
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0answers
16 views

Extending the type scheme for Burks/Warren/Wright machine to handle operators on functions?

Some background: Burks, Warren and Wright published a description of simple interpreter for a Logic Machine for unparenthesized Lukasiewicz notation (paper, SO question, c implementation, postscript ...
4
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1answer
34 views

Should we not reuse constants in tableaux proofs?

I am trying to understand the proof of the following using tableaux: $$ \exists x\forall y.r(x,y) \to \forall x \exists y . r(x,y) $$ This is how it works out: $$ (1) \space \exists x \forall y ...
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1answer
37 views

Satisfiabilty 2-sat

Im trying to work out whether the following clause is satisfiable: {x, y},{x,¬y},{¬x, y},{¬x,¬y},{x, z},{x,¬z},{y, z},{y,¬z} My basic understanding is to work ...
2
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1answer
84 views

Create a shallow logic circuit that increments a binary number

This circuit should be reasonably efficient in size and depth, but with priority on depth. If depth was not a concern, then I guess I could make a specialized adder for the least significant bit and ...
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0answers
53 views

Propositional logic of arguments and undercuts [closed]

The setting An argument from a set of formulas $\Delta$ is a pair $\langle \Phi, \alpha \rangle$ such that $\Phi \subseteq \Delta$ $\Phi \nvdash \bot$ $\Phi \vdash \alpha$ (this is what I am ...
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1answer
68 views

Finding a graph-theoretic representation of expressions in Boole's algebra

I just read "Boole's Algebra Isn't Boolean Algebra" by Theodore Halperin (behind a paywall here). I don't have a strong background in abstract algebra, so, frankly, the paper is a bit over my head but ...
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1answer
31 views

Finding a finite model

Hello I am having difficulty with this question, I am not even sure what strategy one would go about proving something like this: Suppose $L$ is a language which includes an infinite list ...
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1answer
30 views

LK-$\phi$ proof of $\exists y Pby$

I am having difficulty with the concept of $LK-\Phi$ proofs, here is a question I have been working on: Let $\Phi = \{Pafa\}$, where $P$ is a binary predicate symbol and $f$ is a unary function ...
0
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1answer
54 views

Odd Parity Function [closed]

I am trying to define a Odd Parity Function that takes three 1 bit inputs and will output a 1 if the 3 bits are odd as a Boolean function. ...
2
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1answer
54 views

FOL substitution - is it possible to substitute two variables with each other? e.g. $\theta=\{x/y,y/x\}$?

Let $C = m(P,X,Y) \leftarrow m(Q,X,Z), m(R,Z,Y)$. Is it possible to do the following substitution? $D = C\theta$ where $\theta = \{Q/R,R/Q\}$ s.t. $D = m(P,X,Y) \leftarrow m(R,X,Z),m(Q,Z,Y)$
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1answer
55 views

3-coloring a graph with propositional formulas

I am trying to tackle a specific problem so any help would be greatly appreciated: Let $G = (V,E)$ be an undirected graph with vertex set $V$ and edge set $E$. A 3-coloring of $G$ is a map ...
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1answer
57 views

Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality [closed]

The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...
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2answers
54 views

Can we move quantifiers to the left in predicate logic?

Say I have part of a query in the form: ∃xa(...)∧∃xb(...)∧∃xc(...), where a, b, and c are attributes and the ellipses can be anything (I'm looking for a general rule). Is this equivalent to saying ...
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0answers
23 views

Are these CNF clauses for at most one and the same correct?

Given Boolean variable Xij that represents whether dog i is kept in kennel j. Encode in CNF clauses: Dogs that cannot be kept together must be kept in separate kennels Here is what I ...
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2answers
58 views

Constructively deciding whether a decidable predicate holds universally

I am trying to obtain the proof of the proposition: $(\forall x \in \mathbb{N}, P(x)) \vee (\neg \forall x, P(x))$ given that the property $P$ is decidable for every $x \in \mathbb{N}$, i.e. ...
3
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1answer
39 views

grammatical complexity of propositional and monadic predicate validities? (and grammars for recursive but not context-sensitive languages?)

Consider two sets: the set of validities of propositional logic and the set of validities of monadic predicate logic. Call the first set $VP$ and the second set $VQM$. Both of these sets are ...
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4answers
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How can I prove $P \rightarrow (Q \rightarrow R)$ is equivalent to $(P \wedge Q) \rightarrow R$

I'm a freshman CS student at my university and i'm struggling with understanding my professor through his thick accent. I've asked him to explain the proof for this multiple times and still have ...
3
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2answers
61 views

Unification — most specific unifier

In unification, given a set of equations, a standard problem is to compute a most general unifier (mgu). I am interested in a somewhat reversed problem. Imagine having a set of equations that do not ...
4
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2answers
298 views

P, Q, ((P→Q)→R) ⊢ R using only modus ponens

Can $R$ be inferred from $P$, $Q$, and $(P \to Q) \to R$ using only modus ponens? My understanding is that it can, as shown below, but I was told this was incorrect. Proof of ${P, Q, (P \to Q) \to R} ...
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1answer
289 views

How does binary addition work?

I find binary confusing. I have watched minecraft redstone videos on binary adders, real binary adders, diagrams, etc and yet I have not learned much at all. How does electrons flowing through wires ...
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1answer
13 views

Modelling 2 object situation with Propositional Logic

I'm reading up on propositional logic, and I'm completely stuck on this example - spent the past few hours trying to figure it out! Any pointers would be appreciated. There's 2 trees, each with signs ...
4
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2answers
45 views

Mapping intuitionistic logic to the modal logic S4

In his famous Semantical Analysis of Intuitionistic Logic, S. Kripke speaks of the "well-known mappings of intuitionistic logic into the modal system S4". I'm not sure which mappings Kripke means. One ...
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4answers
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Has Anyone Actually Created a System that Writes Computer Programs from specification?

Has anyone ever actually written a system (software or detailed explanation on paper with simple examples) that generates computer programs? I input $Prime(x) \wedge x<10$ and it creates a program ...
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Boolean formula that agrees with most truth assignments

Let $X_1,\dots,X_n$ be $n$ boolean variables. I have an unknown predicate $P(X_1,\dots,X_n)$ on these boolean variables. Of course, I can view the predicate as a function $f_P : \{0,1\}^n \to ...
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1answer
91 views

Karnaugh map - assign variables to the inputs?

I drew the map on the right, but what I drew doesn't work for what the question is asking me. I think I did something very wrong, and I don't really understand what this question is asking me. Am i ...
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1answer
43 views

assertion in first order logic

Can anybody give me an idea how to write this assertion in in first order logic? X has not passed one or more of the prerequisites for A. Here, X is the name of a person and A is a constant ...
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2answers
112 views

Number of instances of SAT (boolean satisfiability) problems of size N?

I assume the size of an instance of the SAT problem is measured by its number of (Boolean) variables. What is total number of instances of SAT problems of size N? I guess that amounts to counting ...
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1answer
39 views

Recognizing Horn clauses

I am currently studying model theory and I am trying to decide if a clause is a Horn Clause. I know that a Horn Clause is a clause with at most one positive literal, but there are some clauses that it ...
2
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1answer
37 views

Finding the minterm expression of F + G [closed]

I have a question that asks me to find the minterm expression of two functions added together. The two functions in this case are F = m(0,4,5,6) G = ...