Questions tagged [propositional-logic]

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Proving unsatisfiability of a propositional formula

I have a propositional formula $F$ and an assignment of truth variables $A$. The assignment $A$ assigns a truth value to each variable in $F$ and then it can be evaluated. I have a function which for ...
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1answer
471 views

Is this possible to solve 4SAT in polynomial time? [closed]

I know and admit that this is long, but please read it slow and understand everything. I think that this is one of the most interesting questions asked in computer science ever. I don't expect for ...
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0answers
70 views

Refutation of pigeonhole using resolution

I am trying to find a way to refute the pigeonhole formula using resolution. Lets assume that $x_{ij}$ is true if $i^{th}$ pigeon is in $j^{th}$ hole. Thus, with $n$ holes and $n+1$ pigeons, the ...
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1answer
261 views

“Implied atomic propositions” in propositional boolean formula

Suppose we have a propositional boolean formula F and let M be the set of all models of that formula. I am wondering if there is an efficient way to find all atomic propositions that are implied by ...
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1answer
47 views

Logic, “and” operator between a set of formulas and a formula

Consider a set $S$ of formulas $\beta_i$ and a formula $\alpha$, if we have a condition such as $S \land \alpha$ is inconsistent what we have to calculate to check the inconsistency of $S \land \alpha$...
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1answer
381 views

Why is learning DNF harder than learning CNF?

In the PAC-learning setting, it is easy to learn a CNF formula when we know each clause has at most $c$ variables. We go through each positive sample, and eliminate every clause that contradicts the ...
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0answers
262 views

Propositional logic in an SMA* algorithm

I'm trying to implement an SMA* graph search algorithm that I found in a paper here(Rong Zhou, Eric A. Hansen: Memory-Bounded A* Graph Search. FLAIRS-02 Proceedings 203-209) and I would like to ...
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2answers
461 views

How is implication same as entailment

In propositional logic (Artificial intelligence to be specific) $\alpha$ entails $\beta$ iff $\alpha\Rightarrow\beta$ is a statement. However if I write the truth table for implies ($\Rightarrow$) if $...
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1answer
58 views

In the Resolution equivalence ($\neg A \implies B, B \implies C \models \neg A \implies C$) must $A$ be negated?

The sheet of equivalences given to us in class provides the the equivalences \begin{array}{|c|c|c|} \hline \text{Resolution} & A \vee B, \neg B \vee C \models A \vee C & \neg A \implies B, B ...
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460 views

What is the simplification of AB + BC + (~B)C?

AB + C is not the answer. The correct answer is AB + BC. HOW?
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35 views

What does the structure ($S_t^x P$) in this Existantial Instantiation imply, Skolemization?

The sheet of equivalences given to us in class provides the two equivalences \begin{array}{|c|c|c|} \hline \text{Universal Instantiation} & \forall x ~ P(x) \implies {S_t}^x P & \exists y ~ \...
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1answer
204 views

Breaking down CNF clauses

I have an assignment where i have to encode certain problem to conjunctive normal form so i can solve it by using SAT solver. I have been able to encode my problem correctly but the solver is quite ...
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2answers
331 views

Does mutual exclusion hold in this case?

I entered into a discussion with a friend on the following question, which asks if mutual exclusion holds: Consider two processes: $s_1$ and $s_2$ are two variables, set equal initially. P1: while $...
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1answer
197 views

Use of hypothesis to prove a tautology

Given $$ (¬(P ∧¬Q)⇒S)∧¬P ∧(R⇒¬S) ⇒ ¬R $$ prove that this is a tautology. One way is to use a hypothesis taken from the proposition itself. As an example: If we want to prove this rule: $P ⇒ (P ∧ Q ≡...
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2answers
65 views

What these simple rules of logical implications mean

Their are many rules of logics, predicate calculus, inferences and syllogisms which haunt me always. It feels better when I find some sensible name to particular rule which also gives me intuition ...
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1answer
101 views

Propositional logic and connectives(translating to english) [closed]

I want to know some stuff about translating compound statements to english: For the propositions below: p: You have the flu. q: You miss the final examination. r: You pass the course. I want to ...
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1answer
98 views

Contingent sentences can always be true

I have a question about contingent sentences. To my knowledge, contingent sentences are sentences that are neither tautologies nor contradictions. In a textbook I read that a sentence might always be ...
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7answers
11k views

Why is A implies B true if A is false and B is false?

It seems to me that the 'implies' in English language does not mean the same thing as the logical operator 'implies', in a similar way how 'OR' word in most cases means 'Exclusive OR' in our everyday ...
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1answer
86 views

How to reduce C′A′B + CAB′ to C′B + CB′?

I faced this Boolean expression: $C'(A'B+A)+C(A'+AB')$ It was solved as follows: $C'(A'B+A)+C(A'+AB')$ $=C'(A+B)+C(A'+B') $ ...by applying absorption laws $(I)$ $=C'A+C'B+CA'+CB'$ $=(C\oplus ...
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1answer
275 views

Show that the following tautology is a theorem of natural deduction

I'm struggling to apply the theorems of natural deduction to these two examples. $$((p \to q) \land (p \to \neg q)) \to \neg p$$ $$((p \to q) \land (p \to r)) \to (p \to (q \land r))$$ Can anyone ...
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1answer
584 views

Deciding which logical proposition is stronger

I am studying propositional logic and I am trying to solve some exercise related to the following definition: Given two logical propositions $\alpha$ and $\beta$, we say that $\alpha$ is stronger ...
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2answers
2k views

Find hamilton cycle in a directed graph reduced to sat problem

I need to find a Hamiltonian cycle in a directed graph using propositional logic, and to solve it by sat solver. So after I couldn't find a working solution, I found a paper that describes how to ...
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1answer
83 views

Is there any precedence for symbols for constructing parse trees? [duplicate]

I am wondering if some symbols such as the ones in propositional logic have precedence over others in drawing parse trees. For example, the sentence: p ∧ q → r, ...
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1answer
342 views

What are the differences between parse and decision trees?

I have created a parse tree for the formula: a∧¬(b∨c)∨¬d∧(¬b) successfully. I am now asked to create a decision tree for the same formula. What are the main differences between a parse tree and a ...
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1answer
301 views

All output functions of a truth table

http://www.cim.mcgill.ca/~langer/273/3-notes.pdf I can name one more: XNOR. But besides these, what other output functions are there?
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2answers
1k views

How to read out a double negation in propositional logic

How would one verbally say ~~R where R = my program is correct? The tildes are negation symbols. I'm not sure if it just cancels out and comes out as 'my program is correct' or if it's something else. ...
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1answer
197 views

How to find a minimum set of axioms within a set of propositions?

I have a set of propositions, for example $\{a_1,a_2,\dots,a_n\}$. Some propositions depend on others (for example, $a_1,a_2\Rightarrow a_3$, means if $a_1,a_2$ are true, then $a_3$ is true). I want ...
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1answer
227 views

Equivalent formulae with different CNF

I was not able to find or come up with two formulae which are equivalent but have different CNF. All my ideas reduce to the same formula after applying transformations. The requirements are the ...
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1answer
33 views

Derivation of implicational propositional axioms

Is there a way to subtract and add properties of axioms to generate new axioms? For example: {L} = {P S K} // natural deduction {P S K} = {P H K I} // natural deduction {S K} = {?} // constructive ...
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1answer
491 views

What are the differences between propositional logic and temporal logic?

I am currently studying temporal logic and find it much similar to propositional logic. However, temporal logic seems to be slightly more specific because it encapsulates more precise meaning symbols, ...
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1answer
76 views

a program discovering himself how to solve propositional calculus

it is well-known that propositional logic problems such as $$ (p\leftrightarrow q) \lor r \quad\overset{?}{\vdash}\quad (((p\lor q)\to(p\land q)) \land \lnot r)\lor r$$ can be simply solved by ...
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1answer
124 views

complexity of modal logic axioms

I am writing a paper in which I want to include complexity results for different modal logics and possibly add a reference to a specific paper. At the moment I have the following: K- no restrictions ...
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2answers
253 views

Which formula corresponds to this K-Map?

I have a K-Map and I need to figure out which expression isn't equivalent to the provided K-Map. $f(w,x,y,z) = \sum(3,7,9,11,13,15) + \Phi(4,5,6)$ We know that both options (a) and (b) are ...
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1answer
368 views

Relations between statements involving universal quantifier, conditional and biconditional

If we consider two predicates: $b(x)$: x is a boy $c(x)$: x is clever Then, there are four statements involving $∀, b(x), c(x), →$ and $↔$ . These are below along with my interpretation of their ...
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1answer
49 views

Transform $A\land B \Leftrightarrow B\lor A$ into conjunctive normal form [closed]

How do I transform the following formula into conjunctive normal form? $$ A \land B \Leftrightarrow B \lor A $$
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1answer
1k views

How to decide if a propositional formula is a well formed?

So, I have the following propositional formula: (((P → Q) ∨ S) ↔ T) How do I decide if it's well formed?
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1answer
84 views

Non-deterministic algorithms and Tautologies

I am studying the lecture The Complexity of Propositional Proofs. ​Here there is a definition together with a discussion (page 3). I don't understand that discussion. Let $F$ denote the set of ...
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1answer
520 views

Size of Propositional Formula

I am study this lecture The Complexity of Propositional Proofs. Here there is a term size of propositional formula (page 2), Every tautology $\tau$ on $n$ variables has an $F$-proof in which thera ...
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1answer
103 views

Complexity of a SAT related problem

Given a set of (propositional) formulae $\Phi$, two formulae $\phi$ and $\xi$, determine whether there exists $\Psi\subseteq \Phi$ such that $\Psi\models \phi$ and $\Psi\not\models \xi$. Question: ...
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2answers
11k views

Operator precedence in propositional logic

there is some kind of priorities for the elements in propositional logic ? for example : p ∧¬q → r , given this ,we there may be two options (p ∧¬q) → r OR p ∧ (¬q → r) , which one is the correct ?...
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1answer
114 views

Learning a small disjunction

I have a boolean function $f: \{0,1\}^n \to \{0,1\}$ that I know takes the form $$f(x_1,\dots,x_n) = x_{i_1} \lor x_{i_2} \lor \dots \lor x_{i_k},$$ but I don't know the values of $i_1,\dots,i_k$. ...
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1answer
156 views

Is there a known way to convert any $QBF_2$-formula into an equisatisfiable $QBF_2$-formula in CNF in polynomial time?

It is easy to turn any boolean formula and any quantified boolean formula into an equisatisfiable formula in CNF using Tseitin transformation: $$ Q_1 z_1 Q_2 z_2 \ldots Q_n z_n \Phi \Rightarrow Q_1 ...
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0answers
75 views

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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1answer
60 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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2answers
523 views

Getting a variable assignment of a Tseitin transformed formula

Let $\phi$ be a Boolean formula and $\mathrm{Tseitin}(\phi)$ the corresponding Tseitin transformed equisatifiable formula. It is well-known that one can get a variable assignment for $\phi$ by ...
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1answer
284 views

FP^NP-complete problems

Is there any other standard FP^NP-complete problem other than the Traveling Salesman Problem? For instance, in the canonical propositional logic?
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1answer
78 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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0answers
61 views

What is the difference between the situation calculus and the propositional logic of context?

I have been reading a lot of papers by John McCarthy lately. As early as 1963, he developed the situation calculus, but then starting in 1987, he developed what came to be known as the propositional ...
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1answer
40 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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1answer
947 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 \...