Questions tagged [propositional-logic]

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25
votes
7answers
13k 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 ...
22
votes
11answers
4k views

Why is $A \lor (A \land \neg B) \equiv A$?

I would like to know if there is a rule to prove this. For example, if I use the distributive law I will get only $(A \lor A) \land (A \lor \neg B)$.
19
votes
4answers
6k views

Can proof by contradiction work without the law of excluded middle?

I was recently thinking about the validity of proof by contradiction. I’ve read for the past few days things on intuitionistic logic and Godel’s theorems to see if they would provide me answers to my ...
13
votes
5answers
3k views

Why does soundness imply consistency?

I was reading the question Consistency and completeness imply soundness? and the first statement in it says: I understand that soundness implies consistency. Which I was quite puzzled about ...
7
votes
4answers
6k views

Boolean algebraic expression vs Propositional logic expression

There is a lot of similarity between Propositional logic and Boolean algebraic expressions. Similar aspects : 1) Both has variables of two states. 2) Operations of Boolean algebra and ...
5
votes
2answers
2k 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. ...
5
votes
1answer
408 views

Implementing mathematical theory of arithmetic in Haskell via Curry-Howard correspondence

I have to ask for forgiveness in advance if the whole question doesn't make a lot of sense, but unfortunately, I have no better intuition as of right now and this seems like the best starting point I ...
5
votes
2answers
3k 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 ...
5
votes
2answers
532 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} ...
5
votes
1answer
235 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 ...
4
votes
2answers
96 views

Why algorithms calculating non-tirivial zeros can't be used as proofs of Riemann Hypothesis?

Recently I was reading again this propositions as types paper by Philip Wadler: http://homepages.inf.ed.ac.uk/wadler/papers/propositions-as-types/propositions-as-types.pdf It gives an impression, ...
4
votes
3answers
127 views

Is it a tautology or not? According to my truth table its not

If $\bigr((q\leftrightarrow p)\leftrightarrow s\bigl)$ is a tautology and $p\rightarrow s$ is contingent, does it follow that $q\rightarrow s$ is contingent? Since I can't show $\bigr((q\...
4
votes
1answer
582 views

What is the connection between combinatorial circuits and finite state automata?

The diagram on the Wikipedia page of FSA shows the hierarchy of the computational devices, in that diagram it is denoted that the finite state machines are superior to the combinatorial circuits. ...
4
votes
1answer
199 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 ...
4
votes
1answer
332 views

Knight and knaves

I have these in my lecture notes, its about the rules where knights always tell the truth and knaves always lie: If A says “The statement ‘there is gold on the island’ and the statement ‘I am a ...
4
votes
0answers
58 views

Is there a correspondence of steps between DPLL and sequent-calculus?

Is there a correspondence between the steps in using DPLL to find out that a formula in propositional logic is unsatisfiable and using sequent calculus to prove that its negation is valid? And given ...
3
votes
2answers
379 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 $...
3
votes
1answer
755 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 ...
3
votes
1answer
111 views

What is the purpose of learning propositional logic

I am in an AI class where we have extensively learned about propositional logic. I am wondering what the point of learning this is? Are there any uses for propositional logic? Beyond basic logic gates ...
3
votes
1answer
263 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 ...
3
votes
1answer
112 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 ...
3
votes
1answer
337 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?
3
votes
1answer
291 views

Propositional logic --- syntactical completeness

Lets consider propositional logic. We say a proof system for propositional logic is syntactically (negation) complete if for every $\alpha$, either $\alpha$ or $\neg \alpha$ are provable within the ...
3
votes
1answer
36 views

Is there a $L$-complete variant of SAT?

Many complete problem of different class of complexity has SAT variant. Like 3-SAT or $k$-SAT is $NP$-complete, Horn-SAT is $P$-complete, 2-SAT is $NL$-complete, and so on. So I was wondering if there ...
3
votes
1answer
92 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, \...
3
votes
1answer
37 views

Stalmarck's method: x ≡ x → z, does z have to be true?

I have been researching Ståmarck's method 1. In the paper cited here, some rules are given. Rules are made of triplets (x, y, z) such that: y $\to$ z $\equiv$ x where x, y and z are booleans which ...
3
votes
1answer
114 views

Are these 2 equivalent?

Is ∀x∀y∀z[φ(x,y)∧p(y,z)->p(x,z)] equivalent to ∀x∀y∀z[φ(x,y)∧p(x,z)->p(y,z)] ? The only thing I can think of is that this question can be answered if we show that ...
3
votes
1answer
127 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 ...
3
votes
1answer
519 views

Iota combinator and implicational propositional calculus

There is are two esoteric languages with minimally functionally complete operators, iota and jot, that are closely related to SK combinators. I'm attempting to understand the relationship between ...
3
votes
1answer
531 views

Matrix Representation for logical gates?

I have been trying to find if there are other matrix type representations logical circuits, in the example below, $$\begin{bmatrix} 1 & 0\\ 1 & 1\\ \end{bmatrix} \equiv \, \, \Rightarrow$$ ...
3
votes
1answer
431 views

How to estimate how many assignments satisfy a given DNF formula using Monte Carlo?

Admittedly, homework. For the purpose of this question: $\phi$ is a DNF formula similar to this one: $(x_1 \wedge \neg x_3 \wedge x_4) \vee (\neg x_1 \wedge x_2)$ Also $C_i$ is a clause in this ...
3
votes
2answers
614 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 ...
3
votes
1answer
51 views

"State of the art" algorithms deciding entailment of propositional formulas?

I fail to find much about how to efficiently calculate whether a propositional formula entails another. Considering the following two points... We can check, for each truth assignment which makes the ...
3
votes
2answers
334 views

Model disjunction in a $\{0,1\}$ integer linear program

How can I model logical OR as an integer linear program? $$(y_3 + y_4 + y_5 + y_6 = 2) \lor (y_2 = 1)$$ where $y_i \in \{0, 1\}$, $1$ = True and $0$ = False.
3
votes
3answers
792 views

Does exist any horn formula that is equisatisfiable to the clause that is disjunction of two literals? Does also exist equivalent?

I just and simply want to know if whether or not exist any horn formula that is equisatisfiable to $(p\lor q)$. I would also be interested to know if there also exists equivalent.
2
votes
3answers
465 views

Why is SAT based on the CNF?

I have been reading up on Boolean logic and, specifically, the Boolean satisfiability problem. I have seen several people mention that the expression must be converted to conjunctive normal form (CNF) ...
2
votes
2answers
960 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 $...
2
votes
4answers
1k views

Is it necessary to learn how to prove Mathematical theorems as a CS Student? [closed]

I've just started my undergraduate course and have tried my hands on MIT's OpenCourseWare on Discrete Math on Logic and Proofs. There was a particular question asking to prove Cantor's Theorem: ...
2
votes
3answers
2k views

Resolution and what it means to derive the empty set

When using resolution, if the empty set {Ø} is derived from a formula like {¬x,¬y} {x,y}, does that mean the formula is unsatisfiable? If this is the case, why is ...
2
votes
1answer
50 views

Does it hold that $F \equiv \sigma(F)$ for a CNF formula $F$ and a permutation $\sigma$ s.t. $F \vDash \sigma(F)$?

Suppose we have a CNF formula $F$ and a permutation $\sigma$ of its literals such that for any literal $x, \sigma(\neg x)=\neg \sigma(x)$ and $F \vDash \sigma(F)$. Does it hold that $F \equiv \sigma(...
2
votes
1answer
92 views

Are there any techniques for checking whether a clause is subsumed by another clause when adding it to a cnf formula?

When doing variable elimination on a formula in cnf form, there is created a lot of new clauses. Is there any efficient way to check if these are subsumed by other, already existing clauses?
2
votes
1answer
279 views

Does every 3CNF propositional formula has an equivalent 2CNF propositional formula?

Does every 3CNF propositional formula has an equivalent 2CNF propositional formula? Definitions: 3CNF propositional formula is conjunctive normal form propositional formula, which is just ...
2
votes
1answer
607 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, ...
2
votes
2answers
93 views

Equivalence of Krom formulas tractable?

Assume I have two Krom formulas $\psi_1, \psi_2$. Krom formulas are propositional formulas in CNF that have 2 literals in every clause. Each literal can be negated or unnegated. In other words, $\...
2
votes
1answer
307 views

How to define the Atomic Propositions in Model Checking

The atomic propositions in Symbolic Model Checking form the state in the state-transition graph (the model $\mathcal{M}$ in Model Checking). The other part of Model Checking is the specification, ...
2
votes
1answer
160 views

First Order Logic : Predicates

I have a small problem with the first order logic, in particular, predicate logic Let us take this sentence as an example: ...
2
votes
2answers
105 views

Problem with formalism in first order logic

This is a general question in first order logic. Assume I have alphabet $\Sigma$ that contains one-argument function (among other symbols). I want a new alphabet, $\Sigma'$, which is the same as the ...
2
votes
1answer
23 views

Circuits and formulas for Clique

Is it correct to say that the Clique Problem is in $P$ iff there exists a family of Boolean circuits $C$ to decide Clique whose sizes are bounded by a polynomial? And based on this question, does that ...
2
votes
1answer
50 views

Programming in Propositional Logic article notation question

I was reading this article about propositional logic and transforming problems to SAT. The author often uses the following notation (taken from Dominating set section): I don't understand what $[v,i]$...
2
votes
1answer
178 views

Definition of 2-CNF (a.k.a Krom) formula

In my lecturer's notes, the following definition for a 2-CNF wff is given: A 2-CNF formula, or Krom formula is a CNF formula F such that every clause has at most two literals. However, there is ...