Questions tagged [lo.logic]
Computational and mathematical logic.
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What is the model of computation that corresponds (in the manner of Curry-Howard) to the deduction rule of resolution?
The Curry-Howard Correspondence is well-documented for the isomorphism which associates the intuitionistic natural deduction proof calculus (logic side) with the type system for the simply typed ...
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Relationship between natural deduction refutation and tableaux for propositional logic
Which kind of relationship is there between natural deduction refutations of a set f propositional logic assumptions, and the corresponding tableaux?
For example, consider the unsatisfiable set $\...
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Are there classes for that FO-model checking is FPT on hypergraphs?
For graphs, there are many classes that admit FPT-algorithms for model checking of first order logic, e.g. the class of nowhere dense graphs by Grohe et. al.
Are there similar results for ($k$-uniform)...
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Efficient transformation into CNF preserving entailment
Suppose you have two propositional formulas $\varphi$ and $\psi$, not necessarily in CNF. I want to convert them to 3CNF efficiently (hence introducing auxiliary variables) in such a way that $\varphi ...
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Are there logical devices similar to "existential variables" or "blank nodes" of Semantic Web?
In Semantic Web, alongside permanent names of things also "temporary names" named "existential variables" or "blank nodes" denoted as "_:label" are used. All ...
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Examples of simulations in proof complexity that are not p-simulations
I am writing a paper on the complexity of some unorthodox proof systems, where I have two systems $P$ and $Q$ such that $P$ simulates $Q$ in the sense of it being possible to translate a $Q$-proof ...
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Satisfiability and a Galois Theory Analog
Let $v(a, b)$ be a binary predicate, and define $\phi$ as follows:
$$\phi: v(a_1, b_1) \land v(a_1, b_2) \land (a_1, b_3)$$
where our universe consists of two sorts $A: \{a_1, a_2, a_3\}$ and $B: \{...
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Defining functions on non-inductive types using LEM in Coq
I'm trying to prove statements about homomorphisms in Coq. Specifically, about in which cases the existence of some set of homomorphisms implies the existence of a specific other homomorphism. I'm ...
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What's the logical counterpart to jumps with arguments on CPS terms?
It's well known that the CPS (continuation-passing style) translation often employed in compilers corresponds to double negation translation under the Curry-Howard isomorphism. Though often the target ...
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Validity problem of intuitionistic two-variable logic
The two-variable fragment $\mathrm{FO}^2$ consist of those sentences of first-order logic $\mathrm{FO}$ in which precisely two variables occur (e.g. $\exists x \exists y \exists z R(x,y,z)$ is not a ...
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Recovering the inputs to Boolean circuits after partial evaluation
This question discusses Boolean Circuits and Boolean functions from $n>1$ inputs to one Boolean output. Notation: $\textit{arity}(\mathcal{C})=n$ if $\mathcal{C}$ takes $n$ inputs, similarly for ...
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Is there a fundamental link between Nash's equilibrium and Turing's halting problem?
Since Nash equilibrium exists, is there a computational analogue of this equilibrium point? I am trying to approach Nash equilibrium from computational point of view to see if the equilibrium point ...
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Where does "Quine's Method" in propositional logic originate?
Hein (407-408) states that Quine's method "...uses these (14) properties together with basic equivalences to determine whether a wff is a tautology, a contradiction, or a contingency." The ...
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Is relativization statement-dependent or proof-dependent?
I'm relearning some computability theory, and have encountered the idea of relativization of results to arbitrary subsets of $\omega$ and the subtlety of figuring out what the correct relativized ...
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Constructive Strong Normalization of the Extended Calculus of Constructions
The extended calculus of constructions (ECC) is basically the calculus of constructions with cumulative universes. I use the definition which Zhaohui Luo used in his PhD theses which contained a proof ...
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Alternatives to Normalization by Evaluation
Reading about lambda calculus I got the impression that normalization is evaluation.
So I don't understand what is meant by Normalization by Evaluation (used e.g. in several publications of A. Abel).
...
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Linear Integer Arithmetic Satisfiability with Three Literals [closed]
I'm stuck on trying to find an unsatisfiable conjunction of the form $a \wedge b \wedge c$ where:
$a \wedge b$ is satisfiable
$a \wedge c$ is satisfiable
$b \wedge c$ is satisfiable
$a, b, c$ are ...
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Boltzmann sampling for containers/dependent polynomials?
I’d like to randomly sample from dependently-typed data structures.
Has anyone looked at extending Boltzmann sampling to containers or dependent polynomials?
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Proof and computational complexity
I couldn't find documents elaborating on this: if the Curry Howard correspondence is to be interpreted as establishing a strong relation between proofs and programs, should there not be a strong ...
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How long does it take at most for $k$ boolean variables to map back to themselves with a positive disjunctive update rule?
I have a vector of boolean variables $v=(x_1,\dots,x_k)$. In each step each variable is updated according to a positive disjunction like so:
$x_1 \leftarrow x_i \vee \dots \vee x_j$
$\dots$
$x_k \...
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Looking for some lecture videos on logic, models of computation and computational complexity/tcs fundamentals [closed]
Looking for some lecture videos (introductory level) on logic, models of computation as well as computational complexity/ other theoretical computer science fundamentals
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Variable wire weights in DLOGTIME-uniform circuits
The definition of a $DLOGTIME$-uniform circuit family is based on a Turing machine that accepts the language $\langle t, a, b \rangle$, where gate $a$ is of type $t$ and has gate $b$ as a child, ...
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Complexity class name for the class of languages that are $\Sigma^1_1$-definable over finite domains
Let ${\cal L}=\{Y_1,..., Y_k, X\}$ be a finite relational language such that $X$ is a unary relation name. Let $\phi(X,\bar{Y})\in{\cal L}$ be a first-order formula (the formula can have the equality ...
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Decidability of diophantine equations over {=, +, gcd}
It is well-known that polynomial diophantine equations are undecidable (Hilbert's 10th problem): that is, given a quantifier-free formula over the language $\{=, +, \cdot, 1\}$ (of equality, addition, ...
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Proof of $DLOGTIME-CC^0 = MOD[<,bit]$
Let $CC^0[m]$ be the class of constant-depth, polynomial-sized circuits consisting entirely of $MOD_m$ gates, which put out $1$ iff the sum of their inputs $\equiv 0~(\textrm{mod}~m)$. In the same way ...
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$\mathit{FO}[+,\times]$ seems more powerful than $\mathit{DLOGTIME}$-uniform $\mathit{AC}^0$?
I’ve been reading up on the connection between first order logic and small circuit complexity classes, and specifically Barrington, Immerman, and Straubing’s paper “On Uniformity Within $\mathit{NC}^1$...
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Is every countable, finite-branching LTS bisimilar to a tree?
Let $L$ be a finite set of labels, and let $\mathcal{C}$ be the set of finitely-branching transition systems labeled by $L$ and with a countable set of states. Let $\sim$ denote the bisimulation ...
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Is modal $\mu$-calculus "equivalent" to bisimulation?
I know that propositional modal $\mu$-calculus $L\mu$ is bisimulation-invariant. However, I'm curious to what degree it captures bisimulation.
Q1: Given two labeled transition systems $T_1$, $T_2$ ...
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Alternative exponential definition of Kolmogorov complexity
In Kikuchi's paper Kolmogorov complexity and the second incompleteness theorem the Kolmogorov Complexity (KC) of $x$ is defined as
$$ K(x) = \mu e (\varphi_e(0) \simeq x) \, , $$
the smallest $e$ such ...
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λProlog vs HiLog
λProlog is a well-known higher-order logic programming language.
On the other hand, HiLog is described as a logic programming language with higher-order syntax, but first-order model theory.
Do I ...
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Uncountability in intuitionistic logic
I've read snippets here and there that inside intuitionistic logic, uncountable can be a subset of the naturals ?
What is the correct intuition to think about this? Andrej Bauer replied above, saying ...
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Order-invariant conjunctive queries are FO-definable without the order
I'm looking for a reference for Exercise 6.11 from Libkin's FMT book:
Prove that an order-invariant conjunctive query is FO-definable without the order relation.
All help is appreciated.
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Is Scott's reduction sound for $\mathrm{FO}^2$ with equality?
As per this paper by Grädel, Kolaitis and Moshe Vardi, they discuss computational complexity of satisfiability problem in $\mathrm{FO^2}$, In order to do this they use Scott's reduction. Which is the ...
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Does focused proof search ever have to backtrack across the choice of focus formula?
There are a lot of different "focused" sequent calculi for lots of different logics, but my understanding is that many or most of them have the following flavor. First one divides the ...
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Normal forms for counting quantifiers?
In the paper by [Erich Grädel and Martin Otto], the authors state that any formula in First Order Logic with two variables with counting quantifiers can be reduced to a formula of the form
$$ \forall ...
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Categorical equivalent of higher order logic
From Simply typed lambda calculus and higher order logic, I get the impression that HOL is STLC + equality + equality axioms. I was wondering if there is a particular kind of category modelling this.
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Is there a language of first-order logic such that every r.e. set is Turing-equivalent to some finitely axiomatizable theory in that language?
I hope that mathematical logic / recursion theory type questions are welcome here. I am sorry this question is so long and technical, but I believe that if you read it you will find that it is well-...
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Why REFL rule is primitive in HOL Light?
HOL Light
assumed REFL as a primitive.
Why does it need to do so?
Can't REFL rule be deduced in this way using ...
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Logic of learning
Does Robust logic (Leslie Valiant), Default logic (Raymond Reiter) and Circumscription logic (John McCarthy) have any relation?
I was Mathematician and Computer Science (dual degree undergraduate) ...
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What is FO(REGULAR)? (The descriptive complexity equivalent of NC1)
According to Immerman's Descriptive Complexity diagram, there is a logic called $\mathsf{FO(REGULAR)}$ which captures $\mathsf{NC}^1$. However, I can't find the reference where this logic is defined. ...
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FO(TC) lower bounding games?
Is anyone aware of any games/algebraic structures that provide lower-bounding methodologies for $FO(TC)$ formulae? I am aware of EF games as they apply to first-order and second-order statements, but ...
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What is the complexity of checking equivalence of two boolean formulae without NOT symbol?
Suppose I have two boolean formulae (propositions) $P_1$, and $P_2$ (can be assumed to be in CNF) over the same variables and such that there are no "NOT" symbols used. I.e. only conjunction and ...
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How do computers check if two functions are the same?
To prove that two given functions are the same involves proving infinitely many statements. I wonder how to implement so that a computer can check such a statement?
An easy example is the following: ...
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Fagin's Theorem implications
I was going through Fagin's Theorem (summary on wiki) and if I understood correctly Existential Second Order Logic (ESO) can be used to represent any NP problem, the same can be said for a Non-...
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Cover set of Boolean formulas with conjunctions
I want to cover a set of Boolean formulas (over the same variables) with disjunctive conjunctions. Here's an example with two formulas $p_1$ and $p_2$ over the set of variables $\{A, B, X, Y\}$:
I ...
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On Courcelle's question about Monadic second-order logic with cardinality predicates
I have found the following question at openproblemgarden.org:
The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the ...
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Is unary $\Pi_2$-SUBSETSUM coNP-complete?
Consider the following problem:
for given integers $a_1, \ldots, a_{2n}$ and $A$ that are given in unary representation
define is it true that
for every $S \subseteq \{1, ..., 2n \}$ such that $|...
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Are there languages require many variables to achieve $\Sigma_n^0$ completeness?
The proof of Post's Theorem that I am familiar with assumes you have access to as many variables as you wish in your language. Matiyasevich's Theorem by contrast gives a $\Sigma_n^0$-complete formula ...
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Counting solutions to extended MSO formulas, and sampling -- do these appear in the literature?
I am trying to determine if the literature contains various extensions of Courcelle's theorem. Since I haven't been able to find these in the literature, I guess that these are folklore results, or ...
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Forward chaining algorithms
I am interested in learning about the current state of the art regarding forward chaining production systems. I understand that things haven't changed much (regarding the basic algorithms) since 1995'...
