Questions tagged [lo.logic]

Computational and mathematical logic.

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57 views

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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86 views

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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1answer
148 views

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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1answer
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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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1answer
252 views

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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1answer
94 views

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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135 views

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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106 views

Functional abbreviation for Inst expression in Turing's 1936 paper

In Turing's 1936 paper "On Computable Numbers", For a Turing Machine $M$, $Inst(q_i S_j S_k L q_l ) $ means that if $M$ scans symbol $S_j $ under $m-configuration$ $q_i$, then the symbol ...
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1answer
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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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91 views

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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286 views

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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1answer
149 views

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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1answer
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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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80 views

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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1answer
153 views

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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1answer
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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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1answer
299 views

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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206 views

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'...
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285 views

Reverse Skolemization?

I'm wondering if there are any references on "reverse skolemization", that is, converting a formula with functions into one purely consisting of quantifiers by eliminating function applications. I'm ...
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1answer
446 views

How to build comparison operator (comparator) in an arithmetic circuit

I am trying to convert a basic program into an arithmetic circuit. I am stuck on the step of converting the greater than operator into an arithmetic circuit. To be specific, I do not know how to ...
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462 views

How do continuations represent negations (under the Curry–Howard correspondence)?

Under the Curry–Howard correspondence, types can be thought of as propositions, and values inhabiting a type can be thought of as proofs that the corresponding proposition is true. (E.g., the ...
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Extending the sequential calculus (logic over words) to allow a hierarchy of languages like the arithmetical hierarchy

Let $\Sigma$ be some finite alphabet. Then consider the logical language $\mathcal L = \{ R_a : a \in \Sigma \} \cup \{ <,= \}$ and first order formulas. For a given first order formula $\varphi$ a ...
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1answer
165 views

Reductions in Descriptive Complexity

Reducing one problem to another are well known in various settings, such as many-one, randomized, truth-table, logspace or a whole slew of other reductions. Descriptive complexity can alternately ...
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Notion of "quotient" or "inverse" for recognizable tree languages?

Related to my previous question but this time I have a better idea of what I'm actually asking. I'm looking at the following operation on recognizable tree languages (i.e. regular tree grammars, ...
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895 views

What logic correponds via Curry-Howard to a Monad?

According to Moggi's 1991 paper "Notions of computation and monads" one can represent monadic equational logic with the well known monad $(T, \eta, \mu)$ with T an functor and the two natural ...
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192 views

Compactness of domino tilings

I've read in Lemma 2 of the paper 1 that if every square region of the plane admits a tiling, then the whole plain admits a tiling, but the proof is omitted. This sounds like a compactness property, ...
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135 views

Dependent C-style types with subtyping rule

I'm looking for previous work regarding an extension of a C-style type system in which types may have constraints and have a defined subtyping rule. In particular, I'm interested in defining algebra-...
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290 views

What is the reference for the proof Gödel's first incompleteness theorem based on the undecidability of the halting problem?

A weaker form of Gödel's First Incompleteness Theorem, direct proofs of which in Gödel's manner are lengthy, involved and at some place rather counter-intuitive, has a simple and intuitive proof based ...

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