Questions tagged [lo.logic]

Computational and mathematical logic.

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Model-checking for three-variable logics and restricted structures

Denote the $k$-variable fragment of logic $L$ by $L^{(k)}$. The model-checking problem for a logic $L$ with respect to a class of structures $C$, denoted $MC(L,C)$, is the decision problem $MC(L,C)...
András Salamon's user avatar
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536 views

Courcelle's theorem for bounded clique-width graphs

Courcelle's theorem states that "Every graph property which is expressible in monadic second order logic is decidable in linear time for bounded tree-width graphs". Later it was extended to bounded ...
Kumar's user avatar
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To what extent MSO = WS1S, when adding relations?

[This question has been asked on MathOverflow with no luck a month ago.] Let me first clarify my definitions. For a word $w \in \Sigma^*$, with $\Sigma =\{a_1, \ldots, a_n\}$, I define two ...
Michaël Cadilhac's user avatar
18 votes
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Descriptive complexity of communication complexity classes

It is well known that some major complexity classes, like P or NP, admit a full logical characterization (e.g NP = existential 2nd order logic by Fagin's theorem). On the other hand, one can also ...
Marcin Kotowski's user avatar
15 votes
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Proof assistant formalizations of Finite Model Theory

I'm wondering if anyone knows of a formalization (even limited) of any part of finite model theory in any of the major proof assistants. (I'm most familiar with Coq, but Isabelle, Agda, etc. would ...
Marc Hamann's user avatar
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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 ...
Elle Najt's user avatar
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Categorical semantics for S5 modal logic?

Does anyone know where I can look to find out what the generally categorical semantics of S5 is? For S4, the answer is well-known: we want a Cartesian closed category with a product-preserving ...
Neel Krishnaswami's user avatar
12 votes
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Is the theory of asymptotic bounds finitely axiomatizable?

Let $F$ be the set of functions over real numbers. Consider the structure $M = \langle F, <, \leq, =, \geq, > \rangle$ where the $<, \leq, =, \geq, >$ are defined as asymptotic notions $o$,...
Kaveh's user avatar
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12 votes
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Combinator logic and unification

Summary: if we are trying to use combinator logic to solve first-order logic type problems, is the best method to feed in free variables and use the standard first-order unification algorithm? In ...
rwallace's user avatar
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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 $|...
Alexey Milovanov's user avatar
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Do Banach spaces and linear contraction maps form a model of ILL with an exponential?

Recently, I read on the nLab that the category of Banach spaces and linear contractions is small complete, small cocomplete, and monoidal closed. This means that Banach spaces and short linear maps ...
Neel Krishnaswami's user avatar
10 votes
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264 views

Is there a text that discusses both the “lambda cube” of pure type theories and Martin-Löf's intuitionistic type theories, and compares them?

I am lost in a maze of twisty little type theories, all different. There are a number of works (textbooks and papers) that discuss pure type theories, and specifically the ones constituting the ...
Gro-Tsen's user avatar
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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 ...
Alexey Milovanov's user avatar
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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 ...
Emre Yolcu's user avatar
9 votes
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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 ...
helmut's user avatar
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Results in denotational semantics from model theory?

Denotational semantics interpret the theories of various lambda calculi in various (set-theoretic, domain-theoretic, category-theoretic, game...) models. Let $T$ be the theory of one such lambda ...
xrq's user avatar
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Relationship between lambda-definability, specification and definability in model theory

I am new to lambda calculus and definability theory, and I am trying to clarify my understanding of the relationship among the following concepts: An element $a$ in the domain of a type $A_\sigma$ is ...
Y.Z.'s user avatar
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Relationship between Pataraia's theorem and inductive-recursive definitions?

Pataraia's fixed point theorem gives a constructive proof of the fact that if you have a monotone function $f$ on a DCPO, then it has a least fixed point. I've frequently used this fixed point theorem ...
Neel Krishnaswami's user avatar
7 votes
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293 views

MLTT vs. [weak] MSOL

I've noted that both Martin-Lof type theory and [Weak] Monadic Second-Order logic (eg over trees) enjoy the ability to express basically any finite computer program, in a decidable manner. I was ...
Troy McClure's user avatar
6 votes
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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 $\...
Nicola Gigante's user avatar
6 votes
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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: \{...
Steven Schaefer's user avatar
6 votes
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399 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 ...
Joey Eremondi's user avatar
6 votes
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294 views

Fischer and Rabin's theorem (1974) for theories of "additive" structures

Fischer and Rabin's Super-Exponential Complexity of Presburger Arithmetic (1974) has the following theorem. (Theorem 12) Let $U$ be any class of additive structures, so if $A = (A, +) \in U$, ...
Pteromys's user avatar
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Complexity of validity of first-order logic over finite words with bounded quantifier alternation?

I'm concerned with the validity problem for sentences of first-order logic over finite words, i.e. $FO[\le]$ interpreted over finite subsets of $\mathbb{N}$. AFAIK it should be nonelementary. However,...
Nicola Gigante's user avatar
6 votes
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177 views

What is the Kolmogorov complexity of arithmetic?

Chaitin's incompleteness theorem says no sufficiently strong theory of arithmetic can prove $K(n) > L$ where $K(n)$ is the Kolmogorov complexity of the number $n$ and $L$ is a sufficiently large ...
Russell Easterly's user avatar
6 votes
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148 views

Comparing the Kolmogorov complexity of theories - Part 2

Chaitin's incompleteness theorem says no sufficiently strong theory of arithmetic can prove $K(x) > L$ where $K(x)$ is the Kolmogorov complexity of natural number $x$ and $L$ is a sufficiently ...
Russell Easterly's user avatar
6 votes
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707 views

Generalized sequential machine synthesis subject to language equivalence/inclusion and reachability

A generalized sequential machine (GSM) is a generalization of a Mealy machine where on each transition one input symbol is read and 0 or more output symbols are written. As in a Mealy machine, we ...
Dave Lang's user avatar
5 votes
0 answers
43 views

Original formulation of Spira's Lemma

I'm currently reading the book "Proof Complexity" by Jan Krajíček (2019), where Spira's Lemma is mentioned: Let $T$ be a finite $k$-ary tree and $|T| > 1$. Then there is a node $a \in T$ ...
user11718766's user avatar
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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 ...
Stella Biderman's user avatar
5 votes
0 answers
163 views

Salomaa's axiomatisation of regular languages and the use of regular expression in it

I am reading the classical article of A. Salomaa where he gives two axiom systems for regular sets and proofs consistency and completeness. As I have understood it, an axiomatic system in some logic (...
StefanH's user avatar
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The evaluation problem for AC$^0_d$ formulas is in FO

Let $d \in \mathbb{N}$ be arbitrary. Let $\mathsf{AC^0_d}$-Eval be the following promise problem: Input: A depth $d$ formula $\varphi(x)$ and a binary string $a$. Output: $\varphi(a)$ I am looking for ...
Kaveh's user avatar
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Are there any logics to formalize understanding?

Are there any logics (modal-based or others) to formalize the following statement: "agent A understands p". For example "agent A understands that Titanic sank because it hit an iceberg" Where "...
user avatar
5 votes
0 answers
201 views

Logic capturing automorphism-invariant $\mathsf{AC^0}$ properties

Q1. Is there a logic that is computable in polynomial-time which contains all order-invariant properties computable in smaller classes like $\mathsf{AC^0}$ (or $\mathsf{TC^0}$)? Motivation As you ...
Kaveh's user avatar
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4 votes
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77 views

Lower bound on the size of Skolem functions

Consider a quantified Boolean formula $f$. We can convert it into Skolem Normal Form formula $f^*$ such that $f$ is satisfiable iff $f^*$ is satisfiable, by replacing variables that are existentially ...
veryinteresting's user avatar
4 votes
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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)...
embie27's user avatar
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4 votes
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172 views

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 ...
Jori's user avatar
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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) ...
Mahdi Heidarpoor's user avatar
4 votes
0 answers
146 views

Formalization of proofs and CC paradox? - Part II

This was the second part of my previous question. It is very similar, and probably it has a similar answer (as Emil said in a comment), but I thought it was worth to separate it and ask it as a new ...
Marzio De Biasi's user avatar
4 votes
0 answers
65 views

Finite intersection property of polymorphic type families

Let $\Phi$ be a type functor definable in polymorphic lambda calculus: $$ \alpha : * \vdash \Phi(\alpha) : * $$ $$ f : A \to B \vdash \mathsf{Map}^{A,B}_\Phi(f) : \Phi(A) \to \Phi(B)$$ Suppose further ...
Andrew Polonsky's user avatar
4 votes
0 answers
98 views

Coherence spaces and full completeness for the implicative fragment of linear logic

Linear logic isn't complete for coherence space semantics since $1$ and $\top$ get identified. But it is, I believe, complete for the fragment of linear logic whose only connective is $\multimap$. I ...
Andrew Bacon's user avatar
4 votes
0 answers
117 views

Transferring results on coalgebras in one category to another

Let $F$ and $G$ be endofunctors over categories $C$ and $ D$, respectively. Suppose that there is a forgetful functor $C \to D$ that has a left adjoint. Can we infer properties of $F$-coalgebras ...
Pteromys's user avatar
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3 votes
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143 views

Is it possible to recover the set of derivation trees of a fact from its semiring provenance in Datalog?

Background: In the context of Datalog, Green et. al (2007) introduce the notion of the Datalog provenance semiring, a generalization of why-provenance as well as bag and probabilistic database ...
Justin Lubin's user avatar
3 votes
0 answers
68 views

What's the difference between "modular" and "compositional"?

When talking about reducing complexity in a software system, we often talk about making it "modular" by breaking it up into multiple modules that are all linked together to form the overall ...
Jonathan Schuster's user avatar
3 votes
0 answers
44 views

Can you compute Shannon expansion of a Boolean formula more efficiently by using a QBF solver?

Maybe this is not enough research level, but I've been scratching my head on it for a while... I'm interested in the Shannon expansion of an existentially quantified Boolean formula of the form: $$ \...
Nicola Gigante's user avatar
3 votes
0 answers
60 views

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 ...
Alex Mine's user avatar
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3 votes
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95 views

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?
Neel Krishnaswami's user avatar
3 votes
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47 views

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 ...
StefanH's user avatar
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3 votes
0 answers
117 views

Decomposition of rectangular relations

Let $\alpha$ be a binary relation from $\gamma$ to $\chi$ and $\beta$ a binary relation from $\chi$ to $\rho$. If both $\alpha$ and $\beta$ are rectangular, i.e., they satisfy $\alpha \alpha^{-1} \...
delete000's user avatar
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3 votes
0 answers
129 views

Family of formulas for which Gabbay's separation algorithm explodes nonelementarily

It is repeated throughout the literature that Gabbay's algorithm for separation of LTL with Since and Until can produce nonelementary blow-up of the size of the formula, but I have never seen a proof ...
Daniel Oliveira's user avatar
3 votes
0 answers
152 views

Efficiently computing the union of all minimal unsatisfiable constraint sets in a first-order unification problem

Suppose we are given a standard first-order unification problem, represented as a set $D$ of term equality constraints, such that the system $D$ as a whole is unsatisfiable. Consider the minimal ...
Aaron Rotenberg's user avatar