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# Questions tagged [lo.logic]

Computational and mathematical logic.

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87 views

### Temporal Logic - Until [closed]

I have a doubt, in Linear Temporal Logic LTL, does the Until operator require that the first occurrence is the first term of the formula? ex: a U b does require ...
439 views

### Comparing the Kolmogorov complexity of theories

Chaitin's incompleteness theorem says no sufficiently strong theory of arithmetic can prove $K(s) > L$ where $K(s)$ is the Kolmogorov complexity of string $s$ and $L$ is a sufficiently large ...
467 views

### Practical applications of parity games

Are there examples of practical applications of parity games, ie systems, in the real world, that can be represented as parity games ? Usually related documentation on parity games has almost never a ...
264 views

### What is the complexity of the equivalence problem for read-once decision trees?

A read-once decision tree is defined as follows: $True$ and $False$ are read-once decision trees. If $A$ and $B$ are read-once decision trees and $x$ is a variable not occurring in $A$ and $B$, ...
255 views

### Reference for the fact that (0=1) implies false requires a universe in MLTT

It's a fairly well-known fact that deriving a contradiction from a disequality (for example, $(0=1) \to \bot$) in Martin-Loef type theory requires a universe. The proof is also fairly ...
819 views

### Law of excluded middle in MLTT

Is it possible to add law of excluded middle to Martin-Löf type theory as an axiom? It seems to me, that it's possible to add it to Coq since Coq has a module for non-constructive reasoning. Also, it ...
579 views

### Uses of $\infty$-categories in TCS

I'm not a theoretical computer scientist. I'm a stable homotopy theorist using $\infty$-categories. I've seen applications of category theory and topos theory to theoretical computer science, and I ...
189 views

### 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 "...
123 views

### Can we verify satisfiability of first order statements via saturation in sub-exponential time?

In first order logic, we can prove satisfiability several ways: Finite model generation, truthful monadic abstractions, and also saturation. With finite model generation techniques, we can verify the ...
117 views

### Logics for timed resource control

I'm studying proof theory and I've seen that linear logic can be used as a way to control resource usage, since by the propositions-as-types it is equivalent to the linear lambda calculus. Is there a ...
153 views

### Recommendations for References on undecidability of First Order Logic

I am currently reading Computability and Logic by Boolos Burgess Geoffrey for the proof on "undecidability of first order logic". however, I find the notations a bit confusing. Can anyone recommend ...
223 views

### Logic with Linear Programming

Can first-order logic be modeled/simulated as linear programming or integer programming? What about other forms of logic (say second order)? Update: am actually not a theory person, but more on the ...
162 views

### Presburger Arithmetic Decision Procedures

What are good textbook references for Presburger Arithmetic decision procedures?
576 views

### Simply typed lambda calculus and higher order logic

What is the relation between simply typed lambda calculus and higher order logic? Under Curry-Howard it seems that simply typed lambda calculus corresponds to propositional logic. How is it related ...
53 views

### what is the meaning of “indicate substitution by priming metavariables”

This is a question about the usage of "indicate substitution by priming metavariables". For example, in the separation logic(it is not important whether familiar to this logic), I can understand the ...
790 views

### Fixed points in computability and logic

This question has also been posted on Math.SE, https://math.stackexchange.com/questions/1002540/fixed-points-in-computability-nd-logic I hope it is ok to also post it here. If not, or if it is too ...
220 views

### What is a term of the type $\bot\rightarrow A$?

The sentence $\bot\rightarrow A$ is provable in intuitionistic logic for any type $A$. The proof is trivial: \begin{align} \bot&\vdash\bot \\ \hline \bot&\vdash A \\ \hline &\vdash\bot\...
789 views

### Scott's stochastic lambda calculi

Recently, Dana Scott proposed stochastic lambda calculus, an attempt to introduce probabilistic elements into (untyped) lambda calculus based on a semantics called graph model. You can find his ...
322 views

### First order satisfiability that doesn't have finite models

We know from Church's theorem that determining first order satisfiability is undecidable in general, but there are several techniques we can use to determine first order satisfiability. The most ...
2k views

### Are there any annotated formal verification systems for pure functional programming languages?

ACSL (Ansi C Specification Language), is a specification for C code, annotated with special comments, that allows C code to be formally verified. I have not looked into it, but I imagine that the ...
107 views

### Consistency and completeness of any arbitrary 3-valued logic? [closed]

Based on the explanations here  I know that 3-valued first order logic has many different versions, which differ in the definition of their operations (e.g. implication). All of these (as far as I ...
333 views

### Three-valued logic solver?

This is not my area, so apologizes if I am asking nonsense! I know that there are very good solver/theorem provers for solving 1st order logic. Now I have a problem, using 3-valued logic, but I am ...
176 views

### Decidability of first-order theory of real closed fields with functions

By a famous theorem of Tarski, the first-order theory of real closed fields is decidable, as it admits quantifier elimination. Can this result be extended so that propositions can be quantified over ...
367 views

### Theorem prover fails to find simple set theory proof?

I am trying to use an automated theorem prover (SNARK) to prove a theorem in first-order logic. Tarski claims in his "a work on mereology" that the goal is provable from assertions 1-3 but he does ...
539 views

### Which formalism is best suited for automated theorem proving in set theory?

Abbreviations - FOL is first-order logic; NBG is Von Neumann–Bernays–Gödel set theory; SEP is Stanford Encyclopedia of Philosophy; HOL is higher-order logic; ATP is automated theorem proving. Context ...
295 views

### About the position of side conditions in an inference rule

Sometimes I see people put side conditions above the inference line as if they were premises of an inference rule. This feels strange. My understanding (which may be wrong) is that a side condition ...
236 views

### Solving problems by deciding a logic

I am curious to know when open problems have been solved by expressing them in a specific logic, and then showing that this logic is decidable. I have two distinct cases in mind: The problem is ...
127 views

### Hypersequents: proof term assigments or translations to hybrid logic

I've been looking at a modal logic with the axiom $$(\Diamond A \land \Diamond B) \to \Diamond((A \land \Diamond B) \vee (A \land B) \vee (A \land \Diamond B))$$ Roughly, this says that the ...
127 views

### A function is lambda-2-definable iff it is HG computable and provably type correct in lambda-PRED2

I'm having a problem regarding Theorem 5.4.40.3 of Barendregt's Lambda calculi with types (1992), a chapter in Handbook in logic in computer science. (I'm referring to the PostScript version available ...
181 views

### Expressiveness of Infinitary Logic

I'm trying to put together a general picture of the expressiveness of some logics: First-Order Logics, Fixed-Point Logics, (Finite Variable) Infinitary Logics and the respected versions with Counting. ...
928 views

### What paradigm of automated theorem proving is appropriate for Principia Mathematica-style formalization?

I am in possession of a book, which, inspired by Russell's Principia Mathematica (PM) and logical positivism, attempts to formalize a specific domain by determining axioms and deducing theorems from ...
108 views

### (co-)Horn formulation of Frankl's union-closed sets conjecture

Based on comments on MO there is simple forumlation of Frankl's union-closed sets conjecture in terms of (co-)Horn. In co-Horn CNF at most one literal is negated in every clause (Horn CNF where every ...
69 views

### Infinitary Counting Logics: 1-sorted vs. 2-sorted framework

There are two ways to extend infinitary logic with counting: Grädel's way (cf. p. 11): We extend $L_{\infty\omega}$ by introducing a counting existential quantifier:  \mathcal{A} \models \exists^{...
3k views

### Does there exist a sentence of first-order logic that is satisfiable only in infinite models that do not have a finite algorithmic representation?

There exist sentences of first-order logic that are satisfiable and are satisfiable only by models of infinite size. However, all such sentences I can think of are satisfied by infinite models that ...
254 views

### How to translate general recursion into a set of $\mu$-recursive operator applications?

I'm trying to find a scheme to translate a functional language with let rec into a set of primitives called "generalized arrows", i.e. $\kappa$-calculus with ...
304 views

### Can factorial be encoded in the Kappa-calculus with fixed point operator?

Suppose we have a $\kappa$-calculus with operator $fix$, that could be used to transform function with type $(1 \rightarrow a) \rightarrow a$ to a value of type $1 \rightarrow a$. We use a normal ...
95 views

### How to develop an effective notation for a partially ordered logic?

I am developing a logic for reasoning about programs in a resource-constrained environment. My starting point is intuitionistic linear logic, but I made the following changes: In intuitionistic ...
367 views

### Rules about Prop and Set in UTT

In Luo's UTT (type theory which is used in Agda, Idris, and other dependently typed programming languages), there're are two rules for $\Pi$ types. One for $\mathsf{Prop}$ and one for $\mathsf{Set}$. ...
517 views

### Types which correspond to sets of cardinality of continuum

Are types which correspond to sets with cardinality of continuum possible in MLTT (or in any other constructive theory)? On the first sight, they aren't, since elements of types are terms and we ...
1k views

### Monomorphic vs Polymorphic type theory

I am currently reading the book Programming in Martin-Löf type theory by Nordström et al. In the book they have two important parts, one about monomorphic set theory and the other about polymorphic ...
88 views

### Do algorithms for solving parity games ignore other possible strategies for player V after finding one?

Solving parity games (from a "start" node) relies on the existence of a history-free winning strategy for player 0 (V) or 1 (R). Whether there is more than one such strategy is probably never ...
236 views

### What does consistency mean for “computational theories” corresponding to inductive types?

I am currently reading the book by Luo on computation and reasoning. In the book he contrasts inductive types considered as computational theories with axiomatic theories widespread in "standard" ...
2k views

### W-types vs Inductive types

Martin-Löf type theory uses W-types to define inductive structures like integers, lists, etc. However, calculus of inductive constructions doesn't use them in the same way, inductive types there seems ...
1k views

### Homotopy type theory and Gödel's incompleteness theorems

Kurt Gödel's incompleteness theorems establish the "inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic". Homotopy Type Theory provides an alternative ...
321 views

### Equational Logic and First Order Predicate Logic

I am interested in using Equational Theories (ET) together with Equational Logic (EL) found in algebraic specification languages such as CafeOBJ . I wish to use ET+EL to represent and prove sentences ...
334 views

### What is logic programming and does it really add anything new to the logic?

I am acquinted with the basics of such notions as logic programming, monotonic and non-monotonic reasoning, modal logic (especially dynamic logic) and now I am wondering - does logic programming ...
181 views

### Decidability of inductive invariant existence in Presburger arithmetic

Problem: Consider a finite number of control states (including an "initial" and a "bad" state), a finite number of integer variables, and for each ordered pair of states a transition relation ...
423 views

### “Correctness” of type theory

How to "proof" that type theory is correct? Or at least explain that it's meaningful in some sense. In what extent is this a mathematical question and in what is a philosophical one? When type ...