31

HoTT "suffers" from Gödel incompleteness, of course, since it has a computably enumerable language and rules of inference, and we can formalize arithmetic in it. The authors of the HoTT book were perfectly aware of its incompletness. (In fact, this is quite obvious, especially when half of the authors are logicians of some sort). But does incompleteness "...


26

I am not aware of a paper concerned with the comparison between symbolic execution and abstract interpretation. Nor do I think one is needed. Reading the original descriptions of these two techniques should be enough. King, Symbolic Execution and Program Testing, 1976 Cousot, Cousot, Abstract Interpretation: a Unified Lattice Model for Static Analysis of ...


26

He wanted to create a formal system for the foundations of logic and mathematics that was simpler than Russell's type theory and Zermelo's set theory. The basic idea was to add a constant $\Xi$ to the untyped lambda calculus (or combinatory logic) and interpret $XZ$ as expressing "$Z$ satisfies the predicate $X$" and $\Xi XY$ as expressing "$X\subseteq Y$". ...


25

As Steven notes, the canonical example is $\mathsf{IP} = \mathsf{PSPACE}$. This collapse does not relativize, in the sense that there is an oracle $A$, subject to which $\mathsf{IP}^A \ne \mathsf{PSPACE}^A$. The intuition why the known proof of this result avoids the relativization barrier is that it uses arithmetization (Yonatan alluded to this in a comment)...


24

Very briefly: the simply-typed $\lambda$-calculus does not have dependent types. Dependent types were proposed by de Bruijn and Howard who wanted to extend the Curry-Howard correspondence from propositional to first-order logic. The key contribution of Martin-Löf's is a novel analysis of equality. There are two main ways of giving Curry-Howard style ...


21

($=$ is a logical symbol, hence I will not write it as part of the signature.) The satisfiability problem is decidable, as $\gcd$ has both a universal and an existential definition in terms of $|$, $+$, and $\le$: $$\begin{align*} \gcd(a,b)=c&\iff c\ge0\land c\mid a\land c\mid b\land\forall d\:(d\mid a\land d\mid b\to d\mid c)\\ &\iff c\ge0\land c\...


20

You must be careful here. You are using set-theoretic concepts (cardinal, continuum) outside set theory. There is potential for confusion. Your question can be understood in several ways. Maybe you are asking whether there can be uncountably many terms of a given type. The answer is: obviously not since there are only countably many finite strings, and ...


19

The basic sum-of-squares proof system, introduced under the name of Positivstellensatz refutations by Grigoriev and Vorobjov, is a “static” proof system for showing that a set of polynomial equations and inequations $$S=\{f_1=0,\dots,f_k=0,h_1\ge0,\dots,h_m\ge0\},$$ where $f_1,\dots,f_k,h_1,\dots,h_m\in\mathbb R[x_1,\dots,x_n]$, has no common solution in $\...


19

Constructive mathematics is not just a formal system but rather an understanding of what mathematics is about. Or to put it differently, not every kind of semantics is accepted by a constructive mathematician. To a constructive mathematician call/cc looks like cheating. Consider how we witness $p \lor \lnot p$ using call/cc: We provide a function $f$ which ...


17

The quick summary is that LTL with only past and no future modalities defines properties expressed over finite-words and these are the star-free subset of the regular languages. Standard LTL when extended with past-time modalities does not have more logical expressive power than LTL with only future modalities but properties can be defined in an ...


17

TL;DR. The metamathematics of binding are subtle: they seem trivial but aren't — whether you deal with (higher-order) logics or 𝜆-calculus. They're so subtle that binding representations form an open research field, with a competition (the POPLmark challenge) some years ago. There are even jokes by people in the field about the complexity of approaches to ...


17

The two words do not refer to the same thing. Hilbert's Entscheidungsproblem was the question whether there is an algorithm that decides the universal truth of first-order logical sentences, which was answered negatively by Turing in his famous 1936 paper "On Computable Numbers, with an application to the Entscheidungsproblem". The word literally means ...


15

You might want to check out Liquid Haskell, which allow working with type refinements rather than dependent types. Type refinements can be seen as a restricted logical language that allow you to express Hoare-style properties of the inhabitants of various datatypes. Another possible candidate is the $F^*$ language, which offers similar constructs. There ...


15

One apparent strength of his approach is that it allows higher-order functions (i.e. lambda terms) to be observable outcomes, which measure theory generally makes quite tricky. (The basic problem is that spaces of measurable functions generally have no Borel $\sigma$-algebra for which the application function - sometimes called "eval" - is measurable; see ...


14

No, you cannot strictly distinguish syntactic from semantic methods, but the distinction still ends up making sense. Structural operational semantics is not denotational, because it is not a compositional method of giving semantics to a programming language. However, you can build denotational models out of a structural operational semantics by using a ...


14

In general, what we usually call the logical relations argument isn't really linked to impredicativity: the main idea is simply to interpret terms in some abstract algebra $\cal A$, and to represent types as a ($n$-ary) relation $R \subseteq \cal A^n$. This works perfectly fine for all kinds of type theories, including dependently typed theories, see e.g. ...


14

EDIT: Adding the caveat that Roger's fixed-point theorem may not be a special case of Lawvere's. Here is a proof that may be "close"... It uses Roger's fixed-point theorem instead of Lawvere's theorem. (See comment section below for further discussion.) Let $K(x)$ be the Kolmogorov complexity of string $x$. lemma. $K$ is not computable. Proof. ...


13

Honda and Yoshida's A Compositional Program Logic for Polymorphic Higher-Order Functions (probably) pioneered Hoare logics for purely functional languages. This work is based on Hennessy-Milner logic and Milner's encoding of functions into processes, as described here: From Process Logic to Program Logic The work by Régis-Gianas et al mentioned in ...


13

I'm going to elaborate my comments into an answer. The origins of predicative type theory are almost as old as type theory itself, since one of Russel's motivations was to ban "circular" definitions which were identified as part of the source of the XIXth century inconsistencies and paradoxes. Thierry Coquand gives an enlightened overview here. In this ...


13

First, you are confusing consistency of CIC as an equational theory with consistency of CIC as a logical theory. The first means that not all terms of CIC (of the same type) are $\beta\eta$-equivalent. The second means that the type $\bot$ is not inhabited. CR implies the first kind of consistency, not the second. This, as has been pointed out in the ...


13

Because one of the principal applications of Type Theory in formalizations has been to study programing languages and computation in general, a lot of thought has gone into ways of representing possibly non-terminating programs. I won't make a complete survey here, but I'll try and give pointers to the main thrusts of different directions. The "...


12

SOS can be considered as a proof system where lines are of the form $p(\vec{x}) \geq 0$ where $p(\vec{x})$ is a polynomial in variables $\vec{x}$. The inference rules are: $\over x^2-x \geq 0$ $\over x-x^2 \geq 0$ $\over p(\vec{x})^2\geq 0$ $p(\vec{x}) \geq 0 \over p(\vec{x})x \geq 0$ $p(\vec{x}) \geq 0 \over p(\vec{x})(1-x) \geq 0$ $p_1(\vec{x}) \geq 0, \...


12

It depends on what kind of a "beginner" level you wish to have. I don't think there is a real good undergraduate level text on proof complexity (this is probably true for most specialized sub-areas in complexity). But for a beginner (graduate level) sources, I would recommend, something like understanding well the basic exponential size lower bound on ...


12

There are several ways of writing such a term, depending on how we write the proof terms for the elimination rule for $\bot$, which is $$\frac{\quad\bot\quad}{A}$$ The corresponding rule in $\lambda$-calculus is $$\frac{\Gamma \vdash e : \bot}{\Gamma \vdash \mathtt{absurd}_A(e) : A}.$$ (We call $\mathtt{absurd}_A$ an eliminator.) Thus, the term of type $\bot ...


12

Summary. A logical framework is a meta-language for the formalisation of deductive systems, where deductions become syntactic objects. Of course what counts as a meta-language is quite vague, and it is helpful to understand the historical development of logical frameworks. The first logical framework was de Bruijn's Automath (1), which is based on $\lambda$-...


12

The following category theory inspired analysis (adapted from Plotkin's A Note on Inductive Generalization) explains a sense in which unification and anti-unification are dual concepts. As notation, let's write $$t \underset{\sigma}{\Longrightarrow} u$$ for two terms $t$ and $u$ and a substitution $\sigma$ whenever $t\sigma = u$. The existence of such a ...


12

The theory of any finite structure is model complete. In fact, it is easy to see that any formula is equivalent to an existential formula with one quantifier per each element of the structure, after which all quantifiers of the original formula can be simulated by conjunctions and disjunctions. In particular, the number of quantifiers (hence quantifier rank) ...


12

You are correct when you observe that for any particular terminating loop $L$ we may simply define the invariant "we're getting one step closer to termination". But proving that this is indeed a valid invariant may require transfinite induction! For instance, we could write a loop that computes the Goodstein sequence. This is particular sequence of numbers ...


12

Note that formulas using $\land$ and $\lor$ gates (and possibly the constants $0$ and $1$) are known as monotone. The complexity of monotone formula equivalence depends on how complex formulas are allowed. Let me start with the related problem of implication (entailment) between two formulas $\phi\vdash\psi$, which is easier to classify. Note that $\phi$ ...


11

Assuming that the complexity of the provability problem would satisfy you, the landscape of complexities of substructural logics with and without contraction is somewhat complex. I'll try to survey here what is known for propositional linear logic and propositional logic. The short answer is that contraction sometimes helps (e.g. LLC is decidable, while LL ...


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