Computational and mathematical logic.

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Incomplete basis of combinators

This is inspired by this question. Let $\mathcal{C}$ be the collection of all combinators which only have two bound variables. Is $\mathcal{C}$ combinatorially complete? I believe the answer is ...
2
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1answer
117 views

Damas-Milner-like subset of the calculus of constructions with global type inference

Damas-Milner is a subset of System Fω that gives up expressivity (type-level computation) for usability (type inference). The experience with Haskell and ML attests to the practical value of this ...
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93 views

Consistency of MSOL over trees

I couldn't find any source speaking about the consistency of [Weak] Monadic Second Order Logic over Binary Trees (or over graphs with finite tree/clique width). I did find decidability results, and it ...
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229 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 ...
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1answer
242 views

Standard reference for basic model theory definitions

I am trying to give a formal presentation of the model-theoretical semantics of a language and I am a bit lost in the terminology. In particular, could somebody clarify the exact definitions of model-...
8
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1answer
185 views

Is there a good notion of non-termination and halting proofs in type theory?

Constructive type theory with its basic interpretation under the curry howard correspondence consists only of total, computable functions. In the literature, some has been said on using "computational ...
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1answer
182 views

Can you explain an intuition behind Coherent Spaces?

Linear Logic is interpreted using Coherent spaces, and they feature prominently in Girard's papers. I know all the three main ways to formally define them, and they don't really pose any problem to ...
8
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1answer
176 views

What is the difference between unification and anti-unification?

I understand that in Unification we try to find a general solution to an equation between two terms, but what is anti-unification, and how is it different?
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1answer
94 views

In regards to the tautologies of a polynomially-bounded propositional proof system

In the book 'Logical Foundations of Proof Complexity', co-authored by Stephen Cook, the following definition is given: A proof-system $F$ is said to be polynomially-bounded if there is a polynomial p(...
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1answer
58 views

How to model degree variable in logic (new type of modal logic?)?

I am trying to model domain in logic (first order logic or some of modal logics) and I have variable which is degree and not true-false variable. There can be different conclusions depending on the ...
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1answer
71 views

On the difference between propositional proof system and polynomially-bounded proof system

For the definition of a propositional proof system we have: An abstract proof system is a polynomial time function f whose range is equal to the set of tautologies. If τ is a tautology, then an f-...
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1answer
67 views

How to specify and verify Horn clauses (logic programming programs)? Semantics of Horn clauses

There are lot of applications of Horn clauses (notable examples include use of rules in cognitive architectures and knowledge bases, as well as use of rules in business rules programs). Are there ...
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130 views

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 ...
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1answer
256 views

Minor closed properties that are explicitly MSO expressible

Below, MSO denotes the monadic second order logic of graphs with vertex-set and edge-set quantifications. Let $\mathcal{F}$ be a minor closed family of graphs. It follows from Robertson and Seymour'...
9
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1answer
302 views

Contradiction between Gödel's Second Incompleteness Theorem and the Church-Rosser's property of CIC?

On one hand, Gödel's Second Incompleteness Theorem states that any consistent formal theory that is strong enough to express any basic arithmetical statements can't prove its own consistency. On the ...
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2answers
286 views

Are there any propositional proof systems which are not Cook-Reckhow proof systems?

An abstract proof system is a polynomial time function $f$ whose range is equal to the set of tautologies. If $\tau$ is a tautology, then an $f$-proof of $\tau$ is any value $\pi$ such that $f(\pi) =...
7
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1answer
122 views

What is the significance of nominal techniques?

What is the significance of nominal techniques, as far as their application to the formal theory of bound variables is concerned? I have been reading M. J. Gabbay's expository work on the topic that ...
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0answers
27 views

Abduction in a Herbrand Constraint System

I have a simple constraint system with a finite set $C$ of constant symbols, an infinite set $V$ of variables, and two relation symbols - $R_1$, a preorder, and $R_2$, an equivalence relation. ...
9
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1answer
133 views

Logical Reations for an Impredicative System in a Predicative MetaTheory

Logical Relations for Impredicative languages like System F seem to rely critically on impredicativity of the ambient logic. Specifically, the interpretation for the forall-type will be defined in ...
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0answers
102 views

Is Logic Done on Superpositional Bit Values Useful?

Let's say I have a way to represent $N$ bits such that those bits are in a superposition of the $2^N$ possible states those bits can have and that I can do XOR and AND on those superpositional bits to ...
6
votes
3answers
299 views

Universal and existential types

I'm trying to wrap my head around the concepts of universal and existential types but everywhere I look, I see either logical or operational intuitions (or implementations) (e.g. TAPL book by B. ...
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1answer
208 views

Is `sort` typeable on elementary affine logic?

The following λ-term, here in normal form: ...
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1answer
209 views

What's wrong with this LEAN proof? [closed]

I'm learning to use the LEAN theorem prover and I got stuck in a proof of a simple fact in first-order logic: $$ p(x) \rightarrow \forall x p(x) $$ My code is the following: variables (A : Type) (...
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1answer
226 views

An example where smallest normal lambda term is not fastest

Let the $size$ of $\lambda$-terms be defined as follows: $size(x) = 1$, $size(λx.t) = size(t) + 1$, $size(t s) = size(t) + size(s) + 1$. Let the complexity of a $\lambda$-term $t$ be defined as ...
6
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1answer
145 views

Concatenative binary lambda calculus/combinatory logic

John Tromp defines a version of the lambda calculus that is encoded in binary: https://tromp.github.io/cl/cl.html a) Does there exist a concatenative version of this language (or its combinatory ...
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50 views

applications of institution-independent model theory

To quote wikipedia, The notion of institution has been created by Joseph Goguen and Rod Burstall in the late 1970s in order to deal with the "population explosion among the logical systems used in ...
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1answer
152 views

Would a proof that the traveling salesman algorithm can't be encoded on LAL also prove P!=NP?

An answer to the traveling salesman (and similar) problems can be easily verified on light lambda-calculi. Also, if I understand correctly, the light lambda-calculi can compute every polinomial-time ...
4
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1answer
126 views

Example of $MSO_2$ definable NP-hard problem on bounded clique-width graphs

All $MSO_1$ and $MSO_2$ definable graph problems can be solved in linear time on bounded tree-width graphs by Courcelle's theorem. But it seems this theorem doesn't work for $MSO_2$ definable graph ...
13
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2answers
443 views

Smallest possible universal combinator

I am looking for the smallest possible universal combinator, measured by the number of abstractions and applications required to specify such a combinator in the lambda calculus. Examples of universal ...
7
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1answer
132 views

Formalized priority argument

A priority argument, an important proof technique in recursion theory, was introduced by Friedberg and Muchnik, to solve Post's Problem, i.e., the existence of two r.e. sets that do not Turing reduce ...
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1answer
193 views

Reference for CTL* logic

I need a reference for CTL* logic (preferably easy to understand). I have gathered some disperse information regarding temporal and CTL logic but I need a more orderly coverage. A chapter of the ...
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216 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 ...
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97 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 ...
4
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1answer
124 views

Calculating least fixed points in equations

Given a set of relations/equations in some logic like LTL would it be possible to find a least fixed point for those equations? For example take the equations in LTL $$ P = Play ~\land X(P \cup S)\\ ...
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3answers
496 views

Why do constructivists not seem to care too much about call/cc

So a little while back I first had someone tell me that call/cc could allow proof objects for classical proofs by implementing Peirce's law. I did some thinking about the topic recently and I can't ...
2
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1answer
146 views

Deciding satisfiability and non-validity

For propositional logic, a decision procedure for satisfiability can be turned into a decision procedure for non-validity by giving it the negated version of a formula. Does this hold for all logics ...
9
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3answers
369 views

Ramification of An Impredicative Type Theory

Most type theories that I'm aware of are predicative by which I mean that Void : Prop Void = (x : Prop) -> x isn't well-typed in most theorem provers as this ...
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1answer
39 views

Is there a rule-based implementation of weighted abduction other than the PTTP?

Is there an implementation of weighted abduction other than the PT theorem prover? A Google search reveals the PTTP and a few statistical approaches but I'm interested in a rule-based approach. The ...
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3answers
262 views

Distinguishing semantics vs syntactic techniques and the syntax of your semantic domains

Consider a denotational semantics from simply-typed $\lambda$-calculus into dependent type theory. Is that actually a (trivial) term transformation into that dependent type theory? After all, type ...
11
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1answer
991 views

Why was there a need for Martin-Löf to create intuitionistic type theory?

I've been reading up on Intuitionistic Type Theory (ITT) and it does make sense. But what I'm struggling to understand is "why" was it created in the first place? Intuitionistic Logic (IL) and Simply-...
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129 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 ...
6
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1answer
391 views

Squash type vs Propositional truncation type

Homotopy type theory has a notion of propositional truncation type. It seems to me that it's strongly related to a notion of squash types. (See https://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ttfp.pdf ...
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1answer
391 views

Logical framework vs type theory

What is the difference between logical framework and type theory? Both of them have types, terms, and are based on dependently typed lambda calculus. We have Edinburg LF which is based on lambda-pi ...
12
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1answer
368 views

Why was Schönfinkel's work on eliminating “bound variables” in logic so crucial?

AFAIK, The first evidence of using higher order functions goes back to Schönfinkel's 1924 paper: "On the Building Blocks of Mathematical Logic" - where he allowed one to pass functions as ...
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2answers
2k views

What was the original intent for the creation of Lambda calculus?

I've read that initially Church proposed the $\lambda$-calculus as part of his Postulates of Logic paper (which is a dense read). But Kleene proved his "system" inconsistent after which, Church ...
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2answers
355 views

Do past time LTL and future time LTL have the same expressiveness?

I would like to ask if anybody is aware of a paper comparing the expressiveness of past time LTL and that of (future time) LTL.
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62 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 ...
12
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1answer
372 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 ...
11
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1answer
292 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 ...
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240 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 $B$, ...