Questions tagged [lo.logic]
Computational and mathematical logic.
465
questions
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A Game-Theoretical Semantics of Software Properties
There are many works on software verification using mathematical logic. For example, using the mathematical logic to verify the liveness property of software. On the other hand, we have a game-...
2
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159 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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90 views
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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79 views
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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172 views
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 ...
10
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295 views
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 $|...
5
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0answers
75 views
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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78 views
Transforming Lambda Calculus syntax into generic relations between finite strings
I am trying to validate the simplest possibly notion of a formal system as relations between finite strings. I know that Lambda Calculus has the expressive power of a Turing Machine:
<λexp> ::= &...
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170 views
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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87 views
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'...
7
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240 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 ...
0
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1answer
135 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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70 views
What is sequence unification?
And why is it interesting? Please provide some examples. This text is here to circumvent the anti-spam filter which thinks my question is bad.
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2answers
226 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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44 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 ...
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1answer
138 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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97 views
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, ...
7
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2answers
655 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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2answers
156 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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1answer
114 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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1answer
254 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 ...
10
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1answer
190 views
P and Descriptive Complexity
In the Complexity Zoo, it says [1] that, in descriptive complexity, $P$ can be defined by three different kind of formulae, $FO(LFP)$ which is also $FO(n^{O(1)})$, and also as $SO(HORN)$.
However, ...
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2answers
239 views
Is case analysis on normal forms of lambda terms sufficient to prove parametricity results?
There are many closed terms of a given type. For instance, both of these terms:
$$ \lambda x . x $$
$$ \lambda x . (\lambda y . y) x $$
have a type of a polymorphic identity function:
$$ \forall X ....
7
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3answers
317 views
When a type is a value?
In functional programming and in the theoretical setting of the $\lambda$-calculus it is standard to consider a lambda abstraction $\lambda x.M$ as a value. In my understanding, the intuitive reason ...
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146 views
Busy Beaver Equivalent for the Untyped Lambda Calculus
In the same way that the Busy Beaver function is defined for Turing Machines, we could define a similar function for the untyped lambda calculus:
Over all terms in the ULC composed of ...
11
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2answers
323 views
Hereditary substitution with a universe hierarchy
I've read about hereditary substitution for the Simple Lambda Calculus and for The Logical Framework with distinct terms and types.
I'm wondering, are there any examples of hereditary substitution in ...
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101 views
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 ...
9
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1answer
196 views
Understanding the Proof of Strong Normalization of the Calculus of Constructions
I have difficulties in understanding the proof of strong normalization for the calculus of constructions. I try to follow the proof in the paper of Herman Geuvers "A short and flexible proof of Strong ...
1
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1answer
72 views
Soft Truth Values in the PSL model
This might sound like a trivial question. But since am starting out with my research in an area that is entirely new to me, I would really appreciate it if someone could kindly elucidate what Soft ...
10
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0answers
133 views
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 ...
4
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2answers
185 views
What does the category of RDF models look like in Institution Theory?
The Question in short
Here is the question in its pure form. Details of my reasoning can be found below.
The RDF1.1 spec semantics defines a model to consist of a set IR of objects and IP of ...
6
votes
3answers
215 views
Why/when do we ever need transfinite loop variants?
I do not understand why one would ever need a transfinite loop variant.
Why do natural-number-valued variants not suffice? My simple (but perhaps too naive) argument is: if a loop $L$ terminates ...
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44 views
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 ...
3
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1answer
76 views
Multimodal logic with quantification over modal operators
I take a multimodal logic to be a logic with multiple (potentially infinitely many) primitive modal operators. I am curious if anyone has studied a logic that allows one to quantify over the modal ...
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5answers
397 views
Representing bound variables with a function from uses to binders
The problem of representing bound variables in syntax, and in particular that of capture-avoiding substitution, is well-known and has a number of solutions: named variables with alpha-equivalence, de ...
4
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1answer
160 views
Is graph connectivity definable in existential MSO with vertices and edges?
Can $\exists$MSO$_2$ express graph connectivity?
Monadic SO (MSO) is the fragment of second-order logic in which the second-order quantifiers range over relations of arity 1 only.
$\exists$MSO is the ...
2
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1answer
93 views
Representable functions on System T
In Proof and Types by Girard et alii. Section 7.4.2, I think that the authors want to show that:
(1) The set of functions definable in System T coincides with the set of
recursive functions whose ...
1
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1answer
97 views
What's the expressive/compressive power of strongly normalizing subset of untyped lambda calculus?
Let $\Lambda$ be a set of strongly normalizing lambda terms.
Let $\mathtt{NF} : \Lambda \rightarrow \Lambda$ be evaluation to the normal form.
Let $ \lvert x\rvert : \Lambda \rightarrow \mathbb{N}$ be ...
7
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1answer
553 views
Is Church-pentation implementable in Agda?
Inspired by suggestion in this question, I've implemented predicative Church encoding of Peano arithmetic. Exponentiation works fine, unfortunately tetration requires the level of one of the arguments ...
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2answers
114 views
A stronger multiplexing rule for soft linear logic?
In (intuitionistic) linear logic the usual rules for the storage modality $!$ are promotion, dereliction, contraction, and weakening:
$$\frac{!\Gamma\vdash A}{!\Gamma\vdash !A}(prom) \qquad \frac{\...
4
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1answer
112 views
What's the difference between proving weak normalization and implementing evaluator?
Implementing an normalization (cut elimination) procedure for a type system A in a language with a total type system B, automatically proves cut elimination for type system A since the implementation ...
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0answers
223 views
Is it possible to type Ackermann function with (stratified variant of) System F?
I was surprised to find no open-source implementation of Ackermann function in pure System F as an illustrative example. I finally managed to implement it myself in Haskell using Church encoding:
<...
2
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1answer
106 views
Is there a formalization of normalization of impredicative system F?
In particular Agda seems not strong enough to prove that.
Is the predicative Calculus of Inductive Constructions universes (Coq without Prop) sufficient?
How about with the impredicative Prop?
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1answer
52 views
Is there a standard format for Dependent QBF?
I know there is a standard input format DIMACS for a formula is in conjunctive normal form (CNF) and QDIMACS for quantified Boolean formulas. Is there a similar standard format for the Dependent-QBF (...
7
votes
1answer
124 views
Does this variant of Multiplicative Linear Logic with mix rule enjoy cut elimination?
In Multiplicative Linear Logic (MLL), addition of the mix rule eliminates 'connectedness' from Danos-Regnier criterion. I'm investigating how the criterion changes if we do not distinguish between ...
3
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1answer
47 views
Algorithm for finding smallest set and instanciation for a given constraint system
I have a system of constraints described by a set of clauses of the form
$x_1 \neq x_2 \lor \dots \lor x_{i-1} \neq x_i$, for instance:
...
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62 views
To what extent supervised learning ERM learn first-order knowledge
Suppose I have a collection of (hidden) first-order rules:
$$
\mathcal{R}: \{ Q_i(x) => P_i(x) \}_{i=1}^{k}
$$
all defined over $x \in \mathcal{X}$.
I can use these rules and (automatically) ...
3
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0answers
112 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} \...
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142 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 ...
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1answer
348 views
Formalization of proofs and computational complexity paradox?
While reading some articles on formal proofs (see also my previous question on cstheory about the length of ZFC proofs versus human written proofs), I came up with this apparent paradox.
Let $M_{...
