Questions tagged [lo.logic]
Computational and mathematical logic.
-1
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0answers
26 views
Encoding naturals in the calculus of constructions and in a language like Idris
I'm learning some type theory and trying to relate that to what I already know about proving things in Idris and similar languages.
So, if I were to encode natural numbers in CoC, I'd probably have ...
9
votes
1answer
203 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 ...
7
votes
1answer
146 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, ...
6
votes
2answers
190 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 ....
6
votes
3answers
267 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 ...
3
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0answers
105 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 ...
9
votes
2answers
277 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 ...
9
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0answers
77 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 ...
8
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1answer
167 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
vote
1answer
48 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
votes
0answers
118 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
votes
2answers
152 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
196 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 ...
8
votes
0answers
41 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
votes
1answer
67 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 ...
11
votes
5answers
351 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
votes
1answer
116 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
votes
1answer
84 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
vote
1answer
81 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
votes
1answer
474 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 ...
5
votes
2answers
92 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
votes
1answer
93 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 ...
1
vote
0answers
152 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
votes
1answer
93 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?
1
vote
1answer
48 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
104 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
votes
1answer
46 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:
...
1
vote
0answers
54 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
votes
0answers
108 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} \...
3
votes
0answers
134 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 ...
-1
votes
0answers
33 views
closed Horn clause Datalog fragment identification
I am trying to identify the appropriate fragment of Datalog containing rules of the type:
A(x,y):-B(x,z),c(z,y)
where B(x,z) ...
6
votes
1answer
335 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_{...
5
votes
0answers
156 views
Is Agda sound as a proof system? [closed]
I've asked the same question on CSSE but with no luck (https://cs.stackexchange.com/questions/89611/is-agda-sound-as-a-proof-system). Therefore I ask it again here in cstheory and hope that more ...
4
votes
1answer
282 views
Sparsification Lemma for k-SAT and Exponential Time Hypothesis
According to R. Impagliazzo, R. Paturi and F. Zane, 2001 an instance of $k$-SAT is called sparse if $m = O(n)$ where $m$ denotes the number of clauses and $n$ the number of variables. The ...
3
votes
0answers
84 views
Looking for a specific tree automata model
is there any tree automata model over unranked trees (that is with unbounded number of children for each node), such that:
Checking non-emptiness and universality is decidable in elementary time,
...
2
votes
1answer
88 views
Is Eulerian Path (or Eulerian Cycle) definable in Monadic Second Order Logic?
Does there exist a monadic second order logic formula which is satisfied by a graph if and only if it has an Eulerian path (or Eulerian cycle).
I am looking for properties of graphs which are ...
6
votes
0answers
214 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 $A = (A, +) \in U$,
...
2
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0answers
39 views
Is there a complete and finite axiom scheme for conditional independence? (Graphoids)
Note: This is a better-written version of an unanswered question asked before on MathOverflow.
Question: Is there a complete and finite axiom scheme for conditional probability?
If so, is ...
3
votes
1answer
104 views
Connection between algebraic logic and computational complexity of logics?
I'm learning a bit about algebraic logic and I was wondering how knowing the algebraic semantics of a given logic might help the study of the logic itself from a computational point of view.
In ...
8
votes
1answer
371 views
Complexity of modal logic IK5
What is the complexity of local satisfiability problem for modal logic $\mathit{IK5}$? Herein we denote by $IK5$ the modal logic over euclidean frames extended with inverse modality. Could you provide ...
3
votes
4answers
281 views
Philosophy behind monotonicity requirement for inductive types
Is there a good philosophical reason for why inductive types with negative occurrences or non-monotonicity should not be considered valid constructions? According to my understanding of the Bishop/...
0
votes
1answer
48 views
Head variables of terms after application
We work in the Church-style simply typed lambda calculus. All terms shall be considered in long normal form. Any term of type $A_1\rightarrow A_2\ldots\rightarrow A_n \rightarrow 0$ is of the form $\...
4
votes
2answers
339 views
Efficiently modeling Turing machines in Peano Arithmetic
The (undecidable) Peano Arithmetic (PA) is powerful enough to model Turing machines.
Consider a standard first order axiomatization of Peano Arithmetic and a standard Hilbert-style proof system $\...
3
votes
0answers
122 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 ...
5
votes
1answer
138 views
Normal forms for intuitionistic formulas?
It is well-known that DNFs/CNFs and prenex normal forms generally do not exist for intuitionistic logic.
Are there any nice results about formula normalization for IL? I've tried googling this but ...
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0answers
57 views
On collapsing the Exponential time hierarchy
Define $\Sigma^E_0 = \Pi^E_0=E$,
for every $n>0$, define $\Sigma^E_n=NE^{\Sigma^p_{n-1}}$,
for every $n>0$, define $\Pi^E_n=CoNE^{\Sigma^p_{n-1}}$.
Define the Exponential time hierarchy by $EH=\...
10
votes
2answers
384 views
State of the Art for the Monadic Class?
Monadic First Order Logic, also known as the Monadic Class of the Decision Problem, is where all predicates take one argument. It was shown to be decidable by Ackermann, and is NEXPTIME-complete.
...
6
votes
1answer
204 views
Algebraic account of Gaussian elimination?
For fun, I've been looking at the interpretation of linear logic in terms of finite-dimensional vector spaces, and ran into an interesting
question about the interpretation of double-negation-...
13
votes
1answer
183 views
Conclusions from reverse mathematical strength of graph minor theorem
Say we have a graph property which can be checked in nondeterministic polynomial time, and a proof in a weak formal system (say RCA0) that the property is minor closed. Can we say anything about the ...
1
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0answers
72 views
What logic(s) exist for attributing belief?
I'm looking for an appropriate formalism to represent "traceability" in claims, especially connecting conclusions to source materials in a rigorous way. For example, I'd like to be able to represent ...
