# Questions tagged [lo.logic]

Computational and mathematical logic.

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### Intuition behind UTT's internal logic

The "internal logic" of type theory UTT is defined in LF as follows: What's the intuition behind this definition? I can kind of understand the declaration of the the first three constants - ...
1 vote
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### Confusion about $T$ and $El$ when defining universes in LF

This is a technical follow-up question to Formulation of Tarski-style universes in LF Consider LF, the logical framework used to define UTT (unified theory of dependent types). The next two quotes ...
1 vote
59 views

### Formulation of Tarski-style universes in LF

Lately I've been asking questions on type theory on MSE, and I've been getting great answers, but I decided to give a try to this site and see if it will be helpful as well. I'm looking at this note ...
1 vote
216 views

### Turing Machines and Logic

It is well known that Monadic Second Order Logic (over words) and finite automata can express the same set of languages. Is there a logic over words (perhaps a nth order logic) such that it and turing ...
4k views

### Theoretical Computer Science vs other Sciences?

So I‘m in my third semester studying Computer Science at a German university, so I‘ve only scratched the surface of Theoretical Computer Science, namely Logic, Formal Languages, Automata Theory, ...
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### Can you define recursive predicates in 2nd order intuitionistic logic?

This is a purely logical question, but I think it's adjacent enough to CS that it's worth a shot here. Take 2nd order Heyting Arithmetic, say Heyting Arithmetic with an extra sort of (unary) ...
1 vote
167 views

### Complexity of "discrete-time" SAT

I'm interested in the complexity of deciding satisfiability of the following family of formulae: $\exists j. I[j(0)] \land \forall t. S[j(t),j(t+1)]$ where: $j:\mathbb{N} \to \{0,1\}^n$ has finite ...
1 vote
70 views

### On completeness of LTL

I am reading the seminal paper "On the temporal analysis of fairness" by Gabbay, Pnueli, Shelah, Stavi, available at shelah.logic.at/papers/134/ In Section 3, completeness of a set of axioms ...
2k views

### Translating SAT to HornSAT

Is it possible to translate a boolean formula B into an equivalent conjunction of Horn clauses? The Wikipedia article about HornSAT seems to imply that it is, but I have not been able to chase down ...
1 vote
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### Gurevich's theorem on primitive recursive functions being logspace-computable

I recently came across the following result attributed to Gurevich, according to which I understood that the class of problems solvable by primitive recursive functions is precisely the class L of ...
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### Are there classes for that FO-model checking is FPT on hypergraphs?

For graphs, there are many classes that admit FPT-algorithms for model checking of first order logic, e.g. the class of nowhere dense graphs by Grohe et. al. Are there similar results for ($k$-uniform)...
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### Combinators for the Primitive Recursive Functions

It is well-known that the S and K combinators are Turing Complete. Are there combinators that suffice to yield (only) the primitive recursive functions?
1 vote
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### Defining functions on non-inductive types using LEM in Coq

I'm trying to prove statements about homomorphisms in Coq. Specifically, about in which cases the existence of some set of homomorphisms implies the existence of a specific other homomorphism. I'm ...
688 views

### What's the logical counterpart to jumps with arguments on CPS terms?

It's well known that the CPS (continuation-passing style) translation often employed in compilers corresponds to double negation translation under the Curry-Howard isomorphism. Though often the target ...
141 views

### Validity problem of intuitionistic two-variable logic

The two-variable fragment $\mathrm{FO}^2$ consist of those sentences of first-order logic $\mathrm{FO}$ in which precisely two variables occur (e.g. $\exists x \exists y \exists z R(x,y,z)$ is not a ...
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### 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.
4k views

### Prove proof irrelevance in Coq?

Is there a way to prove the following theorem in Coq? Theorem bool_pirrel : forall (b : bool) (p1 p2 : b = true), p1 = p2. EDIT: An attempt to give a brief ...
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### Recovering the inputs to Boolean circuits after partial evaluation

This question discusses Boolean Circuits and Boolean functions from $n>1$ inputs to one Boolean output. Notation: $\textit{arity}(\mathcal{C})=n$ if $\mathcal{C}$ takes $n$ inputs, similarly for ...
131 views

### Proof of $DLOGTIME-CC^0 = MOD[<,bit]$

Let $CC^0[m]$ be the class of constant-depth, polynomial-sized circuits consisting entirely of $MOD_m$ gates, which put out $1$ iff the sum of their inputs $\equiv 0~(\textrm{mod}~m)$. In the same way ...
407 views

### Does the order of declarations in an inductive type matter?

I was wondering if the order of declarations of an inductive type can matter. For example in Coq you can define Nat either by: ...
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### Is there a fundamental link between Nash's equilibrium and Turing's halting problem?

Since Nash equilibrium exists, is there a computational analogue of this equilibrium point? I am trying to approach Nash equilibrium from computational point of view to see if the equilibrium point ...
467 views

### 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 ...
4k views

### How would I go about learning the underlying theory of the Coq proof assistant?

I'm going over the course notes at CIS 500: Software Foundations and the exercises are a lot of fun. I'm only at the third exercise set but I would like to know more about what's happening when I use ...
1 vote
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### Where does "Quine's Method" in propositional logic originate?

Hein (407-408) states that Quine's method "...uses these (14) properties together with basic equivalences to determine whether a wff is a tautology, a contradiction, or a contingency." The ...
I'm relearning some computability theory, and have encountered the idea of relativization of results to arbitrary subsets of $\omega$ and the subtlety of figuring out what the correct relativized ...