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Questions tagged [model-theory]

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8
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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 ...
4
votes
1answer
111 views

Descriptive model theory classification of Counting hierarchy

Descriptive model theory uses logic to characterize complexity classes How to model Counting Hierarchy PSPACE in descriptive model theory?
3
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0answers
109 views

Applications of the monoidal closed structure in LTL?

A simple model of temporal logic is via time-indexed truth functions. This lets us model the Boolean connectives, as well as the next-step operator and modal always operator: $$ \begin{array}{lclll} ...
5
votes
0answers
236 views

Modeling union types using sum types

It is trivial to model sum types using only union types and product types: simply add a discriminant. $A + B \cong (0 \times A) \cup (1 \times B)$. What I am wondering is whether or not there is a ...
5
votes
1answer
229 views

Proof that the theory of rationals is convex

In Example 10.12 of the book The calculus of computation by Bradley and Manna, it is said The theory of rationals is convex, as it is convex in a geometric sense. How does the geometric sense of ...
7
votes
1answer
323 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-...
2
votes
0answers
59 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 ...
4
votes
1answer
197 views

Is it decidable that a computable analytic function over $\mathbb{R,C} ,$ equals $0$

Is it decidable whether a computable analytic function $f(x_1,x_2,\dots,x_n)$ over $\mathbb{R}$, $\mathbb{C}$ in a semi-algebraic or semi-analytic domain is identically zero? Is there any algorithm? ...
16
votes
1answer
719 views

To what extent can the mathematics of Reals be applied to Computable Reals?

Is there a general theorem that would state, with proper sanitization, that most known results regarding the use of real numbers can actually be used when considering only computable reals? Or is ...
3
votes
1answer
156 views

Decidability of first-order theory of real closed fields with functions

By a famous theorem of Tarski, the first-order theory of real closed fields is decidable, as it admits quantifier elimination. Can this result be extended so that propositions can be quantified over ...
8
votes
1answer
1k views

What is the axiomatic (set theory) context of the P vs NP and NP=EXPTIME conjectures?

When the conjecture $\mathbf{P} = \mathbf{NP}$ or $\mathbf{P} \neq \mathbf{NP}$ is set (e.g. by the Clay Mathematical Institute by S. Cook, see here) what mathematical axiomatic system is assumed? In ...
5
votes
0answers
232 views

Is there a Galois correspondence between a Haskell class hierarchy and its instance hierarchy?

Can we consider a Haskell class as a loose signature-only-specification (denoting a theory) and an instance as an implementation (denoting a model)? In the example below the specification of the class ...
14
votes
1answer
313 views

How to show that a type in a system with dependent types is not inhabited (i.e. formula not provable)?

For systems without dependent types, like Hindley-Milner type system, the types correspond to formulas of intuitionistic logic. There we know that its models are Heyting algebras, and in particular, ...
1
vote
2answers
244 views

Any Graph is a Model (! or ?)

I know this could be considered a pointless question. However despite I am quite convinced that any possible model (i.e. UML, SysML, natural language, math, etc.) can be defined by means of a graph I ...
8
votes
1answer
131 views

Time/Space Requirements of Verifying or Falsifying a First-Order Statement

Though L.Berman proved that the problem of verifying or falsifying any first-order statement about real numbers that uses addition and comparison but not multiplication is in EXPSPACE. Has it been ...