scholarly journals Descriptive Complexity of Probabilistic Complexity Classes through Second Order Generalized Quantifiers

2018 ◽  
Author(s):  
Thiago Alves Rocha ◽  
Ana Teresa Martins
Keyword(s):  
Second Order ◽  
Complexity Classes ◽  
Generalized Quantifiers ◽  
Descriptive Complexity

Complexidade Descritiva lida com a relação entre definibilidade lógica e complexidade computational em estruturas finitas. Como exemplo no caso de classes de complexidade probabilísticas, temos que BPP é equivalente à classe de problemas definíveis por uma versão randômica da lógica de ponto-fixo infracionário com contagem BPIFP(C). Neste artigo, nós mostramos que podemos definir lógicas com quantificadores generalizados de segunda ordem equivalentes à classes de complexidade probabilísticas. Estes quantificadores são usados para simular o comportamento de máquinas de Turing probabilísticas.  

Semantics of templates in a compositional framework for building logics

2015 ◽  
Vol 15 (4-5) ◽  
pp. 681-695 ◽  
Author(s):  
INGMAR DASSEVILLE ◽  
MATTHIAS VAN DER HALLEN ◽  
GERDA JANSSENS ◽  
MARC DENECKER
Keyword(s):  
Second Order ◽  
Alternative View ◽  
Specification Languages ◽  
Descriptive Complexity ◽  
Base Logic

AbstractThere is a growing need for abstractions in logic specification languages such as FO(·) and ASP. One technique to achieve these abstractions are templates (sometimes called macros). While the semantics of templates are virtually always described through a syntactical rewriting scheme, we present an alternative view on templates as second order definitions. To extend the existing definition construct of FO(·) to second order, we introduce a powerful compositional framework for defining logics by modular integration of logic constructs specified as pairs of one syntactical and one semantical inductive rule. We use the framework to build a logic of nested second order definitions suitable to express templates. We show that under suitable restrictions, the view of templates as macros is semantically correct and that adding them does not extend the descriptive complexity of the base logic, which is in line with results of existing approaches.

Infinitary logic and admissible sets

1969 ◽  
Vol 34 (2) ◽  
pp. 226-252 ◽  
Author(s):  
Jon Barwise
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Predicate Calculus ◽  
Generalized Quantifiers ◽  
Infinitary Logic ◽  
First Order ◽  
Admissible Sets ◽  
Classical Language ◽  
Second Order Logic ◽  
Order Predicate Calculus

In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is(I) strong enough to express interesting properties not expressible by the classical language, but(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.

Delineating classes of computational complexity via second order theories with weak set existence principles. I

1995 ◽  
Vol 60 (1) ◽  
pp. 103-121 ◽  
Author(s):  
Aleksandar Ignjatović
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Second Order ◽  
Complexity Classes ◽  
Bounded Arithmetic ◽  
Recursive Functions ◽  
Speed Up ◽  
Bounded Complexity ◽  
Polynomial Time Hierarchy ◽  
Comprehension Axiom

AbstractIn this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

scholarly journals Descriptive Complexity, Computational Tractability, and the Logical and Cognitive Foundations of Mathematics

2020 ◽  
Author(s):  
Markus Pantsar
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Theoretical Computer Science ◽  
Human Cognition ◽  
Complexity Classes ◽  
Foundations Of Mathematics ◽  
Descriptive Complexity ◽  
Cognitive Capacities ◽  
Computational Complexity Theory ◽  
First Order

Abstract In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes.

The expressive power of higher-order types or, life without CONS

2001 ◽  
Vol 11 (1) ◽  
pp. 55-94 ◽  
Author(s):  
NEIL D. JONES
Keyword(s):  
Programming Languages ◽  
Expressive Power ◽  
Higher Order ◽  
Second Order ◽  
Complexity Classes ◽  
Functional Program ◽  
Control Structures ◽  
First Order ◽  
Partial Recursive Functions ◽  
Turing Complete

Compare first-order functional programs with higher-order programs allowing functions as function parameters. Can the the first program class solve fewer problems than the second? The answer is no: both classes are Turing complete, meaning that they can compute all partial recursive functions. In particular, higher-order values may be first-order simulated by use of the list constructor ‘cons’ to build function closures. This paper uses complexity theory to prove some expressivity results about small programming languages that are less than Turing complete. Complexity classes of decision problems are used to characterize the expressive power of functional programming language features. An example: second-order programs are more powerful than first-order, since a function f of type [Bool]-〉Bool is computable by a cons-free first-order functional program if and only if f is in PTIME, whereas f is computable by a cons-free second-order program if and only if f is in EXPTIME. Exact characterizations are given for those problems of type [Bool]-〉Bool solvable by programs with several combinations of operations on data: presence or absence of constructors; the order of data values: 0, 1, or higher; and program control structures: general recursion, tail recursion, primitive recursion.

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