By Mauricio Ayala-Rincón, Flávio L. C. de Moura
This publication offers an creation to common sense and mathematical induction that are the foundation of any deductive computational framework. a robust mathematical beginning of the logical engines on hand in glossy facts assistants, resembling the PVS verification process, is key for machine scientists, mathematicians and engineers to increment their services to supply formal proofs of theorems and to certify the robustness of software program and structures.
The authors current a concise evaluation of the mandatory computational and mathematical facets of ‘logic’, putting emphasis on either typical deduction and sequent calculus. adjustments among optimistic and classical common sense are highlighted via a number of examples and routines. with out neglecting classical points of computational good judgment, the authors additionally spotlight the connections among logical deduction principles and facts instructions in evidence assistants, proposing basic examples of formalizations of the correctness of algebraic capabilities and algorithms in PVS.
Applied good judgment for laptop Scientists won't in simple terms profit scholars of desktop technological know-how and arithmetic but additionally software program, undefined, automation, electric and mechatronic engineers who're attracted to the applying of formal tools and the similar computational instruments to supply mathematical certificate of the standard and accuracy in their items and technologies.
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Additional info for Applied Logic for Computer Scientists. Computational Deduction and Formal Proofs
X ) means “for all x” (resp. “there exists a x”), and the formula ϕ is the body of the formula (∀x ϕ) (resp. (∃x ϕ)). Since quantification is restricted to variable terms, the defined language corresponds to a so-called firstorder language. The set of formulas of the predicate logic have the following syntax: ϕ:: = p(t, . . , t) || ⊥ || || (¬ϕ) || (ϕ ∧ ϕ) || (ϕ ∨ ϕ) || (ϕ → ϕ) || (∀x ϕ) || (∃x ϕ) Formulas of the form p(t1 ,. , tn ) are called atomic formulas because they cannot be decomposed into simpler formulas.
T) || ⊥ || || (¬ϕ) || (ϕ ∧ ϕ) || (ϕ ∨ ϕ) || (ϕ → ϕ) || (∀x ϕ) || (∃x ϕ) Formulas of the form p(t1 ,. , tn ) are called atomic formulas because they cannot be decomposed into simpler formulas. As usual, parenthesis are used to avoid ambiguities and the external ones will be omitted. The quantifiers ∀x and ∃x bind the variable x in the body of the formula. This idea is formalized by the notion of scope of a quantifier: Definition 17 (Scope of quantifiers, free and bound variables) The scope of ∀x (resp.
In order to overcome these limitations of the expressive power of the propositional logic, we extend its language with variables which range over individuals, and quantification over these variables. Thus, in this chapter we present the predicate logic, also known as first-order logic. In order to obtain a language with abilities to identify the required additional information, we need to extend the propositional language and provide a more expressive deductive calculus. 2 Syntax of the Predicate Logic The language of the first-order predicate logic has two kinds of expressions: terms and formulas.