On the Algebraization of Many-Sorted Logics

Caleiro, Carlos; Gonçalves, Ricardo

doi:10.1007/978-3-540-71998-4_2

Cited by 5 publications

(6 citation statements)

References 18 publications

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“…With respect to many-sortedness, some work has already been presented in [7]. Our aim here is to go further ahead and to capture also logics that are not algebraizable in the standard sense (although they still seem to be sufficiently well behaved to be studied in algebraic terms).…”

Section: Behavioral Algebraizationmentioning

confidence: 99%

“…We know that we have a distinguished sort φ of formulas. In the many-sorted approach to AAL presented in [7] the theory was developed by replacing the role of unsorted equational logic by many-sorted behavioral logic over the same signature and taking the sort φ as the unique visible sort. Despite the success of this generalization to cope with manysorted logics, a lot of non-algebraizable logics could still not be captured.…”

Section: Behavioral Algebraizationmentioning

confidence: 99%

“…As we will show later, if we take Γ = Σ we get ordinary many-sorted algebraization. For details about many-sorted algebraization we point to [7]. Furthermore, when φ is the only sort we get in the realm of the standard tools of AAL.…”

Section: Behavioral Algebraizationmentioning

confidence: 99%

“…Many-sorted algebraization, as introduced in [7], is just a particular case of the behavioral approach, again obtained by taking Γ = Σ.…”

Section: Standard Algebraizationmentioning

confidence: 99%

“…Having this in mind it should be natural to consider a many-sorted framework to study the algebraizability of first-order logic. We point to [7] for a many-sorted approach to the algebraization of FOL and recall that, as explained in Example 3.3.1, our behavioral approach is a generalization of this many-sorted approach to algebraizability. The idea is simply to use our many-sorted framework to handle first-order logic as a two-sorted logic, with a sort for terms and a sort for formulas.…”

Section: First-order Logicmentioning

confidence: 99%

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Behavioral Algebraization of Logics

2009

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Abstract. We introduce and study a new approach to the theory of abstract algebraic logic (AAL) that explores the use of many-sorted behavioral logic in the role traditionally played by unsorted equational logic. Our aim is to extend the range of applicability of AAL towards providing a meaningful algebraic counterpart also to logics with a manysorted language, and possibly including non-truth-functional connectives. The proposed behavioral approach covers logics which are not algebraizable according to the standard approach, while also bringing a new algebraic perspective to logics which are algebraizable using the standard tools of AAL. Furthermore, we pave the way towards a robust behavioral theory of AAL, namely by providing a behavioral version of the Leibniz operator which allows us to generalize the traditional Leibniz hierarchy, as well as several well-known characterization results. A number of meaningful examples will be used to illustrate the novelties and advantages of the approach.Keywords: Abstract algebraic logic, many-sorted behavioral logic, non-truth-functionality. IntroductionThe general theory of abstract algebraic logic (AAL, from now on) was first introduced in [3] with the aim of extending the so-called Lindenbaum-Tarski method, as used for instance to establish the relationship between classical propositional logic and Boolean algebras, to the systematic study of the connection between a given logic and a suitable equational theory. This connection enables one to use the powerful tools of universal algebra to study the metalogical properties of the logic being algebraized, namely with respect to its axiomatizability, definability aspects, the deduction theorem, or interpolation properties [11,13]. Despite of its success, the scope of applicability of the standard tools of AAL is relatively limited. Logics with a many-sorted language, even if well behaved, are good examples of logics that fall out of their scope. It goes without saying that rich logics, with many-sorted languages, are essential to specify and reason about complex systems, as also argued and justified by the theory of combined logics [40]. However, even in the case of propositional based (single-sorted) logics, many interesting examples simply fall out of the scope of the standard tools of AAL. With the proliferation of logical systems, with applications ranging from computer science, to mathematics and philosophy, the examples of nonalgebraizable logics that, therefore, lack from a meaningful and insightful Studia Logica (2008) 0: 1-49

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Section: Behavioral Algebraizationmentioning

confidence: 99%

Section: Behavioral Algebraizationmentioning

confidence: 99%

Section: Behavioral Algebraizationmentioning

confidence: 99%

“…Many-sorted algebraization, as introduced in [7], is just a particular case of the behavioral approach, again obtained by taking Γ = Σ.…”

Section: Standard Algebraizationmentioning

confidence: 99%

Section: First-order Logicmentioning

confidence: 99%

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Behavioral Algebraization of Logics

2009

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Certifying Safety and Termination Proofs for Integer Transition Systems

Brockschmidt

Joosten

Thiemann

et al. 2017

Automated Deduction – CADE 26

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Modern program analyzers translate imperative programs to an intermediate formal language like integer transition systems (ITSs), and then analyze properties of ITSs. Because of the high complexity of the task, a number of incorrect proofs are revealed annually in the Software Verification Competitions. In this paper, we establish the trustworthiness of termination and safety proofs for ITSs. To this end we extend our Isabelle/HOL formalization IsaFoR by formalizing several verification techniques for ITSs, such as invariant checking, ranking functions, etc. Consequently the extracted certifier CeTA can now (in)validate safety and termination proofs for ITSs. We also adapted the program analyzers T2 and AProVE to produce machine-readable proof certificates, and as a result, most termination proofs generated by these tools on a standard benchmark set are now certified.

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Categorical Abstract Algebraic Logic: Behavioral π-Institutions

Voutsadakis

2014

Stud Logica

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On the Algebraization of Many-Sorted Logics

Cited by 5 publications

References 18 publications

Behavioral Algebraization of Logics

Behavioral Algebraization of Logics

Certifying Safety and Termination Proofs for Integer Transition Systems

Categorical Abstract Algebraic Logic: Behavioral π-Institutions

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