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QuasiBoolean Encodings and Conditionals in Algebraic Specification
"... We develop a general study of the algebraic specification practice, originating from the OBJ tradition, which encodes atomic sentences in logical specification languages as Boolean terms. This practice originally motivated by operational aspects, but also leading to significant increase in expressiv ..."
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Cited by 3 (3 self)
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We develop a general study of the algebraic specification practice, originating from the OBJ tradition, which encodes atomic sentences in logical specification languages as Boolean terms. This practice originally motivated by operational aspects, but also leading to significant increase in expressivity power, has recently become important within the context of some formal verification methodologies mainly because it allows the use of simple equational reasoning for frameworks based on logics that do not have an equational nature. Our development includes a generic rigorous definition of the logics underlying the above mentioned practice, based on the novel concept of ‘quasiBoolean encoding’, a general result on existence of initial semantics for these logics, and presents a general method for employing Birkhoff calculus of conditional equations as a sound calculus for these logics. The high level of generality of our study means that the concepts are introduced and the results are obtained at the level of abstract institutions (in the sense of Goguen and Burstall [12]) and are therefore applicable to a multitude of logical systems and environments.
Borrowing Interpolation
, 2011
"... We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a ..."
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We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a mathematical concept of ‘homomorphism’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here.
Author manuscript, published in "Computer Science Logic, Czech Republic (2010)" DOI: 10.1007/9783642152054_37 Untyping Typed Algebraic Structures and Colouring Proof Nets of Cyclic Linear Logic ⋆
, 2010
"... Abstract. We prove “untyping ” theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for ..."
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Abstract. We prove “untyping ” theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to the typed settings. Some of these theorems are obtained via a detour through fragments of cyclic linear logic, and give rise to a substantial optimisation of standard proof search algorithms. 1