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InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schem ..."
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Cited by 821 (23 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
EQUIVALENCES AND TRANSFORMATIONS OF REGULAR SYSTEMS  APPLICATIONS TO RECURSIVE PROGRAM SCHEMES AND GRAMMARS
, 1986
"... This work presents a unified theory of recursive program schemes, contextfree grammars, grammars on arbitrary algebraic structures and, in fact, recursive definitions of all kind by means of regular systems. The equivalences of regular systems associated with either all their solutions or their le ..."
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Cited by 29 (5 self)
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This work presents a unified theory of recursive program schemes, contextfree grammars, grammars on arbitrary algebraic structures and, in fact, recursive definitions of all kind by means of regular systems. The equivalences of regular systems associated with either all their solutions or their least solutions (in all domains of appropriate type satisfying a set of algebraic laws expressed by equations) are systematically investigated and characterized (in some cases) in terms of system transformations by folding, unfolding and rewriting according to the equational algebraic laws. Grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union, and the same questions are raised for them. We also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system. This theory applies to grammars of many kinds which generate trees, graphs, etc. We formulate some classical transformations of contextfree grammars in terms of correct transformations which only use folding, unfolding and algebraic laws and we immediately obtain their correctness.
From lambda calculus to universal algebra and back
 33rd International Symposium on Mathematical Foundations of Computer Science, LNCS
, 2008
"... We generalize to universal algebra concepts originating from lambda calculus and programming in order first to prove a new result on the lattice of λtheories, and second a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. The interest ..."
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We generalize to universal algebra concepts originating from lambda calculus and programming in order first to prove a new result on the lattice of λtheories, and second a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. The interest of a systematic study of the lattice λT of λtheories grows out of several open problems on lambda calculus. For example, the failure of certain lattice identities in λT would imply that the problem of the orderincompleteness of lambda calculus raised by Selinger has a negative answer. In this paper we introduce the class of Church algebras (which includes all Boolean algebras, combinatory algebras, rings with unit and the term algebras of all λtheories) to model the ifthenelse instruction of programming and to extend some properties of Boolean algebras to general universal algebras. The interest of Church algebras is that each has a Boolean algebra of central elements, which play the role of the idempotent elements in rings. Central elements are the key tool to represent any Church algebra as a weak Boolean product of directly indecomposable Church algebras and to prove the meta representation theorem mentioned above. We generalize the notion of easy λterm and prove that any Church algebra with an “easy set ” of cardinality n admits (at the top) a lattice interval of congruences isomorphic to the free Boolean algebra with n generators. This theorem has the following consequence for λT: for every recursively enumerable λtheory φ and each n, there is a λtheory φn ≥ φ such that {ψ: ψ ≥ φn} “is ” the Boolean lattice with 2 n elements. 1.
THE SEMANTICS OF DESTRUCTIVE LISPCSLI Lecture Notes Number 5 THE SEMANTICS OF DESTRUCTIVE LISP
"... AND INFORMATIONCSLI was founded early in 1983 by researchers from Stanford University, SRI International, and Xerox PARC to further research and development of integrated theories of language, information, and computation. CSLI headquarters and the publication offices are located at the Stanford sit ..."
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AND INFORMATIONCSLI was founded early in 1983 by researchers from Stanford University, SRI International, and Xerox PARC to further research and development of integrated theories of language, information, and computation. CSLI headquarters and the publication offices are located at the Stanford site.
KJMS LINEAR REDUCTION OF FIRSTORDER LOGIC TO THE IFTHENELSE EQUATIONAL LOGIC
"... Abstract. We show that Hilbert type proof systems for classical firstorder logic can be reduced to ifthenelse equational logic without losing any substantial efficiency: i.e., for any such proof system H, there exist constants k1, k2> 0 such that for any proof ψ with size ℓ in H, there exists a ..."
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Abstract. We show that Hilbert type proof systems for classical firstorder logic can be reduced to ifthenelse equational logic without losing any substantial efficiency: i.e., for any such proof system H, there exist constants k1, k2> 0 such that for any proof ψ with size ℓ in H, there exists a corresponding proof ¯ ψ in the ifthenelse equational proof system, denoted ITE, with size less than or equal to k1 · ℓ + k2. Moreover, the translation ψ ↦ → ¯ ψ is algorithmic, where the number of steps required in the translation is asymptotically bounded by a linear function of size of ψ. 1.
Which TwoSorted Algebras of Booleans and Naturals have a Finite Basis?
"... Abstract. We show that the twosorted algebra of Booleans and naturals with conjunction, addition and inequality is not finitely based. If addition is removed, or negation is included, then the resulting algebra is finitely based. 1. ..."
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Abstract. We show that the twosorted algebra of Booleans and naturals with conjunction, addition and inequality is not finitely based. If addition is removed, or negation is included, then the resulting algebra is finitely based. 1.
Proposition Algebra with Projective Limits
, 807
"... Sequential logic deviates from propositional logic by taking into account that atomic propositions yield different Boolean values at different times during the sequential evaluation of a single proposition. Reactive valuations capture this dynamics of a proposition’s environment. This logic is phras ..."
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Sequential logic deviates from propositional logic by taking into account that atomic propositions yield different Boolean values at different times during the sequential evaluation of a single proposition. Reactive valuations capture this dynamics of a proposition’s environment. This logic is phrased as an equationally specified algebra rather than in the form of proof rules. It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The proposition algebra PRA is developed in a fashion similar to the process algebra ACP and the program algebra PGA via an algebraic specification which has a meaningful initial algebra for which a range of courser congruences are considered important as well. In addition infinite objects (that is propositions, processes and programs respectively) are preferably dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts about infinite ones while reducing all facts about infinite objects to an infinity of facts about finite ones in return. 1
A Rewrite System for Strongly Normalizable Terms
"... In a 2012 paper, Richard Statman exhibited an inference system, based on second order monadic logic and nonterminating rewrite rules, that exactly types all strongly normalizable lambdaterms. In this paper, we show that this system can be simplified to firstorder minimal logic with rewrite rules, ..."
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In a 2012 paper, Richard Statman exhibited an inference system, based on second order monadic logic and nonterminating rewrite rules, that exactly types all strongly normalizable lambdaterms. In this paper, we show that this system can be simplified to firstorder minimal logic with rewrite rules, along the Deduction modulo lines. We show that our rewrite system is terminating and that the conversion rule respects weak versions of invertibility of the arrow and of quantifiers. This requires additional care, in particular in the treatment of the latter. Then we study proof reduction, and show that every typable proof term is strongly normalizable and viceversa.