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GraphBased Proof Counting and Enumeration with Applications for Program Fragment Synthesis
 in &quot;International Symposium on Logicbased Program Synthesis and Transformation 2004 (LOPSTR 2004)&quot;, S. ETALLE (editor)., Lecture Notes in Computer Science
, 2004
"... Abstract. For use in earlier approaches to automated module interface adaptation, we seek a restricted form of program synthesis. Given some typing assumptions and a desired result type, we wish to automatically build a number of program fragments of this chosen typing, using functions and values av ..."
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Abstract. For use in earlier approaches to automated module interface adaptation, we seek a restricted form of program synthesis. Given some typing assumptions and a desired result type, we wish to automatically build a number of program fragments of this chosen typing, using functions and values available in the given typing environment. We call this problem term enumeration. To solve the problem, we use the CurryHoward correspondence (propositionsastypes, proofsasprograms) to transform it into a proof enumeration problem for an intuitionistic logic calculus. We formally study proof enumeration and counting in this calculus. We prove that proof counting is solvable and give an algorithm to solve it. This in turn yields a proof enumeration algorithm. 1
Parallel Reduction in Type Free λµCalculus
 KYUSHU UNIVERSITY
, 2000
"... Typed λµcalculus is known to be strongly normalizing and weakly ChurchRosser, and hence confluent. In fact, Parigot formulated a parallel reduction to prove confluency of typed λµcalculus by "TaitandMartinLöf" method. However, the diamond property does not hold for his parallel reduc ..."
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Typed λµcalculus is known to be strongly normalizing and weakly ChurchRosser, and hence confluent. In fact, Parigot formulated a parallel reduction to prove confluency of typed λµcalculus by "TaitandMartinLöf" method. However, the diamond property does not hold for his parallel reduction. The confluency for typefree λµcalculus cannot be derived from that of typed λµcalculus and is not known. We analyzed granualities of the reduction rules. We consider a renaming and consecutive structural reductions as one step parallel reduction, and show that the new formulation of parallel reduction has the diamond property, which yields the correct proof of confluency of type free λµcalculus. The diamond property of new parallel reduction is also shown for the callbyvalue version of λµcalculus contains the symmetric structural reduction rule.
Strict Intersection Types for the Lambda Calculus
, 2010
"... This paper will show the usefulness and elegance of strict intersection types for the Lambda Calculus; these are strict in the sense that they are the representatives of equivalence classes of types in the BCDsystem [15]. We will focus on the essential intersection type assignment; this system is a ..."
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This paper will show the usefulness and elegance of strict intersection types for the Lambda Calculus; these are strict in the sense that they are the representatives of equivalence classes of types in the BCDsystem [15]. We will focus on the essential intersection type assignment; this system is almost syntax directed, and we will show that all major properties hold that are known to hold for other intersection systems, like the approximation theorem, the characterisation of (head/strong) normalisation, completeness of type assignment using filter semantics, strong normalisation for cutelimination and the principal pair property. In part, the proofs for these properties are new; we will briefly compare the essential system with other existing systems.
The gamut of dynamic logics
 Handbook of the History of Logic
, 2006
"... Dynamic logic, broadly conceived, is the logic that analyses change by decomposing actions into their basic building blocks and by describing the results of performing actions in given states of the world. The actions studied by dynamic logic can be of various kinds: actions on the memory state of a ..."
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Dynamic logic, broadly conceived, is the logic that analyses change by decomposing actions into their basic building blocks and by describing the results of performing actions in given states of the world. The actions studied by dynamic logic can be of various kinds: actions on the memory state of a computer, actions of a moving robot in a closed world, interactions between cognitive agents performing given communication protocols, actions that change the common ground between speaker and hearer in a conversation, actions that change the contextually available referents in a conversation, and so on. In each of these application areas, dynamic logics can be used to model the states involved and the transitions that occur between them. Dynamic logic is a tool for both state description and action description. Formulae describe states, while actions or programs express state change. The levels of state descriptions and transition characterisations are connected by suitable operations that allow reasoning about pre and postconditions of particular changes.
Polymorphic type inference for the Relational Algebra
 Journal of Computer and System Sciences
, 2002
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SecondOrder Abstract Categorial Grammars as Hyperedge Replacement Grammars
"... Abstract. Secondorder abstract categorial grammars (de Groote 2001) and hyperedge replacement grammars (see Engelfriet 1997) are two natural ways of generalizing “contextfree ” grammar formalisms for string and tree languages. It is known that the string generating power of both formalisms is equi ..."
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Abstract. Secondorder abstract categorial grammars (de Groote 2001) and hyperedge replacement grammars (see Engelfriet 1997) are two natural ways of generalizing “contextfree ” grammar formalisms for string and tree languages. It is known that the string generating power of both formalisms is equivalent to (nonerasing) multiple contextfree grammars (Seki et al. 1991) or linear contextfree rewriting systems (Weir 1988). In this paper, we give a simple, direct proof of the fact that secondorder ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for treegenerating hyperedge replacement grammars given by Engelfriet and Maneth (2000) can then be used to show that the tree generating power of secondorder ACGs is exactly the same as that of hyperedge replacement grammars. 1
doi:10.1093/comjnl/bxn029 Enumerating Proofs of Positive Formulae
, 2007
"... We provide a semigrammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This method is complete in the sense that each normal proofterm ..."
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We provide a semigrammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This method is complete in the sense that each normal proofterm of the formula is produced by some scheme generated by the grammar. As a corollary, we get a similar description of the set of normal proofs of positive formulae for a large class of theories including simple type theory and System F.
Weaker DComplete Logics
 University of Wollongong Department
"... BB 0 IW logic (or T! ) is known to be Dcomplete. This paper shows that there are infinitely many weaker Dcomplete logics and it also examines how certain Dincomplete logics can be made complete by altering their axioms using simple substitutions. Keywords: condensed detachment 1 Introduction ..."
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BB 0 IW logic (or T! ) is known to be Dcomplete. This paper shows that there are infinitely many weaker Dcomplete logics and it also examines how certain Dincomplete logics can be made complete by altering their axioms using simple substitutions. Keywords: condensed detachment 1 Introduction The condensed detachment rule, first proposed by C. A. Meredith in Lemmon et al [5], is a form of modus ponens preceded by `just enough' substitution to make the modus ponens possible. The substitution mechanism, for implicational formulas, was a precursor to Robinson's unification algorithm [8]. Roughly, a system of implicational logic is Dcomplete if the system with the same axioms, but with condensed detachment (D) instead of modus ponens and substitution, has the same theorems. To show that a logic is Dcomplete it is sufficient to show that all the substitution instances of its axioms are deducible in the corresponding condensed logic (i.e. the logic with rule D only). It is well know...
Lambek calculus proofs and tree automata
 Logical Aspects of Computational Linguistics Third International Conference, LACL '98, Selected Papers, volume 2014 of Lecture Notes in Artificial Intelligence
, 2001
"... Abstract. We investigate natural deduction proofs of the Lambek calculus from the point of view of tree automata. The main result is that the set of proofs of the Lambek calculus cannot be accepted by a finite tree automaton. The proof is extended to cover the proofs used by grammars based on the La ..."
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Abstract. We investigate natural deduction proofs of the Lambek calculus from the point of view of tree automata. The main result is that the set of proofs of the Lambek calculus cannot be accepted by a finite tree automaton. The proof is extended to cover the proofs used by grammars based on the Lambek calculus, which typically use only a subset of the set of all proofs. While Lambek grammars can assign regular tree languages as structural descriptions, there exist Lambek grammars that assign nonregular structural descriptions, both when considering normal and nonnormal proof trees. Combining the results of Pentus (1993) and Thatcher (1967), we can conclude that Lambek grammars, although generating only contextfree languages, can extend the strong generative capacity of contextfree grammars. Furthermore, we show that structural descriptions that disregard the use of introduction rules cannot be used for a compositional semantics following the CurryHoward isomorphism. 1