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Formal mathematics for verifiably correct program synthesis
 Journal of the IGPL
, 1996
"... We describe a formalization of the metamathematics of programming in a higherorder logical calculus as a means to create verifiably correct implementations of program synthesis tools. Using reflected notions of programming concepts we can specify the actions of synthesis methods within the object ..."
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Cited by 8 (5 self)
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We describe a formalization of the metamathematics of programming in a higherorder logical calculus as a means to create verifiably correct implementations of program synthesis tools. Using reflected notions of programming concepts we can specify the actions of synthesis methods within the object language of the calculus and prove formal theorems about their behavior. The theorems serve as derived inference rules implementing the kernel of these methods in a flexible, safe, efficient and comprehensible way. We demonstrate the advantages of using formal mathematics in support of program development systems through an example in which we formalize a strategy for deriving global search algorithms from formal specifications.
On the unusual effectiveness of Logic in computer science
 Bulletin of Symbolic Logic
"... Effectiveness of Mathematics in the Natural Sciences [Wig60]. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effecti ..."
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Cited by 7 (0 self)
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Effectiveness of Mathematics in the Natural Sciences [Wig60]. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effectiveness of
iRho: an imperative rewriting calculus
, 2008
"... We propose an imperative version of the Rewriting Calculus, a calculus based on pattern matching, pattern abstraction and side effects, which we call iRho. We formulate both a static and bigstep callbyvalue operational semantics of iRho. The operational semantics is deterministic, and immediately ..."
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Cited by 2 (1 self)
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We propose an imperative version of the Rewriting Calculus, a calculus based on pattern matching, pattern abstraction and side effects, which we call iRho. We formulate both a static and bigstep callbyvalue operational semantics of iRho. The operational semantics is deterministic, and immediately suggests how an interpreter for the calculus may be built. The static semantics is given using a firstorder type system based on a form of product types, which can be assigned to termlike structures (that is, records). The calculus is à la Church, that is, pattern abstractions are decorated with the types of the free variables of the pattern. iRho is a good candidate for the core of a patternmatching imperative language, where a (monomorphic) typed store can be safely manipulated and where fixed points are built into the language itself. Properties such as determinism of the interpreter and subjectreduction have been completely checked using a machineassisted approach with the Coq proof assistant. Progress and decidability of type checking are proved using pen and paper.
U.U.D.M. Report 2008:42 Setoids and universes
"... Abstract. Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with as small universes as possible (and thus, the type theory as weak as possible). Particularly, we will consider epimorphisms and disj ..."
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Abstract. Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with as small universes as possible (and thus, the type theory as weak as possible). Particularly, we will consider epimorphisms and disjoint sums. It is shown that, given the minimal type universe, all epimorphisms are surjections, and disjoint sums exist. Further, without universes, there are countermodels for these statements, and if we use the Logical Framework formulation of type theory, these statements are provably nonderivable. 1.
Machine Assisted Proofs for Generic Semantics to Compiler Transformation Correctness Theorems
"... This thesis investigates the issues involved in the creation of a “general theory of operational semantics ” in LEGO, a typetheoretic theorem proving environment implementing a constructionist logic. Such a general theory permits the ability to manipulate and reason about operational semantics both ..."
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This thesis investigates the issues involved in the creation of a “general theory of operational semantics ” in LEGO, a typetheoretic theorem proving environment implementing a constructionist logic. Such a general theory permits the ability to manipulate and reason about operational semantics both individually and as a class. The motivation for this lies in the studies of semantics directed compiler generation in which a set of generic semantics transforming functions can help convert arbitrary semantic definitions to abstract machines. Such transformations require correctness theorems that quantify over the class of operational semantics. In implementation terms this indicates the need to ensure both the class of operational semantics and the means of inferring results thereon remain at the theorem prover level. The endeavour of this thesis can be seen as assessing both the requirements that general theories of semantics impose on proof assistants and the efficacy of proof assistants in modelling such theories. Acknowledgements First and foremost I would like to thank Kevin Mitchell who supervised me for my first four years, supplying me with many helpful hints and constructive criticisms. He also bore with me at a period of my life when my mental health deteriorated for which I am eternally grateful. Secondly I would like to thank Stuart Anderson an ever present of my life at the University since I first arrived in 1988, for taking over the supervision of my work when it was seemingly near its conclusion. The help and encouragement I received meant I was able to (finally!) complete this thesis. Special mention must go to Rod Burstall, my mentor through the entirety of my postgraduate studies. My all too brief encounters with him lifted my spirits at a time when they were desperately in need of a boost. I would also like to especially thank Thomas Kleymann (formerly Schreiber) for the many times he aided me in my Lego miseries. I also thank James Hugh McKinna, Randy Pollack and other members of the Lego club for their helpful ideas, various helpful officemates
Non determinism through type isomorphism
"... We define an equivalence relation on propositions and a proof system where equivalent propositions have the same proofs. The system obtained this way resembles several known nondeterministic and algebraic lambdacalculi. 1 ..."
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We define an equivalence relation on propositions and a proof system where equivalent propositions have the same proofs. The system obtained this way resembles several known nondeterministic and algebraic lambdacalculi. 1
The dynamics of sense . . .
, 2013
"... This thesis is a both a descriptive and theoretical examination of implicatures, parts of the contextual meanings of utterances that are separate from their sense, their main point. The empirical taxonomy I describe draws on the work of Grice (1975), but fleshes out two important subcategories of im ..."
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This thesis is a both a descriptive and theoretical examination of implicatures, parts of the contextual meanings of utterances that are separate from their sense, their main point. The empirical taxonomy I describe draws on the work of Grice (1975), but fleshes out two important subcategories of implicature that he did not discuss in detail. One of these subcategories is the conventional implicatures, which contains definite anaphora, iterative adverbs, honorifics, and Potts’s (2005) “CIs”: nominal appositives, nonrestrictive relative clauses, asparentheticals, and expressives. The other is the nonconventional implicatures apart from Grice’s conversational implicatures, which contains lexical items often construed as bearing presuppositions, such as socalled factive verbs, aspectuals, and achievements. I offer evidence that what distinguishes the class of anaphora from other conventional implicatures is that the use of a definite must be anchored to the speaker, since it bears the implication that an antecedent is retrievable in the discourse context.
Formal semantics for perceptual classification PREPRINT VERSION
"... A formal semantics for lowlevel perceptual aspects of meaning is presented, tying these together with the logicalinferential aspects of meaning traditionally studied in formal semantics. The key idea is to model perceptual meanings as classifiers of perceptual input. Furthermore, we show how perce ..."
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A formal semantics for lowlevel perceptual aspects of meaning is presented, tying these together with the logicalinferential aspects of meaning traditionally studied in formal semantics. The key idea is to model perceptual meanings as classifiers of perceptual input. Furthermore, we show how perceptual aspects of meaning can be updated as a result of observing language use in interaction, thereby enabling finegrained semantic plasticity and semantic coordination. This requires a framework where intensions are (1) represented independently of extensions, and (2) structured objects which can be modified as a result of learning. We use Type Theory with Records (TTR), a formal semantics framework which starts from the idea that information and meaning is founded on our ability to perceive and classify the world, i.e., to perceive objects and situations as being of types. As an example of our approach, we show how a simple classifier of spatial information based on the Perceptron can be cast in TTR.