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Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 183 (21 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
Representation and Inference for Natural language  A First Course in . . .
, 1999
"... 3.672> X with the complex term 1 + 1, not with 2, which, for people unused to Prolog's little ways, tends to come as a bit of a surprise. If we want to carry out the actual arithmetic involved, we have to explicitly force evaluation by making use of the very special inbuilt `operator' is/2. This ca ..."
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Cited by 86 (11 self)
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3.672> X with the complex term 1 + 1, not with 2, which, for people unused to Prolog's little ways, tends to come as a bit of a surprise. If we want to carry out the actual arithmetic involved, we have to explicitly force evaluation by making use of the very special inbuilt `operator' is/2. This calls an inbuilt mechanism which carries out the arithmetic evaluation of its second argument, and then unication plays no role here!). On the other hand, \== checks whether its argument are not identical. Arithmetic Prolog contains some builtin operators for handling integer arithmetic. These include *, / +,  (for multiplication, division, addition, and subtraction, respectively) and >, < for comparing numbers. These symbols, however, are just ordinary Prolog operators. That is, they are just a user friendly notation for writing
Computational types from a logical perspective
 Journal of Functional Programming
, 1998
"... Moggi’s computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus ..."
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Cited by 54 (6 self)
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Moggi’s computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus also arises naturally as the term calculus corresponding (by the CurryHoward correspondence) to a novel intuitionistic modal propositional logic. We give natural deduction, sequent calculus and Hilbertstyle presentations of this logic and prove strong normalisation and confluence results. 1
Canonical Propositional GentzenType Systems
 in Proceedings of the 1st International Joint Conference on Automated Reasoning (IJCAR 2001) (R. Goré, A Leitsch, T. Nipkow, Eds), LNAI 2083
, 2001
"... . Canonical propositional Gentzentype systems are systems which in addition to the standard axioms and structural rules have only pure logical rules which have the subformula property, introduce exactly one occurrence of a connective in their conclusion, and no other occurrence of any connectiv ..."
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Cited by 29 (16 self)
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. Canonical propositional Gentzentype systems are systems which in addition to the standard axioms and structural rules have only pure logical rules which have the subformula property, introduce exactly one occurrence of a connective in their conclusion, and no other occurrence of any connective is mentioned anywhere else in their formulation. We provide a constructive coherence criterion for the nontriviality of such systems, and show that a system of this kind admits cut elimination i it is coherent. We show also that the semantics of such systems is provided by nondeterministic twovalued matrices (2Nmatrices). 2Nmatrices form a natural generalization of the classical twovalued matrix, and every coherent canonical system is sound and complete for one of them. Conversely, with any 2Nmatrix it is possible to associate a coherent canonical Gentzentype system which has for each connective at most one introduction rule for each side, and is sound and complete for th...
A Computational Model of Belief
, 2000
"... We propose a logic of belief in which the expansion of beliefs beyond what has been explicitly learned is modeled as a finite computational process. The logic does not impose a particular computational mechanism; rather, the mechanism is a parameter of the logic, and we show that as long as the mech ..."
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Cited by 15 (5 self)
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We propose a logic of belief in which the expansion of beliefs beyond what has been explicitly learned is modeled as a finite computational process. The logic does not impose a particular computational mechanism; rather, the mechanism is a parameter of the logic, and we show that as long as the mechanism meets a particular set of constraints, the resulting logic has certain desirable properties. Chief among these is the property that one can reason soundly about another agent's beliefs by simulating its computational mechanism with one's own. We also give a detailed comparison of our model with Konolige's deduction model, another model of belief in which the believer's reasoning mechanism is a parameter. 2000 Elsevier Science B.V. All rights reserved. Keywords: Belief; Belief inference; Belief ascription; Omniscience; ASK/TELL mechanism; Simulative reasoning; Computational model of belief; Deduction model of belief 1. Introduction If an observer sees dark clouds and hears thunder,...
A formal calculus for informal equality with binding
 In WoLLIC’07: 14th Workshop on Logic, Language, Information and Computation, volume 4576 of LNCS
, 2007
"... Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing captureavoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along w ..."
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Cited by 13 (2 self)
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Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing captureavoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along with a direct formalisation of their constraints. We base our logic on equality, probably the simplest available judgement form. In spite of this, we can axiomatise systems of logic and computation such as firstorder logic or the lambdacalculus in a very direct and natural way. We investigate the theory of derivations, prove a suitable semantics sound and complete, and discuss existing and future research. 1
A.: Nominal algebra
, 2006
"... Abstract. Nominal terms are a termlanguage used to accurately and expressively represent systems with binding. We present Nominal Algebra (NA), a theory of algebraic equality on nominal terms. Builtin support for binding in the presence of metavariables allows NA to closely mirror informal mathem ..."
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Cited by 7 (2 self)
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Abstract. Nominal terms are a termlanguage used to accurately and expressively represent systems with binding. We present Nominal Algebra (NA), a theory of algebraic equality on nominal terms. Builtin support for binding in the presence of metavariables allows NA to closely mirror informal mathematical usage and notation, where expressions such as λa.t or ∀a.φ are common, in which metavariables t and φ explicitly occur in the scope of a variable a. We describe the syntax and semantics of NA, and provide a sound and complete proof system for it. We also give some examples of axioms; other work has considered sets of axioms of particular interest in some detail. 1.
Basic Quantifier Theory
"... ing from the domain of discourse, we can say that determiner interpretations (henceforth: determiners) pick out binary relations on sets of individuals, on arbitrary universes (or: domains of discourse) E. Notation: DEAB. We call A the restriction of the quantifier and B its body. If DEAB is the tra ..."
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Cited by 4 (0 self)
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ing from the domain of discourse, we can say that determiner interpretations (henceforth: determiners) pick out binary relations on sets of individuals, on arbitrary universes (or: domains of discourse) E. Notation: DEAB. We call A the restriction of the quantifier and B its body. If DEAB is the translation of a simple sentence consisting of a quantified noun phrase with an intransitive verb phrase then the noun denotation is the restriction and the verb phrase denotation the body. See figure 3 for a graphical representation. E &% '$ A &% '$ B FIGURE 3 Quantifiers as Relations A simple binary quantifier D on a domain E is a relation between subsets of E: DE 2 ((E) \Theta (E)) The trivial quantifiers are ?E and ?E , which hold of all and of no pairs of sets, respectively. Not all elements in ((E)\Theta(E)) serve as natural language determiner denotations. In fact, one of the first insights provided by quantification Basic Quantifier Theory / 7 theory is that such determiners hav...
Simulative Inference in a Computational Model of Belief
 Muskens (Eds.), Computing Meaning, Studies in Linguistics and Philosophy, Kluwer
, 1997
"... We propose a semantics for belief in which the derivation of new beliefs from old ones is modeled as a computational process. Using this model, we characterize conditions under which it is appropriate to reason about other agents by simulating their inference processes with one's own. Keywords: bel ..."
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Cited by 4 (3 self)
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We propose a semantics for belief in which the derivation of new beliefs from old ones is modeled as a computational process. Using this model, we characterize conditions under which it is appropriate to reason about other agents by simulating their inference processes with one's own. Keywords: belief, simulative inference, computational model of belief 1 Introduction If I see a glass begin to fall from a shelf, I can infer that the glass will probably break, by applying pieces of world knowledge such as: 1. When things fall, they collide with whatever is below them. 2. Glasses are fragile; floors are hard. 3. When a fragile thing and a hard thing collide, the fragile thing often breaks. This paper appeared in the Second International Workshop on Computational Semantics, Tilburg, The Netherlands, January 1997. Any citations should be to the published version. y This work was supported in part by ARPA grant F306029510025 and NSF grant IRI9503312. Rajesh Rao contributed severa...