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62
Naming and Diagonalization, from Cantor to Gödel to Kleene
 in Logic Journal of the IGPL, 22 pages, and on Gaifman’s website
, 2006
"... Gödel’s incompleteness results apply to formal theories for which syntactic constructs can be given names, in the same language, so that some basic syntactic operations are representable in the theory. It is now customary to derive these results from the fixed point theorem (also known as the reflec ..."
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Gödel’s incompleteness results apply to formal theories for which syntactic constructs can be given names, in the same language, so that some basic syntactic operations are representable in the theory. It is now customary to derive these results from the fixed point theorem (also known as the reflection theorem), which asserts the existence of sentences that “speak about themselves”. Let T be the theory and, for each wff φ, letpφqbe the term that serves as its name. Then the theorem says that, for any wff α(v) (with one free variable), there exists a sentence β for which: T ` β ↔ α(pβq) β is sometimes called the fixed point of α(v). All that is needed for the fixed point theorem is that the diagonal function, the one that maps each φ(v) toφ(p(φ(v)q)), be representable in T. The construction of β is more transparent if we assume that the functions is represented by a term of the language, diag(x). This means that the following holds for each φ(v): T ` diag(pφ(v)q) =pφ(pφ(v)q)q (Here ‘= ’ is the equality sign of the formal language; we use it also to denote equality in our metalanguage.) In other words, we can prove in T, for each particular argument, what the
Agents in Logic Programming
, 1997
"... The objective of this thesis is to explore ways of describing agents in logical theories. The contribution is that the logical theories we build are a generalised form of logic programs. Like normal logic programs, these theories have an intuitive declarative reading and a procedural interpretation ..."
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The objective of this thesis is to explore ways of describing agents in logical theories. The contribution is that the logical theories we build are a generalised form of logic programs. Like normal logic programs, these theories have an intuitive declarative reading and a procedural interpretation to guide the implementation of automatic devices and software. Both human beings and machines can reason about these logical theories. We employ the amalgamation of object and metalogic programs to model notions such as beliefs, goals and agent's "mental" activities. But we also accommodate less usual notions such as reactivity, openness, activation of goals and preference encoding, that have proved to be essential in realistic models of agents. Four logic programming languages to program agent with those features are introduced. We use an eventbased approach to model dynamic universes with changing properties, concurrency and synergistic effects. NOTE: This is a copy of the thesis with si...
Parsimony Hierarchies for Inductive Inference
 Journal of Symbolic Logic
"... Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requi ..."
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Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A limcomputable function is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be "notsonearly" minimal size, e.g., to be within a limcomputable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are limcomputable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of limcomputability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the limcomputable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.
On a generalized notion of mistake bounds
 Information and Computation
"... This paper proposes the use of constructive ordinals as mistake bounds in the online learning model. This approach elegantly generalizes the applicability of the online mistake bound model to learnability analysis of very expressive concept classes like pattern languages, unions of pattern languag ..."
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This paper proposes the use of constructive ordinals as mistake bounds in the online learning model. This approach elegantly generalizes the applicability of the online mistake bound model to learnability analysis of very expressive concept classes like pattern languages, unions of pattern languages, elementary formal systems, and minimal models of logic programs. The main result in the paper shows that the topological property of effective finite bounded thickness is a sufficient condition for online learnability with a certain ordinal mistake bound. An interesting characterization of the online learning model is shown in terms of the identification in the limit framework. It is established that the classes of languages learnable in the online model with a mistake bound of α are exactly the same as the classes of languages learnable in the limit from both positive and negative data by a Popperian, consistent learner with a mind change bound of α. This result nicely builds a bridge between the two models. 1
Process Algebra with FiveValued Conditions
, 1999
"... . We propose a vevalued logic that can be motivated from an algorithmic point of view and from a logical perspective. This logic is combined with process algebra. For process algebra with vevalued logic we present an operational semantics in SOSstyle and a completeness result. Finally, we discuss ..."
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. We propose a vevalued logic that can be motivated from an algorithmic point of view and from a logical perspective. This logic is combined with process algebra. For process algebra with vevalued logic we present an operational semantics in SOSstyle and a completeness result. Finally, we discuss some generalizations. Key words & Phrases: Concurrency, process algebra, manyvalued logic, conditional guard construct, conditional composition. 1991 CR Categories: F.3, F.4.3, I.1. 1 Introduction Assume P is some simple program or algorithm. Then the initial behaviour of if then P else P depends on evaluation of the condition : either it yields an immediate error, or it starts performing P , or it diverges in evaluation of . Note that the second possibility only requires that is either true or false. The following three nonclassical truth values accommodate these intuitions: Meaningless. Typical examples are errors that are detectable during execution such as a typeclash or...
Nontruthfunctional manyvaluedness
 Aspects of Universal Logic
"... Manyvalued logics are standardly defined by logical matrices. They are truthfunctional. In this paper non truthfunctional manyvalued semantics are presented, in a philosophical and mathematical perspective. ..."
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Manyvalued logics are standardly defined by logical matrices. They are truthfunctional. In this paper non truthfunctional manyvalued semantics are presented, in a philosophical and mathematical perspective.
The theory of the metarecursively enumerable degrees
"... Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equival ..."
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Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK
Quaternary VoltageMode Logic Cells and FixedPoint Multiplication Circuits
, 2010
"... Fixedpoint multiplication architectures are designed and evaluated using a set of logic cells based on a radix4, quaternary number system. The library of logic circuits is based on Field Effect Transistors (FETs) that have different voltage threshold levels. The resulting logic cell library is suf ..."
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Fixedpoint multiplication architectures are designed and evaluated using a set of logic cells based on a radix4, quaternary number system. The library of logic circuits is based on Field Effect Transistors (FETs) that have different voltage threshold levels. The resulting logic cell library is sufficient to implement all possible quaternary switching functions. The logic circuits operate in voltage mode where different ranges of voltages encode the logic levels. Voltage mode circuitry is used to minimize overall power dissipation characteristics. Analysis of the resulting multiplication circuits indicates that power dissipation characteristics are advantageous when compared to equivalent wordsized binary voltage mode configurations with no decrease in performance.
Infinitary queries in spatial databases
, 2007
"... We describe the use of infinitary logics computable over the real numbers (i.e. in the sense of Blum–Shub–Smale, with fullprecision arithmetic) as a constraint query language for spatial databases. We give a characterization of the sets definable in various syntactic classes corresponding to the cl ..."
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We describe the use of infinitary logics computable over the real numbers (i.e. in the sense of Blum–Shub–Smale, with fullprecision arithmetic) as a constraint query language for spatial databases. We give a characterization of the sets definable in various syntactic classes corresponding to the classical hyperarithmetical hierarchy. 1
Probabilistic inductive inference: a survey
 Theoretical Computer Science
, 2001
"... Inductive inference is a recursiontheoretic theory of learning, first developed by E. M. Gold (1967). This paper surveys developments in probabilistic inductive inference. We mainly focus on finite inference of recursive functions, since this simple paradigm has produced the most interesting (and m ..."
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Inductive inference is a recursiontheoretic theory of learning, first developed by E. M. Gold (1967). This paper surveys developments in probabilistic inductive inference. We mainly focus on finite inference of recursive functions, since this simple paradigm has produced the most interesting (and most complex) results. 1