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Notes on Polynomially Bounded Arithmetic
"... We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polyno ..."
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We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polynomially bounded hierarchy. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2 The axioms of secondorder bounded arithmetic. : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.3 Rudimentary functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.4 Other fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.5 Polynomial time computable functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.6 Relations among fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.7 Relations with Buss' bounded arithmetic. : : : :...
A simple proof of Parsons' theorem
"... Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is ..."
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Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the of universal theories. We give a selfcontained proof requiring only basic knowledge of mathematical logic.
Unfolding finitist arithmetic
, 2010
"... The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significan ..."
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Cited by 3 (3 self)
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The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significance was previously carried out for a system of nonfinitist arithmetic, NFA; it was shown that U(NFA) is prooftheoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system of finitist arithmetic, FA, and for an extension of that by a form BR of the socalled Bar Rule. It is shown that U(FA) and U(FA + BR) are prooftheoretically equivalent, respectively, to Primitive Recursive Arithmetic, PRA, and to Peano Arithmetic, PA.
Theorems of Péter and Parsons in Computer Programming
 Proceedings of CSL'98, number 1584 in LNCS
, 1999
"... This paper describes principles behind a declarative programming language CL (Clausal Language) which comes with its own proof system for proving properties of defined functions and predicates. We use our own implementation of CL in three courses in the first and second years of undergraduate study. ..."
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This paper describes principles behind a declarative programming language CL (Clausal Language) which comes with its own proof system for proving properties of defined functions and predicates. We use our own implementation of CL in three courses in the first and second years of undergraduate study. By unifying the domain of LISP's Sexpressions with the domain N of natural numbers we have combined the LISPlike simplicity of coding with the simplicity of semantics. We deal just with functions over N within the framework of formal Peano arithmetic. We believe that most of the time this is as much as is needed. CL is thus an extremely simple language which is completely based in mathematics.
Primitive Recursive Selection Functions for Existential Assertions over Abstract Algebras
"... Abstract. We generalize to abstract manysorted algebras the classical prooftheoretic result due to Parsons, Mints and Takeuti that an assertion ∀x ∃y P(x,y) (where P is Σ0 1), provable in Peano arithmetic with Σ0 1 induction, has a primitive recursive selection function. This involves a correspond ..."
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Abstract. We generalize to abstract manysorted algebras the classical prooftheoretic result due to Parsons, Mints and Takeuti that an assertion ∀x ∃y P(x,y) (where P is Σ0 1), provable in Peano arithmetic with Σ0 1 induction, has a primitive recursive selection function. This involves a corresponding generalization to such algebras of the notion of primitive recursiveness. The main difficulty encountered in carrying out this generalization turns out to be the fact that equality over these algebras may not be computable, and hence atomic formulae in their signatures may not be decidable. The solution given here is to develop an appropriate concept of realizability of existential assertions over such algebras, generalized to realizability of sequents of existential assertions. In this way, the results can be seen to hold for classical proof systems. This investigation may give some insight into the relationship between specifiability and computability for data types such as the reals, where the atomic formulae, i.e., equations between terms of type real, are not computable. Key words and phrases: generalized computability, realizability, selection function2 1
Abstract Syntactic Finitism in the Metatheory of Programming Languages
, 2010
"... One of the central goals of programminglanguage research is to develop mathematically sound formal methods for precisely specifying and reasoning about the behavior of programs. However, just as software developers sometimes make mistakes when programming, researchers sometimes make mistakes when p ..."
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One of the central goals of programminglanguage research is to develop mathematically sound formal methods for precisely specifying and reasoning about the behavior of programs. However, just as software developers sometimes make mistakes when programming, researchers sometimes make mistakes when proving that a formal method is mathematically sound. As the field of programminglanguage research has grown, these proofs have become larger and more complex, and thus harder to verify on paper. This phenomenon has motivated a great deal of research into the development of logical systems that provide an automated means to apply— and verify the application of—trusted reasoning principles to concrete proofs. The boundary between trusted and untrusted reasoning principles is inherently blurry, and different researchers draw the line in different places. However, just as certain principles are widely recognized to allow the proofs of contradictory statements, others are so uncontroversially ubiquitous in practice that they can be considered beyond reproach. We posit the following questions: (1) what are these principles and (2) how much can we do with them?