Loading

Error: Cannot Load Popup Box

Hit List

Title:

Proof Search in Minimal Logic

Description:

this paper was the necessity to have a guide for our implementation, we have paid particular attention to write at least the parts of the proofs with algorithmic content as clear and complete as possible. The paper is organized as follows. Section 2 defines the pattern unification algorithm, and in section 3 its correctness and completeness is p...

this paper was the necessity to have a guide for our implementation, we have paid particular attention to write at least the parts of the proofs with algorithmic content as clear and complete as possible. The paper is organized as follows. Section 2 defines the pattern unification algorithm, and in section 3 its correctness and completeness is proved. Section 4 presents the proof search algorithm, and again its correctness and completeness is proved. The final section 5 contains what we have to say about extensions to and 9. 2 The unification algorithm Unif We work in the simply typed -calculus, with the usual conventions. For instance, whenever we write a term we assume that it is correctly typed. Substitutions are denoted by '; /; ae. The result of applying a substitution ' to a term t or a formula A is written as t' or A', with the understanding that after the substitution all terms are brought into long normal form. Q always denotes a 898-prefix, say 8~x9~y8~z, with distinct variables. We call ~x the signature variables, ~y the flexible variables and ~z the forbidden variables of Q, and write Q 9 for the existential part 9~y of Q. Q-terms are inductively defined by the following clauses. ffl If u is a universally quantified variable in Q or a constant, and ~r are Q-terms, then u~r is a Q-term. ffl For any flexible variable y and distinct forbidden variables ~z from Q, y~z is a Q-term. ffl If r is a Q8z-term, then zr is a Q-term. Explicitely, r is a Q-term iff all its free variables are in Q, and for every subterm y~r of r with y free in r and flexible in Q, the ~r are distinct variables either -bound in r (such that y~r is in the scope of this ) or else forbidden in Q. Q-goals and Q-clauses are simultaneously defined by ffl If ~r are Q-terms, then P~r is. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-16

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/search00/s.ps

http://www.mathematik.uni-muenchen.de/~schwicht/papers/search00/s.ps Minimize

Document Type:

text

Language:

en

DDC:

005 Computer programming, programs & data *(computed)*

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Title:

Program development by proof transformation

Author:

Description:

We begin by reviewing the natural deduction rules for the!^8-fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. We also consider two alternative methods to dea...

We begin by reviewing the natural deduction rules for the!^8-fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. We also consider two alternative methods to deal with the strong or constructive existential quantifier 9\Lambda. Finally we discuss the well-known notion of an extracted program of a derivation involving 9\Lambda, in order to set up a relation between the two alternatives. Section 2 deals with the computational content of classical proofs. As is well-known a proof of a 89-theorem with a quantifier-free kernel-- where 9 is viewed as defined by:8:-- can be used as a program. We describe a "direct method " to use such a proof as a program, and compare it with Harvey Friedman's A-translation [3] followed by program extraction from the resulting constructive proof. It is shown that both algorithms coincide. In section 3 Goad's method of pruning of proof trees is introduced. It is shown how a proof can be simplified after addition of some further assumptions. In a first step some subproofs are replaced by different ones using the additional assumptions. In a second step parts of the proof tree are pruned, i.e. cut out. Note that the first step involves searching for new proofs-- using the new assumptions-- of formulas in the proof tree. Hence we also have to discuss proof search in minimal logic. Finally section 4 treats an example already considered by Goad in his thesis [5], the binpacking problem. The main difference to Goad's work is that he used a logic with the strong existential quantifier, whereas we work within the!8-fragment. This example is particularly well-suited to demonstrate that the pruning method can be applied to adapt programs to particular situations, and moreover that pruning can change the functions computed by programs. In this sense this method is essentially different from program development by program transformation. We would like to thank Michael Bopp and Karl-Heinz Niggl for their help in preparing these notes. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-08-12

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mod93/mod93.ps

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mod93/mod93.ps Minimize

Document Type:

text

Language:

en

DDC:

005 Computer programming, programs & data *(computed)*

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Title:

Content in Proofs of List Reversal

Description:

Berger [2] observed that the well-known linear list reversal algorithm can be obtained as the computational content of a weak (or “classical”) existence proof. The necessary tools are a refinement [3] of the Dragalin/Friedman [4, 5] A-translation, and uniform (or “non-computational”) quantifiers [1]. Both tools are implemented in the Minlog proo...

Berger [2] observed that the well-known linear list reversal algorithm can be obtained as the computational content of a weak (or “classical”) existence proof. The necessary tools are a refinement [3] of the Dragalin/Friedman [4, 5] A-translation, and uniform (or “non-computational”) quantifiers [1]. Both tools are implemented in the Minlog proof assistant (www.minlog-system. de), in addition to the more standard realizability interpretation. The aim of the present paper is to give an introduction into the theory underlying these tools, and to explain their usage in Minlog, using list reversal as a running example. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-07

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mod07/mod07.pdf

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mod07/mod07.pdf Minimize

Document Type:

text

Language:

en

Subjects:

R τ L(ρ

R τ L(ρ Minimize

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Title:

A Syntactical Analysis of Non-Size-Increasing Polynomial Time Computation

Description:

A purely syntactical proof is given that all functions definable in a certain affine linear typed lambda-calculus with iteration in all types are polynomial time computable. The proof also gives explicit polynomial bounds that can easily be calculated.

A purely syntactical proof is given that all functions definable in a certain affine linear typed lambda-calculus with iteration in all types are polynomial time computable. The proof also gives explicit polynomial bounds that can easily be calculated. Minimize

Publisher:

Society Press

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-10-10

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/lics00/main.ps.Z

http://www.mathematik.uni-muenchen.de/~schwicht/papers/lics00/main.ps.Z Minimize

Document Type:

text

Language:

en

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Title:

An Arithmetic for Polynomial-Time Computation

Description:

We de ne a restriction LHA of Heyting arithmetic HA with the property that all extracted programs are feasible. The restrictions consist in linearity and ramification requirements.

We de ne a restriction LHA of Heyting arithmetic HA with the property that all extracted programs are feasible. The restrictions consist in linearity and ramification requirements. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-16

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/lha02/lha.ps.Z

http://www.mathematik.uni-muenchen.de/~schwicht/papers/lha02/lha.ps.Z Minimize

Document Type:

text

Language:

en

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Title:

Monotone Majorizable Functionals

Description:

Several properties of monotone functionals (MF) and monotone majorizable functionals (MMF) used in the earlier work by the author and van de Pol are proved. It turns out that the terms of the simply typed lambda-calculus define MF, but adding primitive recursion, and even monotonic primitive recursion changes the situation: already Z:Z(1 \Gamma ...

Several properties of monotone functionals (MF) and monotone majorizable functionals (MMF) used in the earlier work by the author and van de Pol are proved. It turns out that the terms of the simply typed lambda-calculus define MF, but adding primitive recursion, and even monotonic primitive recursion changes the situation: already Z:Z(1 \Gamma sg) is not MMF. It is proved that extensionality is not Dialectica-realizable by MMF, and a simple example of a MF which is not hereditarily majorizable is given. Keywords: Monotone functionals, monotone majorizable functionals, hereditarily majorizable functionals, simply typed lambda-calculus, extensionality, Dialectica interpretation 1 Introduction For higher order functionals there is a rather natural notion of monotonicity. In the present paper we adress a question asked by Mints (at the International Congress for Logic, Methodology and the Philosophy of Science in Florence 1995): How do monotone functionals relate to Howard's hereditari. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-12

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/casari96/mon3.ps.Z

http://www.mathematik.uni-muenchen.de/~schwicht/papers/casari96/mon3.ps.Z Minimize

Document Type:

text

Language:

en

Subjects:

Monotone functionals ; monotone majorizable functionals ; hereditarily majorizable functionals ; simply typed lambda-calculus ; extensionality ; Dialectica interpretation

Monotone functionals ; monotone majorizable functionals ; hereditarily majorizable functionals ; simply typed lambda-calculus ; extensionality ; Dialectica interpretation Minimize

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Title:

A syntactical analysis of non-size-increasing polynomial time computation

Description:

A syntactical proof is given that all functions definable in a certain affine linear typed λ-calculus with iteration in all types are polynomial time computable. The proof provides explicit polynomial bounds that can easily be calculated.

A syntactical proof is given that all functions definable in a certain affine linear typed λ-calculus with iteration in all types are polynomial time computable. The proof provides explicit polynomial bounds that can easily be calculated. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-10-10

Source:

http://www-cgi.cs.cmu.edu/~fp/courses/15816-s12/misc/aehlig02tocl.pdf

http://www-cgi.cs.cmu.edu/~fp/courses/15816-s12/misc/aehlig02tocl.pdf Minimize

Document Type:

text

Language:

en

Subjects:

Categories and Subject Descriptors ; F.4.1 [Mathematical Logic and Formal Languages ; Mathematical Logic—Lambda calculus and related systems ; F.2.2 [Analysis of Algorithms and Problem Complexity ; Nonnumerical Algorithms and Problems General Terms ; Theory ; Languages Additional Key Words and Phrases ; Complexity ; lambda calculus ; linear logic

Categories and Subject Descriptors ; F.4.1 [Mathematical Logic and Formal Languages ; Mathematical Logic—Lambda calculus and related systems ; F.2.2 [Analysis of Algorithms and Problem Complexity ; Nonnumerical Algorithms and Problems General Terms ; Theory ; Languages Additional Key Words and Phrases ; Complexity ; lambda calculus ; linear logic Minimize

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Title:

Proof Theory

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-04-28

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/lectures/proofth/ss06/s.pdf

http://www.mathematik.uni-muenchen.de/~schwicht/lectures/proofth/ss06/s.pdf Minimize

Document Type:

text

Language:

en

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Title:

Constructive Analysis with Witnesses

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-08-11

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mod03/modart03.ps

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mod03/modart03.ps Minimize

Document Type:

text

Language:

en

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Title:

∀x(A → ∃xA)

Description:

Dedicated to Grigori Mints on occasion of his 70th birthday We consider proofs in minimal logic, written in natural deduction style. The only rules are introduction and elimination for implication and the universal quantifier. The logical connectives ∃, ∧ are seen as special cases of inductively defined predicates, and hence are defined by the i...

Dedicated to Grigori Mints on occasion of his 70th birthday We consider proofs in minimal logic, written in natural deduction style. The only rules are introduction and elimination for implication and the universal quantifier. The logical connectives ∃, ∧ are seen as special cases of inductively defined predicates, and hence are defined by the introduction and elimination schemes Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-04-12

Source:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mints09/deco.pdf

http://www.mathematik.uni-muenchen.de/~schwicht/papers/mints09/deco.pdf Minimize

Document Type:

text

Language:

en

Rights:

Metadata may be used without restrictions as long as the oai identifier remains attached to it.

URL:

Content Provider:

My Lists:

My Tags:

Notes:

Currently in BASE: 69,675,194 Documents of 3,352 Content Sources

http://www.base-search.net