## Mathematical Logic and Its ApplicationsThe Summer School and Conference on Mathematical Logic and its Applications, September 24 - October 4, 1986, Druzhba, Bulgaria, was honourably dedicated to the 80-th anniversary of Kurt Godel (1906 - 1978), one of the greatest scientists of this (and not only of this) century. The main topics of the Meeting were: Logic and the Foundation of Mathematics; Logic and Computer Science; Logic, Philosophy, and the Study of Language; Kurt Godel's life and deed. The scientific program comprised 5 kinds of activities, namely: a) a Godel Session with 3 invited lecturers b) a Summer School with 17 invited lecturers c) a Conference with 13 contributed talks d) Seminar talks (one invited and 12 with no preliminary selection) e) three discussions The present volume reflects an essential part of this program, namely 14 of the invited lectures and all of the contributed talks. Not presented in the volltme remai ned si x of the i nvi ted lecturers who di d not submi t texts: Yu. Ershov - The Language of!:-expressions and its Semantics; S. Goncharov - Mathematical Foundations of Semantic Programming; Y. Moschovakis - Foundations of the Theory of Algorithms; N. Nagornyj - Is Realizability of Propositional Formulae a GBdelean Property; N. Shanin - Some Approaches to Finitization of Mathematical Analysis; V. Uspensky - Algorithms and Randomness - joint with A.N. |

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### Contents

4 | |

Gödels Life and Work | 23 |

Categorial Grammar and Lambda Calculus | 39 |

A Constructive Morse Theory of Sets | 61 |

Differentiation of Constructive Functions of a Real Variable | 81 |

Reasoning in Trees | 125 |

NonDeterministic Program Schemata and their Relation to Dynamic Logic | 137 |

The Parallel Evaluation of Functional Programs | 149 |

Modal Environment for Boolean Speculations | 253 |

Distributive Spaces | 265 |

Approximating the Projective Model | 273 |

Projection Complete Graph Problems Corresponding to a BranchingProgram | 283 |

Constructive Theories with Abstract Data Types for Program Synthesis | 293 |

A FirstOrder Logic for Logic Programming | 303 |

The Relational Semantics for Branched Quantifiers | 315 |

Propositional Dynamic Logic in Two and More Dimensions | 323 |

Logic Approximating Sequences of Sets | 167 |

Intuitionistic Formal Spaces A First Communication | 187 |

On the Logic of Small Changes in Theories II | 205 |

On Logical Relations in Program Semantics | 213 |

Search Computability and Computability with Numberings Are Equivalent | 233 |

CutElimination Theorem for HigherUrder Classical Logic | 243 |

On a NonConstructive Type Theory and Program Derivation | 331 |

Prime Computability on Partial Structures | 341 |

Complexity Bounded MartinLöf Tests | 351 |

Participants and Contributors | 361 |

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### Common terms and phrases

0]-function abstract algebra algorithm Anton Ivanov arbitrary atomic axiom of choice axiomatized axioms Benthem binary Boolean branched quantifiers branching program canonical value Categorial Grammar CFRV classical complete Heyting algebra completeness theorem consider consistent constant continuum hypothesis defined definition denote derivation disjunctive domain DPDL dynamic logic elements equivalent example exists expression fact first-order formal topology formula function f given hence infinite interpretation intuitionistic isomorphic König's lemma Kurt Gödel Lambek language Lemma logic programming logical relations Martin-Löf Math Mathematical Logic means modal logic monotone natural numbers node notation notion obtain operator paper polynomial-size prime computable problem proof properties Proposition provable prove quantifiers real numbers recursive function respect result rules scheme semantics semicomputable sequence sequent tree set of ARNs space structure subset symbols type theory u'xy valid value of type