Some General Incompleteness Results for Partial Correctness Logics

It is known that incompleteness of Hoare's logic relative to certain data type specifications can occur due to the ability of partial correctness assertions to code unsolvable problems; cf. Andreka, Nemeti, and Sain (1979, Lecture Notes in Computer Science Vol. 74, pp. 208–218, Springer-Verlag, New York/Berlin) and Bergstra and Tucker (1982, Theoret. Comput. Sci. 17, 303–315). We improve what we think are the main known theorems of this kind, showing that they depend only on very weak assumptions on the data type specification (ensuring the ability to simulate arbitrarily long finite initial segments of the natural numbers with successor), and pointing out that the recursion theoretic strength of the obtained results can be increased.

[1]  Ildikó Sain,et al.  A Complete Logic for Reasoning about Programs via Nonstandard Model Theory I , 1982, Theor. Comput. Sci..

[2]  Jan A. Bergstra,et al.  Another Incompleteness Result for Hoare's Logic , 1982, Inf. Control..

[3]  Jan A. Bergstra,et al.  Recursive Assertions are not enough - or are they? , 1979, Theor. Comput. Sci..

[4]  Mario Rodrı́guez Artalejo.  Some questions about expressiveness and relative completeness in Hoare's logic , 1985 .

[5]  James R. Bell,et al.  A new method for determining linear precedence functions for precedence grammars , 1969, CACM.

[6]  Jan A. Bergstra,et al.  Two Theorems About the Completeness of Hoare's Logic , 1982, Inf. Process. Lett..

[7]  Krzysztof R. Apt,et al.  Ten Years of Hoare's Logic: A Survey—Part I , 1981, TOPL.

[8]  Ana Pasztor.  Non-Standard Algorithmic and Dynamic Logic , 1986, J. Symb. Comput..

[9]  Jan A. Bergstra,et al.  Expressiveness and the Completeness of Hoare's Logic , 1982, J. Comput. Syst. Sci..

[10]  Jan A. Bergstra,et al.  PC-Compactness, a Necessary Condition for the Existence of Sound and Complete Logics of Partial Correctness , 1983, Logic of Programs.

[11]  Mario Rodríguez-Artalejo.  Some Questions about Expressiveness and Relative Completeness in Hoare's Logic , 1985, Theor. Comput. Sci..

[12]  C. A. R. HOARE,et al.  An axiomatic basis for computer programming , 1969, CACM.

[13]  John C. Shepherdson,et al.  Computability of Recursive Functions , 1963, JACM.

[14]  M. Rabin.  Decidability of second-order theories and automata on infinite trees. , 1969 .

[15]  Wanda Szmielew.  Elementary properties of Abelian groups , 1955 .

[16]  David Harel,et al.  First-Order Dynamic Logic , 1979, Lecture Notes in Computer Science.

[17]  Stephen A. Cook,et al.  Soundness and Completeness of an Axiom System for Program Verification , 1978, SIAM J. Comput..

[18]  A. I. Malcev.  Axiomatizable classes of locally free algebras of various types , 1971 .

[19]  Ernst-Rüdiger Olderog,et al.  On the Notion of Expressiveness and the Rule of Adaption , 1983, Theor. Comput. Sci..

[20]  Edmund M. Clarke.  The characterization problem for Hoare logics , 1984 .

[21]  Hartley Rogers,et al.  Gödel numberings of partial recursive functions , 1958, Journal of Symbolic Logic.

[22]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[23]  Jan A. Bergstra,et al.  Some Natural Structures which Fail to Possess a Sound and Decidable Hoare-Like Logic for their While-Programs , 1982, Theor. Comput. Sci..

[24]  Ana Pasztor,et al.  Recursive Programs and Denotational Semantics in Absolute Logics of Programs , 1990, Theor. Comput. Sci..

[25]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[26]  Raymond M. Smullyan,et al.  Theory of Formal Systems. (AM-47) , 1961 .

[27]  Daniel Leivant,et al.  Partial-Correctness Theories as First-Order Theories , 1985, Logic of Programs.

[28]  Ildikó Sain,et al.  Completeness Problems in Verification of Programs and Program Schemes , 1979, MFCS.

[29]  I. Sain.  A COMPLETE LOGIC FO PROGRAMS VISA NONSTANDAR , 1982 .

[30]  Mario Rodríguez-Artalejo,et al.  Hoare's Logic for Nondeterministic Regular Programs: A Nonstandard Completeness Theorem , 1985, ICALP.