Weak second order characterizations of various program verification systems

AbstractWe show the equivalence of Leivant's characterization of Floyd-Hoare Logic in weak second order logic (Leivant (1985)) with both Csirmaz's (1980) and Sain's (1985) characterizations of Floyd-Hoare logic in Nonstandard Logics of Programs. Our method allows us to spell out the precise role of the comprehension axiom in weak second order logic. We then prove similar results for other program verification systems (suggested by Burstall and Pnueli) and identify exactly the comprehension axioms corresponding to those systems

Some general incompleteness results for partial correctness logics

AbstractIt 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. Andréka, Németi, 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

Input Synthesis for Sampled Data Systems by Program Logic

Inspired by a concrete industry problem we consider the input synthesis problem for hybrid systems: given a hybrid system that is subject to input from outside (also called disturbance or noise), find an input sequence that steers the system to the desired postcondition. In this paper we focus on sampled data systems--systems in which a digital controller interrupts a physical plant in a periodic manner, a class commonly known in control theory--and furthermore assume that a controller is given in the form of an imperative program. We develop a structural approach to input synthesis that features forward and backward reasoning in program logic for the purpose of reducing a search space. Although the examples we cover are limited both in size and in structure, experiments with a prototype implementation suggest potential of our program logic based approach.Comment: In Proceedings HAS 2014, arXiv:1501.0540

An observationally complete program logic for imperative higher-order functions

We establish a strong completeness property called observational completeness of the program logic for imperative, higher-order functions introduced in [1]. Observational completeness states that valid assertions characterise program behaviour up to observational congruence, giving a precise correspondence between operational and axiomatic semantics. The proof layout for the observational completeness which uses a restricted syntactic structure called finite canonical forms originally introduced in game-based semantics, and characteristic formulae originally introduced in the process calculi, is generally applicable for a precise axiomatic characterisation of more complex program behaviour, such as aliasing and local state

Proving acceptability properties of relaxed nondeterministic approximate programs

Approximate program transformations such as skipping tasks [29, 30], loop perforation [21, 22, 35], reduction sampling [38], multiple selectable implementations [3, 4, 16, 38], dynamic knobs [16], synchronization elimination [20, 32], approximate function memoization [11],and approximate data types [34] produce programs that can execute at a variety of points in an underlying performance versus accuracy tradeoff space. These transformed programs have the ability to trade accuracy of their results for increased performance by dynamically and nondeterministically modifying variables that control their execution. We call such transformed programs relaxed programs because they have been extended with additional nondeterminism to relax their semantics and enable greater flexibility in their execution. We present language constructs for developing and specifying relaxed programs. We also present proof rules for reasoning about properties [28] which the program must satisfy to be acceptable. Our proof rules work with two kinds of acceptability properties: acceptability properties [28], which characterize desired relationships between the values of variables in the original and relaxed programs, and unary acceptability properties, which involve values only from a single (original or relaxed) program. The proof rules support a staged reasoning approach in which the majority of the reasoning effort works with the original program. Exploiting the common structure that the original and relaxed programs share, relational reasoning transfers reasoning effort from the original program to prove properties of the relaxed program. We have formalized the dynamic semantics of our target programming language and the proof rules in Coq and verified that the proof rules are sound with respect to the dynamic semantics. Our Coq implementation enables developers to obtain fully machine-checked verifications of their relaxed programs.National Science Foundation (U.S.). (Grant number CCF-0811397)National Science Foundation (U.S.). (Grant number CCF-0905244)National Science Foundation (U.S.). (Grant number CCF-1036241)National Science Foundation (U.S.). (Grant number IIS-0835652)United States. Defense Advanced Research Projects Agency (Grant number FA8650-11-C-7192)United States. Defense Advanced Research Projects Agency (Grant number FA8750-12-2-0110)United States. Dept. of Energy. (Grant Number DE-SC0005288