Completeness of Hoare Logic over Nonstandard Models

The nonstandard approach to program semantics has successfully resolved the completeness problem of Floyd-Hoare logic. The known versions of nonstandard semantics, the Hungary semantics and axiomatic semantics, are so general that they are absent either from mathematical elegance or from practical usefulness. The aim of this paper is to exhibit a not only mathematically elegant but also practically useful nonstandard semantics. A basic property of computable functions in the standard model $N$ of Peano arithmetic $PA$ is $\Sigma_1$-definability. However, the functions induced by the standard interpretation of while-programs $S$ in nonstandard models $M$ of $PA$ are not always arithmetical. The problem consists in that the standard termination of $S$ in $M$ uses the finiteness in $N$, which is not the finiteness in $M$. To this end, we shall give a new interpretation of $S$ in $M$ such that the termination of $S$ uses $M$-finiteness, and the functions produced by $S$ in all models of $PA$ have the uniform $\Sigma_1$-definability. Then we define, based on the new semantics of while-programs, a new semantics of Hoare logic in nonstandard models of $PA$, and show that the standard axiom system of Hoare logic is sound and complete w.r.t. the new semantics. It will be established, in $PA$, that the Hungary semantics and axiomatic semantics coincide with the new semantics of while-programs. Moreover, various comparisons with the previous results, usefulness of the nonstandard semantics, and remarks on the completeness issues are presented

On Completeness Results of Hoare Logic Relative to the Standard Model

The general completeness problem of Hoare logic relative to the standard model $N$ of Peano arithmetic has been studied by Cook, and it allows for the use of arbitrary arithmetical formulas as assertions. In practice, the assertions would be simple arithmetical formulas, e.g. of a low level in the arithmetical hierarchy. In addition, we find that, by restricting inputs to $N$, the complexity of the minimal assertion theory for the completeness of Hoare logic to hold can be reduced. This paper further studies the completeness of Hoare Logic relative to $N$ by restricting assertions to subclasses of arithmetical formulas (and by restricting inputs to $N$). Our completeness results refine Cook's result by reducing the complexity of the assertion theory

SOFSEM 2017: Theory and Practice of Computer Science [electronic resource] : 43rd International Conference on Current Trends in Theory and Practice of Computer Science, Limerick, Ireland, January 16-20, 2017, Proceedings /

This book constitutes the refereed proceedings of the 43rd International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2017, held in Limerick, Ireland, in January 2017. The 34 papers presented in this volume were carefully reviewed and selected from 41 submissions. They were organized in topical sections named: foundations in computer science; semantics, specification and compositionality; theory of mobile and distributed systems; verification and automated system analysis; petri nets, games and relaxed data structures; graph theory and scheduling algorithms; quantum and matrix algorithms; planar and molecular graphs; coloring and vertex covers; algorithms for strings and formal languages; data, information and knowledge engineering; and software engineering: methods, tools, applications.Dependable and Optimal Cyber-Physical Systems -- Verifying Parametric Thread Creation -- Network Constructors: A Model for Programmable Matter -- Logical characterisations and compositionality of input-output conformance simulation -- A Linear-Time Branching-Time Spectrum of Behavioral Specification Theories -- Symbolic semantics for multiparty interactions in the link-calculus -- Different Speeds Suffice for Rendezvous of Two Agents on Arbitrary Graphs -- Deciding structural liveness of Petri nets -- Distributed Network Generation based on Preferential Attachment in ABS -- Completeness of Hoare Logic Relative to the Standard Model -- Configuration- and Residual-Based Transition Systems for Event Structures with Asymmetric Conflict -- Hardness of deriving invertible sequences from finite state machines -- A Graph-theoretical Characterisation of State Separation -- Selfish Transportation Games -- Decomposable Relaxation for Concurrent Data Structures -- Sufficient Conditions for a Connected Graph to have a Hamiltonian Path -- Enumerating Minimal Tropical Connected Sets -- Bamboo Garden Trimming Problem (Perpetual maintenance of machines with different attendance urgency factors) -- Exact quantum query complexity of EXACT_{k,l}^n -- Adjacent vertices can be hard to find by quantum walks -- Matrix semigroup freeness problems in SL(2,Z) -- Order-preserving 1-string representations of planar graphs -- How to Draw a Planarization -- Finding Largest Common Substructures of Molecules in Quadratic Time -- Lower Bounds for On-line Interval Coloring with Vector and Cardinality Constraints -- Parameterized and Exact Algorithms for Class Domination Coloring -- The Approximability of Partial Vertex Covers in Trees -- Longest Common Subsequence in at Least k Length Order-isomorphic Substrings -- Computing longest single-arm-gapped palindromes in a string -- Edit-Distance between Visibly Pushdown Languages -- Trends and Challenges in Predictive Analytics -- Model-driven Development in Practice: From Requirements to Code -- Webpage Menu Detection Based on DOM -- A Hybrid Model for Linking Multiple Social Identities across Heterogeneous Online Social Networks -- Eco Data Warehouse Design Through Logical Variability -- On Featured Transition Systems -- Domain-Specific Languages: A Systematic Mapping Study -- Characterising Malicious Software with High-Level Behavioural Patterns -- AErlang at work -- Software Systems Migration towards Cloud-native Architectures for SME-sized Software Vendors -- Using n-grams for the Automated Clustering of Structural Models.This book constitutes the refereed proceedings of the 43rd International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2017, held in Limerick, Ireland, in January 2017. The 34 papers presented in this volume were carefully reviewed and selected from 41 submissions. They were organized in topical sections named: foundations in computer science; semantics, specification and compositionality; theory of mobile and distributed systems; verification and automated system analysis; petri nets, games and relaxed data structures; graph theory and scheduling algorithms; quantum and matrix algorithms; planar and molecular graphs; coloring and vertex covers; algorithms for strings and formal languages; data, information and knowledge engineering; and software engineering: methods, tools, applications