Unifying theories distil common features of programming languages and design methods by means of algebraic operators and their laws. Several practical concerns---e.g., improvement of a program, conformance of code with design, correctness with respect to specified requirements---are subsumed by the beautiful notion that programs and designs are special forms of specification and their relationships are instances of logical implication between specifications. Mathematical development of this idea has been very successful but limited to an impoverished notion of specification: trace properties. Some mathematically precise properties of programs---dubbed hyperproperties---refer to traces collectively. For example, confidentiality involves knowledge of possible traces. This article reports on both obvious and surprising results about lifting algebras of programming to hyperproperties, especially in connection with loops, and suggests directions for further research. The main results are: the first compositional semantics of imperative programs with loops, at the hyper level, and a proof that this semantics is the same as the direct image of a standard semantics, for subset closed hyperproperties.
A peculiarity of these works is the treatment of conditionals. Assaf take the equation @math as the definition of @math , but this makes the definition of @math non-compositional and thus the inductive proofs a little sketchy. We do not discern an explicit definition in @cite_29 , but do find remarks like this: The definition of the collecting hypersemantics is just the additive lift for every statement, except for the while case.'' As a description of fact, it is true for subset closed hyperproperties (see our Theorem ). But it is unsuitable as a definition. We show that a proper definition is possible.
@cite_29
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Abstract This paper presents a logical approach to nonmonotonic reasoning based on the notion of a nonmonotonic consequence relation. A conditional knowledge base, consisting of a set of conditional assertions of the type if … then … , represents the explicit defeasible knowledge an agent has about the way the world generally behaves. We look for a plausible definition of the set of all conditional assertions entailed by a conditional knowledge base. In a previous paper, Kraus and the authors defined and studied preferential consequence relations. They noticed that not all preferential relations could be considered as reasonable inference procedures. This paper studies a more restricted class of consequence relations, rational relations. It is argued that any reasonable nonmonotonic inference procedure should define a rational relation. It is shown that the rational relations are exactly those that may be represented by a ranked preferential model, or by a (nonstandard) probabilistic model. The rational closure of a conditional knowledge base is defined and shown to provide an attractive answer to the question of the title. Global properties of this closure operation are proved: it is a cumulative operation. It is also computationally tractable. This paper assumes the underlying language is propositional. We show that any language in nondeterministic time @math , where the number of iterated exponentials is an arbitrary function @math , can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness @math and soundness @math , where the number of iterated exponentials is @math and @math is a universal constant. The result was previously known for @math and @math ; we obtain it for any time-constructible function @math . The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC'17). As a separate consequence of this technique we obtain a different proof of Slofstra's recent result (unpublished) on the uncomputability of the entangled value of multiprover games. Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP* contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson's problem on the relation between the commuting operator and tensor product models for quantum correlations. It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is a tautology. Here “reduced” means, roughly speaking, that the first problem can be solved deterministically in polynomial time provided an oracle is available for solving the second. From this notion of reducible, polynomial degrees of difficulty are defined, and it is shown that the problem of determining tautologyhood has the same polynomial degree as the problem of determining whether the first of two given graphs is isomorphic to a subgraph of the second. Other examples are discussed. A method of measuring the complexity of proof procedures for the predicate calculus is introduced and discussed. In this paper we consider particular equational formulae where equality = AC is the congruence induced by a set of associative-commutative axioms. The formulae we are interested in have the form t ? ACt1 ??t ? ACtn and are usually known as complement problems. To solve a complement problem is to find an instance of t which is not an instance of any ti modulo associativity-commutativity. We give a decision procedure based on tree automata which solves these formulae when all the ti1s are linear. We show that this solution also gives a decision of inductive reducibility modulo associativity and commutativity in the linear case and we give several extensions of this approach. Then, we define a new class of tree automata, called conditional tree automata which recognize sets of generalized terms, i.e terms written as multisets, and where the application of a rule depends on the satisfiability of a formula of Presburger's arithmetic. This class of tree automata allows one to solve non-linear complement problems when all occurrences of a non-linear variable occur under the same node (in flattened terms). This solution also provides a procedure to decide inductive reducibility modulo associativity-commutativity in the same case.