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Pointer Logic. Daniel Kroening and Ofer Strichman Decision Procedure. Why pointer logic?. A pointer is a program variable whose sole purpose is to refer to some other program construct. Other program construct - variable, procedures, or another pointer
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Pointer Logic Daniel Kroening and OferStrichman Decision Procedure
Why pointer logic? • A pointer is a program variable whose sole purpose is to refer to some other program construct. • Other program construct - variable, procedures, or another pointer • A Pointer is useful but common source of programming errors. • Aliasing • Null pointer dereference error
Table of Contents • Introduction • Pointer logic • Modeling Heap-Allocated Data Structures • A Decision Procedure • Conclusion
Memory model • The implementation of pointer relies on the fact that the memory cells of a computer have addresses, i.e., each cell has a unique number. • Definition 1. Memory model • A memory model describes the assumptions that are made about the way memory cells are addressed. We assume that the architecture provides a continuous, uniform address space. Each address corresponds to a memory cell that is able to store one data word. A memory valuation M: A D is a mapping from a set of address A into the domain D of data words.
Memory layout • The compiler assigns a particular memory location to each static variable. • The mapping is called memory layout. • Definition 2. Memory layout • A memory layoutL: V A is a mapping from each variable v∊ Vto an address a∊A, where V denote the set of variables and A denote the set of addresses. • Memory layout is nonoverlappingwhich means that the memory locations of the statically allocated variables are distinct. • How about dynamic memory allocation?
Dynamic memory allocation • Dynamic data structures are created at the run time of the program. • A runtime library maintains a list of the memory regions that are unused. • A function, which is part of this library, allocates a region of given size and returns a pointer to the beginning of the region. ( malloc(), …) • The memory layout changes during the runtime of the program. • The lifetime of a dynamic object is the time between its allocation and its deallocation.
Analysis of programs with pointers • Aliasing • a situation in which a data location in memory can be accessed through different symbolic names in the program. • Modifying the data through one name implicitly modifies the values associated to all aliased names, which is not expected by the programmer. • Dereferencing • a situation which a pointer doesn’t point to a proper object. • Example
Table of Contents • Introduction • Pointer logic • Modeling Heap-Allocated Data Structures • A Decision Procedure • Conclusion
Syntax • The syntax of a formula in pointer logic is defined by the following rules: • pointer-identifier is of pointer type • identifier is of integers or an array of integers
Semantics (1/2) • Define the semantics by referring to • memory layout L • L: V A is a mapping from each variable v∊ Vto an address a∊A • memory valuation M • M: A D, mapping from a set of address A into the domain D of data words • Pointer logic formulas are predicates on M, L pairs and the definition of semantics uses a reduction to integer arithmetic and array logic. • We treat M and L as array types.
Semantics (2/2) • Definition 3. Semantics of pointer logic • Let LP denote the set of pointer logic expressions, and let LD denote the set of expressions permitted by the logic for the data words. • The function [⋅] is defined [⋅] : LP LDfor e ∊LP. The function [e] is defined recursively. The expression e∊ LP is valid if and only if [e] is valid.
Example • a is an array identifier. • The semantic definition of the expression expands as follows: • Last equation is obviously valid, and thus so is original expression.
Table of Contents • Introduction • Pointer logic • Modeling Heap-Allocated Data Structures • A Decision Procedure • Conclusion
Lists (1/2) • The simplest dynamically allocated data structure is the linked list. • Structure type • contains fields for a next pointer and the data. • So, How to model linked list in pointer logic formulas?
Lists (2/2) • We need to model the field for a next pointer. • Recursive definition • We need to add the property that the last element of the acyclic list should point a NULL. • Add another definition
Table of Contents • Introduction • Pointer logic • Modeling Heap-Allocated Data Structures • A Decision Procedure • Conclusion
Applying the semantic translation • The semantic translation • assigns meaning to the pointer logic formulas • gives rise to a simple decision procedure • The formula generated by semantic translation contain • array read operator (recall that memory layout L and memory valuation M are considered as array read operation) • linear arithmetic for the index. • equality over the type for modeling the contents of the memory cells • Decision procedure for pointer logic • Apply the semantic translation to a pointer formula to obtain a formula in the combined logic of linear arithmetic and array. • Pass the formula to the decision procedure for the combined logic.
Examples • We want to check validity of following pointer logic formula: • The semantic translation expands as follows: • A decision procedure for array logic and equality logic easily concludes that the formula above is valid.
Pure variables (1/2) • Sometimes the semantic translation can place an undue burden on the underlying decision procedure, as illustrated by the following example: • A decision procedure for array logic and equality logic is certainly able to deduce that the original formula is valid. • Therefore, the semantic translation process is not required in this kind of situation.
Pure variables (2/2) • Definition 4. Pure variables • Given a formula with a set of variables V, let denote the subset of ’s variables that are not used within an argument of the “&” operator within . These variables are called pure. • Definition 5. [∙]P • A new translation function [e]P is identical to the definition [e] unless e denotes a variable . The new definition is:
Example • This no longer burdens the decision procedure for array logic:
Reachability predicate (1/2) • Definition 6. • This stands for the pointer that is obtained by starting from q and following the field f, n times: • Definition 7. reachability predicate, • We call a predicate as reachability predicate which satisfies following two conditions: • There exists some n such that q is reachable from p by following f for n times. • x is not reachable in fewer than n steps from p following f.
Reachability predicate (2/2) • We say that a formula is a reachability predicate formula if it contains the reachability predicate. • Reachability predicate formulas often have quantifiers, which make decision problems much harder. • There is no automatic decision procedure for a logic that includes a reachability predicate.
Table of Contents • Introduction • Pointer logic • Modeling Heap-Allocated Data Structures • A Decision Procedure • Conclusion
Conclusion • The semantic translation not only assigns meaning to the pointer logic formulas, but also gives rise to a simple decision procedure. • Deciding pointer logic including reachability predicate is still an open problem.
