Lambda Calculus

SEP entry on Combinatory Logic

There's a new entry on Combinatory Logic in the Stanford Encyclopedia of Philosophy.

I haven't perused it yet, so please share opinions and comments.

Unchecked Exceptions can be Strictly More Powerful than Call/CC

Here's a little light reading for your day-after-Labor-Day (or whatever yesterday was where you live): Unchecked Exceptions can be Strictly More Powerful than Call/CC, Mark Lillibridge and Olivier Danvy, 1999, Higher-Order and Symbolic Computation.

We demonstrate that in the context of statically-typed purely-functional lambda calculi without recursion, unchecked exceptions (e.g., SML exceptions) can be strictly more powerful than call/cc. More precisely, we prove that a natural extension of the simply-typed lambda calculus with unchecked exceptions is strictly more powerful than all known sound extensions of Girard’s Fω (a superset of the simply-typed lambda calculus) with call/cc. This result is established by showing that the first language is Turing complete while the later languages permit only a subset of the recursive functions to be written.

I have to say that on seeing the title I was surprised: I cut my functional teeth on Scheme and every baby Schemer sucks up the knowledge that call/cc lets you create all manner of flow control including exceptions. But, as the paper makes clear, that's not necessarily the case in a statically-typed context.

Edit: Citeseerx was not responding very well, here's an alternative URL for the paper.

A located lambda calculus

A located lambda calculus. Ezra Cooper and Philip Wadler. Submitted to ICFP 2008.

Several recent language designs have offered a unified language for programming a distributed system; we call these "location-aware" languages. These languages provide constructs that allow the programmer to control the location (the choice of host, for example) where a piece of code should run, which can be useful for security or performance reasons. On the other hand, a central mantra of web engineering insists that web servers should be "stateless": that no "session state" should be maintained on behalf of individual clients---that is, no state that pertains to the particular point of the interaction at which a client program resides. Thus far, most implementations of unified location-aware languages have ignored this precept, usually keeping a process for each client running on the server, or otherwise storing state information in memory. We show how to implement a location-aware language on top of the stateless-server model.

This paper is technical, and I assume most LtU members will mainly read sections 1, 5 & 6. Figure 5 is definition of the located LC.

Register Allocation by Proof Transformation

Register Allocation by Proof Transformation, Atsushi Ohori. ESOP 2003.

This paper presents a proof-theoretical framework for register allocation that accounts for the entire process of register allocation. Liveness analysis is proof reconstruction (similar to type inference), and register allocation is proof transformation from a proof system with unrestricted variable accesses to the one with restricted variable access. In our framework, the set of registers acts as the ``working set'' of the live variables at each instruction step, which changes during the execution of the code. This eliminates the ad-hoc notion of ``spilling''. The necessary memory-register moves are systematically incorporated in the proof transformation process. Its correctness is a simple corollary of our construction; the resulting proof is equivalent to the proof of the original code modulo the treatment of structural rules. This yields a simple yet powerful register allocation algorithm. The algorithm has been implemented, demonstrating the feasibility of the framework.

The idea that making uses of the structural rules explicit gives you proof terms to model register-memory moves is very pretty. Another fun thing to do would be to take a continuation calculus and apply the ideas here to see if it produces a good low-level IR.

EDIT: Ehud asked for some further exposition, so here goes.

At a high level, you can think of the need to do register allocation as arising from a mismatch between a programming language and the hardware. In a language, we refer to data using variables, and in any given expression, we can use as many variables as we like. However, when a CPU does stuff, it wants the data to be in registers -- and it has only a small, finite set of them. So when a program is compiled, some variables can be represented by registers, and the rest must be represented by locations in memory (usually on the stack). Whenever the CPU needs to use a variable in memory, there must be explicit code to move it from memory into a register.

The idea in this paper is to take the typing derivation of a program with an unbounded variable set, and then divide the context into two parts, one representing the register file and the other representing variables in memory. In terms of the implementation, moves between these two zones correspond to register-memory moves; and in terms of logic, this is a use of the structural rule of Exchange, which permutes the order of variables in a context.

So this gives us a high-level, machine-independent characterization of the register allocation problem: take a one-zone derivation and convert it to a two-zone derivation. But it gets even better: as long as the register allocation algorithm only adds uses of the structural rules in its transformation, we know that the meaning of the original program is unchanged -- so this method also yields a simple way of proving that a register allocation pass is semantics-preserving. (The fact that this is an easy proof is one indication of the power of this idea.)

History of Lambda-Calculus and Combinatory logic

F. Cardone and J. R. Hindley. History of Lambda-Calculus and Combinatory logic. To appear as a chapter in Volume 5 of the Handbook of the History of Logic.

From the introduction:

Seen in outline, the history of LC and CL splits into three main periods: first, several years of intensive and very fruitful study in the 1920s and ’30s; next, a middle period of nearly 30 years of relative quiet; then in the late 1960s an upsurge of activity stimulated by developments in higher-order function theory, by connections with programming languages, and by new technical discoveries. The fruits of the first period included the first-ever proof that predicate logic is undecidable. The results of the second attracted very little non-specialist interest, but included completeness, cut-elimination and standardization theorems (for example) that found many uses later. The achievements of the third, from the 1960s onward, included constructions and analyses of models, development of polymorphic type systems, deep analyses of the reduction process, and many others probably well known to the reader. The high level of activity of this period continues today.

Beware: This is a long paper (but less than you might expect it to be by looking at the page count: about half the pages are dedicated to the bibliography).

In the announcement on the TYPES Forum the authors invited comments, suggestions and additional information on the topics of the paper, namely the development of lambda-calculi and combinatory logic from the prehistory (Frege, Peano and Russell) to the end of 20th century.

Project LambdaCan

You can get soup in a can. You can get bread in a can. Now the long wait is over! You can finally get Lambda Calculus in a can...Project LambdaCan takes [the Lambda Calculus] and implements it on a microcontroller better suited to the most mundane of tasks, like running a vending machine or microwave oven. And it sticks the microcontroller in a can that you can connect to your PC using a USB cable.

For those that are both language geeks and hardware geeks...

Relating Complexity and Precision in Control Flow Analysis

Relating Complexity and Precision in Control Flow Analysis, David Van Horn and Harry Mairson. ICFP 2007.

We analyze the computational complexity of kCFA, a hierarchy of control flow analyses that determine which functions may be applied at a given call-site. This hierarchy specifies related decision problems, quite apart from any algorithms that may implement their solutions. We identify a simple decision problem answered by this analysis and prove that in the 0CFA case, the problem is complete for polynomial time. The proof is based on a nonstandard, symmetric implementation of Boolean logic within multiplicative linear logic (MLL). We also identify a simpler version of 0CFA related to eta-expansion, and prove that it is complete for logarithmic space, using arguments based on computing paths and permutations.

For any fixed k > 0, it is known that kCFA (and the analogous decision problem) can be computed in time exponential in the program size. For k = 1, we show that the decision problem is NP-hard, and sketch why this remains true for larger fixed values of k. The proof technique depends on using the approximation of CFA as an essentially nondeterministic computing mechanism, as distinct from the exactness of normalization. When k = n, so that the "depth" of the control flow analysis grows linearly in the program length, we show that the decision problem is complete for exponential time.

In addition, we sketch how the analysis presented here may be extended naturally to languages with control operators. All of the insights presented give clear examples of how straightforward observations about linearity, and linear logic, may in turn be used to give a greater understanding of functional programming and program analysis.

There's ton of really good stuff in here; I was particularly fascinated by the fact that 0-CFA is exact for multiplicatively linear programs (ie, that use variables at most once), because linearity guarantees that every lambda can flow to at most one use site.

Call-by-value Termination in the Untyped Lambda-calculus

From Arxiv:

A fully-automated algorithm is developed able to show that evaluation of a given untyped lambda-expression will terminate under CBV (call-by-value). The ``size-change principle'' from first-order programs is extended to arbitrary untyped lambda-expressions in two steps. The first step suffices to show CBV termination of a single, stand-alone lambda-expression. The second suffices to show CBV termination of any member of a regular set of lambda-expressions, defined by a tree grammar. (A simple example is a minimum function, when applied to arbitrary Church numerals.) The algorithm is sound and proven so in this paper. The Halting Problem's undecidability implies that any sound algorithm is necessarily incomplete: some lambda-expressions may in fact terminate under CBV evaluation, but not be recognised as terminating.
The intensional power of the termination algorithm is reasonably high. It certifies as terminating many interesting and useful general recursive algorithms including programs with mutual recursion and parameter exchanges, and Colson's ``minimum'' algorithm. Further, our type-free approach allows use of the Y combinator, and so can identify as terminating a substantial subset of PCF.

To renew the discussion on Total Functional Programming, this paper is an alternative to Termination Checking with Types.

Theorem proving support in programming language semantics

We describe several views of the semantics of a simple programming language as formal documents in the calculus of inductive constructions that can be verified by the Coq proof system. Covered aspects are natural semantics, denotational semantics, axiomatic semantics, and abstract interpretation. Descriptions as recursive functions are also provided whenever suitable, thus yielding a a verification condition generator and a static analyser that can be run inside the theorem prover for use in reflective proofs. Extraction of an interpreter from the denotational semantics is also described. All different aspects are formally proved sound with respect to the natural semantics specification.

More work on mechanized metatheory with an eye towards naturalness of representation and automation. This seems to me to relate to Adam Chlipala's work on A Certified Type-Preserving Compiler from Lambda Calculus to Assembly Language, which relies on denotational semantics and proof by reflection, in crucial ways. More generally, it seems to reinforce an important trend in using type-theory-based theorem provers to tackle programming language design from the semantic point of view (see also A Very Modal Model of a Modern, Major, General Type System and Verifying Semantic Type Soundness of a Simple Compiler). I find the trend exciting, but of course I also wonder how far we can practically go with it today, given that the overwhelming majority of the literature, including our beloved Types and Programming Languages, is based on A Syntactic Approach to Type Soundness. Even the upcoming How to write your next POPL paper in Coq at POPL '08 centers on the syntactic approach.

Is the syntactic approach barking up the wrong tree, in the long term? The semantic approach? Both? Neither?

Natural Deduction for Intuitionistic Non-Commutative Linear Logic

Natural Deduction for Intuitionistic Non-Commutative Linear Logic, Jeff Polakow and Frank Pfenning. TLCA 1999.

Intuitionistic logic captures functional programming in a logical way, as can be seen from the Curry-Howard isomorphism between constructive proofs and functional programs. However, there are many structural properties of programs that are not captured within the intuitionistic framework, such as resource usage, computational complexity, and sequentiality. Intuitionistic linear logic can be thought of as a refinement of intuitionistic logic in which the resource consumption properties of functions can be expressed internally. Here, we further refine it to allow the expression of sequencing of computations. We achieve this by controlling the use of the structural rule of exchange to arrive at intuitionistic non-commutative linear logic.

My earlier post on linguistics reminded me of the Lambek calculus, which is an ordered logic invented in 1958(!) to model how to parse sentences. So I wanted to find a paper on ordered logic (ie, you can't freely swap the order of hypotheses in a context) and link to that.

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