income tax news | Schoppe's Tax Service, Inc.
WASHINGTON — The Internal Revenue Service today advised those now receiving tax bills because they filed on time but didn't pay in full that there are many easy options for paying what they owe to the IRS.
If a tax return was filed but the balance due remains unpaid, the taxpayer will receive a letter or notice in the mail from the IRS, usually within a few weeks. These notices, including the CP14 and CP501, both of which notify taxpayers that they have a balance due, are frequently mailed in the months of June and July.
Taxpayers can use Direct Pay to pay directly from a checking or savings account. This service is free.
Taxpayers can take advantage of the Electronic Federal Tax Payment System (EFTPS) to pay by phone or online. EFTPS® is a free service of the U.S. Department of Treasury.
Taxpayers may also initiate a debit or credit card payment. The IRS doesn't charge a fee for this service but the processing company may. Fees vary by company.
Taxpayers may pay by check or money order made payable to the United States Treasury (or U.S. Treasury) either in person or through the mail.
Taxpayers should not send cash through the mail. They can pay cash at some IRS offices or at a participating PayNearMe location. Some restrictions apply.
Online Payment Agreement — Individuals who owe $50,000 or less in combined income tax, penalties and interest and businesses that owe $25,000 or less in payroll tax and have filed all tax returns may qualify for an Online Payment Agreement. Most taxpayers qualify for this option, and an agreement can usually be set up in a matter of minutes. Online applications to establish tax payment plans, like online payment agreements and installment agreements, are available Monday – Friday., 6 a.m. to 12:30 a.m.; Saturday., 6 a.m. to 10 p.m.; Sunday, 6 p.m. to midnight.
Installment Agreement — Installment agreements paid by direct deposit from a bank account or a payroll deduction will help taxpayers avoid default on their agreements. It also reduces the burden of mailing payments and saves postage costs. Taxpayers who don't qualify for a payment agreement may still pay by installment. Certain fees apply.
Offer in Compromise — Some struggling taxpayers qualify to settle their tax bill for less than the amount they owe by submitting an offer in compromise. To help determine eligibility, use the Offer in Compromise Pre-Qualifier tool.
Even if a taxpayer works out a payment solution with the IRS, the agency may still need to file a Notice of Federal Tax Lien to secure the government's interest. Federal law requires the lien to establish priority as a creditor in competition with other creditors in certain situations, such as bankruptcy proceedings or sales of real estate. Once the IRS files a lien, it may appear on a taxpayer's credit report and harm their credit rating. Therefore, it's important that they work to resolve a tax liability as quickly as possible before lien filing becomes necessary. Once the IRS files a lien, the agency generally cannot issue a Certificate of Release of Federal Tax Lien until the taxpayer pays taxes, penalties, interest and recording fees in full.
Taxpayers can take steps now to make sure they don't fall behind on their taxes in the future. The IRS encourages several key groups of taxpayers to perform a "paycheck checkup" to check if they are having the right amount of tax withholding following recent tax-law changes.
Employees can increase their tax withholding by filling out a new Form W-4, Employee's Withholding Allowance Certificate, and giving it to their employer. To have more tax withheld, claim fewer withholding allowances or ask the employer to take out a fixed amount of additional tax each pay period. To help figure the right amount to withhold, use the IRS Withholding Calculator on IRS.gov.
The Tax Cuts and Jobs Act makes sweeping tax changes that impact virtually all taxpayers. For individual taxpayers and their families, changes include a decrease in the tax rates, repeal of the personal exemption, increase in the standard deduction, modification to itemized deductions, and doubling of the child tax credit.
Under the Tax Cuts and Jobs Act, personal exemptions are repealed ($4,050 in 2017) for 2018 through 2025. Instead, the Tax Cuts and Jobs Act provides for a near doubling of the standard deduction. For tax year 2018, it increases the standard deduction from $13,000 to $24,000 for married individuals filing a joint return; $9,550 to $18,000 for head-of-household filers; and $6,500 to $12,000 for all other individuals. These standard deduction amounts are indexed for inflation for tax years beginning after 2018. The additional standard deduction for the elderly and the blind ($1,300 for married taxpayers, $1,600 for single taxpayers) is retained.
Mortgage interest deduction. The Tax Cuts and Jobs Act limits the mortgage interest deduction to interest on $750,000 of acquisition indebtedness ($375,000 in the case of married taxpayers filing separately), for tax years beginning 2018 through 2025. For acquisition indebtedness incurred before December 15, 2017, the Tax Cuts and Jobs Act allows current homeowners to keep the current limitation of $1 million ($500,000 in the case of married taxpayers filing separately). Taxpayers may continue to include mortgage interest on second homes, but within those lower dollar caps. However, no interest deduction will be allowed for interest on home equity indebtedness.
State and local taxes. The Tax Cuts and Jobs Act limits annual itemized deductions for all nonbusiness state and local taxes deductions, including property taxes, to $10,000 ($5,000 for married taxpayer filing a separate return) for 2018 through 2025. Sales taxes may be included as an alternative to claiming state and local income taxes.
Miscellaneous itemized deductions. The Tax Cuts and Jobs Act repeals all miscellaneous itemized deductions for tax years 2018 through 2025 that are subject to the two-percent floor under current law.
Medical expenses. The Tax Cuts and Jobs Act lowers the threshold for the deduction to 7.5 percent of adjusted gross income (AGI) for tax years 2017 and 2018.
Casualty losses. For tax years 2018 through 2025, a casualty loss will only be allowed to the extent it is attributable to a federally declared disaster.
The phaseout of itemized deductions is suspended for tax years 2018 through 2025.
The doubling of the standard deduction and modifications to itemized deductions effectively eliminates many individuals from claiming itemized deductions other than higher-income taxpayers. For example, for the vast majority of married taxpayers filing jointly, only those with total allowable mortgage interest, state income and local income/property taxes (up to $10,000), and charitable deductions exceeding $24,000, would claim them as itemized deductions (absent extraordinary medical expenses).
In contrast, the enhanced child credit has been highlighted as one of the provisions that will lower overall tax liability for middle-class families. The Tax Cuts and Jobs Act temporarily increases the current child tax credit from $1,000 to $2,000 per qualifying child. Up to $1,400 of that amount is refundable. The child tax credit is also expanded to provide for a $500 nonrefundable credit for qualifying dependents other than qualifying children. More families will be able to take advantage of the credit due to an increase in the adjusted gross income phaseout thresholds, starting at $400,000 for joint filers ($200,000 for all others).
In addition to changes related to exemptions, the child tax credit and the standard and itemized deduction discussed above, the Tax Cuts and Jobs Act also makes changes to alternative minimum tax and the individual tax brackets. Because these tax provisions are interrelated, estimating the impact of these changes to the tax liability for any particular family is challenging. However, as with any tax reform, there will be winners and losers.
Total itemized deductions in 2018 would remain the same at $14,500 which is less than the new standard deduction of $24,000.
Deduction for total amount of property tax and sales tax would be limited to $10,000 in 2018. Therefore, total allowed itemized deductions of $21,000 would be less than the standard deduction of $24,000.
Deduction for total amount of property tax and sales tax would be limited to $10,000 in 2018. Total allowed itemized deductions of $30,000 are not phased out and is more than the $24,000 standard deduction.
Keep in mind that many of the changes to the Internal Revenue Code in the Tax Cuts and Jobs Act are temporary. This is true especially with respect to the provisions impacting individuals. This decision was made in order to keep the tax reform within budgetary parameters, but with no guarantees that a future Congress would extend them. In future years, as the tax reform provisions expire, tax liability for individuals may be negatively affected.
The Tax Cuts and Jobs Act was signed by President Trump on December 22. The Act makes sweeping changes to the U.S. tax code and impacts virtually every taxpayer. For businesses, tax benefits include a reduction in the corporate tax rate, increase in the bonus depreciation allowance, an enhancement to the Code Sec. 179 expense and repeal of the alternative minimum tax. Owners of partnerships, S corporations, and sole proprietorships are allowed a temporary deduction as a percentage of qualified income of pass-through entities, subject to a number of limitations and qualifications. On the other hand, numerous business tax preferences are eliminated.
A reduced 21-percent corporate tax rate is permanent beginning in 2018. Also, the 80-percent and 70-percent dividends received deductions under current law are reduced to 65-percent and 50-percent, respectively. The Tax Cuts and Jobs Act also repeals the alternative minimum tax on corporations.
The bonus depreciation rate has fluctuated wildly over the last 15 years, from as low as zero percent to as high as 100 percent. It is often seen as a means to incentivize business growth and job creation. The Tax Cuts and Jobs Act temporarily increases the 50-percent "bonus depreciation" allowance to 100 percent. It also removes the requirement that the original use of qualified property must commence with the taxpayer, thus allowing bonus depreciation on the purchase of used property.
The Tax Cuts and Jobs Act sets the Code Sec. 179 dollar limitation at $1 million and the investment limitation at $2.5 million. Although the differences between bonus depreciation and Code Sec. 179 expensing would now be narrowed if both offer 100-percent write-offs for new or used property, some advantages and disadvantages for each will remain. For example, Code Sec. 179 property is subject to recapture if business use of the property during a tax year falls to 50 percent or less; but Code Sec. 179 allows a taxpayer to elect to expense only particular qualifying assets within any asset class.
Numerous business tax preferences are eliminated. These include the Code Sec. 199 domestic production activities deduction, non-real property like-kind exchanges, and more. Additionally, the rules for business meals are revised, as are the rules for the rehabilitation credit. However, the Tax Cuts and Jobs Act leaves the research and development credit in place, but requires five-year amortization of research and development expenditures. It also creates a temporary credit for employers paying employees who are on family and medical leave.
In an attempt to "level the playing field" between businesses that capitalize through equity and those that borrow, the Tax Cuts and Jobs Act generally caps the deduction for net interest expenses at 30 percent of adjusted taxable income, among other criteria. Exceptions exist for small businesses, including an exemption for businesses with average gross receipts of $25 million or less.
Currently, up to the end of 2017, owners of partnerships, S corporations, and sole proprietorships – as "pass-through" entities – pay tax at the individual rates, with the highest rate at 39.6 percent. The Tax Cuts and Jobs Act allows a temporary deduction in an amount equal to 20 percent of qualified income of pass-through entities, subject to a number of limitations and qualifications.
The Tax Cuts and Jobs Act contains rules that will prevent pass-through owners—particularly service providers such as accountants, doctors, lawyers, etc.—from converting their compensation income taxed at higher rates into profits taxed at the lower rate.
The Tax Cuts and Jobs Act modifies current rules for net operating losses (NOLs). Generally, NOLs will be limited to 80 percent of taxable income for losses arising in tax years beginning after December 31, 2017. It also denies the carryback for NOLs in most cases while providing for an indefinite carryforward, subject to the percentage limitation.
These are just highlights of the changes and impact of the Tax Cuts and Jobs Act. There is much more than can be covered in this post.
We know that you have worked hard for your money and would like to reap the benefits to the greatest extent possible. Your ultimate goal is to sustain a successful wealth-building strategy while avoiding unnecessary and expensive tax consequences. We are interested in helping you achieve these objectives.
For the last few years, there has been talk of major tax reform that would place an increased tax burden on higher income individuals. President Trump proposed a tax reform plan that would reduce individual tax rates, abolish the alternative minimum tax (AMT) and federal estate tax, and more. Individual rates under the President's proposal would be 10, 25 and 35 percent. At the same time, the President proposed to double the standard deduction and protect the home ownership and charitable gift tax deductions. The President also proposed to provide unspecified tax relief to families with children and dependents. The President's proposal calls for a 15 percent corporate tax rate. The 15 percent rate would also be available to small and mid-size pass-through businesses, White House officials said.
Further, the President called for elimination of unspecified tax breaks for special interests. Democrats in Congress said the President's plan favored high-income taxpayers and did not deliver enough tax breaks to lower and middle-income taxpayers. As tax reform moves into the latter half of 2017, the chances of a retroactive tax cut to include the 2017 tax year are lessened but not entirely removed from consideration. Tax planning for the balance of the year therefore should remain flexible.
As you can see, the more complex issues faced by higher-income individuals create a challenging planning environment for the 2017 tax filing season.
When natural disasters occur, as has happened recently with Hurricanes Harvey and Irma, the last thing you want to stress over is filing taxes. The Comptroller's office understands, and they have compiled information for evacuees and relief workers on how to request an extension for reporting and paying taxes.
The Internal Revenue Service today warned people to beware of a new scam linked to the Electronic Federal Tax Payment System (EFTPS), where fraudsters call to demand an immediate tax payment through a prepaid debit card. This scam is being reported across the country, so taxpayers should be alert to the details.
The IRS does not use email, text messages or social media to discuss personal tax issues, such as those involving bills or refunds.
For more information, visit the "Tax Scams and Consumer Alerts" page on IRS.gov. Additional information about tax scams is available on IRS social media sites, including YouTube videos.
Federal Student Tax? No such thing!!!
The Federal Trade Commission recently released a statement warning college students not to fall for a new scam: the federal student tax scam.
Students from many colleges are telling the FTC that the calls go something like this: the so-called IRS agent tells you that you owe a "federal student tax," and often has some piece of information that makes the call seem legit. Sometimes it's the name of your school, or another piece of information about you. The caller demands that you wire money immediately, by MoneyGram or another untraceable method. And, if you don't act quickly enough, the caller might threaten to report you to the police. If you hang up on the caller, they might make follow-up calls with spoofed caller-ID information. So, while caller ID might say it's 911 or the U.S. Government calling, it's not. It's all fake.
No one from the IRS will ever ask you to wire money, or pay by sending any sort of reloadable prepaid cards.
The IRS will never contact you by phone first. If you owe money for an actual tax, the IRS will send a letter first.
If you get one of these calls, hang up immediately. Do not give out any personal information — no matter how mundane it may seem. Hang up, and report the call to the FTC.
For the news release from the FTC on this, visit here.
On Sunday, Forbes posted an article explaining a part of the new Protecting Americans from Tax Hikes (PATH) Act of 2015 that will affect many taxpayers: their refunds will be delayed next year.
As part of the "Protecting Americans from Tax Hikes (PATH ) Act of 2015," (P.L. 114-113) signed into law on December 18, 2015, the IRS must wait until February 15 to issue refunds to taxpayers who claimed the earned-income tax credit (EITC) or the child tax credit (CTC).
The Internal Revenue Service has launched a new initiative designed to more quickly identify employers who are falling behind on their payroll or employment taxes and then help them get caught up on their payment and reporting responsibilities. The effort is called the Early Interaction Initiative.
In the past, the first attempt by the IRS to contact an employer having payment difficulties often did not occur until much later in the process, after the employment return was filed and the employer's unpaid tax obligation had already begun to spiral out of control.
Two‐thirds of federal taxes are collected through the payroll tax system. By law, employers must withhold federal income, Social Security and Medicare taxes from employees' wages.
Shortly after employees are paid, employers typically must turn over withheld amounts, along with employer‐matching contributions, to the federal government. Though payment schedules vary, these payments, known as federal tax deposits (FTDs), are made electronically through the Electronic Federal Tax Payment System (EFTPS). These FTDs are later reported on a return, usually filed quarterly, with the IRS.
Employers, especially those facing liquidity difficulties, sometimes inappropriately divert funds withheld from employees' pay for working capital or other purposes. Even when well‐intentioned, such diversions can quickly result in mounting tax liabilities for the employer, along with interest and penalties, potentially threatening the employer's financial viability.
Also, employers may have a payroll processor or others handling their payroll, withholding, matching, remittance, and/or reporting responsibilities, which sometimes leads to miscommunication between the parties and may result in tax deposits and reporting not being made as required. Such miscommunication may also quickly result in mounting tax liabilities, interest and penalties that are costly and risky to the business.
To help employers avoid these problems, the new IRS initiative will monitor deposit patterns and identify employers whose payments decline or are late. Employers identified under this initiative may receive a letter reminding them of their payroll tax responsibilities and asking that they contact the IRS to discuss the situation. In addition, some employers may receive automated phone messages from the IRS providing information and assistance. Where appropriate, an IRS revenue officer will also contact some of these employers at their place of business.
The IRS has seen a surge of phone scams in recent months as scam artists call unsuspecting taxpayers impersonating an IRS agent and proceed to threaten police arrest, deportation, license revocation and other things if the taxpayer does not pay a tax amount.
The IRS reminds taxpayers to guard against all sorts of con games that arise during any filing season.
Scammers are able to alter caller ID numbers to make it look like the IRS is calling. They use fake names and bogus IRS badge numbers. They often leave "urgent" callback requests. They prey on the most vulnerable people, such as the elderly, newly arrived immigrants and those whose first language is not English. Scammers have been known to impersonate agents from IRS Criminal Investigation as well.
The Treasury Inspector General for Tax Administration (TIGTA) has received reports of roughly 290,000 contacts since October 2013 and has become aware of nearly 3,000 victims who have collectively paid over $14 million as a result of the scam, in which individuals make unsolicited calls to taxpayers fraudulently claiming to be IRS officials and demanding that they send them cash via prepaid debit cards.
As telephone scams continue across the country, the IRS recently put out a new YouTube video with a renewed warning to taxpayers not to be fooled by imposters posing as tax agency representatives. The new Tax Scams video describes some basic tips to help protect taxpayers from tax scams.
These callers may demand money or may say you have a refund due and try to trick you into sharing private information. These con artists can sound convincing when they call. They may know a lot about you.
The IRS reminds people that they can know pretty easily when a supposed IRS caller is a fake. Here are five things the scammers often do but the IRS will not do. Any one of these five things is a tell-tale sign of a scam.
Remember, too, the IRS does not use email, text messages or any social media to discuss your personal tax issue involving bills or refunds. For more information on reporting tax scams, go to www.irs.gov and type "scam" in the search box.
Additional information about tax scams is available on IRS social media sites, including YouTube http://www.youtube.com/irsvideos and Tumblr http://internalrevenueservice.tumblr.com, where people can search "scam" to find all the scam-related posts.


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John earned his BS in Engineering Physics from the University of Arizona; his BS in Electrical Engineering from the University of Arizona and his MS in Electrical Engineering from the University of Arizona.


1. The מִהפֶּכתֵ, mahpeketh (Jer 20:2; Jer 29:26; 2Ch 16:10), is supposed by some to have been rather a sort of pillory in which the head and hands were fastened than an instrument for fastening by the feet; yet, as the word, is derived from הָפִך, to twist, it may properly represent the rack for wrenching apart the joints of the entire person (see Scheid, in the Diss. Lugd. p. 986; Bochart, Hieroz. 1, 694). It may perhaps be compared with the Greek κύφων, as described in the Scholia ad Aristoph. Plut. 476; the latter with the Roman nervus (Plaut. Asin. 3, 2, 5; Capt. 5, 3, 40), which admitted, however, of being converted into a species of torture, as the legs could be drawn asunder at the will of the jailer (Biscoe, On Acts, p. 229). The prophet Jeremiah was confined in an instrument of this sort (Jer 20:2), which appears to have been a common mode of punishment in his day (29:26; A.V. "prison"), as the prisons contained a chamber for the special purpose, termed "the house of the pillory" (2Ch 16:10; A.V. "prison house").
2. סִד, sad (Job 13:27; Job 33:11), which is expressly described as a fetter for the feet, and therefore perhaps answered to our stocks.
3. עֶכֶס, ekes (Pr 7:22), was probably a fetter fastened round the ankle. The same word is used for an anklet (Isa 3:18; A.V. "tinkling ornament").
4. צַינֹק, tsinok (Jer 29:26), is, according to the Sept. and Vulg., merely a prison, but is rather the stocks proper, or some other confinement of the limbs; so Symmachus and the Hebrew interpreters generally (comp. the Arab zanak, a fetter, and the root צָנִק, which seems to signify to be straitened).
5. The ξύλον, literally wood, to which Paul and Silas were made fast (Ac 16:24) may have been " stocks" (as in Lucan, Tox, 29; Plato, De Genesis Socratis, 32), but was possibly simply a bar of wood to which they were chained by the feet. SEE PRISON.
What kind of stocks were used by the Jews, especially in the case of Jeremiah (as above), it is difficult to conjecture; whether they were encumbering clogs or fetters that did not absolutely prevent, but only embarrassed motion, or were fixed frames that kept the prisoner stationary. Both kinds were in use very anciently. The fixed kinds, properly called stocks, were of different sorts, being frames of wood with holes either for the feet only, or for the feet, the hands, and the neck at once. At Pompeii stocks have been so contrived that ten prisoners might be chained by the leg, each leg separately, by the sliding of a bar. Some of these forms of confinement particularly that which combined, in some sort, the pillory with the stocks were very painful, and are mentioned in the accounts of the sufferings of the early Christian martyrs (see Newman, Callista, p. 363. sq., where, however, the lignum of the Vulg. is confounded with the robur, or interior cell). SEE PUNISHMENT.


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text:
Process Algebra with Strategic Interleaving
J. A. Bergstra1 &
C. A. Middelburg ORCID: orcid.org/0000-0002-8725-01971
Theory of Computing Systems volume 63, pages 488–505 (2019)Cite this article
In process algebras such as ACP (Algebra of Communicating Processes), parallel processes are considered to be interleaved in an arbitrary way. In the case of multi-threading as found in contemporary programming languages, parallel processes are actually interleaved according to some interleaving strategy. An interleaving strategy is what is called a process-scheduling policy in the field of operating systems. In many systems, for instance hardware/software systems, we have to do with both parallel processes that may best be considered to be interleaved in an arbitrary way and parallel processes that may best be considered to be interleaved according to some interleaving strategy. Therefore, we extend ACP in this paper with the latter form of interleaving. The established properties of the extension concerned include an elimination property, a conservative extension property, and a unique expansion property.
In algebraic theories of processes, such as ACP [5, 7], CCS [18, 21] and CSP [12, 19], processes are discrete behaviours that proceed by doing steps in a sequential fashion. The parallel composition of two processes is usually considered to incorporate all conceivable interleavings of their steps. In each interleaving, the steps of both processes occur in some order where each time one step is taken from either of the processes. According to many, this interpretation of parallel composition, called arbitrary interleaving, is a plausible, general, if not idealized interpretation. Underlying the usual justification of this claim is the assumption that at most one step is done at each point in time. However, others contend that interpretations in which this simplifying assumption is fulfilled are not faithful. Be that as it may, arbitrary interleaving turns out to be appropriate for many applications and to facilitate formal algebraic reasoning.
Multi-threading as found in programming languages such as Java [16] and C# [17], gives rise to parallel composition of processes. In the case of multi-threading, however, the steps of the processes concerned are interleaved according to a process-scheduling policy. We use the term strategic interleaving for this more constrained form of interleaving; and we further use the term interleaving strategy instead of process-scheduling policy. Arbitrary interleaving and strategic interleaving are quite different. The following points illustrate this: (a) whether the interleaving of certain processes leads to inactiveness depends on the interleaving strategy used; (b) sometimes inactiveness occurs with a particular interleaving strategy whereas arbitrary interleaving would not lead to inactiveness and vice versa.
In previous work, we studied strategic interleaving in the setting of thread algebra, which is built on a specialized algebraic theory of processes devoted to the behaviours produced by instruction sequences under execution (see e.g. [8,9,10]). We have, for instance, given demonstrations of points (a) and (b) above in this setting. Nowadays, multi-threading is often used in the implementation of systems. Because of this, in many systems, for instance hardware/software systems, we have to do with parallel processes that may best be considered to be interleaved in an arbitrary way as well as parallel processes that may best be considered to be interleaved according to some interleaving strategy. This is what motivated us to do the work presented in this paper, namely extending ACP such that it supports both arbitrary interleaving and strategic interleaving.
To our knowledge, there exists no work on strategic interleaving in the setting of a general algebraic theory of processes like ACP, CCS and CSP. In the work presented in this paper, we consider strategic interleaving where process creation is taken into account. The approach to process creation followed in this paper originates from the one first followed in [6] to extend ACP with process creation and later followed in [2, 3, 11] to extend different timed versions of ACP with process creation. The only other approach that we know of is the approach, based on [1], that has for instance been followed in [4, 14]. However, with that approach, it is most unlikely that data about the creation of processes can be made available for the decision making concerning the strategic interleaving of processes.
The extension of ACP presented in this paper covers a generic interleaving strategy that can be instantiated with different specific interleaving strategies. We found two plausible ways to deal with inactiveness of a process whose steps are being interleaved with steps of other processes in the case of strategic interleaving. This gives rise to two plausible extensions of ACP. We will treat only one of them in detail.
The rest of this paper is organized as follows. In Section 2, we review ACP (Section 2.1), guarded recursion in the setting of ACP (Section 2.2), and some relevant results about the latter (Section 2.3). In Section 3, we extend ACP with strategic interleaving (Section 3.1) and establish some important properties of the extension (Section 3.2). In Section 4, we make some concluding remarks.
ACP with Guarded Recursion
In this section, we give a survey of ACP (Algebra of Communicating Processes), guarded recursion in the setting of ACP, and some relevant results about the extension of ACP with guarded recursion. For a comprehensive overview, the reader is referred to [5, 13].
In ACP, it is assumed that a fixed but arbitrary set A of actions, with δ∉A, has been given. We write Aδ for A ∪{δ}. It is further assumed that a fixed but arbitrary commutative and associative communication function γ: Aδ ×Aδ →Aδ, with γ(δ,a) = δ for all a ∈Aδ, has been given. The function γ is regarded to give the result of synchronously performing any two actions for which this is possible, and to give δ otherwise.
The signature of ACP consists of the following constants and operators:
for each a ∈A, the action constant a ;
the inaction constant δ ;
the binary alternative composition operator _+_ ;
the binary sequential composition operator _ ⋅_ ;
the binary parallel composition operator _∥_ ;
the binary left merge operator ;
the binary communication merge operator _∣_ ;
for each H ⊆A, the unary encapsulation operator ∂H.
We assume that there are infinitely many variables, including x, y, z. Terms are built as usual. We use infix notation for the binary operators. The precedence conventions used with respect to the operators of ACP are as follows: + binds weaker than all others, ⋅ binds stronger than all others, and the remaining operators bind equally strong.
The constants and operators of ACP can be explained as follows:
the constant a denotes the process that is only capable of first performing action a and next terminating successfully;
the constant δ denotes the process that is not capable of doing anything;
a closed term of the form t + t′ denotes the process that behaves either as the process denoted by t or as the process denoted by t′, but not both;
a closed term of the form t ⋅ t′ denotes the process that first behaves as the process denoted by t and on successful termination of that process it next behaves as the process denoted by t′;
a closed term of the form t∥t′ denotes the process that behaves as the process that proceeds with the processes denoted by t and t′ in parallel;
a closed term of the form denotes the process that behaves the same as the process denoted by t∥t′, except that it starts with performing an action of the process denoted by t;
a closed term of the form t∣t′ denotes the process that behaves the same as the process denoted by t∥t′, except that it starts with performing an action of the process denoted by t and an action of the process denoted by t′ synchronously;
a closed term of the form ∂H(t) denotes the process that behaves the same as the process denoted by t, except that actions from H are blocked.
The operators and ∣ are of an auxiliary nature. They are needed to axiomatize ACP.
The axioms of ACP are the equations given in Table 1. In these equations, a, b and c stand for arbitrary constants of ACP, and H stands for an arbitrary subset of A. Moreover, γ(a,b) stands for the action constant for the action γ(a,b). In D1 and D2, side conditions restrict what a and H stand for.
Table 1 Axioms of ACP
In other presentations of ACP, γ(a,b) is regularly replaced by a∣b in CM5–CM7. By CM12, which is more often called CF, these replacements give rise to an equivalent axiomatization. In other presentations of ACP, CM10 and CM11 are usually absent. These equations are not derivable from the other axioms, but all there closed substitution instances are derivable from the other axioms. Moreover, CM10 and CM11 hold in virtually all models of ACP that have been devised.
In the sequel, we will use the sum notation \({\sum }_{i<n} t_{i}\). For each \(i \in \mathbb {N}\), let ti be a term of ACP or an extension of ACP . Then \({\sum }_{i<0} t_{i} = \delta \) and, for each \(n \in \mathbb {N}_1\),Footnote 1 the term \({\sum }_{i<n} t_{i}\) is defined by induction on n as follows: \({\sum }_{i<1} t_{i} = t_{0}\) and \({\sum }_{i<n + 1} t_{i} ={\sum }_{i<n} t_{i} {+} t_{n}\).
Guarded Recursion
A closed ACP term denotes a process with a finite upper bound to the number of actions that it can perform. Guarded recursion allows the description of processes without a finite upper bound to the number of actions that it can perform.
Let T be ACP or a concrete extensions of ACP,Footnote 2 and let t be a T term containing a variable X. Then an occurrence of X in t is guarded if t has a subterm of the form a ⋅ t′ where a ∈A and t′ is a T term containing this occurrence of X.
Let T be ACP or a concrete extension of ACP. Then a T term t is a guardedT term if all occurrences of variables in t are guarded.
Let T be ACP or a concrete extension of ACP. Then a guarded recursive specification over T is a finite or countably infinite set of recursion equations E = {X = tX∣X ∈ V }, where V is a set of variables and each tX is either a guarded T term in which variables other than the variables from V do not occur or a T term rewritable to such a term using the axioms of T in either direction and/or the equations in E, except the equation X = tX, from left to right. We write V(E) for the set of all variables that occur in E. A solution of E in some model of T is a set {PX∣X ∈V(E)} of elements of the carrier of that model such that the equations of E hold if, for all X ∈V(E), X is assigned PX. We are only interested models of ACP and concrete extensions of ACP in which guarded recursive specifications have unique solutions.
Let T be ACP or a concrete extension of ACP. We extend T with guarded recursion by adding constants for solutions of guarded recursive specifications over T and axioms concerning these additional constants. For each guarded recursive specification E over T and each X ∈V(E), we add a constant standing for the unique solution of E for X to the constants of T. The constant standing for the unique solution of E for X is denoted by 〈X|E〉. We use the following notation. Let t be a T term and E be a guarded recursive specification. Then we write 〈t|E〉 for t with, for all X ∈V(E), all occurrences of X in t replaced by 〈X|E〉. We add the equation RDP and the conditional equation RSP given in Table 2 to the axioms of T. In RDP and RSP, X stands for an arbitrary variable, tX stands for an arbitrary T term, and E stands for an arbitrary guarded recursive specification over T. Side conditions restrict what X, tX and E stand for. We write Trec for the resulting theory.
Table 2 Axioms for guarded recursion
The equations 〈X|E〉 = 〈tX|E〉 for a fixed E express that the constants 〈X|E〉 make up a solution of E. The conditional equations E ⇒ X = 〈X|E〉 express that this solution is the only one.
In extensions of ACP whose axioms include RSP, we have to deal with conditional equational formulas with a countably infinite number of premises. Therefore, infinitary conditional equational logic is used in deriving equations from the axioms of extensions of ACP whose axioms include RSP. A complete inference system for infinitary conditional equational logic can be found in, for example, [15]. It is noteworthy that derivations are allowed to be of countably infinite length in infinitary conditional equational logic.
Some Results about Guarded Recursion
This section is concerned with legitimate ways of manipulating guarded recursive specifications and with guarded terms of a special form.
Let T be ACP or a concrete extension of ACP. Then, each guarded recursive specification over T can be manipulated in several ways that are justified by RDP and RSP.
Proposition 1 (Manipulation)
Let T be ACP or a concrete extension of ACP. Then, for all guarded recursive specifications E over T, for allX ∈V(E):
ifY = tY ∈ Eand\(t_{Y} = t^{\prime }_{Y}\)isderivable from the axioms of T, then\(\langle X \vert E \rangle = \langle X \vert (E \setminus \nolinebreak [4]\left \{ Y = t_{Y} \right \}) \cup \left \{ Y = t^{\prime }_{Y} \right \} \rangle \)isderivable from the axioms of T, RDP and RSP;
ifY = tY ∈ E,Z = tZ ∈ E,and\(t^{\prime }_{Y}\)istYwith some occurrence of Z intYreplaced bytZ,then\(\langle X \vert E \rangle = \langle X \vert (E \setminus \left \{ Y = t_{Y} \right \}) \cup \left \{ Y = t^{\prime }_{Y} \right \} \rangle \)isderivable from the axioms of T, RDP and RSP;
ifY ∉V(E) andtYis a guarded T term in which variables other than the variables from V(E) do not occur, then 〈X|E〉 = 〈X|E ∪{Y = tY}〉 is derivable from the axioms of T, RDP and RSP.
In case (1), first we apply RDP for each recursion equation in E, next we apply \(t_{Y} = t^{\prime }_{Y}\) to 〈Y |E〉 = 〈tY|E〉, and finally we apply RSP to the resulting set of equations. In case (2), first we apply RDP for each recursion equation in E, next we apply 〈Z|E〉 = 〈tZ|E〉 to 〈Y |E〉 = 〈tY|E〉, and finally we apply RSP to the resulting set of equations. In case (3), we first apply RDP for each recursion equation in E ∪{Y = tY} and then we apply RSP to the resulting set of equations.Footnote 3 □
Proposition 1 will be used in the proof of Theorem 1 in Section 3.2.
Let T be ACP or a concrete extension of ACP. Then the set HNF of head normal forms ofT is inductively defined by the following rules:
δ ∈HNF;
if a ∈A, then a ∈HNF;
if a ∈A and t is a T term, then a ⋅ t ∈HNF;
if t,t′∈HNF, then t + t′∈HNF.
Each head normal form of T is derivably equal to a head normal form of the form \({\sum }_{i<n} a_{i} \cdot t_{i} {+} {\sum }_{j<m} b_{i}\), where \(n,m \in \mathbb {N}\), for all \(i \in \mathbb {N}\) with i < n, ai ∈A and ti is a T term, and, for all \(j \in \mathbb {N}\) with j < m, bj ∈A.
It is well-known that each guarded ACPrec term is derivably equal to a head normal form of ACPrec (see also Lemma 2.4.7 in [5]).
Proposition 2 (Head normal form)
For each guarded ACPrecterm t, there exists a head normal formt′of ACP such thatt = t′is derivable from the axioms of ACPrec.
The proof is analogous to the proof of Proposition 3 in Section 3.2. □
Strategic Interleaving
In this section, we extend ACP with strategic interleaving, i.e. interleaving according to some interleaving strategy. Interleaving strategies are abstractions of scheduling algorithms. Interleaving according to some interleaving strategy is what really happens in the case of multi-threading as found in contemporary programming languages.
ACP with Strategic Interleaving
In the extension of ACP with strategic interleaving presented below, it is expected that an interleaving strategy uses the interleaving history in one way or another to make process-scheduling decisions.
The set \(\mathcal {H}\) of interleaving histories is the subset of \({(\mathbb {N}_1 \times \mathbb {N}_1)}^{\ast }\) that is inductively defined by the following rules:
\({\langle \rangle } \in \mathcal {H}\);
if i ≤ n, then \(\left (i,n \right ) \in \mathcal {H}\);
j ≤ n, and n − 1 ≤ m ≤ n + 1, then h ↷ (i,n) ↷ (j,m) ∈. Footnote 4
The intuition concerning interleaving histories is as follows: if the k th pair of an interleaving history is (i,n), then the i th process got a turn in the k th interleaving step and after its turn there were n processes to be interleaved. The number of processes to be interleaved may increase due to process creation (introduced below) and decrease due to successful termination of processes.
The presented extension of ACP is called ACP + SI (ACP with Strategic Interleaving). It covers a generic interleaving strategy that can be instantiated with different specific interleaving strategies that can be represented in the way that is explained below.
In ACP + SI, it is assumed that the following has been given:
a fixed but arbitrary set S;
for each \(n \in \mathbb {N}_1\), a fixed but arbitrary function \(\sigma _{n} {:} \mathcal {H} \times S \to \left \{ 1,\ldots ,n \right \}\);
for each \(n \in \mathbb {N}_1\), a fixed but arbitrary function \(\vartheta _{n} {:} \mathcal {H} \times S \times \left \{ 1,\ldots ,n \right \} \times \mathsf {A} \to S\).
The elements of S are called control states, σn is called an abstract scheduler (for n processes), and 𝜗n is called a control state transformer (for n processes). The intuition concerning S, σn, and 𝜗n is as follows:
the control states from S encode data that are relevant to the interleaving strategy, but not derivable from the interleaving history;
if σn(h,s) = i, then the i th process gets the next turn after interleaving history h in control state s;
if 𝜗n(h,s,i,a) = s′, then s′ is the control state that arises from the i th process doing a after interleaving history h in control state s.
Thus, S, \({\left \langle \sigma _{n} \right \rangle }_{n \in \mathbb {N}_1}\), and \({\left \langle \vartheta _{n} \right \rangle }_{n \in \mathbb {N}_1}\) make up a way to represent an interleaving strategy. This way to represent an interleaving strategy is engrafted on [22].
Consider the case where S is a singleton set, for each \(n \in \mathbb {N}_1\), σn is defined by
and, for each \(n \in \mathbb {N}_1\), 𝜗n is defined by
$$\vartheta{n}(h,s,i,a) = s . $$
In this case, the interleaving strategy corresponds to the round-robin scheduling algorithm. More advanced strategies can be obtained if the scheduling makes more advanced use of the interleaving history and the control state. The interleaving history may, for example, be used to factor the individual lifetimes of the processes to be interleaved and their creation hierarchy into the process-scheduling decision making. Individual properties of the processes to be interleaved that depend on the actions performed by them can be taken into account by making use of the control state. The control state may, for example, be used to factor the processes being interleaved that currently wait to acquire a lock from a process that manages a shared resource into the process-scheduling decision making.Footnote 5
In ACP + SI, it is also assumed that a fixed but arbitrary set D of data and a fixed but arbitrary function ϕ:D → P, where P is the set of all closed terms over the signature of ACP + SI (given below), have been given and that, for each d ∈ D and a,b ∈A, \(\mathsf {cr}(d),\overline {\mathsf {cr}}(d) \in \mathsf {A}\), γ(cr(d),a) = δ, and γ(a,b)≠cr(d). The action cr(d) can be considered a process creation request and the action \(\overline {\mathsf {cr}}(d)\) can be considered a process creation act. They represent the request to start the process denoted by ϕ(d) in parallel with the requesting process and the act of carrying out that request, respectively.
The signature of ACP + SI consists of the constants and operators from the signature of ACP and in addition the following operators:
for each \(n \in \mathbb {N}_1\), \(h \in \mathcal {H}\), and s ∈ S, the n-ary strategic interleaving operator \(\parallel ^{n}_{h,s}\);
for each \(n,i \in \mathbb {N}_1\) with i ≤ n, \(h \in \mathcal {H}\), and s ∈ S, the n-ary positional strategic interleaving operator .
The strategic interleaving operators can be explained as follows:
a closed term of the form \(\parallel ^{n}_{h,s}(t_{1},\ldots ,t_{n})\) denotes the process that results from interleaving of the n processes denoted by t1,…,tn after interleaving history h in control state s, according to the interleaving strategy represented by S, \({\left \langle \sigma _{n} \right \rangle }_{n \in \mathbb {N}_1}\), and \({\left \langle \vartheta _{n} \right \rangle }_{n \in \mathbb {N}_1}\).
The positional strategic interleaving operators are auxiliary operators used to axiomatize the strategic interleaving operators. The role of the positional strategic interleaving operators in the axiomatization is similar to the role of the left merge operator found in ACP.
The axioms of ACP + SI are the axioms of ACP and in addition the equations given in Table 3. In the additional equations, n and i stand for arbitrary numbers from \(\mathbb {N}_1\) with i ≤ n, h stands for an arbitrary interleaving history from \(\mathcal {H}\), s stands for an arbitrary control state from S, a stands for an arbitrary action constant that is not of the form cr(d) or \(\overline {\mathsf {cr}}(d)\), and d stands for an arbitrary datum d from D.
Table 3 Axioms for strategic interleaving
Axiom SI2 expresses that, in the event of inactiveness of the process whose turn it is, the whole becomes inactive immediately. A plausible alternative is that, in the event of inactiveness of the process whose turn it is, the whole becomes inactive only after all other processes have terminated or become inactive. In that case, the functions \(\vartheta _{n} {:}~ \mathcal {H} \times S \times \left \{ 1,\ldots ,n \right \} \times \mathsf {A} \to S\) must be extended to functions 𝜗n : × S ×{1,…,n} × (A ∪{δ}) → S and axiom SI2 must be replaced by the axioms in Table 4.
Table 4 Alternative axioms for SI2
In (ACP + SI)rec, i.e. ACP + SI extended with guarded recursion in the way described in Section 2, the processes that can be created are restricted to the ones denotable by a closed ACP + SI term. This restriction stems from the requirement that ϕ is a function from D to the set of all closed ACP + SI terms. The restriction can be removed by relaxing this requirement to the requirement that ϕ is a function from D to the set of all closed (ACP + SI)rec terms. We write (ACP + SI)\(_{\text {rec}}^{+}\) for the theory resulting from this relaxation. In other words, (ACP + SI)\(_{\text {rec}}^{+}\) differs from (ACP + SI)rec in that it is assumed that a fixed but arbitrary function ϕ : D → P, where P is the set of all closed terms over the signature of (ACP + SI)rec, has been given.
It is customary to associate transition systems with closed terms of the language of an ACP-like theory of processes by means of structural operational semantics and to use this to construct a model in which closed terms are identified if their associated transition systems are bisimilar. The structural operational semantics of ACP can be found in [5, 13]. The additional transition rules for the strategic interleaving operators and the positional strategic interleaving operators are given in Appendix A.
Basic Properties of ACP with Strategic Interleaving
In this section, the subject of concern is the connection between ACP and ACP + SI. The main results are an elimination result and a conservative extension result. We begin with establishing some results that will be used in the proof of those main results.
Each guarded ACP + SI term is derivably equal to a head normal form of ACP + SI.
For each guarded ACP + SIterm t, there exists a head normal formt′of ACP + SIsuch thatt = t′is derivable from the axioms of ACP + SI.
The proof is straightforward by induction on the structure of t. The case where t is of the form δ and the case where t is of the form a (a ∈A) are trivial. The case where t is of the form t1 ⋅ t2 follows immediately from the induction hypothesis and the claim that, for all head normal forms t1 and t2 of ACP + SI, there exists a head normal form t′ of ACP + SI such that t1 ⋅ t2 = t′ is derivable from the axioms of ACP + SI. This claim is easily proved by induction on the structure of t1. The case where t is of the form t1 + t2 follows immediately from the induction hypothesis. The cases where t is of one of the forms are proved along the same lines as the case where t is of the form t1 ⋅ t2. In the case that t is of the form t1 ∣ t2, each of the cases to be considered in the inductive proof of the claim demands a proof by induction on the structure of t2. In the case that t is of the form the claim is of course proved by induction on the structure of ti instead of t1. The case that t is of the form t1 ∥ t2 follows immediately from the case that t is of the form and the case that t is of the form t1 | t2. The case that t is of the form \(\parallel ^{n}_{h,s}{(t_1,\ldots ,t_n)}\) follows immediately from the case that t is of the form . Because t is a guarded ACP+SI term, the case where t is a variable cannot occur. □
Each of the four theorems to come refer to several process algebras. It is implicit that the same set A of actions and the same communication function γ are assumed in the process algebras referred to.
Each guarded recursive specification over ACP + SI can be reduced to a guarded recursive specification over ACP.
Theorem 1 (Reduction)
For each guarded recursive specification E over ACP + SI and eachX ∈V(E),there exists a guarded recursive specificationE′over ACP such that 〈X|E〉 = 〈X|E′〉 is derivable from the axioms of ( ACP + SI)\(_{\text {rec}}^{+}\).
Let E be a guarded recursive specification over ACP + SI. Assume that, for each equation X = tX from E, tX is a guarded ACP + SI term. It follows from Proposition 1 that this assumption does not lead to loss of generality.
Let X = tX be an equation from E. Now, by Proposition 3, there exist \(n,m \in \mathbb {N}\) such that, for each \(i \in \mathbb {N}\) with i < n and \(j \in \mathbb {N}\) with j < m, there exist an ai ∈A, an ACP + SI term ti, and a bj ∈A such that \(t_{X} = {\sum }_{i<n} a_{i} \cdot t_{i} {+} {\sum }_{j<m} b_{j}\) is derivable from the axioms of ACP+SI. For each \(i \in \mathbb {N}\) with i < n, let \(t^{\prime }_{i}\) be ti with, for each equation Y = tY from E, each unguarded occurrence of Y in ti replaced by the guarded ACP + SI term tY. For each \(i \in \mathbb {N}\) with i < n, by its construction, the term \(t^{\prime }_{i}\) is a guarded ACP + SI terms in which variables other than the ones from V(E) do not occur. Now, by Proposition 1, the equation \(X_{i} = t^{\prime }_{i}\), where Xi is a fresh variable, can be added to E for each \(i \in \mathbb {N}\) with i < n and the equation X = tX can be replaced by the equation \(X = {\sum }_{i<n} a_{i} \cdot X_{i} {+} {\sum }_{j<m} b_{j}\) in E. The other equations from E can be replaced by a set of equations in the same way as the equation X = tX.
The set of equations so obtained can be manipulated following the same procedure as in the case of E, but the manipulation can be restricted to the added equations. Repeating this procedure, perhaps countably infinitely many times, we obtain a guarded recursive specification E′ over ACP for which 〈X|E〉 = 〈X|E′〉 is derivable from the axioms of (ACP+SI)\(_{\text {rec}}^{+}\). □
The next three theorems will be proved by means of term rewriting systems. In Appendix B, basic definitions and results regarding term rewriting systems are collected. This appendix also serves to fix the terminology on term rewriting systems used in the proofs of the next three theorems.
Each closed (ACP+SI)\(_{\text {rec}}^{+}\) term is derivably equal to a closed ACPrec term.
Theorem 2 (Elimination)
For each closed ( ACP + SI)\(_{\text {rec}}^{+}\)term t, there exists a closed ACPrectermt′such thatt = t′is derivable from the axioms of ( ACP + SI)\(_{\text {rec}}^{+}\).
We prove this by means of a term rewriting system that takes equational axioms of (ACP+SI)\(_{\text {rec}}^{+}\) and equations derivable from the axioms of (ACP+SI)\(_{\text {rec}}^{+}\) as rewrite rules. Thus, the proof boils down to showing that (a) the term rewriting system concerned has the property that each (ACP+SI)\(_{\text {rec}}^{+}\) term has a unique normal form modulo axioms A1 and A2 and (b) each closed (ACP+SI)\(_{\text {rec}}^{+}\) term that is a normal form modulo axioms A1 and A2 is a closed ACPrec term. Henceforth, we will write AC for the set of equations that consists of axioms A1 and A2.
Let R be a set of equations that contains for each guarded recursive specification E over ACP + SI and X ∈V(E) an equation 〈X|E〉 = 〈X|E′〉, where E′ is a guarded recursive specification over ACP, that is derivable from the axioms of (ACP + SI)\(_{\text {rec}}^{+}\). Such a set R exists by Theorem 1. Consider the term rewriting system \(\mathcal {R}((\text {ACP}+\text {SI})_{\text {rec}}^{+})\) that consists of the axioms of (ACP + SI)\(_{\text {rec}}^{+}\), with the exception of A1, A2, RDP, and RSP, and the equations from R taken as rewrite rules.
We show that \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) has the property that each (ACP + SI)\(_{\text {rec}}^{+}\) term has a unique normal form modulo AC by proving that \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+})\) is terminating modulo AC and confluent modulo AC.
First, we show that \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+})\) is terminating modulo AC. This can be proved by the reduction ordering > induced by the extended integer polynomials 𝜃(t) associated with (ACP + SI)\(_{\text {rec}}^{+}\) terms t as follows:Footnote 6
where it is assumed that, for each variable X over processes, X is a variable over integers. The following is easy to see: (a) t > t′ for all rewrite rules t = t′ of \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) and (b) t > t′ implies s > s′ for all (ACP + SI)\(_{\text {rec}}^{+}\) terms s and s′ for which t = s and t′ = s′ are derivable from AC.Footnote 7 Hence, \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) is terminating modulo AC.
Next, we show that \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) is confluent modulo AC. It follows from Theorems 5 and 16 in [20] and the fact that \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) is terminating modulo AC that \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) is confluent modulo AC if it does not give rise to critical pairs modulo AC that are not convergent. It is easy to see that all critical pairs modulo AC arise from overlappings of (a) A3 on A4, CM4, CM8, CM9, D3, and SI8, (b) A6 on A4, CM4, CM8, CM9, D3, and SI8, (c) A7 on CM3, CM5, CM6, CM7, D4, and SI5, (d) CM10 on CM9, and (e) CM11 on CM8. It is straightforward to check that all critical pairs concerned are convergent. Hence, \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) is confluent modulo AC.
Above, we have shown that \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) is terminating modulo AC and confluent modulo AC and by this that it has the property that each (ACP + SI)\(_{\text {rec}}^{+}\) term has a unique normal form modulo AC. It remains to be shown that each closed (ACP + SI)\(_{\text {rec}}^{+}\) term that is a normal form modulo AC is a closed ACPrec term. It is not hard to see that, for each closed (ACP + SI)\(_{\text {rec}}^{+}\) term in which other operators than + and ⋅ occur, a reduction step modulo AC is still possible in \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)). Because a reduction step modulo AC is impossible for a normal form modulo AC, no other operators than + or ⋅ can occur in a closed (ACP + SI)\(_{\text {rec}}^{+}\) term that is a normal form modulo AC. Hence, each closed (ACP + SI)\(_{\text {rec}}^{+}\) term that is a normal form modulo AC is a closed ACPrec term. □
Each equation between closed ACP terms that is derivable in ACP + SI is also derivable in ACP.
Theorem 3 (Conservative extension)
For each two closed ACP terms t andt′,t = t′is derivable from the axioms of ACP + SI only ift = t′is derivable from the axioms of ACP.
We prove this by means of a restriction of the term rewriting system from the proof of Theorem 2. Consider the term rewriting system \(\mathcal {R}\)(ACP + SI) that consists of the axioms of ACP + SI, with the exception of A1 and A2. \(\mathcal {R}\)(ACP + SI) is \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) restricted to ACP + SI terms. Just like \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)), \(\mathcal {R}\)(ACP + SI) is terminating modulo AC and confluent modulo AC. The proofs of these properties for \(\mathcal {R}\)((ACP + SI)\(_{\text {rec}}^{+}\)) carry over to \(\mathcal {R}\)(ACP + SI).
Let t and t′ be two closed ACP terms such that t = t′ is derivable from the axioms of ACP + SI. Reduce t and t′ to normal forms s and s′, respectively, by means of the term rewriting system \(\mathcal {R}\)(ACP + SI). By Theorem 5 in [20], being confluent modulo AC is equivalent to being Church-Rosser modulo AC for a term rewriting system that is terminating modulo AC. This means that t and t′ have the same normal form modulo AC. In other words, s = s′ is derivable from axioms A1 and A2. Because (a) no other operators than + and ⋅ occur in t and t′ and (b) no rewrite rule introduces one or more of the other operators if one or more of the other operators was not already in its left-hand side, each rewrite rule applied in the reduction from t to s or the reduction from t′ to s′ is one of the axioms of ACP. Therefore, each rewrite rule involved in the reduction from t to s or the reduction from t′ to s′ is an axiom of ACP. Hence, the reduction from t to s shows that t = s is derivable from the axioms of ACP and the reduction from t′ to s′ shows that t′ = s′ is derivable from the axioms of ACP. From this and the fact that s = s′ is derivable from axioms A1 and A2, it follows t = t′ is derivable from the axioms of ACP. □
The following theorem concerns the expansion of minimal models of ACP to models of ACP + SI.
Theorem 4 (Unique expansion)
Each minimal model of ACP has a unique expansion to a model of ACP + SI.
We write \(f^{\mathcal {A}}\), where \(\mathcal {A}\) is a model of ACP or ACP + SI and f is a constant or operator from the signature of \(\mathcal {A}\), for the interpretation of f in \(\mathcal {A}\). We write \(t^{\mathcal {A}}\), where \(\mathcal {A}\) is a model of ACP or ACP + SI and t is a closed term over the signature of \(\mathcal {A}\), for the interpretation of t in \(\mathcal {A}\).
Let \(\mathcal {A}\) be a minimal model of ACP. Let CT be a function from the carrier of \(\mathcal {A}\) to the set of all closed ACP terms such that, for each element p of the carrier of \(\mathcal {A}\), \(\mathit {CT}(p)^{\mathcal {A}} = p\). Because \(\mathcal {A}\) is a minimal model of ACP, CT(p) is a total function. We write \(\underline {p}\), where p is an element of the carrier of \(\mathcal {A}\), for CT(p). Let NF be a function from the set of all closed ACP + SI terms to the set of all closed ACP terms such that, for each closed ACP + SI term t, NF(t) is one of the normal forms that t can be reduced to by means of the term rewriting system \(\mathcal {R}\)(ACP + SI) from the proof of Theorem 3.
We start with constructing an expansion of \(\mathcal {A}\) with interpretations of the additional operators of ACP + SI. Let \(\mathcal {B}\) be the expansion of \(\mathcal {A}\) with interpretations of the additional operators of ACP + SI where these interpretations are defined as follows:
for all p1,…,pn from the carrier of \(\mathcal {A}\).
We proceed with proving that \(\mathcal {B}\) is a model of ACP + SI. By Theorem 3, it is sufficient to prove that \(\mathcal {B}\) satisfies axioms SI1–SI8. By its construction, \(\mathcal {B}\) is a minimal algebra and consequently it is sufficient to prove that \(\mathcal {B}\) satisfies all closed substitution instances of SI1–SI8. We use the following three claims to prove this:
for all closed substitution instances t = t′ of SI1–SI8, \(t^{\mathcal {B}} = {\mathit {NF}(t)}^{\mathcal {A}}\);
for all closed substitution instances t = t′ of SI1–SI8, \(t^{\prime \mathcal {B}} = {\mathit {NF}(t^{\prime })}^{\mathcal {A}}\);
for all closed substitution instances t = t′ of SI1–SI8, \(\mathit {NF}(t)^{\mathcal {A}} = \mathit {NF}(t^{\prime })^{\mathcal {A}}\).
The first claim follows easily from the definitions of the interpretations of the additional operators of ACP + SI given above. The second claim follows easily from these definitions and the proof of the first claim. Because \(\mathcal {R}\)(ACP + SI) is Church-Rosser modulo AC (see the proof of Theorem 3), we have that NF(t) = NF(t′) is derivable from axioms A1 and A2. From this, the third claim follows immediately. It is an immediate consequence of the three claims that \(\mathcal {B}\) satisfies all closed substitution instances of SI1–SI8.
We still have to prove that \(\mathcal {B}\) is the only expansion of \(\mathcal {A}\) to a model of ACP + SI. We can prove this by contradiction. Assume that \(\mathcal {C}\) is an expansion of \(\mathcal {A}\) to a model of ACP + SI that differs from \(\mathcal {B}\). Then at least one of the additional operators of ACP + SI has different interpretations in \(\mathcal {B}\) and \(\mathcal {C}\). By the definitions of the interpretations of the additional operators of ACP + SI in \(\mathcal {B}\), this means that there exists a closed ACP + SI term t such that \(t^{\mathcal {C}} \neq \mathit {NF}(t)^{\mathcal {A}}\). Moreover, because because t = NF(t) is derivable from the axioms of ACP + SI, \(t^{\mathcal {C}} = \mathit {NF}(t)^{\mathcal {C}}\). Hence, \(\mathit {NF}(t)^{\mathcal {C}} \neq \mathit {NF}(t)^{\mathcal {A}}\). Because NF(t) is a closed ACP term, this contradicts the fact that \(\mathcal {C}\) is an expansion of \(\mathcal {A}\). □
We have extended the algebraic theory of processes known as ACP with the form of interleaving that underlies multi-threading as found in contemporary programming languages. We have also established some basic properties of the resulting theory. It remains an open question whether strategic interleaving is definable in an established extension of ACP.
We write \(\mathbb {N}_1\) for the set \(\left \{ n \in \mathbb {N} \mid n \geq 1 \right \}\) of positive natural numbers.
A concrete extension of ACP is an extension of ACP that does not offer the possibility of abstraction from certain actions. All extensions of ACP introduced in this paper are concrete extensions.
Further details on cases (1) and (2) can be found in the proof of Theorem 4.3.2 from [13].
We write 〈〉 for the empty sequence, d for the sequence having d as sole element, and for the concatenation of sequences α and α′. We assume that the usual identities, such as 〈 〉↷ α = α and , hold.
In [8], various examples of interleaving strategies are given in the setting of the relatively unknown thread algebra. The representation of the more serious of these examples in the current setting demands nontrivial use of the control state.
Here, extended polynomials differ from polynomials in that both variables and expressions of the form 2X, where X is a variable, are allowed where only variables are allowed in polynomials.
We do not have that t > t′ for all rewrite rules t = s if SI2 is replaced by SI2a and SI2b (see Table 4).
See e.g. Definition 10 in [20] for the definitions of most general unifier and complete set of unifiers modulo E.
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We thank an anonymous referee for carefully reading a preliminary version of this paper, for pointing out an error in one of the proofs, and for suggesting improvements of the presentation.
Informatics Institute, Faculty of Science, University of Amsterdam, Science Park 904, 1098, XH, Amsterdam, The Netherlands
J. A. Bergstra & C. A. Middelburg
J. A. Bergstra
C. A. Middelburg
Correspondence to C. A. Middelburg.
Appendix A: Structural Operational Semantics of ACP + SI
It is customary to associate transition systems with closed terms of the language of an ACP-like theory about processes by means of structural operational semantics and to use this to construct a model in which closed terms are identified if their associated transition systems are bisimilar. The structural operational semantics of ACP can be found in [5, 13]. The additional transition rules for the strategic interleaving operators and the positional strategic interleaving operators are given in Table 5. In this table,
\(t \overset {a}{\rightarrow } \surd \) indicates that t is capable of performing action a and then terminating successfully;
\(t \overset {a}{\rightarrow } t^{\prime }\) indicates that t is capable of performing action a and then behaving as t′.
The transition rules for the strategic interleaving operator are similar to the transition rules for the positional strategic interleaving operators. However, each transition rule for the strategic interleaving operator has the side-condition i = σn(h,s).
Table 5 Transition rules for strategic interleaving
Appendix B: Term Rewriting Systems
In this appendix, basic definitions and results regarding term rewriting systems are collected. This appendix also serves to fix the terminology on term rewriting systems used in the proofs that make use of term rewriting systems.
We assume that a set of constants, a set of operators with fixed arities, and a set of variables have been given; and we consider an arbitrary term rewriting system \(\mathcal {R}\) for terms that can be built from the constants, operators, and variables in these sets.
A rewrite rule is a pair of terms t → s, where t is not a variable and each variable occurring in s occurs in t as well. A term rewriting system is a set of rewrite rules.
A reduction step of \(\mathcal {R}\) is a pair t → s such that for some substitution instance t′→ s′ of a rewrite rule of \(\mathcal {R}\), t′ is a subterm of t, and s is t with t′ replaced by s′. Here, t′ is called the redex of the reduction step. A reduction of \(\mathcal {R}\) is a pair \(t \twoheadrightarrow s\) such that either t ≡ s or there exists a finite sequence t1 → t2, …, tn → tn+ 1 of consecutive reduction steps of \(\mathcal {R}\) such that t ≡ t1 and s ≡ tn+ 1.
A term tis a normal form of\(\mathcal {R}\) if there does not exist a term s such that t → s is a reduction step of \(\mathcal {R}\). A term thas a normal form in\(\mathcal {R}\) if there exists a reduction \(t \twoheadrightarrow s\) of \(\mathcal {R}\) and s is a normal form of \(\mathcal {R}\). \(\mathcal {R}\)is terminating on term t if there does not exist an infinite sequence t → t1, t1 → t2, t2 → t3,… of consecutive reduction steps of \(\mathcal {R}\). \(\mathcal {R}\)is terminating if \(\mathcal {R}\) is terminating on all terms. \(\mathcal {R}\)is confluent if for all reductions \(t \twoheadrightarrow s_{1}\) and \(t \twoheadrightarrow s_{2}\) of \(\mathcal {R}\) there exist reductions \(s_{1} \twoheadrightarrow s\) and \(s_{2} \twoheadrightarrow s\) of \(\mathcal {R}\). If \(\mathcal {R}\) is terminating and confluent, then each term has a unique normal form in \(\mathcal {R}\).
A reduction ordering for \(\mathcal {R}\) is a well-founded ordering on terms that is closed under substitutions and contexts. \(\mathcal {R}\) is terminating if and only if there exists a reduction ordering > for \(\mathcal {R}\) such that t > s for each rewrite rule t → s of \(\mathcal {R}\).
A unifier of two terms s and t is a substitution σ such that σ(s) ≡ σ(t). A critical pair of \(\mathcal {R}\) is a pair (t1,t2) of terms for which there exist rewrite rules s → s′ and t → t′ of \(\mathcal {R}\) and a 'most general unifier' σ of s and a non-variable subterm of t such that t1 ≡ σ(t″) and t2 ≡ σ(t′), where t″ is t with σ(s) replaced by σ(s′).Footnote 8 A critical pair (t1,t2) of \(\mathcal {R}\)is convergent if there exist reductions \(t_{1} \twoheadrightarrow s\) and \(t_{2} \twoheadrightarrow s\) of \(\mathcal {R}\). If \(\mathcal {R}\) is terminating, then \(\mathcal {R}\) is confluent if and only if all critical pairs of \(\mathcal {R}\) are convergent.
Henceforth, we consider an arbitrary set E of equations between terms.
A reduction step moduloE of \(\mathcal {R}\) is a pair t →Es such that there exists a reduction step t′→ s′ of \(\mathcal {R}\) such that t = t′ and s = s′ are derivable from E. A reduction moduloE of \(\mathcal {R}\) is pair \(t \twoheadrightarrow _{E} s\) such that either t = s is derivable from E or there exists a finite sequence t1 →Et2,…,tn →Etn+ 1 of consecutive reduction steps modulo E of \(\mathcal {R}\) such that t ≡ t1 and s ≡ tn+ 1.
A term tis a normal form moduloEof\(\mathcal {R}\) if there does not exist a term s such that t →Es is a reduction step modulo E of \(\mathcal {R}\). A term thas a normal form moduloE in \(\mathcal {R}\) if there exists a reduction modulo E\(t \twoheadrightarrow _{E} s\) of \(\mathcal {R}\) and s is a normal form modulo E of \(\mathcal {R}\). \(\mathcal {R}\)is terminating modulo E on term t if there does not exist an infinite sequence t →Et1, t1 →Et2, t2 →Et3, … of consecutive reduction steps modulo E of \(\mathcal {R}\). \(\mathcal {R}\)is terminating moduloE if \(\mathcal {R}\) is terminating modulo E on all terms. \(\mathcal {R}\)is confluent moduloE if for all reductions modulo E\(t \twoheadrightarrow _{E} s_{1}\) and \(t \twoheadrightarrow _{E} s_{2}\) of \(\mathcal {R}\) there exist reductions modulo E\(s_{1} \twoheadrightarrow _{E} s\) and \(s_{2} \twoheadrightarrow _{E} s\) of \(\mathcal {R}\). If \(\mathcal {R}\) is terminating modulo E and confluent modulo E, then each term has a unique normal form modulo E in \(\mathcal {R}\).
A reduction ordering > for \(\mathcal {R}\)isE-compatible if t > s implies t′ > s′ for all terms t′ and s′ for which t = t′ and s = s′ are derivable from E. \(\mathcal {R}\) is terminating modulo E if and only if there exists an E-compatible reduction ordering > for \(\mathcal {R}\) such that t > s for each rewrite rule t → s of \(\mathcal {R}\).
A unifier moduloE of two terms s and t is a substitution σ such that σ(s) = σ(t) is derivable from E. A critical pair moduloE of \(\mathcal {R}\) is a pair (t1,t2) of terms for which there exist rewrite rules s → s′ and t → t′ of \(\mathcal {R}\) and a substitution σ from a 'complete set of unifiers modulo E′ of s and a non-variable subterm of t such that t1 ≡ σ(t″) and t2 ≡ σ(t′), where t″ is t with σ(s) replaced by σ(s′).8 If \(\mathcal {R}\) is terminating modulo E, then \(\mathcal {R}\) is confluent modulo E if and only if all critical pairs modulo E of \(\mathcal {R}\) are convergent.
An E-equality step is a pair such that, for some substitution instance t′ = s′ of an equation from E, either t′ is a subterm of t and s is t with t′ replaced by s′ or s′ is a subterm of t and s is t with s′ replaced by t′. An \(\mathcal {R}\) ∪ E-equality step is a pair such that t → s is a reduction step of \(\mathcal {R}\) or s → t is a reduction step of \(\mathcal {R}\) or is an E-equality step. An \(\mathcal {R}\) ∪ E-equality is a pair such that either t ≡ s or there exists a finite sequence of consecutive \(\mathcal {R}\) ∪ E-equality steps such that t ≡ t1 and s ≡ tn+ 1. \(\mathcal {R}\) is Church-Rosser moduloE if for all \(\mathcal {R}\) ∪ E-equalities there exist reductions modulo Et ↠Es and t′ ↠Es of \(\mathcal {R}\). If \(\mathcal {R}\) is terminating modulo E, then \(\mathcal {R}\) is Church-Rosser modulo E if and only if \(\mathcal {R}\) is confluent modulo E.
Bergstra, J.A., Middelburg, C.A. Process Algebra with Strategic Interleaving. Theory Comput Syst 63, 488–505 (2019). https://doi.org/10.1007/s00224-018-9873-2
Issue Date: 15 April 2019
Process algebra
Arbitrary interleaving
Abstract scheduler
Interleaving history

redpajama_set_name: RedPajamaCommonCrawl


The release features every known SUN master and outtakes from Elvis, plus two private records and radio / concert performances from the period. This 3-CD package features 10 previously unreleased live cuts.
The accompanying book is a fully-illustrated travelogue. It details the birth of Elvis' career through facts, anecdotes, documentation and rare photos.
In the first hours of that extremely hot summer morning, I was leaning against a old leather chair, placed behind a wooden desk. Everything was very simple, melancholic even. In the desk there was a typewriter, a telephone and a old mechanical fan. I looked to my side, to the iron and glass door. The sunlight was coming thru the blinds and I tough about that poor young guy, full of dreams, entering and nodding to the lady seating on the desk. I imagined other people seating on some chairs, sweating profusely in that hot summer of 53, waiting for their opportunity. I tried to measure the size of a dream, of a desire so big to achieve something that is so distant, almost unreachable.
I thought how a simple gesture, like have the courage to open a door, can change a person's life so radically. And I asked myself if Elvis, later in his life, remembered that moment and seeing everything that surrounded him, felt a bit scared.
It was August 9, 2007, when I entered SUN Records for the very first time and my imagination flew to someday in July 1953 when he entered that same studio for the very first time to listening how he voice sounded like in a record. He paid $ 3.98 for that and changed his life and the life of millions upside down. He changed my life, yours, and the life of a guy named Ernst Jørgensen.
Since he became the official Elvis record producer and researcher in the early 90's he traveled thousands of miles and searched old dust attics, garage sales, high school's files, old newspaper archives and every fragment that could reconstruct what came before, during and in the 29 months after that first summer. Along the way he found rarities, lost memories of those who lived those days, kept in wallets or in forgotten corners of the mind and put it all together in a historic project that couldn't be better titled: A Boy From Tupelo.
There's a certain melancholy when looking at the book's cover, the definitive document of this era, I can say it right here. A young man, with eyes full of hope and innocence, totally far away from the bright lights that would shine upon him for the rest of his life, after he opened that door.
This is the Elvis of dusty highway bars, of parking lot gigs and high school gyms. Of days and nights inside a Cadillac with instruments on the rack, crossing muddy roads in the depths of USA.
In the more than 500 pages of the book, Jørgensen creates a unbelievable and meticulous day by day account of Elvis first years in the road to stardom, trough numerous pictures, newspaper clippings and documents that tells how the biggest revolution in the modern music happened.
A monumental task that took more than six years to be done and demanded more than 1700 hours of studio work. Almost everyday between July 1954 and December 1955 is documented at least by a picture, an article or an story.
The photos, of which more than 200 are never before seen, show a young man sometimes shy and reclusive off-stage and totally at easy and confident upon it. Others show legendary places of Elvis history like Eagle's Nest in Memphis and the Louisiana Hayride. Unthinkable places like the parking lot of a drugstore or the gym of a high school are also documented in pictures.
The research sometimes even revealed the color of Elvis clothing, something that the old black and white pictures can not tell.
Stories told by local DJ's, joint owners, musicians, eye witness and promoters of the shows provides even more detail and take the reader to a wonderful trip to the genesis of Elvis Presley.
"Hi Babies. Here's money to pay the bills. Don't Tell no one how much I sent. I will send more next week. There is a card in the mail. Love, Elvis."
But the almost obsessive research of Ernst Jørgensen didn't only revealed pictures and documents, it also brought up new discoveries registered in acetates and in the old tapes of Sam Phillips.
Spread trough 3 discs are unreleased takes of 'Harbor Lights', 'Blue Moon' and 'When It Rain It Really Pours'. Some are just fragments or false starts, but some, like take number 7 of 'Harbor Lights', gives the listener a wonderful insight of a yet insecure Elvis and his first try in a studio with accompaniment.
Between takes its possible to hear studio chat between the musicians. Some few and precious seconds before Blue Moon Of Kentucky registered the excitement of Elvis, Scott and Bill with the sound they were creating.
The biggest discoveries however, are in the third CD, containing radio broadcasts of the places where Elvis, Bill, and Scotty played. The most important of all is without a doubt 'Little Mama'. For decades, rumors persisted regarding its existence and those who claimed to have heard it made our imagination soar.
'Little Mama' was one of the songs from an acetate that also contained 'Tweedle Dee', 'Money Honey', 'Hearts Of Stone', 'Shake Rattle And Roll' and 'You're Heartbreaker', recorded in March 5, 1955 at the Hayride.
The acetate found the hands of Sherif Hanna, one of the biggest Elvis collectors, and was necessary Ernst tenacity to have it on the project. After years of continuous play and a disastrous attempt to have it digitally transferred, the recording deteriorated to the point of being broke to pieces.
Once in his hand, Ernst and his team dedicated a great deal of effort in its restoration. The sound quality, given the poor conditions, is more than satisfactory, making it possible to enjoy the new song and the first live version of 'You're Heartbreaker' with the same flavor of tasting something new.
The restoration job also improved significantly the sound quality of 'Hearts Of Stone'.
Other acetates also surfaced, containing unreleased performances of 'That's All Right' and 'Shake, Rattle 'n Roll'.
It doesn't matter that these recording are hindered by tons of cracks, pops and hiss. Its simply existence gives the listener the opportunity to peak into a past that was thought to be long lost. In a way the wear these recordings bears gives it a aura of mystic, like if we could enter a time machine that could only take us to a certain point, were we could see the past thru a thin, uncrossable veil. The search also revealed two small interviews, one given to Mae Boren Axton, who in the next year would write the song that would catapult Elvis to worldwide fame.
Ernst couldn't be more right. Ironically, just a few weeks after the announcement of the project, a unreleased, never before heard or know version of 'I Forgot To Remember To Forget' appeared out of nowhere, in no other place other than Youtube. It happened as the most fascinating discoveries happens, by coincidence. A nameless wire recording, long forgotten in a old record player that landed in the hands of a record player collector, clueless of the meaning and value of such treasure.
Some may point it as an unfortunate coincidence for Ernst, but it really should be seen as a glimpse of hope. Hope that not all rocks were turned, that not every corner was searched and that some old chest in a dust attic still holds a unreleased piece of Elvis. A unreleased song, picture, or performance.
We will keep dreaming with discoveries like this and until it not happens, we will keep listening to every know bit of record, everything he left behind. We'll go back to the books, to the pictures,to the movies and to the internet discussions. Most of all, to his music. He will be forever present.
And all of this because one day, he opened that door.


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Mattel is releasing 17 new Barbie dolls honoring historical and modern-day role models from around the world, ahead of International Women's Day on March 8.
Barbie's Shero program expands with 14 modern-day role models, honoring women all over the world. (Note that these Barbie dolls are not for sale, they are one of a kind for now).
Patty Jenkins, Filmmaker, U.S. – The first woman in history to helm a film with a budget in excess of 100 million dollars for Wonder Woman, which broke the record for Biggest Grossing Live-Action Film Directed by a Woman, domestic and worldwide.
Chloe Kim, Snowboarding Champion, U.S. – A first generation Asian-American snowboarding champion and the youngest woman to win a gold medal in snowboarding at the 2018 winter games at age 17.
Nicola Adams, Boxing Champion, the UK – Two-time gold medalist who is Great Britain's most successful female boxer of all time and is the only female boxer in the history of the sport to have held all four amateur titles.
Hélène Darroze, World-Renowned Chef, France – Inspirational fourth-generation French chef with three restaurants and two Michelin stars.
Hui Ruoqi, Volleyball Champion, China – Fifteenth Captain of the Chinese women's national volleyball team who also won gold in the 2016 summer games.
Leyla Piedayesh, Designer and Entrepreneur, Germany – Iranian immigrant and founder of fashion label lala Berlin that incorporates urban-cool and elegant-chic elements in its high-quality knitwear.
Lorena Ochoa, Professional Golfer, Mexico – Athlete, mother, entrepreneur and unbreakable promoter of golf in Mexico.
Sara Gama, Soccer Player, Italy – Captain of Juventus and Italian national soccer team, member of the Federal Board and President of Commission for the development of women's soccer.
Xiaotong Guan, Actress and Philanthropist, China – Ambassador for "World Life Day," a joint campaign by the United Nations Environment Program, International Fund for Animal Welfare and The Nature Conservancy, and known as the "Nation's Daughter" in China.
Yuan Tan, Prima Ballerina, China – Prima ballerina and principal dancer at the San Francisco Ballet and guest principal dancer at the Hong Kong Ballet.
Katherine Johnson – Pioneer in mathematics who broke through barriers of race and gender, and joined a pool of women hired by NASA to work as "human computers" to calculate the trajectory of the first American-manned flight into space.


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Built in Bra. Dry clean only.
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Two Actors and a Camera
Two actors and a camera is just that; two actors and a camera. There is no 40-foot grip truck, generator, or crew of fifty people. There is nothing behind the camera. Everything is in front, on the screen.
Movies are filmed with a small crew, very much in the style of European or U.S. documentary filmmaking.
Films are shot using a modified form of Dogme 95. The camera is mostly handheld, and all scenes are filmed using available light. Movie lights can be used to light a dark corner of a room, but cannot be used as the main source of light.
Actors do their own hair, makeup and wardrobe. They become the characters.
Please visit the pages to the left, to find out about our current and future projects.
Jack Lindauer
Writer/Producer/Director
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Simon Hirst
Director CEO & Founder of Sygnature Discovery
Simon is Sygnature's founder, Chief Executive Officer and chair of the Board of Directors.
He is a medicinal chemist with 30 years' experience in the pharmaceutical industry. Simon has been engaged in the full spectrum of drug discovery activities – from high-throughput screening, to progressing hit-to-lead campaigns and all the way to successful lead optimisation projects and candidate drug selection.
He has a broad experience of research in both a large pharmaceutical company (AstraZeneca) and a mid-size biotech (OSI Pharmaceuticals) as well as in a more commercial environment at CombiPure Ltd., where he was Managing Director and Director of Chemistry.
Simon has a BSc (Hons.) in chemistry and a PhD in synthetic organic chemistry from the University of Nottingham and undertook post-doctoral research at the University of Pittsburgh.
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Our board of directors is made up of seasoned industry investors and executives who provide their leadership and experience as we progress our promising pipeline into the clinic.
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